Theory Unwinding
section ‹Unwinding Proof Method for Relative Security›
text ‹ This theory formalizes the notion of unwinding for relative security,
and proves its soundness. ›
theory Unwinding
imports Relative_Security
begin
subsection ‹ The types and operators underlying unwinding: status, matching operators, etc. ›
context Rel_Sec
begin
datatype status = Eq | Diff
fun newStat :: "status ⇒ bool × 'a ⇒ bool × 'a ⇒ status" where
"newStat Eq (True,a) (True,a') = (if a = a' then Eq else Diff)"
|"newStat stat _ _ = stat"
definition "sstatO' statO sv1 sv2 = newStat statO (isIntV sv1, getObsV sv1) (isIntV sv2, getObsV sv2)"
definition "sstatA' statA s1 s2 = newStat statA (isIntO s1, getObsO s1) (isIntO s2, getObsO s2)"
lemma newStat_EqI:
assumes ‹R = S›
shows ‹newStat Eq (P, R) (Q, S) = Eq›
apply (cases P)
apply (metis assms newStat.simps(1) newStat.simps(4))
by (cases Q) auto
lemma newStat_diff:"newStat stat r r = Diff ⟹ stat = Diff"
by (metis newStat.elims newStat.simps(1))
definition initCond ::
"(enat ⇒ enat ⇒ enat ⇒ 'stateO ⇒ 'stateO ⇒ status ⇒ 'stateV ⇒ 'stateV ⇒ status ⇒ bool) ⇒ bool" where
"initCond Δ ≡ ∀s1 s2.
istateO s1 ∧ istateO s2
⟶
(∃sv1 sv2. istateV sv1 ∧ istateV sv2 ∧ corrState sv1 s1 ∧ corrState sv2 s2
∧ Δ ∞ ∞ ∞ s1 s2 Eq sv1 sv2 Eq)"
definition "match1_1 Δ w1 w2 s1 s1' s2 statA sv1 sv2 statO ≡
∃sv1'. validTransV (sv1,sv1') ∧
Δ ∞ w1 w2 s1' s2 statA sv1' sv2 statO"
definition "match1_12 Δ w1 w2 s1 s1' s2 statA sv1 sv2 statO ≡
(∃sv1' sv2'.
let statO' = sstatO' statO sv1 sv2 in
validTransV (sv1,sv1') ∧
validTransV (sv2,sv2') ∧
Δ ∞ w1 w2 s1' s2 statA sv1' sv2' statO')"
definition "match1 Δ w1 w2 s1 s2 statA sv1 sv2 statO ≡
¬ isIntO s1 ⟶
(∀s1'. validTransO (s1,s1')
⟶
(∃w1'< w1. ∃w2'< w2. ¬ isSecO s1 ∧ Δ ∞ w1' w2' s1' s2 statA sv1 sv2 statO) ∨
(∃w2'< w2. eqSec sv1 s1 ∧ ¬ isIntV sv1 ∧ match1_1 Δ ∞ w2' s1 s1' s2 statA sv1 sv2 statO) ∨
(eqSec sv1 s1 ∧ ¬ isSecV sv2 ∧ Van.eqAct sv1 sv2 ∧ match1_12 Δ ∞ ∞ s1 s1' s2 statA sv1 sv2 statO))"
lemmas match1_defs = match1_def match1_1_def match1_12_def
lemma match1_1_mono:
"Δ ≤ Δ' ⟹ match1_1 Δ w1 w2 s1 s1' s2 statA sv1 sv2 statO ⟹
match1_1 Δ' w1 w2 s1 s1' s2 statA sv1 sv2 statO"
unfolding le_fun_def match1_1_def by auto
lemma match1_12_mono:
"Δ ≤ Δ' ⟹ match1_12 Δ w1 w2 s1 s1' s2 statA sv1 sv2 statO ⟹
match1_12 Δ' w1 w2 s1 s1' s2 statA sv1 sv2 statO"
unfolding le_fun_def match1_12_def by fastforce
lemma match1_mono:
assumes "Δ ≤ Δ'"
shows "match1 Δ w1 w2 s1 s2 statA sv1 sv2 statO ⟹ match1 Δ' w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding match1_def apply clarify subgoal for s1' apply(erule allE[of _ s1'])
using match1_1_mono[OF assms, of _ _ s1 s1' s2 statA sv1 sv2 statO]
match1_12_mono[OF assms, of _ _ s1 s1' s2 statA sv1 sv2 statO]
assms[unfolded le_fun_def, rule_format, of _ _ _ s1' s2 statA sv1 sv2 statO]
by fastforce .
definition "match2_1 Δ w1 w2 s1 s2 s2' statA sv1 sv2 statO ≡
∃sv2'. validTransV (sv2,sv2') ∧
Δ ∞ w1 w2 s1 s2' statA sv1 sv2' statO"
definition "match2_12 Δ w1 w2 s1 s2 s2' statA sv1 sv2 statO ≡
∃sv1' sv2'.
let statO' = sstatO' statO sv1 sv2 in
validTransV (sv1,sv1') ∧
validTransV (sv2,sv2') ∧
Δ ∞ w1 w2 s1 s2' statA sv1' sv2' statO'"
definition "match2 Δ w1 w2 s1 s2 statA sv1 sv2 statO ≡
¬ isIntO s2 ⟶
(∀s2'. validTransO (s2,s2')
⟶
(∃w1'< w1. ∃w2'< w2. ¬ isSecO s2 ∧ Δ ∞ w1' w2' s1 s2' statA sv1 sv2 statO) ∨
(∃w1'< w1. eqSec sv2 s2 ∧ ¬ isIntV sv2 ∧ match2_1 Δ w1' ∞ s1 s2 s2' statA sv1 sv2 statO) ∨
(¬ isSecV sv1 ∧ eqSec sv2 s2 ∧ Van.eqAct sv1 sv2 ∧ match2_12 Δ ∞ ∞ s1 s2 s2' statA sv1 sv2 statO))"
lemmas match2_defs = match2_def match2_1_def match2_12_def
lemma match2_1_mono:
"Δ ≤ Δ' ⟹ match2_1 Δ w1 w2 s1 s1' s2 statA sv1 sv2 statO ⟹ match2_1 Δ' w1 w2 s1 s1' s2 statA sv1 sv2 statO"
unfolding le_fun_def match2_1_def by auto
lemma match2_12_mono:
"Δ ≤ Δ' ⟹ match2_12 Δ w1 w2 s1 s1' s2 statA sv1 sv2 statO ⟹ match2_12 Δ' w1 w2 s1 s1' s2 statA sv1 sv2 statO"
unfolding le_fun_def match2_12_def by fastforce
lemma match2_mono:
assumes "Δ ≤ Δ'"
shows "match2 Δ w1 w2 s1 s2 statA sv1 sv2 statO ⟹ match2 Δ' w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding match2_def apply clarify subgoal for s2' apply(erule allE[of _ s2'])
using match2_1_mono[OF assms, of _ _ s1 s2 s2' statA sv1 sv2 statO]
match2_12_mono[OF assms, of _ _ s1 s2 s2' statA sv1 sv2 statO]
assms[unfolded le_fun_def, rule_format, of _ _ _ s1 s2' statA sv1 sv2 statO]
by fastforce .
definition "match12_1 Δ w1 w2 s1' s2' statA' sv1 sv2 statO ≡
∃sv1'. validTransV (sv1,sv1') ∧
Δ ∞ w1 w2 s1' s2' statA' sv1' sv2 statO"
definition "match12_2 Δ w1 w2 s1' s2' statA' sv1 sv2 statO ≡
∃sv2'. validTransV (sv2,sv2') ∧
Δ ∞ w1 w2 s1' s2' statA' sv1 sv2' statO"
definition "match12_12 Δ w1 w2 s1' s2' statA' sv1 sv2 statO ≡
∃sv1' sv2'.
let statO' = sstatO' statO sv1 sv2 in
validTransV (sv1,sv1') ∧
validTransV (sv2,sv2') ∧
(statA' = Diff ⟶ statO' = Diff) ∧
Δ ∞ w1 w2 s1' s2' statA' sv1' sv2' statO'"
definition "match12 Δ w1 w2 s1 s2 statA sv1 sv2 statO ≡
∀s1' s2'.
let statA' = sstatA' statA s1 s2 in
validTransO (s1,s1') ∧
validTransO (s2,s2') ∧
Opt.eqAct s1 s2 ∧
isIntO s1 ∧ isIntO s2
⟶
(∃w1'< w1. ∃w2'< w2. ¬ isSecO s1 ∧ ¬ isSecO s2 ∧ (statA = statA' ∨ statO = Diff) ∧
Δ ∞ w1' w2' s1' s2' statA' sv1 sv2 statO)
∨
(∃w2'< w2. ¬ isSecO s2 ∧ eqSec sv1 s1 ∧ ¬ isIntV sv1 ∧
(statA = statA' ∨ statO = Diff) ∧
match12_1 Δ ∞ w2' s1' s2' statA' sv1 sv2 statO)
∨
(∃w1'< w1. ¬ isSecO s1 ∧ eqSec sv2 s2 ∧ ¬ isIntV sv2 ∧
(statA = statA' ∨ statO = Diff) ∧
match12_2 Δ w1' ∞ s1' s2' statA' sv1 sv2 statO)
∨
(eqSec sv1 s1 ∧ eqSec sv2 s2 ∧ Van.eqAct sv1 sv2 ∧
match12_12 Δ ∞ ∞ s1' s2' statA' sv1 sv2 statO)"
lemmas match12_defs = match12_def match12_1_def match12_2_def match12_12_def
lemma match12_simpleI:
assumes "⋀s1' s2' statA'.
statA' = sstatA' statA s1 s2 ⟹
validTransO (s1,s1') ⟹
validTransO (s2,s2') ⟹
Opt.eqAct s1 s2 ⟹
(∃w1'< w1. ∃w2'< w2. ¬ isSecO s1 ∧ ¬ isSecO s2 ∧ (statA = statA' ∨ statO = Diff) ∧
Δ ∞ w1' w2' s1' s2' statA' sv1 sv2 statO)
∨
(eqSec sv1 s1 ∧ eqSec sv2 s2 ∧ Van.eqAct sv1 sv2 ∧
match12_12 Δ ∞ ∞ s1' s2' statA' sv1 sv2 statO)"
shows "match12 Δ w1 w2 s1 s2 statA sv1 sv2 statO"
using assms unfolding match12_def Let_def by blast
lemma match12_1_mono:
"Δ ≤ Δ' ⟹ match12_1 Δ w1 w2 s1' s2' statA' sv1 sv2 statO ⟹ match12_1 Δ' w1 w2 s1' s2' statA' sv1 sv2 statO"
unfolding le_fun_def match12_1_def by auto
lemma match12_2_mono:
"Δ ≤ Δ' ⟹ match12_2 Δ w1 w2 s1 s2' statA' sv1 sv2 statO ⟹ match12_2 Δ' w1 w2 s1 s2' statA' sv1 sv2 statO"
unfolding le_fun_def match12_2_def by auto
lemma match12_12_mono:
"Δ ≤ Δ' ⟹ match12_12 Δ w1 w2 s1' s2' statA' sv1 sv2 statO ⟹ match12_12 Δ' w1 w2 s1' s2' statA' sv1 sv2 statO"
unfolding le_fun_def match12_12_def by fastforce
lemma match12_mono:
assumes "Δ ≤ Δ'"
shows "match12 Δ w1 w2 s1 s2 statA sv1 sv2 statO ⟹ match12 Δ' w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding match12_def apply clarify subgoal for s1' s2' apply(erule allE[of _ s1']) apply(erule allE[of _ s2'])
using match12_1_mono[OF assms, of _ _ s1' s2' _ sv1 sv2 statO]
match12_2_mono[OF assms, of _ _ s1' s2' _ sv1 sv2 statO]
match12_12_mono[OF assms, of _ _ s1' s2' _ sv1 sv2 statO]
assms[unfolded le_fun_def, rule_format, of _ _ _ s1' s2'
"sstatA' statA s1 s2" sv1 sv2 statO]
apply simp by blast .
definition "react Δ w1 w2 s1 s2 statA sv1 sv2 statO ≡
match1 Δ w1 w2 s1 s2 statA sv1 sv2 statO
∧
match2 Δ w1 w2 s1 s2 statA sv1 sv2 statO
∧
match12 Δ w1 w2 s1 s2 statA sv1 sv2 statO"
lemmas react_defs = match1_def match2_def match12_def
lemmas match_deep_defs = match1_defs match2_defs match12_defs
lemma match_mono:
assumes "Δ ≤ Δ'"
shows "react Δ w1 w2 s1 s2 statA sv1 sv2 statO ⟹ react Δ' w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding react_def using match1_mono[OF assms] match2_mono[OF assms] match12_mono[OF assms] by auto
definition "move_1 Δ w w1 w2 s1 s2 statA sv1 sv2 statO ≡
∃sv1'. validTransV (sv1,sv1') ∧
Δ w w1 w2 s1 s2 statA sv1' sv2 statO"
definition "move_2 Δ w w1 w2 s1 s2 statA sv1 sv2 statO ≡
∃sv2'. validTransV (sv2,sv2') ∧
Δ w w1 w2 s1 s2 statA sv1 sv2' statO"
definition "move_12 Δ w w1 w2 s1 s2 statA sv1 sv2 statO ≡
∃sv1' sv2'.
let statO' = sstatO' statO sv1 sv2 in
validTransV (sv1,sv1') ∧ validTransV (sv2,sv2') ∧
Δ w w1 w2 s1 s2 statA sv1' sv2' statO'"
definition "proact Δ w w1 w2 s1 s2 statA sv1 sv2 statO ≡
(¬ isSecV sv1 ∧ ¬ isIntV sv1 ∧ move_1 Δ w w1 w2 s1 s2 statA sv1 sv2 statO)
∨
(¬ isSecV sv2 ∧ ¬ isIntV sv2 ∧ move_2 Δ w w1 w2 s1 s2 statA sv1 sv2 statO)
∨
(¬ isSecV sv1 ∧ ¬ isSecV sv2 ∧ Van.eqAct sv1 sv2 ∧ move_12 Δ w w1 w2 s1 s2 statA sv1 sv2 statO)"
lemmas proact_defs = proact_def move_1_def move_2_def move_12_def
lemma move_1_mono:
"Δ ≤ Δ' ⟹ move_1 Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ move_1 Δ' w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding le_fun_def move_1_def by auto
lemma move_2_mono:
"Δ ≤ Δ' ⟹ move_2 Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ move_2 Δ' w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding le_fun_def move_2_def by auto
lemma move_12_mono:
"Δ ≤ Δ' ⟹ move_12 Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ move_12 Δ' w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding le_fun_def move_12_def by fastforce
lemma proact_mono:
assumes "Δ ≤ Δ'"
shows "proact Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ proact Δ' w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding proact_def using move_1_mono[OF assms] move_2_mono[OF assms] move_12_mono[OF assms] by auto
subsection ‹ The definition of unwinding ›
definition unwindCond ::
"(enat ⇒ enat ⇒ enat ⇒ 'stateO ⇒ 'stateO ⇒ status ⇒ 'stateV ⇒ 'stateV ⇒ status ⇒ bool) ⇒ bool"
where
"unwindCond Δ ≡ ∀w w1 w2 s1 s2 statA sv1 sv2 statO.
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
Δ w w1 w2 s1 s2 statA sv1 sv2 statO
⟶
(finalO s1 ⟷ finalO s2) ∧ (finalV sv1 ⟷ finalO s1) ∧ (finalV sv2 ⟷ finalO s2)
∧
(statA = Eq ⟶ (isIntO s1 ⟷ isIntO s2))
∧
((∃v < w. proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO)
∨
react Δ w1 w2 s1 s2 statA sv1 sv2 statO
)"
lemma unwindCond_simpleI:
assumes
"⋀w w1 w2 s1 s2 statA sv1 sv2 statO.
reachO s1 ⟹ reachO s2 ⟹ reachV sv1 ⟹ reachV sv2 ⟹
Δ w w1 w2 s1 s2 statA sv1 sv2 statO
⟹
(finalO s1 ⟷ finalO s2) ∧ (finalV sv1 ⟷ finalO s1) ∧ (finalV sv2 ⟷ finalO s2)"
and
"⋀w w1 w2 s1 s2 statA sv1 sv2 statO.
reachO s1 ⟹ reachO s2 ⟹ reachV sv1 ⟹ reachV sv2 ⟹
Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ statA = Eq
⟹
isIntO s1 ⟷ isIntO s2"
and
"⋀w w1 w2 s1 s2 statA sv1 sv2 statO.
reachO s1 ⟹ reachO s2 ⟹ reachV sv1 ⟹ reachV sv2 ⟹
Δ w w1 w2 s1 s2 statA sv1 sv2 statO
⟹
react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
shows "unwindCond Δ"
using assms unfolding unwindCond_def by auto
subsection ‹ The soundness of unwinding ›
text ‹ The proof of soundness for general unwinding is significantly
more elaborate than that for the finitary case. ›
definition "ψ s1 tr1 s2 tr2 statO sv1 trv1 sv2 trv2 ≡
trv1 ≠ [] ∧ trv2 ≠ [] ∧
Van.validFromS sv1 trv1 ∧
Van.validFromS sv2 trv2 ∧
(finalV (lastt sv1 trv1) ⟷ finalO (lastt s1 tr1)) ∧ (finalV (lastt sv2 trv2) ⟷ finalO (lastt s2 tr2)) ∧
Van.S trv1 = Opt.S tr1 ∧ Van.S trv2 = Opt.S tr2 ∧
Van.A trv1 = Van.A trv2 ∧
(statO = Eq ∧ Opt.O tr1 ≠ Opt.O tr2 ⟶ Van.O trv1 ≠ Van.O trv2)"
lemma ψ_completedFrom: "completedFromO s1 tr1 ⟹ completedFromO s2 tr2 ⟹
ψ s1 tr1 s2 tr2 statO sv1 trv1 sv2 trv2
⟹ completedFromV sv1 trv1 ∧ completedFromV sv2 trv2"
unfolding ψ_def Opt.completedFrom_def Van.completedFrom_def lastt_def
by presburger
lemma completedFromO_lastt: "completedFromO s1 tr1 ⟹ finalO (lastt s1 tr1)"
unfolding Opt.completedFrom_def lastt_def by auto
lemma rsecure_strong:
assumes
"⋀s1 tr1 s2 tr2.
istateO s1 ∧ Opt.validFromS s1 tr1 ∧ completedFromO s1 tr1 ∧
istateO s2 ∧ Opt.validFromS s2 tr2 ∧ completedFromO s2 tr2 ∧
Opt.A tr1 = Opt.A tr2
⟹
∃sv1 trv1 sv2 trv2.
istateV sv1 ∧ istateV sv2 ∧ corrState sv1 s1 ∧ corrState sv2 s2 ∧
ψ s1 tr1 s2 tr2 Eq sv1 trv1 sv2 trv2"
shows rsecure
unfolding rsecure_def2 apply safe
subgoal for s1 tr1 s2 tr2
using assms[of s1 tr1 s2 tr2]
using ψ_completedFrom ψ_def completedFromO_lastt apply clarsimp by metis .
proposition unwindCond_ex_ψ:
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO" and stat: "(statA = Diff ⟶ statO = Diff)"
and v: "Opt.validFromS s1 tr1" "Opt.completedFrom s1 tr1" "Opt.validFromS s2 tr2" "Opt.completedFrom s2 tr2"
and tr14: "Opt.A tr1 = Opt.A tr2"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
shows "∃trv1 trv2. ψ s1 tr1 s2 tr2 statO sv1 trv1 sv2 trv2"
using assms(2-)
proof(induction "length tr1 + length tr2" w
arbitrary: w1 w2 s1 s2 statA sv1 sv2 statO tr1 tr2 rule: less2_induct')
case (less w tr1 tr2 w1 w2 s1 s2 statA sv1 sv2 statO)
note ok = `statA = Diff ⟶ statO = Diff`
note Δ = `Δ w w1 w2 s1 s2 statA sv1 sv2 statO`
note A34 = `Opt.A tr1 = Opt.A tr2`
note r34 = less.prems(8,9) note r12 = less.prems(10,11)
note r = r34 r12
note r3 = r34(1) note r4 = r34(2) note r1 = r12(1) note r2 = r12(2)
have i34: "statA = Eq ⟶ isIntO s1 = isIntO s2"
and f34: "finalO s1 = finalO s2 ∧ finalV sv1 = finalO s1 ∧ finalV sv2 = finalO s2"
using Δ unwind[unfolded unwindCond_def] r by auto
have proact_match: "(∃v<w. proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO) ∨ react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
using Δ unwind[unfolded unwindCond_def] r by auto
show ?case using proact_match proof safe
fix v assume v: "v < w"
assume "proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
thus ?thesis unfolding proact_def proof safe
assume sv1: "¬ isSecV sv1" "¬ isIntV sv1" and "move_1 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv1'
where 0: "validTransV (sv1,sv1')"
and Δ: "Δ v w1 w2 s1 s2 statA sv1' sv2 statO"
unfolding move_1_def by auto
have r1': "reachV sv1'" using r1 0 by (metis Van.reach.Step fst_conv snd_conv)
obtain trv1 trv2 where ψ: "ψ s1 tr1 s2 tr2 statO sv1' trv1 sv2 trv2"
using less(2)[OF v, of tr1 tr2 w1 w2 s1 s2 statA sv1' sv2 statO, simplified, OF Δ ok _ _ _ _ _ r34 r1' r2]
using A34 less.prems(3-6) by blast
show ?thesis apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ trv2])
using ψ ok 0 sv1 unfolding ψ_def Van.completedFrom_def by auto
next
assume sv2: "¬ isSecV sv2" "¬ isIntV sv2" and "move_2 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv2'
where 0: "validTransV (sv2,sv2')"
and Δ: "Δ v w1 w2 s1 s2 statA sv1 sv2' statO"
unfolding move_2_def by auto
have r2': "reachV sv2'" using r2 0 by (metis Van.reach.Step fst_conv snd_conv)
obtain trv1 trv2 where ψ: "ψ s1 tr1 s2 tr2 statO sv1 trv1 sv2' trv2"
using less(2)[OF v, of tr1 tr2 w1 w2 s1 s2 statA sv1 sv2' statO, simplified, OF Δ ok _ _ _ _ _ r34 r1 r2']
using A34 less.prems(3-6) by blast
show ?thesis apply(rule exI[of _ trv1]) apply(rule exI[of _ "sv2 # trv2"])
using ψ ok 0 sv2 unfolding ψ_def Van.completedFrom_def by auto
next
assume sv12: "¬ isSecV sv1" "¬ isSecV sv2" "Van.eqAct sv1 sv2"
and "move_12 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv1' sv2' statO'
where 0: "statO' = sstatO' statO sv1 sv2"
"validTransV (sv1,sv1') " "¬ isSecV sv1"
"validTransV (sv2,sv2')" "¬ isSecV sv2"
"Van.eqAct sv1 sv2"
and Δ: "Δ v w1 w2 s1 s2 statA sv1' sv2' statO'"
unfolding move_12_def by auto
have r12': "reachV sv1'" "reachV sv2'" using r1 r2 0 by (metis Van.reach.Step fst_conv snd_conv)+
have ok': "statA = Diff ⟶ statO' = Diff" using ok 0 unfolding sstatO'_def by (cases statO, auto)
obtain trv1 trv2 where ψ: "ψ s1 tr1 s2 tr2 statO' sv1' trv1 sv2' trv2"
using less(2)[OF v, of tr1 tr2 w1 w2 s1 s2 statA sv1' sv2' statO', simplified, OF Δ ok' _ _ _ _ _ r34 r12']
using A34 less.prems(3-6) by blast
show ?thesis apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "sv2 # trv2"])
using ψ ok' 0 sv12 unfolding ψ_def sstatO'_def Van.completedFrom_def
using Van.A.Cons_unfold Van.eqAct_def completedFromO_lastt less.prems(4)
less.prems(6) by auto
qed
next
assume m: "react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
show ?thesis
proof(cases "length tr1 ≤ Suc 0")
case True note tr1 = True
hence "tr1 = [] ∨ tr1 = [s1]"
by (metis Simple_Transition_System.validFromS_Cons_iff Suc_length_conv le_Suc_eq le_zero_eq length_0_conv less.prems(3))
hence "finalO s1" using less(3-6)
using Opt.completed_Cons Opt.completed_Nil by blast
hence f4: "finalO s2" using f34 by blast
hence tr2: "tr2 = [] ∨ tr2 = [s2]"
by (metis Opt.final_def Simple_Transition_System.validFromS_Cons_iff less.prems(5) neq_Nil_conv)
show ?thesis apply(rule exI[of _ "[sv1]"], rule exI[of _ "[sv2]"]) using tr1 tr2
using f4 f34
using completedFromO_lastt less.prems(4)
by (auto simp add: lastt_def ψ_def)
next
case False
then obtain s13 tr1' where tr1: "tr1 = s13 # tr1'" and tr1'NE: "tr1' ≠ []"
by (cases tr1, auto)
have s13[simp]: "s13 = s1" using `Opt.validFromS s1 tr1`
by (simp add: Opt.validFromS_Cons_iff tr1)
obtain s1' where
trn3: "validTransO (s1,s1')" and
tr1': "Opt.validFromS s1' tr1'" using `Opt.validFromS s1 tr1`
unfolding tr1 s13 by (metis tr1'NE Simple_Transition_System.validFromS_Cons_iff)
have r3': "reachO s1'" using r3 trn3 by (metis Opt.reach.Step fst_conv snd_conv)
have f3: "¬ finalO s1" using Opt.final_def trn3 by blast
hence f4: "¬ finalO s2" using f34 by blast
hence tr2: "¬ length tr2 ≤ Suc 0"
by (metis Opt.completed_Cons Simple_Transition_System.validFromS_Cons_iff
bot_nat_0.extremum completedFromO_def length_Cons less.prems(5) less.prems(6) neq_Nil_conv not_less_eq_eq)
then obtain s24 tr2' where tr2: "tr2 = s24 # tr2'" and tr2'NE: "tr2' ≠ []"
by (cases tr2, auto)
have s24[simp]: "s24 = s2" using `Opt.validFromS s2 tr2`
by (simp add: Opt.validFromS_Cons_iff tr2)
obtain s2' where
trn4: "validTransO (s2,s2') ∨ (s2 = s2' ∧ tr2' = [])" and
tr2': "Opt.validFromS s2' tr2'" using `Opt.validFromS s2 tr2`
unfolding tr2 s24 using Opt.validFromS_Cons_iff by auto
have r34': "reachO s1'" "reachO s2'"
using r3 trn3 r4 trn4 by (metis Opt.reach.Step fst_conv snd_conv)+
note r3' = r34'(1) note r4' = r34'(2)
define statA' where statA': "statA' = sstatA' statA s1 s2"
have "¬ isIntO s1 ∨ ¬ isIntO s2 ∨ (isIntO s1 ∧ isIntO s2)"
by auto
thus ?thesis
proof safe
assume isAO3: "¬ isIntO s1"
have O33': "Opt.O tr1 = Opt.O tr1'" "Opt.A tr1 = Opt.A tr1'"
using isAO3 unfolding tr1 by auto
have A34': "Opt.A tr1' = Opt.A tr2"
using A34 O33'(2) by auto
have m: "match1 Δ w1 w2 s1 s2 statA sv1 sv2 statO" using m unfolding react_def by auto
have "(∃w1'<w1. ∃w2'<w2. ¬ isSecO s1 ∧ Δ ∞ w1' w2' s1' s2 statA sv1 sv2 statO) ∨
(∃w2'<w2. eqSec sv1 s1 ∧ ¬ isIntV sv1 ∧ match1_1 Δ ∞ w2' s1 s1' s2 statA sv1 sv2 statO) ∨
(eqSec sv1 s1 ∧ ¬ isSecV sv2 ∧ Van.eqAct sv1 sv2 ∧ match1_12 Δ ∞ ∞ s1 s1' s2 statA sv1 sv2 statO)"
using m isAO3 trn3 ok unfolding match1_def by auto
thus ?thesis
proof safe
fix w1' w2'
assume "¬ isSecO s1" and Δ: "Δ ∞ w1' w2' s1' s2 statA sv1 sv2 statO"
hence S3: "Opt.S tr1' = Opt.S tr1" unfolding tr1 by auto
obtain trv1 trv2 where ψ: "ψ s1 tr1' s2 tr2 statO sv1 trv1 sv2 trv2"
using less(1)[of tr1' tr2, OF _ Δ _ _ _ _ _ _ r3' r4 r12, unfolded O33', simplified]
using less.prems tr1' ok A34' f3 f4 unfolding tr1 Opt.completedFrom_def
by (auto split: if_splits simp: ψ_def lastt_def)
show ?thesis apply(rule exI[of _ trv1]) apply(rule exI[of _ trv2])
using ψ O33' S3 unfolding ψ_def
using completedFromO_lastt less.prems(4)
by (auto simp add: tr1 tr1'NE)
next
fix w2'
assume trn13: "eqSec sv1 s1" and
Atrn1: "¬ isIntV sv1" and "match1_1 Δ ∞ w2' s1 s1' s2 statA sv1 sv2 statO"
then obtain sv1' where
trn1: "validTransV (sv1,sv1') " and
Δ: "Δ ∞ ∞ w2' s1' s2 statA sv1' sv2 statO"
unfolding match1_1_def by auto
have r1': "reachV sv1'"using r1 trn1 by (metis Van.reach.Step fst_conv snd_conv)
obtain trv1 trv2 where ψ: "ψ s1 tr1' s2 tr2 statO sv1' trv1 sv2 trv2"
using less(1)[of tr1' tr2, OF _ Δ _ _ _ _ _ _ r3' r4 r1' r2, unfolded O33', simplified]
using less.prems tr1' ok A34' f3 f4 unfolding tr1 tr2 Opt.completedFrom_def
by (auto simp: ψ_def lastt_def split: if_splits)
show ?thesis apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ trv2])
using ψ O33' unfolding tr1 tr2 Van.completedFrom_def
using Van.validFromS_Cons trn1 tr1'NE tr2'NE
using isAO3 ok Atrn1 eqSec_S_Cons trn13
unfolding ψ_def using completedFromO_lastt less.prems(4) tr1 by auto
next
assume sv2: "¬ isSecV sv2" and trn13: "eqSec sv1 s1" and
Atrn12: "Van.eqAct sv1 sv2" and "match1_12 Δ ∞ ∞ s1 s1' s2 statA sv1 sv2 statO"
then obtain sv1' sv2' statO' where
statO': "statO' = sstatO' statO sv1 sv2" and
trn1: "validTransV (sv1,sv1') " and
trn2: "validTransV (sv2,sv2')" and
Δ: "Δ ∞ ∞ ∞ s1' s2 statA sv1' sv2' statO'"
unfolding match1_12_def by auto
have r12': "reachV sv1'" "reachV sv2'"
using r1 trn1 r2 trn2 by (metis Van.reach.Step fst_conv snd_conv)+
obtain trv1 trv2 where ψ: "ψ s1' tr1' s2 tr2 statO' sv1' trv1 sv2' trv2"
using less(1)[of tr1' tr2, OF _ Δ _ _ _ _ _ _ r3' r4 r12', unfolded O33', simplified]
using less.prems tr1' ok A34' f3 f4 unfolding tr1 tr2 Opt.completedFrom_def statO' sstatO'_def
by auto presburger+
show ?thesis apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "sv2 # trv2"])
using ψ O33' tr1'NE tr2'NE sv2
using Van.validFromS_Cons trn1 trn2
using isAO3 ok Atrn12 eqSec_S_Cons trn13 f3 f34 s13
unfolding ψ_def tr1 Van.completedFrom_def Van.eqAct_def statO' sstatO'_def
using Van.A.Cons_unfold tr1' trn3 by auto
qed
next
assume isAO4: "¬ isIntO s2"
have O44': "Opt.O tr2 = Opt.O tr2'" "Opt.A tr2 = Opt.A tr2'"
using isAO4 unfolding tr2 by auto
have A34': "Opt.A tr1 = Opt.A tr2'"
using A34 O44'(2) by auto
have m: "match2 Δ w1 w2 s1 s2 statA sv1 sv2 statO" using m unfolding react_def by auto
have "(∃w1'<w1. ∃w2'<w2. ¬ isSecO s2 ∧ Δ ∞ w1' w2' s1 s2' statA sv1 sv2 statO) ∨
(∃w1'<w1. eqSec sv2 s2 ∧ ¬ isIntV sv2 ∧ match2_1 Δ w1' ∞ s1 s2 s2' statA sv1 sv2 statO) ∨
(¬ isSecV sv1 ∧ eqSec sv2 s2 ∧ Van.eqAct sv1 sv2 ∧ match2_12 Δ ∞ ∞ s1 s2 s2' statA sv1 sv2 statO)"
using m isAO4 trn4 ok tr2'NE unfolding match2_def by auto
thus ?thesis
proof safe
fix w1' w2'
assume "¬ isSecO s2" and Δ: "Δ ∞ w1' w2' s1 s2' statA sv1 sv2 statO"
hence S4: "Opt.S tr2' = Opt.S tr2" unfolding tr2 by auto
obtain trv1 trv2 where ψ: "ψ s1 tr1 s2' tr2' statO sv1 trv1 sv2 trv2"
using less(1)[of tr1 tr2', OF _ Δ _ _ _ _ _ _ r3 r4', simplified]
using less.prems tr2' ok A34' tr1'NE tr2'NE unfolding tr1 tr2 Opt.completedFrom_def by (cases "isIntO s2", auto)
show ?thesis apply(rule exI[of _ trv1]) apply(rule exI[of _ trv2])
using ψ O44' S4 unfolding ψ_def
using completedFromO_lastt less.prems(6)
unfolding Opt.completedFrom_def using tr2 tr2'NE by auto
next
fix w1'
assume trn24: "eqSec sv2 s2" and
Atrn2: "¬ isIntV sv2" and "match2_1 Δ w1' ∞ s1 s2 s2' statA sv1 sv2 statO"
then obtain sv2' where trn2: "validTransV (sv2,sv2')" and
Δ: "Δ ∞ w1' ∞ s1 s2' statA sv1 sv2' statO"
unfolding match2_1_def by auto
have r2': "reachV sv2'" using r2 trn2 by (metis Van.reach.Step fst_conv snd_conv)
obtain trv1 trv2 where ψ: "ψ s1 tr1 s2' tr2' statO sv1 trv1 sv2' trv2"
using less(1)[of tr1 tr2', OF _ Δ _ _ _ _ _ _ r3 r4' r1 r2', simplified]
using less.prems tr2' ok A34' tr1'NE tr2'NE unfolding tr1 tr2 Opt.completedFrom_def by (cases "isIntO s2", auto)
show ?thesis apply(rule exI[of _ trv1]) apply(rule exI[of _ "sv2 # trv2"])
using ψ tr1'NE tr2'NE
using Van.validFromS_Cons trn2
using isAO4 ok Atrn2 eqSec_S_Cons trn24
unfolding ψ_def tr1 tr2 s13 s24 Van.completedFrom_def lastt_def by auto
next
assume sv1: "¬ isSecV sv1" and trn24: "eqSec sv2 s2" and
Atrn12: "Van.eqAct sv1 sv2" and "match2_12 Δ ∞ ∞ s1 s2 s2' statA sv1 sv2 statO"
then obtain sv1' sv2' statO' where
statO': "statO' = sstatO' statO sv1 sv2" and
trn1: "validTransV (sv1,sv1')" and
trn2: "validTransV (sv2,sv2')" and
Δ: "Δ ∞ ∞ ∞ s1 s2' statA sv1' sv2' statO'"
unfolding match2_12_def by auto
have r12': "reachV sv1'" "reachV sv2'"
using r1 trn1 r2 trn2 by (metis Van.reach.Step fst_conv snd_conv)+
obtain trv1 trv2 where ψ: "ψ s1 tr1 s2' tr2' statO' sv1' trv1 sv2' trv2"
using less(1)[of tr1 tr2', OF _ Δ _ _ _ _ _ _ r3 r4' r12', simplified]
using less.prems tr2' ok A34' tr1'NE tr2'NE unfolding tr1 tr2 Opt.completedFrom_def statO' sstatO'_def
by (cases "isIntO s2", auto)
show ?thesis apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "sv2 # trv2"])
using ψ O44' tr1'NE tr2'NE sv1
using Van.validFromS_Cons trn1 trn2
using isAO4 ok Atrn12 eqSec_S_Cons trn24
unfolding ψ_def tr2 tr1'NE Van.completedFrom_def Van.eqAct_def
statO' sstatO'_def
using Van.A.Cons_unfold tr2' trn4 by auto
qed
next
assume isAO34: "isIntO s1" "isIntO s2"
have A34': "getActO s1 = getActO s2" "Opt.A tr1' = Opt.A tr2'"
using A34 isAO34 tr1'NE tr2'NE unfolding tr1 tr2 by auto
have O33': "Opt.O tr1 = getObsO s1 # Opt.O tr1'" and
O44': "Opt.O tr2 = getObsO s2 # Opt.O tr2'"
using isAO34 tr1'NE tr2'NE unfolding tr1 s13 tr2 s24 by auto
have m: "match12 Δ w1 w2 s1 s2 statA sv1 sv2 statO" using m unfolding statA' react_def by auto
have trn34: "getObsO s1 = getObsO s2 ∨ statA' = Diff"
using isAO34 unfolding statA' sstatA'_def by (cases statA,auto)
have "(∃w1'<w1. ∃w2'<w2. ¬ isSecO s1 ∧ ¬ isSecO s2 ∧ (statA = statA' ∨ statO = Diff) ∧
Δ ∞ w1' w2' s1' s2' statA' sv1 sv2 statO)
∨
(∃w2'<w2. ¬ isSecO s2 ∧ eqSec sv1 s1 ∧ ¬ isIntV sv1 ∧
(statA = statA' ∨ statO = Diff) ∧
match12_1 Δ ∞ w2' s1' s2' statA' sv1 sv2 statO)
∨
(∃w1'<w1. ¬ isSecO s1 ∧ eqSec sv2 s2 ∧ ¬ isIntV sv2 ∧
(statA = statA' ∨ statO = Diff) ∧
match12_2 Δ w1' ∞ s1' s2' statA' sv1 sv2 statO)
∨
(eqSec sv1 s1 ∧ eqSec sv2 s2 ∧ Van.eqAct sv1 sv2 ∧
match12_12 Δ ∞ ∞ s1' s2' statA' sv1 sv2 statO)"
(is "?K1 ∨ ?K2 ∨ ?K3 ∨ ?K4")
using m[unfolded match12_def, rule_format, of s1' s2']
isAO34 A34' trn3 trn4 tr1'NE tr2'NE
unfolding s13 s24 trn34 statA' Opt.eqAct_def sstatA'_def by auto
thus ?thesis proof (elim disjE)
assume K1: "?K1"
then obtain w1' w2' where Δ: "Δ ∞ w1' w2' s1' s2' statA' sv1 sv2 statO" by auto
have ok': "(statA' = Diff ⟶ statO = Diff)"
using ok K1 unfolding statA' using isAO34 by auto
obtain trv1 trv2 where ψ: "ψ s1' tr1' s2' tr2' statO sv1 trv1 sv2 trv2"
using less(1)[of tr1' tr2', OF _ Δ _ _ _ _ _ _ r34' r12, simplified]
using less.prems tr1' tr2' ok' A34' isAO34 tr1'NE tr2'NE unfolding tr1 tr2 Opt.completedFrom_def by auto
show ?thesis apply(rule exI[of _ trv1]) apply(rule exI[of _ trv2])
using ψ trn34 O33' O44' K1 ok unfolding ψ_def tr1 tr2
using completedFromO_lastt less.prems(4,6)
unfolding Opt.completedFrom_def using tr1 tr2 tr1'NE tr2'NE by auto
next
assume K2: "?K2"
then obtain w2' sv1' where
trn1: "validTransV (sv1,sv1') " and
trn13: "eqSec sv1 s1" and
Atrn1: "¬ isIntV sv1" and ok': "(statA' = statA ∨ statO = Diff)" and
Δ: "Δ ∞ ∞ w2' s1' s2' statA' sv1' sv2 statO"
unfolding match12_1_def by auto
have r1': "reachV sv1'" using r1 trn1 by (metis Van.reach.Step fst_conv snd_conv)
obtain trv1 trv2 where ψ: "ψ s1' tr1' s2' tr2' statO sv1' trv1 sv2 trv2"
using less(1)[of tr1' tr2', OF _ Δ _ _ _ _ _ _ r34' r1' r2, simplified]
using less.prems tr1' tr2' ok' A34' tr1'NE tr2'NE unfolding tr1 tr2 Opt.completedFrom_def by auto
show ?thesis apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ trv2])
using ψ O33' O44' tr1'NE tr2'NE unfolding tr1 tr2
using Van.validFromS_Cons trn1 ok
using K2 ok' Atrn1 eqSec_S_Cons trn13 trn34
unfolding statA' Van.completedFrom_def eqSec_def
using s13 tr1 tr1' tr2' trn3 trn4
by simp (smt (verit, best) Opt.S.Cons_unfold Simple_Transition_System.lastt_Cons
Van.A.Cons_unfold Van.O.Cons_unfold ψ_def completedFromO_lastt f3 f34 lastt_Nil
less.prems(4) status.simps(1))
next
assume K3: "?K3"
then obtain w1' sv2' where
trn2: "validTransV (sv2,sv2')" and
trn24: "eqSec sv2 s2" and
Atrn2: "¬ isIntV sv2" and ok': "(statA' = statA ∨ statO = Diff)" and
Δ: "Δ ∞ w1' ∞ s1' s2' statA' sv1 sv2' statO"
unfolding match12_2_def by auto
have r2': "reachV sv2'" using r2 trn2 by (metis Van.reach.Step fst_conv snd_conv)
obtain trv1 trv2 where ψ: "ψ s1' tr1' s2' tr2' statO sv1 trv1 sv2' trv2"
using less(1)[of tr1' tr2', OF _ Δ _ _ _ _ _ _ r34' r1 r2', simplified]
using less.prems tr1' tr2' ok' A34' tr1'NE tr2'NE unfolding tr1 tr2 Opt.completedFrom_def by auto
show ?thesis apply(rule exI[of _ trv1]) apply(rule exI[of _ "sv2 # trv2"])
using ψ O33' O44' tr1'NE tr2'NE unfolding ψ_def tr1 tr2
using Van.validFromS_Cons trn2 ok
using K3 ok' Atrn2 eqSec_S_Cons trn24 trn34
unfolding statA' Van.completedFrom_def
using tr1' tr2' trn3 trn4 by force
next
assume K4: "?K4"
then obtain sv1' sv2' statO' where 0:
"statO' = sstatO' statO sv1 sv2"
"validTransV (sv1,sv1') "
"eqSec sv1 s1"
"validTransV (sv2,sv2')"
"eqSec sv2 s2"
"Van.eqAct sv1 sv2"
and ok': "statA' = Diff ⟶ statO' = Diff" and Δ: "Δ ∞ ∞ ∞ s1' s2' statA' sv1' sv2' statO'"
unfolding match12_12_def by auto
have r12': "reachV sv1'" "reachV sv2'" using r1 r2 0
by (metis Van.reach.Step fst_conv snd_conv)+
obtain trv1 trv2 where ψ: "ψ s1' tr1' s2' tr2' statO' sv1' trv1 sv2' trv2"
using less(1)[of tr1' tr2', OF _ Δ _ _ _ _ _ _ r34' r12', simplified]
using less.prems tr1' tr2' ok' A34' tr1'NE tr2'NE unfolding tr1 tr2 Opt.completedFrom_def by auto
show ?thesis apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "sv2 # trv2"])
using trn34
using ψ O33' O44' isAO34 tr1'NE tr2'NE unfolding ψ_def tr1 tr2
using Van.validFromS_Cons 0
using K4 eqSec_S_Cons
unfolding statA' Van.eqAct_def Van.completedFrom_def match12_12_def sstatO'_def
by simp (smt (z3) Simple_Transition_System.lastt_Cons Van.A.Cons_unfold Van.O.Cons_unfold list.inject status.exhaust status.simps(1) tr1' tr2' trn3 trn4 newStat.simps(4) newStat_diff)
qed
qed
qed
qed
qed
lemma unwindCond_final:
"unwindCond Δ ⟹ reachO s1 ⟹ reachO s2 ⟹ reachV sv1 ⟹ reachV sv2 ⟹ Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹
(finalV sv1 ⟷ finalO s1) ∧ (finalV sv2 ⟷ finalO s2)"
unfolding unwindCond_def
unfolding proact_def react_def match1_def match1_1_def
by auto
definition "φ Δ w w1 w2 w1' w2' statA s1 tr1 s2 tr2 statAA statO sv1 trv1 sv2 trv2 statOO ≡
trv1 ≠ [] ∧ trv2 ≠ [] ∧
(length trv1 > Suc 0 ∨ w1' ≤ w1) ∧ (length trv2 > Suc 0 ∨ w2' ≤ w2) ∧
Van.validFromS sv1 trv1 ∧
Van.validFromS sv2 trv2 ∧
Van.S trv1 = Opt.S tr1 ∧ Van.S trv2 = Opt.S tr2 ∧
Van.A trv1 = Van.A trv2 ∧
(statO = Eq ⟶ (statOO = Diff ⟷ Van.O trv1 ≠ Van.O trv2)) ∧
(statA = Eq ⟶ (statAA = Diff ⟷ Opt.O tr1 ≠ Opt.O tr2)) ∧
(statO = Diff ⟶ statOO = Diff) ∧
(statAA = Diff ⟶ statOO = Diff) ∧
Δ w w1' w2' (lastt s1 tr1) (lastt s2 tr2) statAA (lastt sv1 trv1) (lastt sv2 trv2) statOO"
lemma φ_final:
assumes unw: "unwindCond Δ"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and vtr14: "Opt.validFromS s1 tr1" "Opt.validFromS s2 tr2"
and φ: "φ Δ w w1 w2 w1' w2' statA s1 tr1 s2 tr2 statAA statO sv1 trv1 sv2 trv2 statOO"
shows "(finalV (lastt sv1 trv1) ⟷ finalO (lastt s1 tr1)) ∧ (finalV (lastt sv2 trv2) ⟷ finalO (lastt s2 tr2))"
proof-
have rsv12: "Van.validFromS sv1 trv1 ⟶ reachV (lastt sv1 trv1)"
"Van.validFromS sv2 trv2 ⟶ reachV (lastt sv2 trv2)" using r
by (simp add: Van.reach_validFromS_reach lastt_def)+
have rs14: "Opt.validFromS s1 tr1 ⟶ reachO (lastt s1 tr1)"
"Opt.validFromS s2 tr2 ⟶ reachO (lastt s2 tr2)" using r
by (simp add: Opt.reach_validFromS_reach lastt_def)+
show ?thesis using φ[unfolded φ_def] rsv12 rs14 using unw[unfolded unwindCond_def, rule_format,
of "lastt s1 tr1" "lastt s2 tr2" "lastt sv1 trv1" "lastt sv2 trv2" w w1' w2' statAA statOO]
using vtr14(1) vtr14(2) by auto
qed
lemma φ_completedFrom: "unwindCond Δ ⟹
reachO s1 ⟹ reachO s2 ⟹ reachV sv1 ⟹ reachV sv2 ⟹
Opt.validFromS s1 tr1 ⟹ completedFromO s1 tr1 ⟹
Opt.validFromS s2 tr2 ⟹ completedFromO s2 tr2 ⟹
φ Δ statA w w1 w2 w1' w2' s1 tr1 s2 tr2 statAA statO sv1 trv1 sv2 trv2 statOO
⟹ completedFromV sv1 trv1 ∧ completedFromV sv2 trv2"
using φ_final
by (metis Van.completedFrom_def completedFromO_lastt lastt_def)
lemma unwindCond_ex_φ:
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "(statA = Diff ⟶ statO = Diff)"
and v: "Opt.validFromS s1 tr1" "Opt.validFromS s2 tr2"
and i: "isIntO (lastt s1 tr1)" "isIntO (lastt s2 tr2)"
and nev: "never isIntO (butlast tr1)" "never isIntO (butlast tr2)"
shows "∃w' w1' w2' trv1 trv2 statAA statOO. φ Δ w' w1 w2 w1' w2' statA s1 tr1 s2 tr2 statAA statO sv1 trv1 sv2 trv2 statOO"
using assms(2-)
proof(induction "length tr1 + length tr2" w
arbitrary: w1 w2 s1 s2 statA sv1 sv2 statO tr1 tr2 rule: less2_induct')
case (less w tr1 tr2 w1 w2 s1 s2 statA sv1 sv2 statO)
note ok = `statA = Diff ⟶ statO = Diff`
note Δ = `Δ w w1 w2 s1 s2 statA sv1 sv2 statO`
note r34 = less(4,5) note r12 = less(6,7)
note r = r34 r12
note r3 = r34(1) note r4 = r34(2) note r1 = r12(1) note r2 = r12(2)
note nev34 = less(13,14)
note nev3 = nev34(1) note nev4 = nev34(2)
have i34: "statA = Eq ⟶ isIntO s1 = isIntO s2"
and f34: "finalO s1 = finalO s2 ∧ finalV sv1 = finalO s1 ∧ finalV sv2 = finalO s2"
using Δ unwind[unfolded unwindCond_def] r by auto
note is1 = `isIntO (lastt s1 tr1)`
note is2 = `isIntO (lastt s2 tr2)`
note vtr1 = `Opt.validFromS s1 tr1`
note vtr2 = `Opt.validFromS s2 tr2`
have proact_match: "(∃v<w. proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO) ∨ react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
using Δ unwind[unfolded unwindCond_def] r by auto
show ?case using proact_match proof safe
fix v assume v: "v < w"
assume "proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
thus ?thesis unfolding proact_def proof safe
assume sv1: "¬ isSecV sv1" "¬ isIntV sv1" and "move_1 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv1'
where 0: "validTransV (sv1,sv1')"
and Δ: "Δ v w1 w2 s1 s2 statA sv1' sv2 statO"
unfolding move_1_def by auto
have r1': "reachV sv1'" using r1 0 by (metis Van.reach.Step fst_conv snd_conv)
obtain w' w1' w2' trv1 trv2 statAA statOO where φ: "φ Δ w' w1 w2 w1' w2' statA s1 tr1 s2 tr2 statAA statO sv1' trv1 sv2 trv2 statOO"
using less(2)[OF v, of tr1 tr2 w1 w2 s1 s2 statA sv1' sv2 statO, simplified, OF Δ r34 r1' r2 ok]
using is1 is2 nev3 nev4 vtr1 vtr2 by blast
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1'])
apply(rule exI[of _ w2']) apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ trv2])
using φ ok 0 sv1 unfolding φ_def by auto
next
assume sv2: "¬ isSecV sv2" "¬ isIntV sv2" and "move_2 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv2'
where 0: "validTransV (sv2,sv2')"
and Δ: "Δ v w1 w2 s1 s2 statA sv1 sv2' statO"
unfolding move_2_def by auto
have r2': "reachV sv2'" using r2 0 by (metis Van.reach.Step fst_conv snd_conv)
obtain w' w1' w2' trv1 trv2 statAA statOO where φ: "φ Δ w' w1 w2 w1' w2' statA s1 tr1 s2 tr2 statAA statO sv1 trv1 sv2' trv2 statOO"
using less(2)[OF v, of tr1 tr2 w1 w2 s1 s2 statA sv1 sv2' statO, simplified, OF Δ r34 r1 r2' ok]
using is1 is2 nev3 nev4 vtr1 vtr2 by blast
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ trv1]) apply(rule exI[of _ "sv2 # trv2"])
using φ ok 0 sv2 unfolding φ_def by auto
next
assume sv12: "¬ isSecV sv1" "¬ isSecV sv2" "Van.eqAct sv1 sv2"
and "move_12 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv1' sv2' statO'
where 0: "statO' = sstatO' statO sv1 sv2"
"validTransV (sv1,sv1') " "¬ isSecV sv1"
"validTransV (sv2,sv2')" "¬ isSecV sv2"
"Van.eqAct sv1 sv2"
and Δ: "Δ v w1 w2 s1 s2 statA sv1' sv2' statO'"
unfolding move_12_def by auto
have r12': "reachV sv1'" "reachV sv2'" using r1 r2 0 by (metis Van.reach.Step fst_conv snd_conv)+
have ok': "statA = Diff ⟶ statO' = Diff"
using ok 0 unfolding sstatO'_def by (cases statO, auto)
obtain w' w1' w2' trv1 trv2 statAA statOO where φ: "φ Δ w' w1 w2 w1' w2' statA s1 tr1 s2 tr2 statAA statO' sv1' trv1 sv2' trv2 statOO"
using less(2)[OF v, of tr1 tr2 w1 w2 s1 s2 statA sv1' sv2' statO', simplified, OF Δ r34 r12' ok']
using is1 is2 nev3 nev4 vtr1 vtr2 by blast
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "sv2 # trv2"])
apply(rule exI[of _ statAA]) apply(rule exI[of _ statOO])
using φ ok' 0 sv12 nev unfolding φ_def sstatO'_def
by simp (smt (verit, ccfv_SIG) Statewise_Attacker_Mod.eqAct_def
Van.A.Cons_unfold Van.O.Cons_unfold Van.Statewise_Attacker_Mod_axioms
Van.validFromS_Cons list.inject newStat.simps(1) newStat.simps(4))
qed
next
assume m: "react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
define statA' where statA': "statA' = sstatA' statA s1 s2"
show ?thesis
proof(cases "length tr1 ≤ Suc 0")
case True
hence tr1e: "tr1 = [] ∨ tr1 = [s1]"
by (metis Opt.validFromS_singl_iff Suc_length_conv le_Suc_eq le_zero_eq length_0_conv vtr1)
hence "Opt.A tr1 = []" by (simp add: True)
hence "Opt.A tr2 = []" using Opt.A.eq_Nil_iff nev4 by blast
show ?thesis
proof(cases "length tr2 ≤ Suc 0")
case True
hence tr2e: "tr2 = [] ∨ tr2 = [s2]"
by (metis Opt.validFromS_def Suc_length_conv le_Suc_eq le_zero_eq length_0_conv list.sel(1) vtr2)
show ?thesis apply(rule exI[of _ w]) apply(rule exI[of _ w1]) apply(rule exI[of _ w2])
apply(rule exI[of _ "[sv1]"], rule exI[of _ "[sv2]"], rule exI[of _ statA], rule exI[of _ statO])
using tr1e tr2e
using f34 Δ apply (clarsimp simp: φ_def lastt_def)
apply(cases statA, simp_all)
apply (metis Opt.O.simps(4) Opt.S.simps(4) last_ConsL)
by (metis Opt.S.simps(4) last.simps ok)
next
case False
then obtain s24 tr2' where tr2: "tr2 = s24 # tr2'" and tr2'NE: "tr2' ≠ []"
by (cases tr2, auto)
have s24[simp]: "s24 = s2" using `Opt.validFromS s2 tr2`
by (simp add: Opt.validFromS_Cons_iff tr2)
obtain s2' where
trn4: "validTransO (s2,s2') ∨ (s2 = s2' ∧ tr2' = [])" and
tr2': "Opt.validFromS s2' tr2'" using `Opt.validFromS s2 tr2`
unfolding tr2 s24 using Opt.validFromS_Cons_iff by auto
have r4': "reachO s2'"
using r4 trn4 by (metis Opt.reach.Step fst_conv snd_conv)+
have nev4': "never isIntO (butlast tr2')"
by (metis Opt.O.Nil_iff Opt.O.eq_Nil_iff nev4 tr2)
have isAO4: "¬ isIntO s2"
using ‹Opt.A tr2 = []› tr2 tr2'NE by auto
have O44': "Opt.O tr2 = Opt.O tr2'" "Opt.A tr2 = Opt.A tr2'"
using isAO4 ‹Opt.A tr2 = []› tr2 by auto
have m: "match2 Δ w1 w2 s1 s2 statA sv1 sv2 statO" using m unfolding react_def by auto
have "(∃w1'<w1. ∃w2'<w2. ¬ isSecO s2 ∧ Δ ∞ w1' w2' s1 s2' statA sv1 sv2 statO) ∨
(∃w1'<w1. eqSec sv2 s2 ∧ ¬ isIntV sv2 ∧ match2_1 Δ w1' ∞ s1 s2 s2' statA sv1 sv2 statO) ∨
(¬ isSecV sv1 ∧ eqSec sv2 s2 ∧ Van.eqAct sv1 sv2 ∧ match2_12 Δ ∞ ∞ s1 s2 s2' statA sv1 sv2 statO)"
using isAO4 trn4 ok tr2'NE
using m[unfolded match2_def, rule_format, of s2'] by auto
thus ?thesis
proof safe
fix w1'' w2'' assume w12': "w1'' < w1" "w2'' < w2"
assume "¬ isSecO s2" and Δ: "Δ ∞ w1'' w2'' s1 s2' statA sv1 sv2 statO"
hence S4: "Opt.S tr2' = Opt.S tr2" unfolding tr2 by auto
obtain w' w1' w2' trv1 trv2 statAA statOO where φ: "φ Δ w' w1'' w2'' w1' w2' statA s1 tr1 s2' tr2' statAA statO sv1 trv1 sv2 trv2 statOO"
using less(1)[of tr1 tr2', OF _ Δ r3 r4' _ _ _ _ _ _ _ nev3 nev4', unfolded tr2, simplified]
using is1 is2 vtr1 vtr2 tr2' ok tr2'NE trn4 r1 r2 tr2 by auto
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ trv1]) apply(rule exI[of _ trv2])
using φ O44' S4 tr2 tr2'NE trn4 tr2' w12' unfolding φ_def by auto
next
fix w1'' assume w1': "w1'' < w1"
assume trn24: "eqSec sv2 s2" and
Atrn2: "¬ isIntV sv2" and "match2_1 Δ w1'' ∞ s1 s2 s2' statA sv1 sv2 statO"
then obtain sv2' where trn2: "validTransV (sv2,sv2')" and
Δ: "Δ ∞ w1'' ∞ s1 s2' statA sv1 sv2' statO"
unfolding match2_1_def by auto
have r2': "reachV sv2'" using r2 trn2 by (metis Van.reach.Step fst_conv snd_conv)
obtain w' w1' w2' trv1 trv2 statAA statOO where φ: "φ Δ w' w1'' ∞ w1' w2' statA s1 tr1 s2' tr2' statAA statO sv1 trv1 sv2' trv2 statOO"
using less(1)[of tr1 tr2', OF _ Δ r3 r4' r1 r2' _ _ _ _ _ nev3 nev4', unfolded tr2, simplified]
using is1 is2 tr2' tr2 vtr1 ok tr2'NE trn4 by auto
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ trv1]) apply(rule exI[of _ "sv2 # trv2"])
using φ tr2'NE
using Van.validFromS_Cons trn2
using isAO4 ok Atrn2 eqSec_S_Cons trn24 tr2' trn4 w1'
unfolding φ_def tr2 s24
by auto
next
assume sv1: "¬ isSecV sv1" and trn24: "eqSec sv2 s2" and
Atrn12: "Van.eqAct sv1 sv2" and "match2_12 Δ ∞ ∞ s1 s2 s2' statA sv1 sv2 statO"
then obtain sv1' sv2' statO' where
statO': "statO' = sstatO' statO sv1 sv2" and
trn1: "validTransV (sv1,sv1')" and
trn2: "validTransV (sv2,sv2')" and
Δ: "Δ ∞ ∞ ∞ s1 s2' statA sv1' sv2' statO'"
unfolding match2_12_def by auto
have r12': "reachV sv1'" "reachV sv2'"
using r1 trn1 r2 trn2 by (metis Van.reach.Step fst_conv snd_conv)+
obtain w' w1' w2' trv1 trv2 statAA statOO where φ: "φ Δ w' ∞ ∞ w1' w2' statA s1 tr1 s2' tr2' statAA statO' sv1' trv1 sv2' trv2 statOO"
using less(1)[of tr1 tr2', OF _ Δ r3 r4' r12' _ _ _ _ _ nev3 nev4', simplified]
using is1 is2 vtr1 tr2 tr2' ok tr2'NE trn4 unfolding tr2 statO' sstatO'_def by auto
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "sv2 # trv2"])
using φ O44' tr2'NE sv1
using Van.validFromS_Cons trn1 trn2
using isAO4 ok Atrn12 eqSec_S_Cons trn24 tr2' trn4
unfolding φ_def tr2 Van.completedFrom_def Van.eqAct_def statO' sstatO'_def
by simp (smt (verit, ccfv_threshold) Van.A.Cons_unfold i34 is1 last_ConsL
lastt_def status.exhaust tr1e newStat.simps(2))
qed
qed
next
case False
then obtain s13 tr1' where tr1: "tr1 = s13 # tr1'" and tr1'NE: "tr1' ≠ []"
by (cases tr1, auto)
have s13[simp]: "s13 = s1" using `Opt.validFromS s1 tr1`
by (simp add: Opt.validFromS_Cons_iff tr1)
obtain s1' where
trn3: "validTransO (s1,s1')" and
tr1': "Opt.validFromS s1' tr1'" using `Opt.validFromS s1 tr1`
unfolding tr1 s13 by (metis tr1'NE Simple_Transition_System.validFromS_Cons_iff)
have r3': "reachO s1'" using r3 trn3 by (metis Opt.reach.Step fst_conv snd_conv)
have f3: "¬ finalO s1" using Opt.final_def trn3 by blast
hence f4: "¬ finalO s2" using f34 by blast
have nev3': "never isIntO (butlast tr1')"
using nev3 tr1 tr1'NE by auto
have isAO3: "¬ isIntO s1" using less.prems(11) tr1 tr1'NE by auto
have O33': "Opt.O tr1 = Opt.O tr1'" "Opt.A tr1 = Opt.A tr1'"
using isAO3 unfolding tr1 by auto
have m: "match1 Δ w1 w2 s1 s2 statA sv1 sv2 statO" using m unfolding react_def by auto
have "(∃w1'<w1. ∃w2'<w2. ¬ isSecO s1 ∧ Δ ∞ w1' w2' s1' s2 statA sv1 sv2 statO) ∨
(∃w2'<w2. eqSec sv1 s1 ∧ ¬ isIntV sv1 ∧ match1_1 Δ ∞ w2' s1 s1' s2 statA sv1 sv2 statO) ∨
(eqSec sv1 s1 ∧ ¬ isSecV sv2 ∧ Van.eqAct sv1 sv2 ∧ match1_12 Δ ∞ ∞ s1 s1' s2 statA sv1 sv2 statO)"
using m isAO3 trn3 ok unfolding match1_def by auto
thus ?thesis
proof safe
fix w1'' w2'' assume w12': "w1'' < w1" "w2'' < w2"
assume "¬ isSecO s1" and Δ: "Δ ∞ w1'' w2'' s1' s2 statA sv1 sv2 statO"
hence S3: "Opt.S tr1' = Opt.S tr1" unfolding tr1 by auto
obtain w' w1' w2' trv1 trv2 statAA statOO where φ: "φ Δ w' w1'' w2'' w1' w2' statA s1' tr1' s2 tr2 statAA statO sv1 trv1 sv2 trv2 statOO"
using less(1)[of tr1' tr2, OF _ Δ r3' r4 r12, unfolded O33', simplified]
using is1 is2 tr1' ok f3 f4 tr1'NE trn3 O33'(1) nev3' nev4 vtr2 unfolding tr1 by auto
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ trv1]) apply(rule exI[of _ trv2])
using φ O33' S3 tr1 tr1'NE tr1' trn3 w12' unfolding φ_def by auto
next
fix w2'' assume w2': "w2'' < w2"
assume trn13: "eqSec sv1 s1" and
Atrn1: "¬ isIntV sv1" and "match1_1 Δ ∞ w2'' s1 s1' s2 statA sv1 sv2 statO"
then obtain sv1' where
trn1: "validTransV (sv1,sv1') " and
Δ: "Δ ∞ ∞ w2'' s1' s2 statA sv1' sv2 statO"
unfolding match1_1_def by auto
have r1': "reachV sv1'"using r1 trn1 by (metis Van.reach.Step fst_conv snd_conv)
obtain w' w1' w2' trv1 trv2 statAA statOO where φ: "φ Δ w' ∞ w2'' w1' w2' statA s1' tr1' s2 tr2 statAA statO sv1' trv1 sv2 trv2 statOO"
using less(1)[of tr1' tr2, OF _ Δ r3' r4 r1' r2, unfolded O33', simplified]
using is1 is2 tr1 nev3' nev4 vtr1 vtr2 tr1' ok f3 f4 tr1'NE trn3 O33'(1)
unfolding tr1 by auto
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ trv2])
using φ O33' unfolding φ_def tr1 Van.completedFrom_def
using Van.validFromS_Cons trn1 tr1'NE tr1' trn3
using isAO3 ok Atrn1 eqSec_S_Cons trn13 w2'
by auto
next
assume sv2: "¬ isSecV sv2" and trn13: "eqSec sv1 s1" and
Atrn12: "Van.eqAct sv1 sv2" and "match1_12 Δ ∞ ∞ s1 s1' s2 statA sv1 sv2 statO"
then obtain sv1' sv2' statO' where
statO': "statO' = sstatO' statO sv1 sv2" and
trn1: "validTransV (sv1,sv1') " and
trn2: "validTransV (sv2,sv2')" and
Δ: "Δ ∞ ∞ ∞ s1' s2 statA sv1' sv2' statO'"
unfolding match1_12_def by auto
have r12': "reachV sv1'" "reachV sv2'"
using r1 trn1 r2 trn2 by (metis Van.reach.Step fst_conv snd_conv)+
obtain w' w1' w2' trv1 trv2 statAA statOO where φ: "φ Δ w' ∞ ∞ w1' w2' statA s1' tr1' s2 tr2 statAA statO' sv1' trv1 sv2' trv2 statOO"
using less(1)[of tr1' tr2, OF _ Δ r3' r4 r12', unfolded O33', simplified]
using less.prems tr1' ok f3 f4 tr1'NE trn3 O33'(1) unfolding tr1 statO' sstatO'_def by auto
have trv1NE: "trv1 ≠ []" and trv2NE: "trv2 ≠ []" using φ unfolding φ_def by auto
have [simp]: "Van.O (sv1 # trv1) = Van.O (sv2 # trv2) ⟷ (isIntV sv1 ⟶ getObsV sv1 = getObsV sv2) ∧ Van.O trv1 = Van.O trv2"
using Atrn12 trv1NE trv2NE unfolding Van.O.map_filter Van.eqAct_def by simp
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "sv2 # trv2"])
using φ O33' tr1'NE sv2
using Van.validFromS_Cons trn1 trn2
using isAO3 ok Atrn12 eqSec_S_Cons trn13 f3 f34 s13 tr1' trn3
unfolding φ_def tr1 Van.completedFrom_def Van.eqAct_def statO' sstatO'_def apply clarsimp
by (smt (verit, ccfv_SIG) Van.A.Cons_unfold newStat.simps(1) newStat.simps(2) newStat.simps(4))
qed
qed
qed
qed
definition "φa Δ w w1 w2 w1' w2' statA s1 tr1 s2 tr2 statAA statO sv1 trv1 sv2 trv2 statOO ≡
trv1 ≠ [] ∧ trv2 ≠ [] ∧
(length trv1 > Suc 0 ∨ w1' < w1) ∧ (length trv2 > Suc 0 ∨ w2' < w2) ∧
Van.validFromS sv1 trv1 ∧
Van.validFromS sv2 trv2 ∧
Van.S trv1 = Opt.S tr1 ∧ Van.S trv2 = Opt.S tr2 ∧
Van.A trv1 = Van.A trv2 ∧
(statO = Eq ⟶ (statOO = Diff ⟷ Van.O trv1 ≠ Van.O trv2)) ∧
(statA = Eq ⟶ (statAA = Diff ⟷ Opt.O tr1 ≠ Opt.O tr2)) ∧
(statO = Diff ⟶ statOO = Diff) ∧
(statAA = Diff ⟶ statOO = Diff) ∧
Δ w w1' w2' (lastt s1 tr1) (lastt s2 tr2) statAA (lastt sv1 trv1) (lastt sv2 trv2) statOO"
lemma unwindCond_ex_φa_getActO:
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO"
and r34: "reachO s1" "reachO s2" and r12: "reachV sv1" "reachV sv2"
and stat: "(statA = Diff ⟶ statO = Diff)"
and v: "validTransO (s1, s1')" "validTransO (s2, s2')"
and i34: "isIntO s1" "isIntO s2" "getActO s1 = getActO s2"
shows "∃w1' w2' trv1 trv2 statOO.
φa Δ ∞ w1 w2 w1' w2' statA s1 [s1, s1'] s2 [s2, s2'] (sstatA' statA s1 s2) statO sv1 trv1 sv2 trv2 statOO"
using Δ r12 stat
proof(induction w arbitrary: w1 w2 sv1 sv2 statO rule: less_induct)
case (less w w1 w2 sv1 sv2 statO)
note Δ = `Δ w w1 w2 s1 s2 statA sv1 sv2 statO`
note r12 = less.prems(2,3)
note r1 = r12(1) note r2 = r12(2)
note r = r34 r12
note stat = `statA = Diff ⟶ statO = Diff`
have f34: "finalO s1 = finalO s2 ∧ finalV sv1 = finalO s1 ∧ finalV sv2 = finalO s2"
using Δ unwind[unfolded unwindCond_def] r by auto
have proact_match: "(∃v<w. proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO) ∨ react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
using Δ unwind[unfolded unwindCond_def] r by auto
show ?case using proact_match proof safe
fix v assume v: "v < w"
assume "proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
thus ?thesis unfolding proact_def proof safe
assume sv1: "¬ isSecV sv1" "¬ isIntV sv1" and "move_1 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv1'
where 0: "validTransV (sv1,sv1')"
and Δ: "Δ v w1 w2 s1 s2 statA sv1' sv2 statO"
unfolding move_1_def by auto
have r1': "reachV sv1'" using r1 0 by (metis Van.reach.Step fst_conv snd_conv)
obtain w1' w2' trv1 trv2 statOO where
φ: "φa Δ ∞ w1 w2 w1' w2' statA s1 [s1, s1'] s2 [s2, s2'] (sstatA' statA s1 s2) statO sv1' trv1 sv2 trv2 statOO"
using less(1)[OF v Δ r1' r2 stat] by auto
show ?thesis apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ trv2])
using φ 0 sv1 unfolding φa_def apply simp
by (metis Van.validFromS_Cons)
next
assume sv2: "¬ isSecV sv2" "¬ isIntV sv2" and "move_2 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv2'
where 0: "validTransV (sv2,sv2')"
and Δ: "Δ v w1 w2 s1 s2 statA sv1 sv2' statO"
unfolding move_2_def by auto
have r2': "reachV sv2'" using r2 0 by (metis Van.reach.Step fst_conv snd_conv)
obtain w1' w2' trv1 trv2 statOO where
φ: "φa Δ ∞ w1 w2 w1' w2' statA s1 [s1, s1'] s2 [s2, s2'] (sstatA' statA s1 s2) statO sv1 trv1 sv2' trv2 statOO"
using less(1)[OF v Δ r1 r2' stat] by auto
show ?thesis apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "trv1"]) apply(rule exI[of _ "sv2 # trv2"])
using φ 0 sv2 unfolding φa_def apply simp by (metis Van.validFromS_Cons)
next
assume sv12: "¬ isSecV sv1" "¬ isSecV sv2" "Van.eqAct sv1 sv2"
and "move_12 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv1' sv2' statO'
where 0: "statO' = sstatO' statO sv1 sv2"
"validTransV (sv1,sv1') " "¬ isSecV sv1"
"validTransV (sv2,sv2')" "¬ isSecV sv2"
"Van.eqAct sv1 sv2"
and Δ: "Δ v w1 w2 s1 s2 statA sv1' sv2' statO'"
unfolding move_12_def by auto
have r12': "reachV sv1'" "reachV sv2'" using r1 r2 0 by (metis Van.reach.Step fst_conv snd_conv)+
have stat': "statA = Diff ⟶ statO' = Diff"
using stat 0 unfolding sstatO'_def by (cases statO, auto)
obtain w1' w2' trv1 trv2 statOO where
φ: "φa Δ ∞ w1 w2 w1' w2' statA s1 [s1, s1'] s2 [s2, s2'] (sstatA' statA s1 s2) statO' sv1' trv1 sv2' trv2 statOO"
using less(1)[OF v Δ r12' stat'] unfolding φa_def apply simp by metis
show ?thesis apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "sv2 # trv2"])
using φ 0 unfolding φa_def sstatO'_def apply clarsimp apply(intro conjI)
subgoal by auto
subgoal by auto
subgoal by (metis Van.A.Cons_unfold Van.eqAct_def)
subgoal apply(rule exI[of _ statOO]) apply simp
by (smt (verit, ccfv_threshold) Van.O.Cons_unfold Van.eqAct_def
list.inject newStat.simps(1) newStat.simps(3)) .
qed
next
assume m: "react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
define statA' where statA': "statA' = sstatA' statA s1 s2"
have m: "match12 Δ w1 w2 s1 s2 statA sv1 sv2 statO" using m unfolding react_def by auto
have "(∃w1' w2'. w1' < w1 ∧ w2'< w2 ∧ ¬ isSecO s1 ∧ ¬ isSecO s2 ∧ (statA = statA' ∨ statO = Diff) ∧
Δ ∞ w1' w2' s1' s2' statA' sv1 sv2 statO)
∨
(∃w2'< w2. ¬ isSecO s2 ∧
eqSec sv1 s1 ∧ ¬ isIntV sv1 ∧ (statA = statA' ∨ statO = Diff) ∧
match12_1 Δ ∞ w2' s1' s2' statA' sv1 sv2 statO)
∨
(∃w1'<w1. ¬ isSecO s1 ∧
eqSec sv2 s2 ∧ ¬ isIntV sv2 ∧ (statA = statA' ∨ statO = Diff) ∧
match12_2 Δ w1' ∞ s1' s2' statA' sv1 sv2 statO)
∨
(eqSec sv1 s1 ∧ eqSec sv2 s2 ∧ Van.eqAct sv1 sv2 ∧
match12_12 Δ ∞ ∞ s1' s2' statA' sv1 sv2 statO)"
using m unfolding match12_def
by (simp add: Opt.eqAct_def i34(1) i34(2) i34(3) statA' v(1) v(2))
thus ?thesis
apply(elim disjE exE)
subgoal for w1' w2' apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ "[sv1]"]) apply(rule exI[of _ "[sv2]"])
apply(rule exI[of _ statO])
using stat unfolding φa_def statA'
by (auto simp add: i34(1) i34(2) sstatA'_def lastt_def)
subgoal for w2' apply(rule exI[of _ ∞]) apply(rule exI[of _ w2'])
unfolding match12_1_def apply(elim conjE exE) subgoal for sv1'
apply(rule exI[of _ "[sv1,sv1']"]) apply(rule exI[of _ "[sv2]"])
apply(rule exI[of _ statO])
using stat unfolding φa_def statA'
by (auto simp add: i34(1) i34(2) sstatA'_def lastt_def) .
subgoal for w1' apply(rule exI[of _ w1']) apply(rule exI[of _ ∞])
unfolding match12_2_def apply(elim conjE exE) subgoal for sv2'
apply(rule exI[of _ "[sv1]"]) apply(rule exI[of _ "[sv2,sv2']"])
apply(rule exI[of _ statO])
using stat unfolding φa_def statA'
by (auto simp add: i34(1) i34(2) sstatA'_def lastt_def) .
subgoal unfolding match12_12_def apply(elim conjE exE) subgoal for sv1' sv2'
apply(rule exI[of _ ∞]) apply(rule exI[of _ ∞])
apply(rule exI[of _ "[sv1,sv1']"]) apply(rule exI[of _ "[sv2,sv2']"])
apply(rule exI[of _ "sstatO' statO sv1 sv2"])
using stat unfolding φa_def statA'
by (auto simp add: i34 i34 sstatA'_def sstatO'_def lastt_def Van.eqAct_def) . .
qed
qed
lemma unwindCond_ex_φa'_aux:
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "(statA = Diff ⟶ statO = Diff)"
and tr14NE: "tr1 ≠ []" "tr2 ≠ []"
and v3': "Opt.validFromS s1 (tr1 ## s1')" and v4': "Opt.validFromS s2 (tr2 ## s2')"
and i: "isIntO (lastt s1 tr1)" "isIntO (lastt s2 tr2)"
and A34: "getActO (lastt s1 tr1) = getActO (lastt s2 tr2)"
and nev: "never isIntO (butlast tr1)" "never isIntO (butlast tr2)"
shows "∃w1' w2' trv1' trv2' statAA' statOO'.
φa Δ ∞ w1 w2 w1' w2' statA s1 (tr1 ## s1') s2 (tr2 ## s2') statAA' statO sv1 trv1' sv2 trv2' statOO'"
proof-
have v3: "Opt.validFromS s1 tr1" and s13': "validTransO (lastt s1 tr1,s1')"
apply (metis v3' Opt.validFromS_def Opt.validS_append1 Nil_is_append_conv hd_append2)
by (metis Opt.validFromS_def Opt.validS_validTrans append_is_Nil_conv lastt_def list.distinct(1) list.sel(1) tr14NE(1) v3')
have v4: "Opt.validFromS s2 tr2" and s24': "validTransO (lastt s2 tr2,s2')"
apply (metis v4' Opt.validFromS_def Opt.validS_append1 Nil_is_append_conv hd_append2)
by (metis Opt.validFromS_def Opt.validS_validTrans append_is_Nil_conv lastt_def list.sel(1) list.simps(3) tr14NE(2) v4')
obtain ww ww1 ww2 trv1 trv2 statAA statOO where φ: "φ Δ ww w1 w2 ww1 ww2 statA s1 tr1 s2 tr2 statAA statO sv1 trv1 sv2 trv2 statOO"
using unwindCond_ex_φ[OF unwind Δ r stat v3 v4 i nev] by auto
have trv12NE: "trv1 ≠ []" "trv2 ≠ []" using φ unfolding φ_def by auto
define ss1 ss2 ssv1 ssv2 where ss1: "ss1 ≡ lastt s1 tr1" and ss2: "ss2 ≡ lastt s2 tr2"
and ssv1: "ssv1 ≡ lastt sv1 trv1" and ssv2: "ssv2 ≡ lastt sv2 trv2"
have ss1l: "ss1 = last tr1" by (simp add: lastt_def ss1 tr14NE(1))
have tr1l: "tr1 = butlast tr1 @ [ss1]" by (simp add: ss1l tr14NE(1))
have ss2l: "ss2 = last tr2" by (simp add: lastt_def ss2 tr14NE(2))
have tr2l: "tr2 = butlast tr2 @ [ss2]" by (simp add: ss2l tr14NE(2))
have ssv1l: "ssv1 = last trv1" using φ unfolding φ_def by (metis lastt_def ssv1)
have trv1l: "trv1 = butlast trv1 @ [ssv1]" by (simp add: ssv1l trv12NE(1))
have ssv2l: "ssv2 = last trv2" using φ unfolding φ_def by (metis lastt_def ssv2)
have trv2l: "trv2 = butlast trv2 @ [ssv2]" by (simp add: ssv2l trv12NE(2))
have iss14[simp]: "isIntO ss1" "isIntO ss2" using i unfolding ss1 ss2 by auto
have giss14[simp]: "getActO ss1 = getActO ss2"
using A34 ss1 ss2 by fastforce
have [simp]: "Opt.O (tr1 ## s1') = Opt.O tr1 ## getObsO ss1"
by (metis Opt.O_def ‹isIntO ss1› holds_filtermap_RCons snoc_eq_iff_butlast tr1l)
have [simp]: "Opt.O (tr2 ## s2') = Opt.O tr2 ## getObsO ss2"
by (metis Opt.O_def ‹isIntO ss2› holds_filtermap_RCons snoc_eq_iff_butlast tr2l)
have [simp]: "Opt.A (tr1 ## s1') = Opt.A tr1 ## getActO ss1"
by (metis Opt.A_def ‹isIntO ss1› holds_filtermap_RCons snoc_eq_iff_butlast tr1l)
have [simp]: "Opt.A (tr2 ## s2') = Opt.A tr2 ## getActO ss2"
by (metis Opt.A_def ‹isIntO ss2› holds_filtermap_RCons snoc_eq_iff_butlast tr2l)
have [simp]: "Opt.A (tr1 ## s1') = Opt.A (tr2 ## s2') ⟷ Opt.A tr1 = Opt.A tr2" by simp
have rss: "reachO ss1" "reachO ss2" "reachV ssv1" "reachV ssv2"
using Opt.reach_validFromS_reach r ss1l tr14NE(1) v3 apply blast
using Opt.reach_validFromS_reach r(2) ss2l tr14NE(2) v4 apply blast
using Van.reach_validFromS_reach φ_def φ r(3) ssv1l
apply (smt (verit, del_insts))
using Van.reach_validFromS_reach φ_def φ r(4) ssv2l
apply (smt (verit, del_insts)) .
have stat: "statAA = Diff ⟶ statOO = Diff"
and Δ: "Δ ww ww1 ww2 ss1 ss2 statAA ssv1 ssv2 statOO"
using φ unfolding φ_def ss1[symmetric] ss2[symmetric] ssv1[symmetric] ssv2[symmetric] by auto
note vs13 = s13'[unfolded ss1[symmetric]] note vs24 = s24'[unfolded ss2[symmetric]]
have "∃ w1' w2' trv1' trv2' statA' statO'.
φa Δ ∞ ww1 ww2 w1' w2' statAA ss1 [ss1,s1'] ss2 [ss2,s2'] (sstatA' statAA ss1 ss2) statOO ssv1 trv1' ssv2 trv2' statO'"
using unwindCond_ex_φa_getActO[OF unwind Δ rss stat vs13 vs24 iss14 giss14]
by blast
then obtain w1' w2' trv1' trv2' statA' statO' where
φ1: "φa Δ ∞ ww1 ww2 w1' w2' statAA ss1 [ss1,s1'] ss2 [ss2,s2'] statA' statOO ssv1 trv1' ssv2 trv2' statO'" by auto
have trv12'NE: "trv1' ≠ []" "trv2' ≠ []" using φ1 unfolding φa_def by auto
have [simp]: "Van.O (butlast trv1 @ trv1') = Van.O trv1 @ Van.O trv1'"
using trv12'NE unfolding φ_def Van.O.map_filter Opt.O.map_filter apply(subst butlast_append) by simp
have [simp]: "Van.O (butlast trv2 @ trv2') = Van.O trv2 @ Van.O trv2'"
using trv12'NE unfolding φ_def Van.O.map_filter Opt.O.map_filter apply(subst butlast_append) by simp
have "Van.A trv1' = Van.A trv2'" using φ1 unfolding φa_def by auto
moreover have "length (Van.O trv1') = length (Van.A trv1') ∧ length (Van.O trv2') = length (Van.A trv2')"
unfolding Van.A.map_filter Van.O.map_filter by auto
ultimately have "length (Van.O trv1') = length (Van.O trv2')" by auto
hence [simp]: "Van.O trv1 @ Van.O trv1' = Van.O trv2 @ Van.O trv2' ⟷
Van.O trv1 = Van.O trv2 ∧ Van.O trv1' = Van.O trv2'" by auto
have len: "trv1 ≠ [] ∧ trv2 ≠ [] ∧ trv1' ≠ [] ∧ trv2' ≠ [] ∧
(Suc 0 < length trv1 ∨ ww1 ≤ w1) ∧
(Suc 0 < length trv1' ∨ w1' < ww1) ∧
(Suc 0 < length trv2 ∨ ww2 ≤ w2) ∧
(Suc 0 < length trv2' ∨ w2' < ww2)"
using φ φ1 unfolding φ_def φa_def by auto
show ?thesis
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ "butlast trv1 @ trv1'"]) apply(rule exI[of _ "butlast trv2 @ trv2'"])
apply(rule exI[of _ statA']) apply(rule exI[of _ statO'])
unfolding φa_def apply(intro conjI)
subgoal using φ φ1 unfolding φ_def φa_def by auto
subgoal using φ φ1 unfolding φ_def φa_def by auto
subgoal using len
by simp (metis Suc_lessI add_is_1 diff_is_0_eq length_greater_0_conv linorder_not_less
order_trans trans_less_add2)
subgoal using len
by simp (metis Suc_leI le_add_diff_inverse2 length_greater_0_conv nless_le order_le_less_trans trans_less_add2)
subgoal using φ φ1 unfolding φ_def φa_def ssv1
using Van.validFromS_append by auto
subgoal using φ φ1 unfolding φ_def φa_def ssv2
using Van.validFromS_append by auto
subgoal using φ φ1 unfolding φ_def φa_def Van.S.map_filter Opt.S.map_filter
apply(subst tr1l) apply(subst butlast_append) by simp
subgoal using φ φ1 unfolding φ_def φa_def Van.S.map_filter Opt.S.map_filter
apply(subst tr2l) apply(subst butlast_append) by simp
subgoal using φ φ1 unfolding φ_def φa_def Van.A.map_filter Opt.A.map_filter
apply(subst trv1l) apply(subst trv2l)
apply(subst butlast_append) apply simp apply(subst butlast_append) by simp
subgoal using φ φ1 unfolding φ_def φa_def apply simp
apply(cases "Opt.O tr1 = Opt.O tr2", simp_all) apply clarify
using status.exhaust by (metis (full_types))+
subgoal using φ φ1 unfolding φ_def φa_def apply simp
apply(cases "Opt.O tr1 = Opt.O tr2", simp_all) apply clarify
apply (smt (verit, del_insts) status.exhaust)
by (metis Opt.O.eq_Nil_iff nev(1) nev(2))
subgoal using φ φ1 unfolding φ_def φa_def by simp
subgoal using φ φ1 unfolding φ_def φa_def by simp
subgoal using φ1 trv12'NE tr14NE unfolding φ_def φa_def lastt_def by simp .
qed
lemma unwindCond_ex_φa_aux2:
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "(statA = Diff ⟶ statO = Diff)"
and v3': "Opt.validFromS s1 (tr1 @ [s1',s1''])" and v4': "Opt.validFromS s2 (tr2 @ [s2',s2''])"
and i: "isIntO s1'" "isIntO s2'"
and A34: "getActO s1' = getActO s2'"
and nev: "never isIntO tr1" "never isIntO tr2"
shows "∃w1' w2' trv1 trv2 statAA statOO.
φa Δ ∞ w1 w2 w1' w2' statA s1 (tr1 @ [s1',s1'']) s2 (tr2 @ [s2',s2'']) statAA statO sv1 trv1 sv2 trv2 statOO"
proof-
have 0: "lastt s1 (tr1 ## s1') = s1'" "lastt s2 (tr2 ## s2') = s2'"
unfolding lastt_def by auto
show ?thesis
apply(rule unwindCond_ex_φa'_aux[OF unwind Δ r stat, of "tr1 ## s1'" "tr2 ## s2'", unfolded 0, simplified])
using assms by auto
qed
lemma lastt_snoc[simp]: "lastt s1 (tr1 @ [s1'']) = s1''"
unfolding lastt_def by auto
lemma lastt_snoc2[simp]: "lastt s1 (tr1 @ [s1', s1'']) = s1''"
unfolding lastt_def by auto
lemma append_snoc2: "tr1 @ [s1', s1''] = (tr1 ## s1') ## s1''"
by auto
definition "φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO ≡
(trv1 ≠ [] ∨ w1' < w1) ∧ (trv2 ≠ [] ∨ w2' < w2) ∧
Van.validFromS sv1 (trv1 ## sv1'') ∧ Van.validFromS sv2 (trv2 ## sv2'') ∧
Van.S (trv1 ## sv1'') = Opt.S ((tr1 ## s1') ## s1'') ∧ Van.S (trv2 ## sv2'') = Opt.S ((tr2 ## s2') ## s2'') ∧
Van.A (trv1 ## sv1'') = Van.A (trv2 ## sv2'') ∧
(statO = Eq ⟶ (statOO = Diff) = (Van.O (trv1 ## sv1'') ≠ Van.O (trv2 ## sv2''))) ∧
(statA = Eq ⟶ (statAA = Diff) = (Opt.O ((tr1 ## s1') ## s1'') ≠ Opt.O ((tr2 ## s2') ## s2''))) ∧
(statO = Diff ⟶ statOO = Diff) ∧ (statAA = Diff ⟶ statOO = Diff) ∧
Δ ∞ w1' w2' s1'' s2'' statAA sv1'' sv2'' statOO"
proposition unwindCond_ex_φ':
assumes unwind: "unwindCond Δ" and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and v3': "Opt.validFromS s1 ((tr1 ## s1') ## s1'')" and v4': "Opt.validFromS s2 ((tr2 ## s2') ## s2'')"
and i: "isIntO s1'" "isIntO s2'"
and A34: "getActO s1' = getActO s2'"
and nev: "never isIntO tr1" "never isIntO tr2"
shows "∃w1' w2' trv1 sv1'' trv2 sv2'' statAA statOO.
φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
using unwindCond_ex_φa_aux2[unfolded φ_def, unfolded lastt_snoc lastt_snoc2 append_snoc2, OF assms]
unfolding φa_def apply(elim exE) subgoal for w1' w2' trv1 trv2 statAA statOO
apply(cases "trv1" rule: rev_cases)
subgoal by auto
apply(cases "trv2" rule: rev_cases)
subgoal by auto
subgoal unfolding φ'_def apply simp by blast . .
definition "χ3 Δ w (w1::enat) w2 w1' w2' s1 tr1 s2 statAA sv1 trv1 sv2 trv2 statOO ≡
trv1 ≠ [] ∧ trv2 ≠ [] ∧ (length trv2 > Suc 0 ∨ w2' ≤ w2) ∧
Van.validFromS sv1 trv1 ∧ Van.validFromS sv2 trv2 ∧
never isSecV (butlast trv1) ∧
isSecV (lastt sv1 trv1) ∧ getSecV (lastt sv1 trv1) = getSecO (lastt s1 tr1) ∧
never isSecV (butlast trv2) ∧
Van.A trv1 = Van.A trv2 ∧
Δ w w1' w2' (lastt s1 tr1) s2 statAA (lastt sv1 trv1) (lastt sv2 trv2) statOO"
lemma χ3_final:
assumes unw: "unwindCond Δ"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and vtr1: "Opt.validFromS s1 tr1"
and χ3: "χ3 Δ w w1 w2 w1' w2' s1 tr1 s2 statAA sv1 trv1 sv2 trv2 statOO"
shows "(finalV (lastt sv1 trv1) ⟷ finalO (lastt s1 tr1)) ∧ (finalV (lastt sv2 trv2) ⟷ finalO s2)"
proof-
have rsv12: "Van.validFromS sv1 trv1 ⟶ reachV (lastt sv1 trv1)"
"Van.validFromS sv2 trv2 ⟶ reachV (lastt sv2 trv2)" using r
by (simp add: Van.reach_validFromS_reach lastt_def)+
have rs1: "Opt.validFromS s1 tr1 ⟶ reachO (lastt s1 tr1)"
using r
by (simp add: Opt.reach_validFromS_reach lastt_def)+
show ?thesis using χ3[unfolded χ3_def] rsv12 rs1 using unw[unfolded unwindCond_def, rule_format,
of "lastt s1 tr1" s2 "lastt sv1 trv1" "lastt sv2 trv2" w w1' w2' statAA statOO]
using vtr1 `reachO s2` by auto
qed
lemma χ3_completedFrom: "unwindCond Δ ⟹
reachO s1 ⟹ reachO s2 ⟹ reachV sv1 ⟹ reachV sv2 ⟹
Opt.validFromS s1 tr1 ⟹ completedFromO s1 tr1 ⟹
χ3 Δ w w1 w2 w1' w2' s1 tr1 s2 statAA sv1 trv1 sv2 trv2 statOO
⟹ completedFromV sv1 trv1 ∧ completedFromV sv2 trv2"
by (metis Van.final_not_isSec χ3_def χ3_final completedFromO_lastt)
lemma unwindCond_ex_χ3:
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and vtr1: "Opt.validFromS s1 tr1"
and nis1: "¬ isIntO s1" and nis2: "¬ isIntO s2"
and inter3: "never isIntO tr1"
and sec: "never isSecO (butlast tr1)" "isSecO (lastt s1 tr1)"
shows "∃w' w1' w2' trv1 trv2 statOO. χ3 Δ w' w1 w2 w1' w2' s1 tr1 s2 statA sv1 trv1 sv2 trv2 statOO"
using assms(2-)
proof(induction "length tr1" w
arbitrary: w1 w2 s1 s2 statA sv1 sv2 statO tr1 rule: less2_induct')
case (less w tr1 w1 w2 s1 s2 statA sv1 sv2 statO)
note vtr1 = less(8)
note Δ = `Δ w w1 w2 s1 s2 statA sv1 sv2 statO`
note nis1 = less(9) note nis2 = less(10)
note inter3 = less(11)
note sec3 = less(12,13)
note r34 = less.prems(2,3) note r12 = less.prems(4,5)
note r = r34 r12
note r3 = r34(1) note r4 = r34(2) note r1 = r12(1) note r2 = r12(2)
have i34: "statA = Eq ⟶ isIntO s1 = isIntO s2"
and f34: "finalO s1 = finalO s2 ∧ finalV sv1 = finalO s1 ∧ finalV sv2 = finalO s2"
using Δ unwind[unfolded unwindCond_def] r by auto
have proact_match: "(∃v<w. proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO) ∨ react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
using Δ unwind[unfolded unwindCond_def] r by auto
show ?case using proact_match proof safe
fix v assume v: "v < w"
assume "proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
thus ?thesis unfolding proact_def proof safe
assume sv1: "¬ isSecV sv1" "¬ isIntV sv1" and "move_1 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv1'
where 0:"validTransV (sv1,sv1')"
and Δ: "Δ v w1 w2 s1 s2 statA sv1' sv2 statO"
unfolding move_1_def by auto
have r1': "reachV sv1'" using r1 0 by (metis Van.reach.Step fst_conv snd_conv)
obtain w' w1' w2' trv1 trv2 statOO where χ3: "χ3 Δ w' w1 w2 w1' w2' s1 tr1 s2 statA sv1' trv1 sv2 trv2 statOO"
using less(2)[OF v, of tr1 w1 w2 s1 s2 statA sv1' sv2 statO,
simplified, OF Δ r34 r1' r2 vtr1 nis1 nis2 inter3 sec3] by auto
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ trv2])
using χ3 0 sv1 unfolding χ3_def by auto
next
assume sv2: "¬ isSecV sv2" "¬ isIntV sv2" and "move_2 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv2'
where 0: "validTransV (sv2,sv2')"
and Δ: "Δ v w1 w2 s1 s2 statA sv1 sv2' statO"
unfolding move_2_def by auto
have r2': "reachV sv2'" using r2 0 by (metis Van.reach.Step fst_conv snd_conv)
obtain w' w1' w2' trv1 trv2 statOO where χ3: "χ3 Δ w' w1 w2 w1' w2' s1 tr1 s2 statA sv1 trv1 sv2' trv2 statOO"
using less(2)[OF v, of tr1 w1 w2 s1 s2 statA sv1 sv2' statO,
simplified, OF Δ r34 r1 r2' vtr1 nis1 nis2 inter3 sec3] by auto
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ trv1]) apply(rule exI[of _ "sv2 # trv2"])
using χ3 0 sv2 unfolding χ3_def by auto
next
assume sv12: "¬ isSecV sv1" "¬ isSecV sv2" "Van.eqAct sv1 sv2"
and "move_12 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv1' sv2' statO'
where 0: "statO' = sstatO' statO sv1 sv2"
"validTransV (sv1,sv1') " "¬ isSecV sv1"
"validTransV (sv2,sv2')" "¬ isSecV sv2"
"Van.eqAct sv1 sv2"
and Δ: "Δ v w1 w2 s1 s2 statA sv1' sv2' statO'"
unfolding move_12_def by auto
have r12': "reachV sv1'" "reachV sv2'" using r1 r2 0 by (metis Van.reach.Step fst_conv snd_conv)+
obtain w' w1' w2' trv1 trv2 statOO where χ3: "χ3 Δ w' w1 w2 w1' w2' s1 tr1 s2 statA sv1' trv1 sv2' trv2 statOO"
using less(2)[OF v, of tr1 w1 w2 s1 s2 statA sv1' sv2' statO',
simplified, OF Δ r34 r12' vtr1 nis1 nis2 inter3 sec3] by auto
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "sv2 # trv2"])
apply(rule exI[of _ statOO])
using χ3 0 sv12 unfolding χ3_def sstatO'_def
by (auto simp: Van.eqAct_def)
qed
next
assume m: "react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
define statA' where statA': "statA' = sstatA' statA s1 s2"
show ?thesis
proof(cases "length tr1 ≤ Suc 0")
case True
hence tr1e: "tr1 = [] ∨ tr1 = [s1]"
by (metis Opt.validFromS_singl_iff Suc_length_conv le_Suc_eq le_zero_eq length_0_conv vtr1)
hence "Opt.A tr1 = []" by (simp add: True)
have is1: "isSecO s1"
by (metis last.simps lastt_def sec3(2) tr1e)
hence "¬ finalO s1" using Opt.final_not_isSec by blast
then obtain s1' where s13': "validTransO (s1, s1')" unfolding Opt.final_def by auto
hence isv1: "isSecV sv1 ∧ getSecV sv1 = getSecO s1" using m is1 nis1
unfolding react_def match1_def eqSec_def by auto
show ?thesis using tr1e isv1 apply-
apply(rule exI[of _ w]) apply(rule exI[of _ w1]) apply(rule exI[of _ w2])
apply(rule exI[of _ "[sv1]"], rule exI[of _ "[sv2]"], rule exI[of _ statO])
using tr1e
using f34 Δ by (clarsimp simp: χ3_def lastt_def)
next
case False
then obtain s13 tr1' where tr1: "tr1 = s13 # tr1'" and tr1'NE: "tr1' ≠ []"
by (cases tr1, auto)
have s13[simp]: "s13 = s1" using `Opt.validFromS s1 tr1`
by (simp add: Opt.validFromS_Cons_iff tr1)
obtain s1' where
trn3: "validTransO (s1,s1')" and
tr1': "Opt.validFromS s1' tr1'" using `Opt.validFromS s1 tr1`
unfolding tr1 s13 by (metis tr1'NE Simple_Transition_System.validFromS_Cons_iff)
have r3': "reachO s1'" using r3 trn3 by (metis Opt.reach.Step fst_conv snd_conv)
have f3: "¬ finalO s1" using Opt.final_def trn3 by blast
hence f4: "¬ finalO s2" using f34 by blast
have nev3': "never isIntO tr1'"
using inter3 tr1 tr1'NE by auto
have isAO3: "¬ isIntO s1" by (simp add: nis1)
have O33': "Opt.O tr1 = Opt.O tr1'" "Opt.A tr1 = Opt.A tr1'"
using isAO3 unfolding tr1 by auto
have m: "match1 Δ w1 w2 s1 s2 statA sv1 sv2 statO" using m unfolding react_def by auto
have "(∃w1'<w1. ∃w2'<w2. ¬ isSecO s1 ∧ Δ ∞ w1' w2' s1' s2 statA sv1 sv2 statO) ∨
(∃w2'<w2. eqSec sv1 s1 ∧ ¬ isIntV sv1 ∧ match1_1 Δ ∞ w2' s1 s1' s2 statA sv1 sv2 statO) ∨
(eqSec sv1 s1 ∧ ¬ isSecV sv2 ∧ Van.eqAct sv1 sv2 ∧ match1_12 Δ ∞ ∞ s1 s1' s2 statA sv1 sv2 statO)"
using m isAO3 trn3 unfolding match1_def by auto
thus ?thesis
proof safe
fix w1'' w2'' assume w12': "w1'' < w1" "w2'' < w2"
assume "¬ isSecO s1" and Δ: "Δ ∞ w1'' w2'' s1' s2 statA sv1 sv2 statO"
hence S3: "Opt.S tr1' = Opt.S tr1" unfolding tr1 by auto
obtain w' w1' w2' trv1 trv2 statOO where χ3: "χ3 Δ w' w1'' w2'' w1' w2' s1' tr1' s2 statA sv1 trv1 sv2 trv2 statOO"
using less(1)[of tr1', OF _ Δ r3' r4 r12 _] unfolding tr1
by simp (metis Opt.S.eq_Nil_iff(2) S3 Opt.validFromS_def ‹¬ isSecO s1› last.simps
lastt_def list_all_hd nev3' nis2 s13 sec3(1) sec3(2) tr1 tr1')
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "trv1"]) apply(rule exI[of _ trv2])
using χ3 O33' unfolding χ3_def tr1 Van.completedFrom_def
using Van.validFromS_Cons tr1'NE tr1' trn3 isAO3 w12' by auto
next
fix w2'' assume w2': "w2'' < w2"
assume trn13: "eqSec sv1 s1" and
Atrn1: "¬ isIntV sv1" and "match1_1 Δ ∞ w2'' s1 s1' s2 statA sv1 sv2 statO"
then obtain sv1' where
trn1: "validTransV (sv1,sv1')" and
Δ: "Δ ∞ ∞ w2'' s1' s2 statA sv1' sv2 statO"
unfolding match1_1_def by auto
have r1': "reachV sv1'"using r1 trn1 by (metis Van.reach.Step fst_conv snd_conv)
obtain w' w1' w2' trv1 trv2 statOO where χ3: "χ3 Δ w' ∞ w2'' w1' w2' s1' tr1' s2 statA sv1' trv1 sv2 trv2 statOO"
using less(1)[of tr1', OF _ Δ r3' r4 r1' r2, unfolded O33', simplified]
using less.prems tr1' f3 f4 tr1'NE trn3 O33'(1)
unfolding tr1
by simp (metis Opt.validFromS_def list_all_hd)
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ trv2])
using χ3 O33' unfolding χ3_def tr1 Van.completedFrom_def
using Van.validFromS_Cons trn1 tr1'NE tr1' trn3
using isAO3 Atrn1 eqSec_S_Cons trn13 w2'
by simp (metis Opt.S.Nil_iff Opt.S.eq_Nil_iff(1) eqSec_def nless_le order_le_less_trans s13 sec3(1) tr1)
next
assume sv2: "¬ isSecV sv2" and trn13: "eqSec sv1 s1" and
Atrn12: "Van.eqAct sv1 sv2" and "match1_12 Δ ∞ ∞ s1 s1' s2 statA sv1 sv2 statO"
then obtain sv1' sv2' statO' where
statO': "statO' = sstatO' statO sv1 sv2" and
trn1: "validTransV (sv1,sv1') " and
trn2: "validTransV (sv2,sv2')" and
Δ: "Δ ∞ ∞ ∞ s1' s2 statA sv1' sv2' statO'"
unfolding match1_12_def by auto
have r12': "reachV sv1'" "reachV sv2'"
using r1 trn1 r2 trn2 by (metis Van.reach.Step fst_conv snd_conv)+
obtain w' w1' w2' trv1 trv2 statOO where χ3: "χ3 Δ w' ∞ ∞ w1' w2' s1' tr1' s2 statA sv1' trv1 sv2' trv2 statOO"
using less(1)[of tr1', OF _ Δ r3' r4 r12', unfolded O33', simplified]
using less.prems tr1' f3 f4 tr1'NE trn3 O33'(1) unfolding tr1 statO' sstatO'_def
by simp (metis Simple_Transition_System.validFromS_def list_all_hd)+
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "sv2 # trv2"])
using χ3 O33' tr1'NE sv2
using Van.validFromS_Cons trn1 trn2
using isAO3 Atrn12 eqSec_S_Cons trn13 f3 f34 s13 tr1' trn3
unfolding χ3_def tr1 Van.completedFrom_def Van.eqAct_def
using Van.A.Cons_unfold eqSec_def sec3(1) tr1 by auto
qed
qed
qed
qed
definition χ3a where "χ3a Δ w (w1::enat) w2 w1' w2' s1 s1' s2 statAA sv1 trv1 sv2 trv2 statOO ≡
trv1 ≠ [] ∧ trv2 ≠ [] ∧ (length trv2 > Suc 0 ∨ w2' < w2) ∧
Van.validFromS sv1 trv1 ∧ Van.validFromS sv2 trv2 ∧
Van.S trv1 = [getSecO s1] ∧
never isSecV (butlast trv2) ∧
Van.A trv1 = Van.A trv2 ∧
Δ w w1' w2' s1' s2 statAA (lastt sv1 trv1) (lastt sv2 trv2) statOO"
lemma unwindCond_ex_χ3a_getSec:
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO"
and r34: "reachO s1" "reachO s2" and r12: "reachV sv1" "reachV sv2"
and v: "validTransO (s1, s1')"
and ii3: "¬ isIntO s1"
and is1: "isSecO s1" and isv13: "isSecV sv1" "getSecO s1 = getSecV sv1"
shows "∃w1' w2' trv1 trv2 statOO.
χ3a Δ ∞ w1 w2 w1' w2' s1 s1' s2 statA sv1 trv1 sv2 trv2 statOO"
using Δ r12 isv13
proof(induction w arbitrary: w1 w2 sv1 sv2 statO rule: less_induct)
case (less w w1 w2 sv1 sv2 statO)
note Δ = `Δ w w1 w2 s1 s2 statA sv1 sv2 statO`
note r12 = less.prems(2,3)
note r1 = r12(1) note r2 = r12(2)
note r = r34 r12
note isv13 = `isSecV sv1` `getSecO s1 = getSecV sv1`
have f34: "finalO s1 = finalO s2 ∧ finalV sv1 = finalO s1 ∧ finalV sv2 = finalO s2"
using Δ unwind[unfolded unwindCond_def] r by auto
have proact_match: "(∃v<w. proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO) ∨ react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
using Δ unwind[unfolded unwindCond_def] r by auto
show ?case using proact_match proof safe
fix v assume v: "v < w"
assume "proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
thus ?thesis unfolding proact_def proof safe
assume sv1: "¬ isSecV sv1" "¬ isIntV sv1" and "move_1 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
hence False using isv13 by blast
thus ?thesis by auto
next
assume sv2: "¬ isSecV sv2" "¬ isIntV sv2" and "move_2 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv2'
where 0: "validTransV (sv2,sv2')"
and Δ: "Δ v w1 w2 s1 s2 statA sv1 sv2' statO"
unfolding move_2_def by auto
have r2': "reachV sv2'" using r2 0 by (metis Van.reach.Step fst_conv snd_conv)
obtain w1' w2' trv1 trv2 statOO where
χ3a: "χ3a Δ ∞ w1 w2 w1' w2' s1 s1' s2 statA sv1 trv1 sv2' trv2 statOO"
using less(1)[OF v Δ r1 r2' isv13] by auto
show ?thesis apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "trv1"]) apply(rule exI[of _ "sv2 # trv2"])
using χ3a 0 sv2 unfolding χ3a_def by auto
next
assume sv12: "¬ isSecV sv1" "¬ isSecV sv2" "Van.eqAct sv1 sv2"
and "move_12 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
hence False using isv13 by blast
thus ?thesis by auto
qed
next
assume m: "react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
have m: "match1 Δ w1 w2 s1 s2 statA sv1 sv2 statO" using m unfolding react_def by auto
have "(∃w1' w2'. w1'<w1 ∧ w2'<w2 ∧ ¬ isSecO s1 ∧ Δ ∞ w1' w2' s1' s2 statA sv1 sv2 statO) ∨
(∃w2'< w2. eqSec sv1 s1 ∧ ¬ isIntV sv1 ∧ match1_1 Δ ∞ w2' s1 s1' s2 statA sv1 sv2 statO) ∨
(eqSec sv1 s1 ∧ ¬ isSecV sv2 ∧ Van.eqAct sv1 sv2 ∧ match1_12 Δ ∞ ∞ s1 s1' s2 statA sv1 sv2 statO)"
using m v ii3 unfolding match1_def by auto
thus ?thesis
apply(elim disjE exE)
subgoal for w1' w2' using is1 by auto
subgoal for w2' apply(rule exI[of _ ∞]) apply(rule exI[of _ w2'])
unfolding match1_1_def apply(elim conjE exE) subgoal for sv1'
apply(rule exI[of _ "[sv1,sv1']"]) apply(rule exI[of _ "[sv2]"])
apply(rule exI[of _ statO])
using is1 isv13 unfolding χ3a_def
by (auto simp : sstatA'_def lastt_def) .
subgoal apply(rule exI[of _ ∞]) apply(rule exI[of _ ∞])
unfolding match1_12_def apply(elim conjE exE) subgoal for sv1' sv2'
apply(rule exI[of _ "[sv1,sv1']"]) apply(rule exI[of _ "[sv2,sv2']"])
apply(rule exI[of _ "sstatO' statO sv1 sv2"])
using is1 isv13 unfolding χ3a_def
by (auto simp : sstatA'_def sstatO'_def lastt_def Van.eqAct_def) . .
qed
qed
definition "χ3b Δ w (w1::enat) w2 w1' w2' s1 tr1 s2 statAA sv1 trv1 sv2 trv2 statOO ≡
trv1 ≠ [] ∧
trv2 ≠ [] ∧ (length trv2 > Suc 0 ∨ w2' < w2) ∧
Van.validFromS sv1 trv1 ∧
Van.validFromS sv2 trv2 ∧
Van.S trv1 = Opt.S tr1 ∧
never isSecV (butlast trv2) ∧ Van.A trv1 = Van.A trv2 ∧
Δ w w1' w2' (lastt s1 tr1) s2 statAA (lastt sv1 trv1) (lastt sv2 trv2) statOO"
lemma unwindCond_ex_χ3b_aux:
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO" and
r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and tr1NE: "tr1 ≠ []"
and v3': "Opt.validFromS s1 (tr1 ## s1')"
and nis1: "¬ isIntO s1" and nis2: "¬ isIntO s2"
and ninter3': "never isIntO (tr1 ## s1')"
and sec: "never isSecO (butlast tr1)" "isSecO (lastt s1 tr1)"
shows "∃w1' w2' trv1 trv2 statOO. χ3b Δ ∞ w1 w2 w1' w2' s1 (tr1 ## s1') s2 statA sv1 trv1 sv2 trv2 statOO"
proof-
have v3: "Opt.validFromS s1 tr1" and s13': "validTransO (lastt s1 tr1,s1')"
apply (metis v3' Opt.validFromS_def Opt.validS_append1 Nil_is_append_conv hd_append2)
by (metis Opt.validFromS_def Opt.validS_validTrans lastt_def list.sel(1) not_Cons_self2 snoc_eq_iff_butlast tr1NE v3')
have ninter3: "never isIntO tr1" and nis1': "¬ isIntO s1'"
using ninter3' by auto
obtain ww ww1 ww2 trv1 trv2 statOO where χ3: "χ3 Δ ww w1 w2 ww1 ww2 s1 tr1 s2 statA sv1 trv1 sv2 trv2 statOO"
using unwindCond_ex_χ3[OF unwind Δ r v3 nis1 nis2 ninter3 sec] by auto
have trv12NE: "trv1 ≠ []" "trv2 ≠ []" using χ3 unfolding χ3_def by auto
define ss1 ssv1 ssv2 where ss1: "ss1 ≡ lastt s1 tr1"
and ssv1: "ssv1 ≡ lastt sv1 trv1" and ssv2: "ssv2 ≡ lastt sv2 trv2"
have ss1l: "ss1 = last tr1" by (simp add: lastt_def ss1 tr1NE)
have tr1l: "tr1 = butlast tr1 @ [ss1]" by (simp add: ss1l tr1NE)
have ssv1l: "ssv1 = last trv1" using χ3 unfolding χ3_def by (metis lastt_def ssv1)
have trv1l: "trv1 = butlast trv1 @ [ssv1]" by (simp add: ssv1l trv12NE(1))
have ssv2l: "ssv2 = last trv2" using χ3 unfolding χ3_def by (metis lastt_def ssv2)
have trv2l: "trv2 = butlast trv2 @ [ssv2]" by (simp add: ssv2l trv12NE(2))
have iss1[simp]: "isSecO ss1" using sec(2) unfolding ss1 by auto
have issv1[simp]: "isSecV ssv1" and gissv13[simp]: "getSecO ss1 = getSecV ssv1"
using χ3 unfolding χ3_def ssv1 ss1 by auto
have niss1: "¬ isIntO ss1"
by (metis list_all_append list_all_simps(1) ninter3 tr1l)
have rss1: "reachO ss1" and rssv12: "reachV ssv1" "reachV ssv2"
using Opt.reach_validFromS_reach r ss1l tr1NE v3 apply blast
apply (metis Van.reach_validFromS_reach χ3_def χ3 r(3) ssv1l)
by (metis Van.reach_validFromS_reach χ3_def χ3 r(4) ssv2l)
have Δ: "Δ ww ww1 ww2 ss1 s2 statA ssv1 ssv2 statOO"
using χ3 unfolding χ3_def ss1[symmetric] ssv1[symmetric] ssv2[symmetric] by auto
have s13': "validTransO (ss1, s1')"
by (simp add: s13' ss1)
note vs13 = s13'[unfolded ss1[symmetric]]
obtain w1' w2' trv1' trv2' statO' where
χ3a: "χ3a Δ ∞ ww1 ww2 w1' w2' ss1 s1' s2 statA ssv1 trv1' ssv2 trv2' statO'"
using unwindCond_ex_χ3a_getSec[OF unwind Δ rss1 r(2) rssv12 s13' niss1 iss1 issv1 gissv13]
by blast
have trv12'NE: "trv1' ≠ []" "trv2' ≠ []" using χ3a unfolding χ3a_def by auto
have [simp]: "Van.O (butlast trv1 @ trv1') = Van.O trv1 @ Van.O trv1'"
using trv12'NE unfolding χ3_def Van.O.map_filter Opt.O.map_filter apply(subst butlast_append) by simp
have [simp]: "Van.O (butlast trv2 @ trv2') = Van.O trv2 @ Van.O trv2'"
using trv12'NE unfolding χ3_def Van.O.map_filter Opt.O.map_filter apply(subst butlast_append) by simp
have "Van.A trv1' = Van.A trv2'" using χ3a unfolding χ3a_def by auto
moreover have "length (Van.O trv1') = length (Van.A trv1') ∧ length (Van.O trv2') = length (Van.A trv2')"
unfolding Van.A.map_filter Van.O.map_filter by auto
ultimately have "length (Van.O trv1') = length (Van.O trv2')" by auto
hence [simp]: "Van.O trv1 @ Van.O trv1' = Van.O trv2 @ Van.O trv2' ⟷
Van.O trv1 = Van.O trv2 ∧ Van.O trv1' = Van.O trv2'" by auto
show ?thesis
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ "butlast trv1 @ trv1'"]) apply(rule exI[of _ "butlast trv2 @ trv2'"])
apply(rule exI[of _ statO'])
unfolding χ3b_def apply(intro conjI)
subgoal using χ3 χ3a unfolding χ3_def χ3a_def by auto
subgoal using χ3 χ3a unfolding χ3_def χ3a_def by auto
subgoal using χ3 χ3a unfolding χ3_def χ3a_def
by simp (metis Simple_Transition_System.fromS_eq_Nil Simple_Transition_System.toS_fromS_nonSingl Van.toS_Nil diff_add_inverse2 linorder_not_less order_le_less_trans trans_less_add2 zero_less_diff)
subgoal using χ3 χ3a unfolding χ3_def χ3a_def ssv1
using Van.validFromS_append by auto
subgoal using χ3 χ3a unfolding χ3_def χ3a_def ssv2
using Van.validFromS_append by auto
subgoal using χ3 χ3a unfolding χ3_def χ3a_def unfolding Van.S.map_filter Opt.S.map_filter
apply(subst tr1l) apply(subst butlast_append)
by simp (metis Opt.S.map_filter Opt.S.eq_Nil_iff(2) Van.S.map_filter Van.S.eq_Nil_iff(2) sec(1))
subgoal using χ3 χ3a unfolding χ3_def χ3a_def
by (simp add: butlast_append)
subgoal using χ3 χ3a unfolding χ3_def χ3a_def Van.A.map_filter Opt.A.map_filter
apply(subst trv1l) apply(subst trv2l) by (simp add: butlast_append)
subgoal using χ3a trv12'NE tr1NE unfolding χ3a_def lastt_def by simp .
qed
lemma unwindCond_ex_χ3b_aux2:
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and v3': "Opt.validFromS s1 (tr1 @ [s1',s1''])"
and nis1: "¬ isIntO s1" and nis2: "¬ isIntO s2"
and ninter3': "never isIntO (tr1 @ [s1',s1''])"
and sec: "never isSecO tr1" "isSecO s1'"
shows "∃w1' w2' trv1 trv2 statOO. χ3b Δ ∞ w1 w2 w1' w2' s1 (tr1 @ [s1',s1'']) s2 statA sv1 trv1 sv2 trv2 statOO"
proof-
have 0: "lastt s1 (tr1 ## s1') = s1'"
unfolding lastt_def by auto
show ?thesis
using unwindCond_ex_χ3b_aux[OF unwind Δ r, of "tr1 ## s1'", unfolded 0, simplified]
using assms by auto
qed
definition "χ3' Δ w1 w2 w1' w2' s1 tr1 s1' s1'' s2 statAA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO ≡
Van.validFromS sv1 (trv1 ## sv1'') ∧ Van.validFromS sv2 (trv2 ## sv2'') ∧
Van.S (trv1 ## sv1'') = Opt.S ((tr1 ## s1') ## s1'') ∧ never isSecV trv2 ∧
Van.A (trv1 ## sv1'') = Van.A (trv2 ## sv2'') ∧
trv1 ≠ [] ∧ (trv2 ≠ [] ∨ w2' < w2) ∧
Δ ∞ w1' w2' s1'' s2 statAA sv1'' sv2'' statOO"
proposition unwindCond_ex_χ3':
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO" and
r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and v3': "Opt.validFromS s1 ((tr1 ## s1') ## s1'')"
and nis1: "¬ isIntO s1" and nis2: "¬ isIntO s2"
and ninter3': "never isIntO ((tr1 ## s1') ## s1'')"
and sec: "never isSecO tr1" "isSecO s1'"
shows "∃w1' w2' trv1 sv1'' trv2 sv2'' statOO. χ3' Δ w1 w2 w1' w2' s1 tr1 s1' s1'' s2 statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
using unwindCond_ex_χ3b_aux2[unfolded φ_def, unfolded lastt_snoc lastt_snoc2 append_snoc2, OF assms]
unfolding χ3b_def apply(elim exE) subgoal for w1' w2' trv1 trv2 statOO
apply(cases "trv1" rule: rev_cases)
subgoal by auto
subgoal for trv1' sv1'' apply(cases "trv2" rule: rev_cases)
subgoal by auto
subgoal for trv2' sv2'' unfolding χ3'_def
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ trv1']) apply(rule exI[of _ sv1''])
apply(rule exI[of _ trv2']) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statOO])
by simp (metis Opt.S.Nil_iff Opt.S.eq_Nil_iff(1) Van.S.simps(4) append_snoc2 list_all_append sec(2)
self_append_conv2 snoc_eq_iff_butlast) . . .
definition "ω3 Δ w1 w2 w1' w2' s1 s1' s2 statAA sv1 trv1 sv1' sv2 trv2 sv2' statOO ≡
Van.validFromS sv1 (trv1 ## sv1') ∧ Van.validFromS sv2 (trv2 ## sv2') ∧
never isSecV trv1 ∧ never isSecV trv2 ∧
Van.A (trv1 ## sv1') = Van.A (trv2 ## sv2') ∧
(trv1 ≠ [] ∨ w1' < w1) ∧ (trv2 ≠ [] ∨ w2' < w2) ∧
Δ ∞ w1' w2' s1' s2 statAA sv1' sv2' statOO"
proposition unwindCond_ex_ω3:
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO"
and r34: "reachO s1" "reachO s2" and r12: "reachV sv1" "reachV sv2"
and v3: "validTransO (s1,s1')"
and nis1: "¬ isIntO s1" "¬ isIntO s1'" "¬ isSecO s1"
and nis2: "¬ isIntO s2"
shows "∃w1' w2' trv1 sv1' trv2 sv2' statOO. ω3 Δ w1 w2 w1' w2' s1 s1' s2 statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
using Δ r12
proof(induction w arbitrary: w1 w2 sv1 sv2 statO rule: less_induct)
case (less w w1 w2 sv1 sv2 statO)
note Δ = `Δ w w1 w2 s1 s2 statA sv1 sv2 statO`
note r12 = less.prems(2,3)
note r1 = r12(1) note r2 = r12(2)
note r = r34 r12
have f34: "finalO s1 = finalO s2 ∧ finalV sv1 = finalO s1 ∧ finalV sv2 = finalO s2"
using Δ unwind[unfolded unwindCond_def] r by auto
have proact_match: "(∃v<w. proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO) ∨ react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
using Δ unwind[unfolded unwindCond_def] r by auto
show ?case using proact_match proof safe
fix v assume v: "v < w"
assume "proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
thus ?thesis unfolding proact_def proof safe
assume sv1: "¬ isSecV sv1" "¬ isIntV sv1" and "move_1 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv1' where 0: "validTransV (sv1, sv1')" and Δ: "Δ v w1 w2 s1 s2 statA sv1' sv2 statO"
unfolding move_1_def by auto
have r1': "reachV sv1'" using r1 0 by (metis Van.reach.Step fst_conv snd_conv)
obtain w1' w2' trv1 sv1'' trv2 sv2' statOO where
ω3: "ω3 Δ w1 w2 w1' w2' s1 s1' s2 statA sv1' trv1 sv1'' sv2 trv2 sv2' statOO"
using less(1)[OF v Δ r1' r2] by auto
show ?thesis apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "sv1''"])
apply(rule exI[of _ "trv2"]) apply(rule exI[of _ "sv2'"])
using ω3 0 sv1 unfolding ω3_def by auto
next
assume sv2: "¬ isSecV sv2" "¬ isIntV sv2" and "move_2 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv2'
where 0: "validTransV (sv2,sv2')"
and Δ: "Δ v w1 w2 s1 s2 statA sv1 sv2' statO"
unfolding move_2_def by auto
have r2': "reachV sv2'" using r2 0 by (metis Van.reach.Step fst_conv snd_conv)
obtain w1' w2' trv1 sv1' trv2 sv2'' statOO where
ω3: "ω3 Δ w1 w2 w1' w2' s1 s1' s2 statA sv1 trv1 sv1' sv2' trv2 sv2'' statOO"
using less(1)[OF v Δ r1 r2'] by auto
show ?thesis apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ "trv1"]) apply(rule exI[of _ "sv1'"])
apply(rule exI[of _ "sv2 # trv2"]) apply(rule exI[of _ "sv2''"])
using ω3 0 sv2 unfolding ω3_def by auto
next
assume sv1: "¬ isSecV sv1" and sv2: "¬ isSecV sv2" and
"move_12 Δ v w1 w2 s1 s2 statA sv1 sv2 statO" and
sv12: "Van.eqAct sv1 sv2"
then obtain sv1' sv2' statO'
where statO': "statO' = sstatO' statO sv1 sv2"
and 0: "validTransV (sv1,sv1')" "validTransV (sv2,sv2')"
and Δ: "Δ v w1 w2 s1 s2 statA sv1' sv2' statO'"
unfolding move_12_def by auto
have r1': "reachV sv1'" and r2': "reachV sv2'" using r1 r2 0
by (metis Van.reach.Step fst_conv snd_conv)+
obtain w1' w2' trv1 sv1'' trv2 sv2'' statOO where
ω3: "ω3 Δ w1 w2 w1' w2' s1 s1' s2 statA sv1' trv1 sv1'' sv2' trv2 sv2'' statOO"
using less(1)[OF v Δ r1' r2'] by auto
show ?thesis apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "sv1''"])
apply(rule exI[of _ "sv2 # trv2"]) apply(rule exI[of _ "sv2''"])
using ω3 0 sv1 sv2 sv12 unfolding ω3_def statO' by (auto simp: Van.eqAct_def)
qed
next
assume m: "react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
have m: "match1 Δ w1 w2 s1 s2 statA sv1 sv2 statO" using m unfolding react_def by auto
have "(∃w1' w2'. w1' < w1 ∧ w2'<w2 ∧ ¬ isSecO s1 ∧ Δ ∞ w1' w2' s1' s2 statA sv1 sv2 statO) ∨
(∃w2'<w2. eqSec sv1 s1 ∧ ¬ isIntV sv1 ∧ match1_1 Δ ∞ w2' s1 s1' s2 statA sv1 sv2 statO) ∨
(eqSec sv1 s1 ∧ ¬ isSecV sv2 ∧ Van.eqAct sv1 sv2 ∧ match1_12 Δ ∞ ∞ s1 s1' s2 statA sv1 sv2 statO)"
using m v3 nis1 unfolding match1_def by auto
thus ?thesis
apply(elim disjE exE)
subgoal for w1' w2'
apply(rule exI[of _ "w1'"]) apply(rule exI[of _ "w2'"])
apply(rule exI[of _ "[]"]) apply(rule exI[of _ "sv1"])
apply(rule exI[of _ "[]"]) apply(rule exI[of _ "sv2"])
apply(rule exI[of _ statO]) unfolding ω3_def
by auto
subgoal for w2'
apply(rule exI[of _ "∞"]) apply(rule exI[of _ "w2'"])
unfolding match1_1_def apply(elim conjE exE) subgoal for sv1'
apply(rule exI[of _ "[sv1]"]) apply(rule exI[of _ "sv1'"])
apply(rule exI[of _ "[]"]) apply(rule exI[of _ "sv2"])
apply(rule exI[of _ statO])
unfolding ω3_def using nis1(3) by (auto simp: eqSec_def) .
subgoal
apply(rule exI[of _ "∞"]) apply(rule exI[of _ "∞"])
unfolding match1_12_def apply(elim conjE exE) subgoal for sv1' sv2'
apply(rule exI[of _ "[sv1]"]) apply(rule exI[of _ "sv1'"])
apply(rule exI[of _ "[sv2]"]) apply(rule exI[of _ "sv2'"])
apply(rule exI[of _ "sstatO' statO sv1 sv2"])
unfolding ω3_def using nis1(3) apply (auto simp: eqSec_def
sstatA'_def sstatO'_def lastt_def Van.eqAct_def) . . .
qed
qed
definition "χ4 Δ w w1 (w2::enat) w1' w2' s1 s2 tr2 statAA sv1 trv1 sv2 trv2 statOO ≡
trv1 ≠ [] ∧ trv2 ≠ [] ∧ (length trv1 > Suc 0 ∨ w1' ≤ w1) ∧
Van.validFromS sv1 trv1 ∧ Van.validFromS sv2 trv2 ∧
never isSecV (butlast trv1) ∧
never isSecV (butlast trv2) ∧
isSecV (lastt sv2 trv2) ∧ getSecV (lastt sv2 trv2) = getSecO (lastt s2 tr2) ∧
Van.A trv1 = Van.A trv2 ∧
Δ w w1' w2' s1 (lastt s2 tr2) statAA (lastt sv1 trv1) (lastt sv2 trv2) statOO"
lemma χ4_final:
assumes unw: "unwindCond Δ"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and vtr2: "Opt.validFromS s2 tr2"
and χ4: "χ4 Δ w w1 w2 w1' w2' s1 s2 tr2 statAA sv1 trv1 sv2 trv2 statOO"
shows "(finalV (lastt sv1 trv1) ⟷ finalO s1) ∧ (finalV (lastt sv2 trv2) ⟷ finalO (lastt s2 tr2))"
proof-
have rsv12: "Van.validFromS sv1 trv1 ⟶ reachV (lastt sv1 trv1)"
"Van.validFromS sv2 trv2 ⟶ reachV (lastt sv2 trv2)" using r
by (simp add: Van.reach_validFromS_reach lastt_def)+
have rs2: "Opt.validFromS s2 tr2 ⟶ reachO (lastt s2 tr2)"
using r
by (simp add: Opt.reach_validFromS_reach lastt_def)+
show ?thesis using χ4[unfolded χ4_def] rsv12 rs2 using unw[unfolded unwindCond_def, rule_format,
of s1 "lastt s2 tr2" "lastt sv1 trv1" "lastt sv2 trv2" w w1' w2' statAA statOO]
using vtr2 `reachO s1` by auto
qed
lemma χ4_completedFrom: "unwindCond Δ ⟹
reachO s1 ⟹ reachO s2 ⟹ reachV sv1 ⟹ reachV sv2 ⟹
Opt.validFromS s2 tr2 ⟹ completedFromO s2 tr2 ⟹
χ4 Δ w w1 w2 w1' w2' s1 s2 tr2 statAA sv1 trv1 sv2 trv2 statOO
⟹ completedFromV sv1 trv1 ∧ completedFromV sv2 trv2"
by (metis Van.final_not_isSec χ4_def χ4_final completedFromO_lastt)
proposition unwindCond_ex_χ4:
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and vtr2: "Opt.validFromS s2 tr2"
and nis2: "¬ isIntO s1" and nis2: "¬ isIntO s2"
and inter4: "never isIntO tr2"
and sec: "never isSecO (butlast tr2)" "isSecO (lastt s2 tr2)"
shows "∃w' w1' w2' trv1 trv2 statOO. χ4 Δ w' w1 w2 w1' w2' s1 s2 tr2 statA sv1 trv1 sv2 trv2 statOO"
using assms(2-)
proof(induction "length tr2" w
arbitrary: w1 w2 s1 s2 statA sv1 sv2 statO tr2 rule: less2_induct')
case (less w tr2 w1 w2 s1 s2 statA sv1 sv2 statO)
note vtr2 = less(8)
note Δ = `Δ w w1 w2 s1 s2 statA sv1 sv2 statO`
note nis1 = less(9) note nis2 = less(10)
note inter4 = less(11)
note sec4 = less(12,13)
note r34 = less.prems(2,3) note r12 = less.prems(4,5)
note r = r34 r12
note r3 = r34(1) note r4 = r34(2) note r1 = r12(1) note r2 = r12(2)
have i34: "statA = Eq ⟶ isIntO s1 = isIntO s2"
and f34: "finalO s1 = finalO s2 ∧ finalV sv1 = finalO s1 ∧ finalV sv2 = finalO s2"
using Δ unwind[unfolded unwindCond_def] r by auto
have proact_match: "(∃v<w. proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO) ∨ react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
using Δ unwind[unfolded unwindCond_def] r by auto
show ?case using proact_match proof safe
fix v assume v: "v < w"
assume "proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
thus ?thesis unfolding proact_def proof safe
assume sv1: "¬ isSecV sv1" "¬ isIntV sv1" and "move_1 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv1'
where 0:"validTransV (sv1,sv1')"
and Δ: "Δ v w1 w2 s1 s2 statA sv1' sv2 statO"
unfolding move_1_def by auto
have r1': "reachV sv1'" using r1 0 by (metis Van.reach.Step fst_conv snd_conv)
obtain w' w1' w2' trv1 trv2 statOO where χ4: "χ4 Δ w' w1 w2 w1' w2' s1 s2 tr2 statA sv1' trv1 sv2 trv2 statOO"
using less(2)[OF v, of tr2 w1 w2 s1 s2 statA sv1' sv2 statO,
simplified, OF Δ r34 r1' r2 vtr2 nis1 nis2 inter4 sec4] by auto
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ trv2])
using χ4 0 sv1 unfolding χ4_def by auto
next
assume sv2: "¬ isSecV sv2" "¬ isIntV sv2" and "move_2 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv2'
where 0: "validTransV (sv2,sv2')"
and Δ: "Δ v w1 w2 s1 s2 statA sv1 sv2' statO"
unfolding move_2_def by auto
have r2': "reachV sv2'" using r2 0 by (metis Van.reach.Step fst_conv snd_conv)
obtain w1' w2' w' trv1 trv2 statOO where χ4: "χ4 Δ w' w1 w2 w1' w2' s1 s2 tr2 statA sv1 trv1 sv2' trv2 statOO"
using less(2)[OF v, of tr2 w1 w2 s1 s2 statA sv1 sv2' statO,
simplified, OF Δ r34 r1 r2' vtr2 nis1 nis2 inter4 sec4] by auto
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ trv1]) apply(rule exI[of _ "sv2 # trv2"])
using χ4 0 sv2 unfolding χ4_def by auto
next
assume sv12: "¬ isSecV sv1" "¬ isSecV sv2" "Van.eqAct sv1 sv2"
and "move_12 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv1' sv2' statO'
where 0: "statO' = sstatO' statO sv1 sv2"
"validTransV (sv1,sv1') " "¬ isSecV sv1"
"validTransV (sv2,sv2')" "¬ isSecV sv2"
"Van.eqAct sv1 sv2"
and Δ: "Δ v w1 w2 s1 s2 statA sv1' sv2' statO'"
unfolding move_12_def by auto
have r12': "reachV sv1'" "reachV sv2'" using r1 r2 0 by (metis Van.reach.Step fst_conv snd_conv)+
obtain w' w1' w2' trv1 trv2 statOO where χ4: "χ4 Δ w' w1 w2 w1' w2' s1 s2 tr2 statA sv1' trv1 sv2' trv2 statOO"
using less(2)[OF v, of tr2 w1 w2 s1 s2 statA sv1' sv2' statO',
simplified, OF Δ r34 r12' vtr2 nis1 nis2 inter4 sec4] by auto
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "sv2 # trv2"])
apply(rule exI[of _ statOO])
using χ4 0 sv12 unfolding χ4_def sstatO'_def
by (auto simp: Van.eqAct_def)
qed
next
assume m: "react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
define statA' where statA': "statA' = sstatA' statA s1 s2"
show ?thesis
proof(cases "length tr2 ≤ Suc 0")
case True
hence tr2e: "tr2 = [] ∨ tr2 = [s2]"
by (metis Opt.validFromS_def Suc_length_conv le_Suc_eq le_zero_eq length_0_conv list.sel(1) vtr2)
hence "Opt.A tr2 = []" by (simp add: True)
have is2: "isSecO s2"
by (metis last.simps lastt_def sec4(2) tr2e)
hence "¬ finalO s2" using Opt.final_not_isSec by blast
then obtain s2' where s24': "validTransO (s2, s2')" unfolding Opt.final_def by auto
hence isv2: "isSecV sv2 ∧ getSecV sv2 = getSecO s2" using m is2 nis2
unfolding react_def match2_def eqSec_def by auto
show ?thesis using tr2e isv2 apply-
apply(rule exI[of _ w]) apply(rule exI[of _ w1]) apply(rule exI[of _ w2])
apply(rule exI[of _ "[sv1]"], rule exI[of _ "[sv2]"], rule exI[of _ statO])
using tr2e
using f34 Δ by(clarsimp simp: χ4_def lastt_def)
next
case False
then obtain s24 tr2' where tr2: "tr2 = s24 # tr2'" and tr2'NE: "tr2' ≠ []"
by (cases tr2, auto)
have s24[simp]: "s24 = s2" using `Opt.validFromS s2 tr2`
by (simp add: Opt.validFromS_Cons_iff tr2)
obtain s2' where
trn4: "validTransO (s2,s2')" and
tr2': "Opt.validFromS s2' tr2'" using `Opt.validFromS s2 tr2`
unfolding tr2 s24 by (metis tr2'NE Simple_Transition_System.validFromS_Cons_iff)
have r4': "reachO s2'" using r4 trn4 by (metis Opt.reach.Step fst_conv snd_conv)
have f4: "¬ finalO s2" using Opt.final_def trn4 by blast
hence f3: "¬ finalO s1" using f34 by blast
have nev4': "never isIntO tr2'"
using inter4 tr2 tr2'NE by auto
have isAO4: "¬ isIntO s2" by (simp add: nis2)
have O44': "Opt.O tr2 = Opt.O tr2'" "Opt.A tr2 = Opt.A tr2'"
using isAO4 unfolding tr2 by auto
have m: "match2 Δ w1 w2 s1 s2 statA sv1 sv2 statO" using m unfolding react_def by auto
have "(∃w1'<w1. ∃w2'<w2. ¬ isSecO s2 ∧ Δ ∞ w1' w2' s1 s2' statA sv1 sv2 statO) ∨
(∃w1'<w1. eqSec sv2 s2 ∧ ¬ isIntV sv2 ∧ match2_1 Δ w1' ∞ s1 s2 s2' statA sv1 sv2 statO) ∨
(eqSec sv2 s2 ∧ ¬ isSecV sv1 ∧ Van.eqAct sv1 sv2 ∧ match2_12 Δ ∞ ∞ s1 s2 s2' statA sv1 sv2 statO)"
using m isAO4 trn4 unfolding match2_def by auto
thus ?thesis
proof safe
fix w1'' w2'' assume w12': "w1'' < w1" "w2'' < w2"
assume "¬ isSecO s2" and Δ: "Δ ∞ w1'' w2'' s1 s2' statA sv1 sv2 statO"
hence S4: "Opt.S tr2' = Opt.S tr2" unfolding tr2 by auto
obtain w' w1' w2' trv1 trv2 statOO where χ4: "χ4 Δ w' w1'' w2'' w1' w2' s1 s2' tr2' statA sv1 trv1 sv2 trv2 statOO"
using less(1)[of tr2', OF _ Δ r3 r4' r12] unfolding tr2
by simp (metis Opt.S.eq_Nil_iff(2) S4 Simple_Transition_System.validFromS_def last.simps lastt_def
list.discI list_all_hd nev4' nis1 sec4(1) sec4(2) tr2 tr2' tr2'NE)
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "trv1"]) apply(rule exI[of _ trv2])
using χ4 O44' unfolding χ4_def tr2 Van.completedFrom_def
using Van.validFromS_Cons tr2'NE tr2' trn4 isAO4 w12' by auto
next
fix w1'' assume w1': "w1'' < w1"
assume trn24: "eqSec sv2 s2" and
Atrn2: "¬ isIntV sv2" and "match2_1 Δ w1'' ∞ s1 s2 s2' statA sv1 sv2 statO"
then obtain sv2' where
trn2: "validTransV (sv2,sv2') " and
Δ: "Δ ∞ w1'' ∞ s1 s2' statA sv1 sv2' statO"
unfolding match2_1_def by auto
have r2': "reachV sv2'"using r2 trn2 by (metis Van.reach.Step fst_conv snd_conv)
obtain w' w1' w2' trv1 trv2 statOO where χ4: "χ4 Δ w' w1'' ∞ w1' w2' s1 s2' tr2' statA sv1 trv1 sv2' trv2 statOO"
using less(1)[of tr2', OF _ Δ r3 r4' r1 r2', unfolded O44', simplified]
using less.prems tr2' f3 f4 tr2'NE trn4 O44'(1)
unfolding tr2
by simp (metis Opt.validFromS_def list_all_hd)
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "trv1"]) apply(rule exI[of _ "sv2 # trv2"])
using χ4 O44' unfolding χ4_def tr2 Van.completedFrom_def
using Van.validFromS_Cons trn2 tr2'NE tr2' trn4
using isAO4 Atrn2 eqSec_S_Cons trn24 w1'
by simp (metis Opt.S.Nil_iff Opt.S.eq_Nil_iff(1) eqSec_def nless_le order_le_less_trans s24 sec4(1) tr2)
next
assume sv1: "¬ isSecV sv1" and trn24: "eqSec sv2 s2" and
Atrn12: "Van.eqAct sv1 sv2" and "match2_12 Δ ∞ ∞ s1 s2 s2' statA sv1 sv2 statO"
then obtain sv1' sv2' statO' where
statO': "statO' = sstatO' statO sv1 sv2" and
trn1: "validTransV (sv1,sv1') " and
trn2: "validTransV (sv2,sv2')" and
Δ: "Δ ∞ ∞ ∞ s1 s2' statA sv1' sv2' statO'"
unfolding match2_12_def by auto
have r12': "reachV sv1'" "reachV sv2'"
using r1 trn1 r2 trn2 by (metis Van.reach.Step fst_conv snd_conv)+
obtain w' w1' w2' trv1 trv2 statOO where χ4: "χ4 Δ w' ∞ ∞ w1' w2' s1 s2' tr2' statA sv1' trv1 sv2' trv2 statOO"
using less(1)[of tr2', OF _ Δ r3 r4' r12', unfolded O44', simplified]
using less.prems tr2' f3 f4 tr2'NE trn4 O44'(1) unfolding tr2 statO' sstatO'_def
by simp (metis Simple_Transition_System.validFromS_def list_all_hd)
show ?thesis apply(rule exI[of _ w']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "sv2 # trv2"])
using χ4 O44' tr2'NE sv1
using Van.validFromS_Cons trn1 trn2
using isAO4 Atrn12 eqSec_S_Cons trn24 f3 f34 s24 tr2' trn4
unfolding χ4_def tr2 Van.completedFrom_def Van.eqAct_def
using Van.A.Cons_unfold eqSec_def sec4(1) tr2 by auto
qed
qed
qed
qed
definition χ4a where "χ4a Δ w w1 (w2::enat) w1' w2' s1 s2 s2' statAA sv1 trv1 sv2 trv2 statOO ≡
trv1 ≠ [] ∧ trv2 ≠ [] ∧ (length trv1 > Suc 0 ∨ w1' < w1) ∧
Van.validFromS sv1 trv1 ∧ Van.validFromS sv2 trv2 ∧
never isSecV (butlast trv1) ∧
Van.S trv2 = [getSecO s2] ∧
Van.A trv1 = Van.A trv2 ∧
Δ w w1' w2' s1 s2' statAA (lastt sv1 trv1) (lastt sv2 trv2) statOO"
lemma unwindCond_ex_χ4a_getSec:
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO"
and r34: "reachO s1" "reachO s2" and r12: "reachV sv1" "reachV sv2"
and v: "validTransO (s2, s2')"
and ii4: "¬ isIntO s2"
and is2: "isSecO s2" and isv24: "isSecV sv2" "getSecO s2 = getSecV sv2"
shows "∃w1' w2' trv1 trv2 statOO.
χ4a Δ ∞ w1 w2 w1' w2' s1 s2 s2' statA sv1 trv1 sv2 trv2 statOO"
using Δ r12 isv24
proof(induction w arbitrary: w1 w2 sv1 sv2 statO rule: less_induct)
case (less w w1 w2 sv1 sv2 statO)
note Δ = `Δ w w1 w2 s1 s2 statA sv1 sv2 statO`
note r12 = less.prems(2,3)
note r1 = r12(1) note r2 = r12(2)
note r = r34 r12
note isv24 = `isSecV sv2` `getSecO s2 = getSecV sv2`
have f34: "finalO s1 = finalO s2 ∧ finalV sv1 = finalO s1 ∧ finalV sv2 = finalO s2"
using Δ unwind[unfolded unwindCond_def] r by auto
have proact_match: "(∃v<w. proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO) ∨ react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
using Δ unwind[unfolded unwindCond_def] r by auto
show ?case using proact_match proof safe
fix v assume v: "v < w"
assume "proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
thus ?thesis unfolding proact_def proof safe
assume sv1: "¬ isSecV sv1" "¬ isIntV sv1" and "move_1 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv1'
where 0: "validTransV (sv1,sv1')"
and Δ: "Δ v w1 w2 s1 s2 statA sv1' sv2 statO"
unfolding move_1_def by auto
have r1': "reachV sv1'" using r1 0 by (metis Van.reach.Step fst_conv snd_conv)
obtain w1' w2' trv1 trv2 statOO where
χ4a: "χ4a Δ ∞ w1 w2 w1' w2' s1 s2 s2' statA sv1' trv1 sv2 trv2 statOO"
using less(1)[OF v Δ r1' r2 isv24] by auto
show ?thesis apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "trv2"])
using χ4a 0 sv1 unfolding χ4a_def by auto
next
assume sv2: "¬ isSecV sv2" "¬ isIntV sv2" and "move_2 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
hence False using isv24 by blast
thus ?thesis by auto
next
assume sv12: "¬ isSecV sv1" "¬ isSecV sv2" "Van.eqAct sv1 sv2"
and "move_12 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
hence False using isv24 by blast
thus ?thesis by auto
qed
next
assume m: "react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
have m: "match2 Δ w1 w2 s1 s2 statA sv1 sv2 statO" using m unfolding react_def by auto
have "(∃w1' w2'. w1'<w1 ∧ w2'<w2 ∧ ¬ isSecO s2 ∧ Δ ∞ w1' w2' s1 s2' statA sv1 sv2 statO) ∨
(∃w1'<w1. eqSec sv2 s2 ∧ ¬ isIntV sv2 ∧ match2_1 Δ w1' ∞ s1 s2 s2' statA sv1 sv2 statO) ∨
(eqSec sv2 s2 ∧ ¬ isSecV sv1 ∧ Van.eqAct sv1 sv2 ∧ match2_12 Δ ∞ ∞ s1 s2 s2' statA sv1 sv2 statO)"
using m v ii4 unfolding match2_def by auto
thus ?thesis
apply(elim disjE exE)
subgoal for w1' w2' using is2 by auto
subgoal for w1' apply(rule exI[of _ w1']) apply(rule exI[of _ ∞])
unfolding match2_1_def apply(elim conjE exE) subgoal for sv2'
apply(rule exI[of _ "[sv1]"]) apply(rule exI[of _ "[sv2,sv2']"])
apply(rule exI[of _ statO])
using is2 isv24 unfolding χ4a_def
by (auto simp : sstatA'_def lastt_def) .
subgoal apply(rule exI[of _ ∞]) apply(rule exI[of _ ∞])
unfolding match2_12_def apply(elim conjE exE) subgoal for sv1' sv2'
apply(rule exI[of _ "[sv1,sv1']"]) apply(rule exI[of _ "[sv2,sv2']"])
apply(rule exI[of _ "sstatO' statO sv1 sv2"])
using is2 isv24 unfolding χ4a_def
by (auto simp : sstatA'_def sstatO'_def lastt_def Van.eqAct_def) . .
qed
qed
definition "χ4b Δ w w1 w2 w1' (w2'::enat) s1 s2 tr2 statAA sv1 trv1 sv2 trv2 statOO ≡
trv1 ≠ [] ∧ trv2 ≠ [] ∧ (length trv1 > Suc 0 ∨ w1' < w1) ∧
Van.validFromS sv1 trv1 ∧ Van.validFromS sv2 trv2 ∧
never isSecV (butlast trv1) ∧
Van.S trv2 = Opt.S tr2 ∧
Van.A trv1 = Van.A trv2 ∧
Δ w w1' w2' s1 (lastt s2 tr2) statAA (lastt sv1 trv1) (lastt sv2 trv2) statOO"
lemma unwindCond_ex_χ4b_aux:
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and tr2NE: "tr2 ≠ []"
and v4': "Opt.validFromS s2 (tr2 ## s2')"
and nis1: "¬ isIntO s1" and nis2: "¬ isIntO s2"
and ninter4': "never isIntO (tr2 ## s2')"
and sec: "never isSecO (butlast tr2)" "isSecO (lastt s2 tr2)"
shows "∃w1' w2' trv1 trv2 statOO. χ4b Δ ∞ w1 w2 w1' w2' s1 s2 (tr2 ## s2') statA sv1 trv1 sv2 trv2 statOO"
proof-
have v4: "Opt.validFromS s2 tr2" and s24': "validTransO (lastt s2 tr2,s2')"
apply (metis v4' Opt.validFromS_def Opt.validS_append1 Nil_is_append_conv hd_append2)
by (metis Opt.validFromS_def Opt.validS_validTrans append_is_Nil_conv lastt_def list.distinct(1) list.sel(1) tr2NE v4')
have ninter4: "never isIntO tr2" and nis2': "¬ isIntO s2'"
using ninter4' by auto
obtain ww ww1 ww2 trv1 trv2 statOO where χ4: "χ4 Δ ww w1 w2 ww1 ww2 s1 s2 tr2 statA sv1 trv1 sv2 trv2 statOO"
using unwindCond_ex_χ4[OF unwind Δ r v4 nis1 nis2 ninter4 sec]
by auto
have trv12NE: "trv1 ≠ []" "trv2 ≠ []" using χ4 unfolding χ4_def by auto
define ss2 ssv1 ssv2 where ss2: "ss2 ≡ lastt s2 tr2"
and ssv1: "ssv1 ≡ lastt sv1 trv1" and ssv2: "ssv2 ≡ lastt sv2 trv2"
have ss2l: "ss2 = last tr2" by (simp add: lastt_def ss2 tr2NE)
have tr2l: "tr2 = butlast tr2 @ [ss2]" by (simp add: ss2l tr2NE)
have ssv1l: "ssv1 = last trv1" using χ4 unfolding χ4_def by (metis lastt_def ssv1)
have trv1l: "trv1 = butlast trv1 @ [ssv1]" by (simp add: ssv1l trv12NE(1))
have ssv2l: "ssv2 = last trv2" using χ4 unfolding χ4_def by (metis lastt_def ssv2)
have trv2l: "trv2 = butlast trv2 @ [ssv2]" by (simp add: ssv2l trv12NE(2))
have iss2[simp]: "isSecO ss2" using sec(2) unfolding ss2 by auto
have issv2[simp]: "isSecV ssv2" and gissv24[simp]: "getSecO ss2 = getSecV ssv2"
using χ4 unfolding χ4_def ssv2 ss2 by auto
have niss2: "¬ isIntO ss2"
by (metis list_all_append list_all_simps(1) ninter4 tr2l)
have rss2: "reachO ss2" and rssv12: "reachV ssv1" "reachV ssv2"
using Opt.reach_validFromS_reach r ss2l tr2NE v4 apply blast
unfolding ssv1 ssv2 using r(3,4) using χ4 unfolding χ4_def
using Van.reach_validFromS_reach ssv1 ssv2 ssv1l ssv2l by auto metis+
have Δ: "Δ ww ww1 ww2 s1 ss2 statA ssv1 ssv2 statOO"
using χ4 unfolding χ4_def ss2[symmetric] ssv1[symmetric] ssv2[symmetric] by auto
have s13': "validTransO (ss2, s2')"
by (simp add: s24' ss2)
note vs24 = s24'[unfolded ss2[symmetric]]
obtain w1' w2' trv1' trv2' statO' where
χ4a: "χ4a Δ ∞ ww1 ww2 w1' w2' s1 ss2 s2' statA ssv1 trv1' ssv2 trv2' statO'"
using unwindCond_ex_χ4a_getSec[OF unwind Δ r(1) rss2 rssv12 s13' niss2 iss2 issv2 gissv24]
by blast
have trv12'NE: "trv1' ≠ []" "trv2' ≠ []" using χ4a unfolding χ4a_def by auto
have [simp]: "Van.O (butlast trv1 @ trv1') = Van.O trv1 @ Van.O trv1'"
using trv12'NE unfolding χ4_def Van.O.map_filter Opt.O.map_filter apply(subst butlast_append) by simp
have [simp]: "Van.O (butlast trv2 @ trv2') = Van.O trv2 @ Van.O trv2'"
using trv12'NE unfolding χ4_def Van.O.map_filter Opt.O.map_filter apply(subst butlast_append) by simp
have "Van.A trv1' = Van.A trv2'" using χ4a unfolding χ4a_def by auto
moreover have "length (Van.O trv1') = length (Van.A trv1') ∧ length (Van.O trv2') = length (Van.A trv2')"
unfolding Van.A.map_filter Van.O.map_filter by auto
ultimately have "length (Van.O trv1') = length (Van.O trv2')" by auto
hence [simp]: "Van.O trv1 @ Van.O trv1' = Van.O trv2 @ Van.O trv2' ⟷
Van.O trv1 = Van.O trv2 ∧ Van.O trv1' = Van.O trv2'" by auto
show ?thesis
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ "butlast trv1 @ trv1'"]) apply(rule exI[of _ "butlast trv2 @ trv2'"])
apply(rule exI[of _ statO'])
unfolding χ4b_def apply(intro conjI)
subgoal using χ4 χ4a unfolding χ4_def χ4a_def by auto
subgoal using χ4 χ4a unfolding χ4_def χ4a_def by auto
subgoal using χ4 χ4a unfolding χ4_def χ4a_def
by simp (metis Simple_Transition_System.fromS_eq_Nil Van.toS_Nil Van.toS_fromS_nonSingl
diff_add_inverse2 linorder_not_less order_le_less_trans trans_less_add2 zero_less_diff)
subgoal using χ4 χ4a unfolding χ4_def χ4a_def ssv1
using Van.validFromS_append by auto
subgoal using χ4 χ4a unfolding χ4_def χ4a_def ssv2
using Van.validFromS_append by auto
subgoal using χ4 χ4a unfolding χ4_def χ4a_def
by (simp add: butlast_append)
subgoal using χ4 χ4a unfolding χ4_def χ4a_def unfolding Van.S.map_filter Opt.S.map_filter
apply(subst tr2l) apply(subst butlast_append)
by simp (metis Opt.S.map_filter Opt.S.eq_Nil_iff Van.S.map_filter Van.S.eq_Nil_iff sec(1))
subgoal using χ4 χ4a unfolding χ4_def χ4a_def Van.A.map_filter Opt.A.map_filter
apply(subst trv1l) apply(subst trv2l)
apply(subst butlast_append) apply simp apply(subst butlast_append) by simp
subgoal using χ4a trv12'NE tr2NE unfolding χ4a_def lastt_def by simp .
qed
lemma unwindCond_ex_χ4b_aux2:
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO" and
r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and v4': "Opt.validFromS s2 (tr2 @ [s2',s2''])"
and nis1: "¬ isIntO s1" and nis2: "¬ isIntO s2"
and ninter4': "never isIntO (tr2 @ [s2',s2''])"
and sec: "never isSecO tr2" "isSecO s2'"
shows "∃w1' w2' trv1 trv2 statOO. χ4b Δ ∞ w1 w2 w1' w2' s1 s2 (tr2 @ [s2',s2'']) statA sv1 trv1 sv2 trv2 statOO"
proof-
have 0: "lastt s2 (tr2 ## s2') = s2'"
unfolding lastt_def by auto
show ?thesis
using unwindCond_ex_χ4b_aux[OF unwind Δ r, of "tr2 ## s2'", unfolded 0, simplified]
using assms by auto
qed
definition "χ4' Δ w1 w2 w1' (w2'::enat) s1 s2 tr2 s2' s2'' statAA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO ≡
Van.validFromS sv1 (trv1 ## sv1'') ∧ Van.validFromS sv2 (trv2 ## sv2'') ∧
never isSecV (butlast (trv1 ## sv1'')) ∧
Van.S (trv2 ## sv2'') = Opt.S ((tr2 ## s2') ## s2'') ∧
Van.A (trv1 ## sv1'') = Van.A (trv2 ## sv2'') ∧
trv2 ≠ [] ∧ (trv1 ≠ [] ∨ w1' < w1) ∧
Δ ∞ w1' w2' s1 s2'' statAA sv1'' sv2'' statOO"
proposition unwindCond_ex_χ4':
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO" and
r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and v4': "Opt.validFromS s2 ((tr2 ## s2') ## s2'')"
and nis1: "¬ isIntO s1" and nis2: "¬ isIntO s2"
and ninter4': "never isIntO ((tr2 ## s2') ## s2'')"
and sec: "never isSecO tr2" "isSecO s2'"
shows "∃w1' w2' trv1 sv1'' trv2 sv2'' statOO. χ4' Δ w1 w2 w1' w2' s1 s2 tr2 s2' s2'' statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
using unwindCond_ex_χ4b_aux2[unfolded φ_def, unfolded lastt_snoc lastt_snoc2 append_snoc2, OF assms]
unfolding χ4b_def apply(elim exE) subgoal for w1' w2' trv1 trv2 statOO
apply(cases "trv1" rule: rev_cases)
subgoal by auto
subgoal for trv1' sv1'' apply(cases "trv2" rule: rev_cases)
subgoal by auto
subgoal for trv2' sv2'' unfolding χ4'_def
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ trv1']) apply(rule exI[of _ sv1''])
apply(rule exI[of _ trv2']) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statOO])
by simp (metis Opt.S.Nil_iff Opt.S.eq_Nil_iff(1) Van.S.simps(4) butlast_append
list.discI list_all_append sec(2) self_append_conv2) . . .
definition "ω4 Δ w1 w2 w1' (w2'::enat) s1 s2 s2' statAA sv1 trv1 sv1' sv2 trv2 sv2' statOO ≡
Van.validFromS sv1 (trv1 ## sv1') ∧ Van.validFromS sv2 (trv2 ## sv2') ∧
never isSecV trv1 ∧ never isSecV trv2 ∧
Van.A (trv1 ## sv1') = Van.A (trv2 ## sv2') ∧
(trv1 ≠ [] ∨ w1' < w1) ∧ (trv2 ≠ [] ∨ w2' < w2) ∧
Δ ∞ w1' w2' s1 s2' statAA sv1' sv2' statOO"
proposition unwindCond_ex_ω4:
assumes unwind: "unwindCond Δ"
and Δ: "Δ w w1 w2 s1 s2 statA sv1 sv2 statO"
and r34: "reachO s1" "reachO s2" and r12: "reachV sv1" "reachV sv2"
and nis1: "¬ isIntO s1"
and v4: "validTransO (s2,s2')"
and nis2: "¬ isIntO s2" "¬ isIntO s2'" "¬ isSecO s2"
shows "∃w1' w2' trv1 sv1' trv2 sv2' statOO. ω4 Δ w1 w2 w1' w2' s1 s2 s2' statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
using Δ r12
proof(induction w arbitrary: w1 w2 sv1 sv2 statO rule: less_induct)
case (less w w1 w2 sv1 sv2 statO)
note Δ = `Δ w w1 w2 s1 s2 statA sv1 sv2 statO`
note r12 = less.prems(2,3)
note r1 = r12(1) note r2 = r12(2)
note r = r34 r12
have f34: "finalO s1 = finalO s2 ∧ finalV sv1 = finalO s1 ∧ finalV sv2 = finalO s2"
using Δ unwind[unfolded unwindCond_def] r by auto
have proact_match: "(∃v<w. proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO) ∨ react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
using Δ unwind[unfolded unwindCond_def] r by auto
show ?case using proact_match proof safe
fix v assume v: "v < w"
assume "proact Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
thus ?thesis unfolding proact_def proof safe
assume sv1: "¬ isSecV sv1" "¬ isIntV sv1" and "move_1 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv1' where 0: "validTransV (sv1, sv1')" and Δ: "Δ v w1 w2 s1 s2 statA sv1' sv2 statO"
unfolding move_1_def by auto
have r1': "reachV sv1'" using r1 0 by (metis Van.reach.Step fst_conv snd_conv)
obtain w1' w2' trv1 sv1'' trv2 sv2' statOO where
ω4: "ω4 Δ w1 w2 w1' w2' s1 s2 s2' statA sv1' trv1 sv1'' sv2 trv2 sv2' statOO"
using less(1)[OF v Δ r1' r2] by auto
show ?thesis apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "sv1''"])
apply(rule exI[of _ "trv2"]) apply(rule exI[of _ "sv2'"])
using ω4 0 sv1 unfolding ω4_def by auto
next
assume sv2: "¬ isSecV sv2" "¬ isIntV sv2" and "move_2 Δ v w1 w2 s1 s2 statA sv1 sv2 statO"
then obtain sv2'
where 0: "validTransV (sv2,sv2')"
and Δ: "Δ v w1 w2 s1 s2 statA sv1 sv2' statO"
unfolding move_2_def by auto
have r2': "reachV sv2'" using r2 0 by (metis Van.reach.Step fst_conv snd_conv)
obtain w1' w2' trv1 sv1' trv2 sv2'' statOO where
ω4: "ω4 Δ w1 w2 w1' w2' s1 s2 s2' statA sv1 trv1 sv1' sv2' trv2 sv2'' statOO"
using less(1)[OF v Δ r1 r2'] by auto
show ?thesis apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ "trv1"]) apply(rule exI[of _ "sv1'"])
apply(rule exI[of _ "sv2 # trv2"]) apply(rule exI[of _ "sv2''"])
using ω4 0 sv2 unfolding ω4_def by auto
next
assume sv1: "¬ isSecV sv1" and sv2: "¬ isSecV sv2" and
"move_12 Δ v w1 w2 s1 s2 statA sv1 sv2 statO" and
sv12: "Van.eqAct sv1 sv2"
then obtain sv1' sv2' statO'
where statO': "statO' = sstatO' statO sv1 sv2"
and 0: "validTransV (sv1,sv1')" "validTransV (sv2,sv2')"
and Δ: "Δ v w1 w2 s1 s2 statA sv1' sv2' statO'"
unfolding move_12_def by auto
have r1': "reachV sv1'" and r2': "reachV sv2'" using r1 r2 0
by (metis Van.reach.Step fst_conv snd_conv)+
obtain w1' w2' trv1 sv1'' trv2 sv2'' statOO where
ω4: "ω4 Δ w1 w2 w1' w2' s1 s2 s2' statA sv1' trv1 sv1'' sv2' trv2 sv2'' statOO"
using less(1)[OF v Δ r1' r2'] by auto
show ?thesis apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ "sv1 # trv1"]) apply(rule exI[of _ "sv1''"])
apply(rule exI[of _ "sv2 # trv2"]) apply(rule exI[of _ "sv2''"])
using ω4 0 sv1 sv2 sv12 unfolding ω4_def statO' by (auto simp: Van.eqAct_def)
qed
next
assume m: "react Δ w1 w2 s1 s2 statA sv1 sv2 statO"
have m: "match2 Δ w1 w2 s1 s2 statA sv1 sv2 statO" using m unfolding react_def by auto
have "(∃w1' w2'. w1'<w1 ∧ w2'<w2 ∧ ¬ isSecO s2 ∧ Δ ∞ w1' w2' s1 s2' statA sv1 sv2 statO) ∨
(∃w1'<w1. eqSec sv2 s2 ∧ ¬ isIntV sv2 ∧ match2_1 Δ w1' ∞ s1 s2 s2' statA sv1 sv2 statO) ∨
¬ isSecV sv1 ∧ eqSec sv2 s2 ∧ Van.eqAct sv1 sv2 ∧ match2_12 Δ ∞ ∞ s1 s2 s2' statA sv1 sv2 statO"
using m v4 nis2 unfolding match2_def by auto
thus ?thesis
apply(elim disjE exE)
subgoal for w1' w2' apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ "[]"]) apply(rule exI[of _ "sv1"])
apply(rule exI[of _ "[]"]) apply(rule exI[of _ "sv2"])
apply(rule exI[of _ statO]) unfolding ω4_def
by auto
subgoal for w1' apply(rule exI[of _ w1']) apply(rule exI[of _ ∞])
unfolding match2_1_def apply(elim conjE exE) subgoal for sv2'
apply(rule exI[of _ "[]"]) apply(rule exI[of _ "sv1"])
apply(rule exI[of _ "[sv2]"]) apply(rule exI[of _ "sv2'"])
apply(rule exI[of _ statO])
unfolding ω4_def using nis2(3) by (auto simp: eqSec_def) .
subgoal apply(rule exI[of _ ∞]) apply(rule exI[of _ ∞])
unfolding match2_12_def apply(elim conjE exE) subgoal for sv1' sv2'
apply(rule exI[of _ "[sv1]"]) apply(rule exI[of _ "sv1'"])
apply(rule exI[of _ "[sv2]"]) apply(rule exI[of _ "sv2'"])
apply(rule exI[of _ "sstatO' statO sv1 sv2"])
unfolding ω4_def using nis2(3) apply (auto simp: eqSec_def
sstatA'_def sstatO'_def lastt_def Van.eqAct_def) . . .
qed
qed
definition "φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2' ≡
ltr1 = lappend (llist_of (tr1 ## s1')) (s1'' $ ltr1') ∧
ltr2 = lappend (llist_of (tr2 ## s2')) (s2'' $ ltr2') ∧
Opt.validFromS s1 ((tr1 ## s1') ## s1'') ∧ Opt.validFromS s2 ((tr2 ## s2') ## s2'') ∧
never isIntO tr1 ∧ never isIntO tr2 ∧
isIntO s1' ∧ isIntO s2' ∧ getActO s1' = getActO s2' ∧
Opt.lvalidFromS s1'' (s1'' $ ltr1') ∧ Opt.lcompletedFrom s1'' (s1'' $ ltr1') ∧
Opt.lvalidFromS s2'' (s2'' $ ltr2') ∧ Opt.lcompletedFrom s2'' (s2'' $ ltr2') ∧
Opt.lA (s1'' $ ltr1') = Opt.lA (s2'' $ ltr2')"
lemma isIntO_φφ:
assumes vltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and vltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and A: "Opt.lA ltr1 = Opt.lA ltr2" and inter3: "¬ lnever isIntO ltr1"
shows "∃tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2'. φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2'"
proof-
have 03: "∃s∈lset ltr1. isIntO s" using inter3 unfolding llist.pred_set by auto
define ttr1 where ttr1: "ttr1 ≡ ltakeUntil isIntO ltr1"
define lltr1' where lltr1': "lltr1' ≡ ldropUntil isIntO ltr1"
have ltr1: "ltr1 = lappend (llist_of ttr1) lltr1'"
unfolding ttr1 lltr1' lappend_ltakeUntil_ldropUntil[OF 03] ..
have 13: "ttr1 ≠ [] ∧ never isIntO (butlast ttr1) ∧ isIntO (last ttr1)"
unfolding ttr1
using ltakeUntil_last[OF 03] ltakeUntil_not_Nil[OF 03] ltakeUntil_never_butlast[OF 03] by simp
then obtain tr1 s1' where ttr1_eq: "ttr1 = tr1 ## s1'"
using rev_exhaust by blast
hence tr1s1': "never isIntO tr1" "isIntO s1'" using 13 by auto
have "lfinite ltr1 ⟹ s1' ≠ llast ltr1"
by (metis Opt.final_not_isInt Opt.lcompletedFrom_def llast_last_llist_of tr1s1'(2) vltr1(2))
hence ne: "lltr1' ≠ [[]]"
using ltr1 unfolding ttr1_eq by auto
then obtain s1'' ltr1' where lltr1': "lltr1' = s1'' $ ltr1'"
by (meson llist.exhaust)
have [simp]: "filter isIntO tr1 = []"
by (metis never_Nil_filter tr1s1'(1))
have clltr1': "Opt.lcompletedFrom s1 lltr1'"
by (metis Opt.lcompletedFrom_def lfinite_lappend lfinite_llist_of llast_lappend_LCons
llast_last_llist_of lltr1' ltr1 ne vltr1(2))
have inter4: "¬ lnever isIntO ltr2" using A inter3
by (metis Opt.lA.eq_LNil_iff Opt.lO Opt.lO.eq_LNil_iff lfiltermap_LNil_never
lfiltermap_lmap_lfilter vltr1(2) vltr2(2))
have 04: "∃s∈lset ltr2. isIntO s" using inter4 unfolding llist.pred_set by auto
define ttr2 where ttr2: "ttr2 ≡ ltakeUntil isIntO ltr2"
define lltr2' where lltr2': "lltr2' ≡ ldropUntil isIntO ltr2"
have ltr2: "ltr2 = lappend (llist_of ttr2) lltr2'"
unfolding ttr2 lltr2' lappend_ltakeUntil_ldropUntil[OF 04] ..
have 14: "ttr2 ≠ [] ∧ never isIntO (butlast ttr2) ∧ isIntO (last ttr2)"
unfolding ttr2
using ltakeUntil_last[OF 04] ltakeUntil_not_Nil[OF 04] ltakeUntil_never_butlast[OF 04] by simp
then obtain tr2 s2' where ttr2_eq: "ttr2 = tr2 ## s2'"
using rev_exhaust by blast
hence tr2s2': "never isIntO tr2" "isIntO s2'" using 14 by auto
have "lfinite ltr2 ⟹ s2' ≠ llast ltr2"
by (metis Opt.final_not_isInt Opt.lcompletedFrom_def llast_last_llist_of tr2s2'(2) vltr2(2))
hence ne: "lltr2' ≠ [[]]"
using ltr2 unfolding ttr2_eq by auto
then obtain s2'' ltr2' where lltr2': "lltr2' = s2'' $ ltr2'"
by (meson llist.exhaust)
have [simp]: "filter isIntO tr2 = []"
by (metis never_Nil_filter tr2s2'(1))
have clltr2': "Opt.lcompletedFrom s2 lltr2'"
by (metis Opt.lcompletedFrom_def lfinite_lappend lfinite_llist_of llast_lappend_LCons
llast_last_llist_of lltr2' ltr2 ne vltr2(2))
have AA: "Opt.lA lltr1' = Opt.lA lltr2'"
unfolding Opt.lA[OF clltr1'] Opt.lA[OF clltr2']
using A[unfolded Opt.lA[OF vltr1(2)] Opt.lA[OF vltr2(2)]] tr1s1' tr2s2'
unfolding ltr1 ltr2 ttr1_eq ttr2_eq
unfolding lfilter_lappend_llist_of by simp
show ?thesis apply(rule exI[of _ tr1]) apply(rule exI[of _ s1'])
apply(rule exI[of _ s1'']) apply(rule exI[of _ ltr1'])
apply(rule exI[of _ tr2]) apply(rule exI[of _ s2'])
apply(rule exI[of _ s2'']) apply(rule exI[of _ ltr2'])
unfolding φφ_def apply(intro conjI)
subgoal unfolding ltr1 ttr1_eq lltr1' ..
subgoal unfolding ltr2 ttr2_eq lltr2' ..
subgoal using vltr1(1) unfolding ltr1 ttr1_eq lltr1'
by (simp add: Opt.lvalidFromS_lappend_finite lappend_llist_of_LCons)
subgoal using vltr2(1) unfolding ltr2 ttr2_eq lltr2'
by (simp add: Opt.lvalidFromS_lappend_finite lappend_llist_of_LCons)
subgoal using tr1s1' by simp
subgoal using tr2s2' by simp
subgoal using tr1s1' by simp
subgoal using tr2s2' by simp
subgoal using A[unfolded Opt.lA[OF vltr1(2)] Opt.lA[OF vltr2(2)]]
tr1s1' tr2s2'
unfolding ltr1 ttr1_eq ltr2 ttr2_eq lltr1' lltr2'
unfolding lfilter_lappend_llist_of by simp
subgoal using vltr1(1) unfolding ltr1 ttr1_eq lltr1'
using Opt.lvalidFromS_lappend_LCons by blast
subgoal using vltr1(2) unfolding ltr1 ttr1_eq lltr1'
by (metis Opt.lcompletedFrom_def lfinite_lappend lfinite_llist_of
llast_lappend_LCons llast_last_llist_of llist.distinct(1))
subgoal using vltr2(1) unfolding ltr2 ttr2_eq lltr2'
using Opt.lvalidFromS_lappend_LCons by blast
subgoal using vltr2(2) unfolding ltr2 ttr2_eq lltr2'
by (metis Opt.lcompletedFrom_def lfinite_lappend lfinite_llist_of
llast_lappend_LCons llast_last_llist_of llist.distinct(1))
subgoal using AA unfolding lltr1' lltr2' . .
qed
definition "χχ s1 ltr1 tr1 s1' s1'' ltr1' ≡
ltr1 = lappend (llist_of (tr1 ## s1')) (s1'' $ ltr1') ∧
Opt.validFromS s1 ((tr1 ## s1') ## s1'') ∧
never isIntO tr1 ∧ ¬ isIntO s1' ∧ ¬ isIntO s1'' ∧
never isSecO tr1 ∧ isSecO s1' ∧
Opt.lvalidFromS s1'' (s1'' $ ltr1') ∧ Opt.lcompletedFrom s1'' (s1'' $ ltr1')"
lemma isSecO_χχ:
assumes vltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and inter: "lnever isIntO ltr1" and isec: "¬ lnever isSecO ltr1"
shows "∃tr1 s1' s1'' ltr1'. χχ s1 ltr1 tr1 s1' s1'' ltr1'"
proof-
have 0: "∃s∈lset ltr1. isSecO s" using isec unfolding llist.pred_set by auto
define ttr1 where ttr1: "ttr1 ≡ ltakeUntil isSecO ltr1"
define lltr1' where lltr1': "lltr1' ≡ ldropUntil isSecO ltr1"
have ltr1: "ltr1 = lappend (llist_of ttr1) lltr1'"
unfolding ttr1 lltr1' lappend_ltakeUntil_ldropUntil[OF 0] ..
have 1: "ttr1 ≠ [] ∧ never isSecO (butlast ttr1) ∧ isSecO (last ttr1)"
unfolding ttr1
using ltakeUntil_last[OF 0] ltakeUntil_not_Nil[OF 0] ltakeUntil_never_butlast[OF 0] by simp
then obtain tr1 s1' where ttr1_eq: "ttr1 = tr1 ## s1'"
using rev_exhaust by blast
hence tr1s1': "never isSecO tr1" "isSecO s1'" using 1 by auto
have 2: "never isIntO tr1 ∧ ¬ isIntO s1' ∧ lnever isIntO lltr1'"
using inter unfolding ltr1 ttr1_eq
unfolding llist_all_lappend_llist_of list_all_append by simp
have "lfinite ltr1 ⟹ s1' ≠ llast ltr1"
by (metis Opt.final_not_isSec Opt.lcompletedFrom_def llast_last_llist_of tr1s1'(2) vltr1(2))
hence ne: "lltr1' ≠ [[]]"
using ltr1 unfolding ttr1_eq by auto
then obtain s1'' ltr1' where lltr1': "lltr1' = s1'' $ ltr1'"
by (meson llist.exhaust)
show ?thesis apply(rule exI[of _ tr1]) apply(rule exI[of _ s1'])
apply(rule exI[of _ s1'']) apply(rule exI[of _ ltr1'])
unfolding χχ_def apply(intro conjI)
subgoal unfolding ltr1 ttr1_eq lltr1' ..
subgoal using vltr1(1) unfolding ltr1 ttr1_eq lltr1'
by (simp add: Opt.lvalidFromS_lappend_finite lappend_llist_of_LCons)
subgoal using 2 by simp
subgoal using 2 by simp
subgoal using 2 unfolding lltr1' by simp
subgoal using tr1s1' by simp
subgoal using tr1s1' by simp
subgoal using vltr1(1) unfolding ltr1 ttr1_eq lltr1'
using Opt.lvalidFromS_lappend_LCons by blast
subgoal using vltr1(2) unfolding ltr1 ttr1_eq lltr1'
by (metis Opt.lcompletedFrom_def ne lfinite_lappend
lfinite_llist_of llast_lappend_LCons llast_last_llist_of lltr1') .
qed
type_synonym ('stA,'stO) tuple34 =
"enat × enat ×
'stA × 'stA llist ×
'stA × 'stA llist ×
status ×
'stO × 'stO × status"
type_synonym ('stA,'stO) tuple12 =
"'stO list × 'stO × 'stO list × 'stO × status × status"
context
fixes Δ :: "enat ⇒ enat ⇒ enat ⇒ 'stateO ⇒ 'stateO ⇒ status ⇒ 'stateV ⇒ 'stateV ⇒ status ⇒ bool"
begin
fun isn :: "turn × ('stateO,'stateV) tuple34 ⇒ bool"
where
"isn (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ⟷ ltr1 = [[]] ∧ ltr2 = [[]]"
fun h_t ::
"turn × ('stateO,'stateV)tuple34 ⇒
('stateO,'stateV)tuple12 ×
turn × ('stateO,'stateV)tuple34"
where
"h_t (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
(if trn = L
then if lnever isSecO ltr1
then let (s1',ltr1') = (lhd (ltl ltr1), ltl ltr1)
in let (w1',w2',trv1,sv1',trv2,sv2',statOO) =
(SOME k. case k of (w1',w2',trv1,sv1',trv2,sv2',statOO) ⇒
ω3 Δ w1 w2 w1' w2' s1 s1' s2 statA sv1 trv1 sv1' sv2 trv2 sv2' statOO)
in ((trv1,sv1',trv2,sv2',statA,statOO),
(if trv1 = [] then L else R,
w1',w2',s1',ltr1',s2,ltr2,statA,sv1',sv2',statOO))
else
let (tr1,s1',s1'',ltr1') =
(SOME k. case k of (tr1,s1',s1'',ltr1') ⇒
χχ s1 ltr1 tr1 s1' s1'' ltr1')
in let (w1',w2',trv1,sv1'',trv2,sv2'',statOO) =
(SOME k'. case k' of (w1',w2',trv1,sv1'',trv2,sv2'',statOO) ⇒
χ3' Δ w1 w2 w1' w2' s1 tr1 s1' s1'' s2 statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO)
in ((trv1,sv1'',trv2,sv2'',statA,statOO),
(R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO))
else if lnever isSecO ltr2
then let (s2',ltr2') = (lhd (ltl ltr2), ltl ltr2)
in let (w1',w2',trv1,sv1',trv2,sv2',statOO) =
(SOME k. case k of (w1',w2',trv1,sv1',trv2,sv2',statOO) ⇒
ω4 Δ w1 w2 w1' w2' s1 s2 s2' statA sv1 trv1 sv1' sv2 trv2 sv2' statOO)
in ((trv1,sv1',trv2,sv2',statA,statOO),
(if trv2 = [] then R else L,
w1',w2',s1,ltr1,s2',ltr2',statA,sv1',sv2',statOO))
else
let (tr2,s2',s2'',ltr2') =
(SOME k. case k of (tr2,s2',s2'',ltr2') ⇒
χχ s2 ltr2 tr2 s2' s2'' ltr2')
in let (w1',w2',trv1,sv1'',trv2,sv2'',statOO) =
(SOME k'. case k' of (w1',w2',trv1,sv1'',trv2,sv2'',statOO) ⇒
χ4' Δ w1 w2 w1' w2' s1 s2 tr2 s2' s2'' statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO)
in ((trv1,sv1'',trv2,sv2'',statA,statOO),
(L,w1',w2',s1, ltr1, s2'',s2'' $ ltr2',statA,sv1'',sv2'',statOO))
)"
declare h_t.simps[simp del]
definition "h ≡ fst o h_t"
definition "t ≡ snd o h_t"
fun econd where "econd (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
(llength ltr1 ≤ Suc 0 ∨ llength ltr2 ≤ Suc 0)"
fun e where "e (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) = [[([sv1],sv1,[sv2],sv2,statA,statO)]]"
definition f :: "turn × ('stateO,'stateV)tuple34 ⇒ ('stateO,'stateV)tuple12 llist"
where "f ≡ ccorec_llist isn h econd e t"
lemma f_LNil:
"ltr1 = [[]] ⟹ ltr2 = [[]] ⟹ f (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) = [[]]"
unfolding f_def apply(subst llist_ccorec(1)) by auto
lemma f_length_1:
assumes "ltr1 ≠ [[]] ∨ ltr2 ≠ [[]]" "llength ltr1 ≤ Suc 0 ∨ llength ltr2 ≤ Suc 0"
shows "f (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) = [[([sv1],sv1,[sv2],sv2,statA,statO)]]"
using assms unfolding f_def apply(subst llist_ccorec(2))
subgoal unfolding e.simps lnull_def by auto
subgoal by auto
subgoal unfolding econd.simps by simp .
lemma f_length_ge1:
assumes "llength ltr1 > Suc 0" "llength ltr2 > Suc 0"
shows "f (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
LCons (h (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)) (f (t (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ))"
proof-
show ?thesis using assms unfolding f_def apply(subst llist_ccorec(2))
subgoal unfolding e.simps lnull_def by auto
subgoal by auto
subgoal unfolding econd.simps by auto .
qed
definition lltrv1 :: "turn × ('stateO,'stateV)tuple34 ⇒ 'stateV llist" where
"lltrv1 trn_tp = lconcat (lmap (λ(trv1,sv1'',trv2,sv2'',statAA,statOO). llist_of trv1) (f trn_tp))"
definition then1 :: "turn × ('stateO,'stateV)tuple34 ⇒ nat" where
"then1 trn_tp = firstNC (lmap (λ(trv1,sv1'',trv2,sv2'',statAA,statOO). trv1) (f trn_tp))"
definition lltrv2 :: "turn × ('stateO,'stateV)tuple34 ⇒ 'stateV llist" where
"lltrv2 trn_tp = lconcat (lmap (λ(trv1,sv1'',trv2,sv2'',statAA,statOO). llist_of trv2) (f trn_tp))"
definition then2 :: "turn × ('stateO,'stateV)tuple34 ⇒ nat" where
"then2 trn_tp = firstNC (lmap (λ(trv1,sv1'',trv2,sv2'',statAA,statOO). trv2) (f trn_tp))"
lemma lltrv1_ne_imp:
assumes "lltrv1 trn_tp ≠ [[]]"
shows "∃trv1 sv1'' trv2 sv2'' statAA statOO. (trv1,sv1'',trv2,sv2'',statAA,statOO) ∈ lset (f trn_tp) ∧
trv1 ≠ []"
using assms unfolding lltrv1_def unfolding lconcat_eq_LNil_iff by force
lemma lltrv2_ne_imp:
assumes "lltrv2 trn_tp ≠ [[]]"
shows "∃trv1 sv1'' trv2 sv2'' statAA statOO. (trv1,sv1'',trv2,sv2'',statAA,statOO) ∈ lset (f trn_tp) ∧
trv2 ≠ []"
using assms unfolding lltrv2_def unfolding lconcat_eq_LNil_iff by force
lemma lltrv1_LNil[simp]:
"ltr1 = [[]] ⟹ ltr2 = [[]] ⟹ lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) = [[]]"
unfolding lltrv1_def f_LNil by simp
lemma lltrv2_LNil[simp]:
"ltr1 = [[]] ⟹ ltr2 = [[]] ⟹ lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) = [[]]"
unfolding lltrv2_def f_LNil by simp
lemma lltrv1_lnever[simp]:
assumes "ltr1 ≠ [[]] ∨ ltr2 ≠ [[]]" "llength ltr1 ≤ Suc 0 ∨ llength ltr2 ≤ Suc 0"
shows "lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) = [[sv1]]"
unfolding lltrv1_def using f_length_1[OF assms] by auto
lemma lltrv2_lnever[simp]:
assumes "ltr1 ≠ [[]] ∨ ltr2 ≠ [[]]" "llength ltr1 ≤ Suc 0 ∨ llength ltr2 ≤ Suc 0"
shows "lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) = [[sv2]]"
unfolding lltrv2_def using f_length_1[OF assms] by auto
lemma h_t_lnever_L:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and l': "lnever isIntO ltr1" "¬ isIntO s2"
and len: "llength ltr1 > Suc 0" "llength ltr2 > Suc 0"
and l: "trn = L" "lnever isSecO ltr1"
shows "∃ w1' w2' s1' ltr1' trv1 sv1' trv2 sv2' statOO.
ltr1 = s1 $ ltr1' ∧ validTransO (s1,s1') ∧
Opt.lvalidFromS s1' ltr1' ∧ Opt.lcompletedFrom s1' ltr1' ∧ lnever isIntO ltr1' ∧
ω3 Δ w1 w2 w1' w2' s1 s1' s2 statA sv1 trv1 sv1' sv2 trv2 sv2' statOO ∧
h_t (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
((trv1,sv1',trv2,sv2',statA,statOO),
(if trv1 = [] then L else R,
w1',w2', s1', ltr1',s2,ltr2,statA,sv1',sv2',statOO))"
proof-
have s1: "¬ isIntO s1" using l' ltr1
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil lhd_LCons
llist.exhaust llist.pred_inject(2))
obtain ltr1' where ltr13: "ltr1 = s1 $ ltr1'"
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil llist.exhaust_sel ltr1(1) ltr1(2))
hence ltr1': "ltr1' = ltl ltr1" by auto
have ltr1'ne: "ltr1' ≠ [[]]" using len(1) unfolding ltr13
by (metis One_nat_def llength_LCons llength_LNil one_eSuc one_enat_def order_less_irrefl)
define s1' where s1': "s1' = lhd (ltl ltr1)"
have v3: "validTransO (s1,s1')" and vv3: "Opt.lvalidFromS s1' ltr1'" "Opt.lcompletedFrom s1' ltr1'"
using ltr1 ltr1'ne unfolding ltr13 s1'
by (metis Opt.lcompletedFrom_LCons Opt.lcompletedFrom_def Opt.lvalidFromS_Cons_iff ltr1' ltr13)+
have is1': "¬ isIntO s1'" and "lnever isIntO ltr1'"
using l'(1) unfolding ltr13
by (metis llist.exhaust_sel llist.pred_inject(2) ltr1' ltr1'ne s1')+
have iss1: "¬ isSecO s1"
using l(2) ltr13 by auto
obtain w1' w2' trv1 sv1' trv2 sv2' statOO
where ω3: "ω3 Δ w1 w2 w1' w2' s1 (lhd (ltl ltr1)) s2 statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
using unwindCond_ex_ω3[OF unw Δ r v3 s1 is1' iss1 l'(2)] s1' by auto
define tp' where
"tp' = (SOME k'. case k' of (w1',w2',trv1,sv1',trv2,sv2',statOO) ⇒
ω3 Δ w1 w2 w1' w2' s1 (lhd (ltl ltr1)) s2 statA sv1 trv1 sv1' sv2 trv2 sv2' statOO)"
have 1: "case tp' of (w1',w2',trv1,sv1',trv2,sv2',statOO) ⇒
ω3 Δ w1 w2 w1' w2' s1 s1' s2 statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
using ω3 unfolding tp'_def s1' apply- apply(rule someI_ex)
apply(rule exI[of _ "(w1',w2',trv1,sv1',trv2,sv2',statOO)"]) by auto
obtain w1' w2' trv1 sv1' trv2 sv2' statOO where
tp': "tp' = (w1',w2',trv1,sv1',trv2,sv2',statOO)" by(cases tp', auto)
have ω3: "ω3 Δ w1 w2 w1' w2' s1 s1' s2 statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
using 1 unfolding tp' by auto
show ?thesis
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1']) apply(rule exI[of _ ltr1'])
apply(rule exI[of _ trv1]) apply(rule exI[of _ sv1'])
apply(rule exI[of _ trv2]) apply(rule exI[of _ sv2'])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal by fact
subgoal by fact
subgoal by fact
subgoal by fact
subgoal by fact
subgoal by fact
subgoal using len l unfolding h_t.simps apply simp
unfolding tp'_def[symmetric] tp' s1' ltr1' by simp .
qed
lemma lltrv1_lltrv2_lnever_L:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and l': "lnever isIntO ltr1" "¬ isIntO s2"
and len: "llength ltr1 > Suc 0" "llength ltr2 > Suc 0"
and l: "trn = L" "lnever isSecO ltr1"
shows "∃ w1' w2' s1' ltr1' trv1 sv1' trv2 sv2' statOO.
ltr1 = s1 $ ltr1' ∧ validTransO (s1,s1') ∧
Opt.lvalidFromS s1' ltr1' ∧ Opt.lcompletedFrom s1' ltr1' ∧ lnever isIntO ltr1' ∧
ω3 Δ w1 w2 w1' w2' s1 s1' s2 statA sv1 trv1 sv1' sv2 trv2 sv2' statOO ∧
lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
lappend (llist_of trv1) (lltrv1 (if trv1 = [] then L else R,
w1',w2',s1',ltr1',s2,ltr2,statA,sv1',sv2',statOO)) ∧
lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
lappend (llist_of trv2) (lltrv2 (if trv1 = [] then L else R,
w1',w2',s1',ltr1',s2,ltr2,statA,sv1',sv2',statOO))"
proof-
show ?thesis
using h_t_lnever_L[OF assms] apply(elim exE)
subgoal for w1' w2' s1' ltr1' trv1 sv1' trv2 sv2' statOO
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1']) apply(rule exI[of _ ltr1'])
apply(rule exI[of _ trv1]) apply(rule exI[of _ sv1'])
apply(rule exI[of _ trv2]) apply(rule exI[of _ sv2'])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal by simp
subgoal by simp
subgoal by simp
subgoal by simp
subgoal by simp
subgoal by simp
subgoal unfolding lltrv1_def apply(subst f_length_ge1[OF len])
unfolding h_def t_def by auto
subgoal unfolding lltrv2_def apply(subst f_length_ge1[OF len])
unfolding h_def t_def by auto . .
qed
lemma h_t_not_lnever_L:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and l': "lnever isIntO ltr1" "¬ isIntO s2"
and len: "llength ltr1 > Suc 0" "llength ltr2 > Suc 0"
and l: "trn = L" "¬ lnever isSecO ltr1"
shows "∃ w1' w2' tr1 s1' s1'' ltr1' trv1 sv1'' trv2 sv2'' statOO.
χχ s1 ltr1 tr1 s1' s1'' ltr1' ∧
χ3' Δ w1 w2 w1' w2' s1 tr1 s1' s1'' s2 statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO ∧
h_t (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
((trv1,sv1'',trv2,sv2'',statA,statOO),
(R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO))"
proof-
have s1: "¬ isIntO s1" using l' ltr1
by (metis Opt.lvalidFromS_def l(2) lhd_LCons llist.exhaust llist.pred_inject(1) llist.pred_inject(2))
obtain tr1 s1' s1'' ltr1'
where χχ: "χχ s1 ltr1 tr1 s1' s1'' ltr1'"
using isSecO_χχ[OF ltr1 l'(1) l(2)] by auto
define tp where
"tp = (SOME k. case k of (tr1,s1',s1'',ltr1') ⇒
χχ s1 ltr1 tr1 s1' s1'' ltr1')"
have 0: "case tp of (tr1,s1',s1'',ltr1') ⇒
χχ s1 ltr1 tr1 s1' s1'' ltr1'"
using χχ unfolding tp_def apply- apply(rule someI_ex)
apply(rule exI[of _ "(tr1,s1',s1'',ltr1')"]) by auto
obtain tr1 s1' s1'' ltr1' where
tp: "tp = (tr1,s1',s1'',ltr1')" by(cases tp, auto)
have χχ: "χχ s1 ltr1 tr1 s1' s1'' ltr1'"
using 0 unfolding tp by auto
obtain w1' w2' trv1 sv1'' trv2 sv2'' statOO
where χ3': "χ3' Δ w1 w2 w1' w2' s1 tr1 s1' s1'' s2 statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
using unwindCond_ex_χ3'[OF unw Δ r, of tr1 s1' s1'']
using χχ l' s1 unfolding χχ_def by auto
define tp' where
"tp' = (SOME k'. case k' of (w1',w2',trv1,sv1'',trv2,sv2'',statOO) ⇒
χ3' Δ w1 w2 w1' w2' s1 tr1 s1' s1'' s2 statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO)"
have 1: "case tp' of (w1',w2',trv1,sv1'',trv2,sv2'',statOO) ⇒
χ3' Δ w1 w2 w1' w2' s1 tr1 s1' s1'' s2 statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
using χ3' unfolding tp'_def apply- apply(rule someI_ex)
apply(rule exI[of _ "(w1',w2',trv1,sv1'',trv2,sv2'',statOO)"]) by auto
obtain w1' w2' trv1 sv1'' trv2 sv2'' statOO where
tp': "tp' = (w1',w2',trv1,sv1'',trv2,sv2'',statOO)" by(cases tp', auto)
have χ3': "χ3' Δ w1 w2 w1' w2' s1 tr1 s1' s1'' s2 statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
using 1 unfolding tp' by auto
show ?thesis
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ tr1]) apply(rule exI[of _ s1']) apply(rule exI[of _ s1'']) apply(rule exI[of _ ltr1'])
apply(rule exI[of _ trv1]) apply(rule exI[of _ sv1'']) apply(rule exI[of _ trv2]) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal using χχ .
subgoal using χ3' .
subgoal using l unfolding h_t.simps
unfolding tp_def[symmetric] tp apply simp
unfolding tp'_def[symmetric] tp' by simp .
qed
lemma lltrv1_lltrv2_not_lnever_L:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and l': "lnever isIntO ltr1" "¬ isIntO s2"
and len: "llength ltr1 > Suc 0" "llength ltr2 > Suc 0"
and l: "trn = L" "¬ lnever isSecO ltr1"
shows "∃ w1' w2' tr1 s1' s1'' ltr1' trv1 sv1'' trv2 sv2'' statOO.
χχ s1 ltr1 tr1 s1' s1'' ltr1' ∧
χ3' Δ w1 w2 w1' w2' s1 tr1 s1' s1'' s2 statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO ∧
lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
lappend (llist_of trv1) (lltrv1 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO)) ∧
lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
lappend (llist_of trv2) (lltrv2 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO)) "
proof-
show ?thesis
using h_t_not_lnever_L[OF assms] apply(elim exE)
subgoal for w1' w2' tr1 s1' s1'' ltr1' trv1 sv1'' trv2 sv2'' statOO
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ tr1]) apply(rule exI[of _ s1']) apply(rule exI[of _ s1'']) apply(rule exI[of _ ltr1'])
apply(rule exI[of _ trv1]) apply(rule exI[of _ sv1'']) apply(rule exI[of _ trv2]) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal by simp
subgoal by simp
subgoal unfolding lltrv1_def apply(subst f_length_ge1[OF len])
unfolding h_def t_def by simp
subgoal unfolding lltrv2_def apply(subst f_length_ge1[OF len])
unfolding h_def t_def by simp . .
qed
lemma h_t_lnever_R:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and l': "¬ isIntO s1" "lnever isIntO ltr2"
and len: "llength ltr1 > Suc 0" "llength ltr2 > Suc 0"
and l: "trn = R" "lnever isSecO ltr2"
shows "∃ w1' w2' s2' ltr2' trv1 sv1' trv2 sv2' statOO.
ltr2 = s2 $ ltr2' ∧ validTransO (s2,s2') ∧
Opt.lvalidFromS s2' ltr2' ∧ Opt.lcompletedFrom s2' ltr2' ∧ lnever isIntO ltr2' ∧
ω4 Δ w1 w2 w1' w2' s1 s2 s2' statA sv1 trv1 sv1' sv2 trv2 sv2' statOO ∧
h_t (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
((trv1,sv1',trv2,sv2',statA,statOO),
(if trv2 = [] then R else L,
w1',w2',s1,ltr1,s2',ltr2',statA,sv1',sv2',statOO))"
proof-
have s2: "¬ isIntO s2" using l' ltr2
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil lhd_LCons
llist.exhaust llist.pred_inject(2))
obtain ltr2' where ltr24: "ltr2 = s2 $ ltr2'"
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil llist.exhaust_sel ltr2(1) ltr2(2))
hence ltr2': "ltr2' = ltl ltr2" by auto
have ltr2'ne: "ltr2' ≠ [[]]" using len(2) unfolding ltr24
by (metis One_nat_def llength_LCons llength_LNil one_eSuc one_enat_def order_less_irrefl)
define s2' where s2': "s2' = lhd (ltl ltr2)"
have v4: "validTransO (s2,s2')" and vv4: "Opt.lvalidFromS s2' ltr2'" "Opt.lcompletedFrom s2' ltr2'"
using ltr2 ltr2'ne unfolding ltr24 s2'
by (metis Opt.lcompletedFrom_LCons Opt.lcompletedFrom_def Opt.lvalidFromS_Cons_iff ltr2' ltr24)+
have is2': "¬ isIntO s2'" and "lnever isIntO ltr2'"
using l'(2) unfolding ltr24
by (metis llist.exhaust_sel llist.pred_inject(2) ltr2' ltr2'ne s2')+
have iss2: "¬ isSecO s2"
using l(2) ltr24 by auto
obtain w1' w2' trv1 sv1' trv2 sv2' statOO
where ω4: "ω4 Δ w1 w2 w1' w2' s1 s2 (lhd (ltl ltr2)) statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
using unwindCond_ex_ω4[OF unw Δ r l'(1) v4 s2 is2' iss2 ] s2' by auto
define tp' where
"tp' = (SOME k'. case k' of (w1',w2',trv1,sv1',trv2,sv2',statOO) ⇒
ω4 Δ w1 w2 w1' w2' s1 s2 (lhd (ltl ltr2)) statA sv1 trv1 sv1' sv2 trv2 sv2' statOO)"
have 1: "case tp' of (w1',w2',trv1,sv1',trv2,sv2',statOO) ⇒
ω4 Δ w1 w2 w1' w2' s1 s2 s2' statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
using ω4 unfolding tp'_def s2' apply- apply(rule someI_ex)
apply(rule exI[of _ "(w1',w2',trv1,sv1',trv2,sv2',statOO)"]) by auto
obtain w1' w2' trv1 sv1' trv2 sv2' statOO where
tp': "tp' = (w1',w2',trv1,sv1',trv2,sv2',statOO)" by(cases tp', auto)
have ω4: "ω4 Δ w1 w2 w1' w2' s1 s2 s2' statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
using 1 unfolding tp' by auto
show ?thesis
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s2']) apply(rule exI[of _ ltr2'])
apply(rule exI[of _ trv1]) apply(rule exI[of _ sv1'])
apply(rule exI[of _ trv2]) apply(rule exI[of _ sv2'])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal by fact
subgoal by fact
subgoal by fact
subgoal by fact
subgoal by fact
subgoal by fact
subgoal using len l unfolding h_t.simps apply simp
unfolding tp'_def[symmetric] tp' s2' ltr2' by simp .
qed
lemma lltrv1_lltrv2_lnever_R:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and l': "¬ isIntO s1" "lnever isIntO ltr2"
and len: "llength ltr1 > Suc 0" "llength ltr2 > Suc 0"
and l: "trn = R" "lnever isSecO ltr2"
shows "∃ w1' w2' s2' ltr2' trv1 sv1' trv2 sv2' statOO.
ltr2 = s2 $ ltr2' ∧ validTransO (s2,s2') ∧
Opt.lvalidFromS s2' ltr2' ∧ Opt.lcompletedFrom s2' ltr2' ∧ lnever isIntO ltr2' ∧
ω4 Δ w1 w2 w1' w2' s1 s2 s2' statA sv1 trv1 sv1' sv2 trv2 sv2' statOO ∧
lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
lappend (llist_of trv1) (lltrv1 (if trv2 = [] then R else L,
w1',w2',s1,ltr1,s2',ltr2',statA,sv1',sv2',statOO)) ∧
lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
lappend (llist_of trv2) (lltrv2 (if trv2 = [] then R else L,
w1',w2',s1,ltr1,s2',ltr2',statA,sv1',sv2',statOO))"
proof-
show ?thesis
using h_t_lnever_R[OF assms] apply(elim exE)
subgoal for w1' w2' s2' ltr2' trv1 sv1' trv2 sv2' statOO
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s2']) apply(rule exI[of _ ltr2'])
apply(rule exI[of _ trv1]) apply(rule exI[of _ sv1'])
apply(rule exI[of _ trv2]) apply(rule exI[of _ sv2'])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal by simp
subgoal by simp
subgoal by simp
subgoal by simp
subgoal by simp
subgoal by simp
subgoal unfolding lltrv1_def apply(subst f_length_ge1[OF len])
unfolding h_def t_def by auto
subgoal unfolding lltrv2_def apply(subst f_length_ge1[OF len])
unfolding h_def t_def by auto . .
qed
lemma h_t_not_lnever_R:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and l': "¬ isIntO s1" "lnever isIntO ltr2"
and len: "llength ltr1 > Suc 0" "llength ltr2 > Suc 0"
and l: "trn = R" "¬ lnever isSecO ltr2"
shows "∃ w1' w2' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statOO.
χχ s2 ltr2 tr2 s2' s2'' ltr2' ∧
χ4' Δ w1 w2 w1' w2' s1 s2 tr2 s2' s2'' statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO ∧
h_t (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
((trv1,sv1'',trv2,sv2'',statA,statOO),
(L,w1',w2',s1,ltr1,s2'',s2'' $ ltr2',statA,sv1'',sv2'',statOO))"
proof-
have s2: "¬ isIntO s2" using l' ltr2
by (metis Simple_Transition_System.lvalidFromS_def l(2) lhd_LCons llist.pred_inject(1)
llist.pred_inject(2) neq_LNil_conv)
obtain tr2 s2' s2'' ltr2'
where χχ: "χχ s2 ltr2 tr2 s2' s2'' ltr2'"
using isSecO_χχ[OF ltr2 l'(2) l(2)] by auto
define tp where
"tp = (SOME k. case k of (tr2,s2',s2'',ltr2') ⇒
χχ s2 ltr2 tr2 s2' s2'' ltr2')"
have 0: "case tp of (tr2,s2',s2'',ltr2') ⇒
χχ s2 ltr2 tr2 s2' s2'' ltr2'"
using χχ unfolding tp_def apply- apply(rule someI_ex)
apply(rule exI[of _ "(tr2,s2',s2'',ltr2')"]) by auto
obtain tr2 s2' s2'' ltr2' where
tp: "tp = (tr2,s2',s2'',ltr2')" by(cases tp, auto)
have χχ: "χχ s2 ltr2 tr2 s2' s2'' ltr2'"
using 0 unfolding tp by auto
obtain w1' w2' trv1 sv1'' trv2 sv2'' statOO
where χ4': "χ4' Δ w1 w2 w1' w2' s1 s2 tr2 s2' s2'' statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
using unwindCond_ex_χ4'[OF unw Δ r, of tr2 s2' s2'']
using χχ l' s2 unfolding χχ_def by auto
define tp' where
"tp' = (SOME k'. case k' of (w1',w2',trv1,sv1'',trv2,sv2'',statOO) ⇒
χ4' Δ w1 w2 w1' w2' s1 s2 tr2 s2' s2'' statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO)"
have 1: "case tp' of (w1',w2',trv1,sv1'',trv2,sv2'',statOO) ⇒
χ4' Δ w1 w2 w1' w2' s1 s2 tr2 s2' s2'' statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
using χ4' unfolding tp'_def apply- apply(rule someI_ex)
apply(rule exI[of _ "(w1',w2',trv1,sv1'',trv2,sv2'',statOO)"]) by auto
obtain w1' w2' trv1 sv1'' trv2 sv2'' statOO where
tp': "tp' = (w1',w2',trv1,sv1'',trv2,sv2'',statOO)" by(cases tp', auto)
have χ4': "χ4' Δ w1 w2 w1' w2' s1 s2 tr2 s2' s2'' statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
using 1 unfolding tp' by auto
show ?thesis
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ tr2]) apply(rule exI[of _ s2']) apply(rule exI[of _ s2'']) apply(rule exI[of _ ltr2'])
apply(rule exI[of _ trv1]) apply(rule exI[of _ sv1'']) apply(rule exI[of _ trv2]) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal using χχ .
subgoal using χ4' .
subgoal using l unfolding h_t.simps
unfolding tp_def[symmetric] tp apply simp
unfolding tp'_def[symmetric] tp' by auto .
qed
lemma lltrv1_lltrv2_not_lnever_R:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and l': "¬ isIntO s1" "lnever isIntO ltr2"
and len: "llength ltr1 > Suc 0" "llength ltr2 > Suc 0"
and l: "trn = R" "¬ lnever isSecO ltr2"
shows "∃ w1' w2' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statOO.
χχ s2 ltr2 tr2 s2' s2'' ltr2' ∧
χ4' Δ w1 w2 w1' w2' s1 s2 tr2 s2' s2'' statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO ∧
lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
lappend (llist_of trv1) (lltrv1 (L,w1',w2',s1,ltr1,s2'',s2'' $ ltr2',statA,sv1'',sv2'',statOO)) ∧
lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
lappend (llist_of trv2) (lltrv2 (L,w1',w2',s1,ltr1,s2'',s2'' $ ltr2',statA,sv1'',sv2'',statOO))"
proof-
show ?thesis
using h_t_not_lnever_R[OF assms] apply(elim exE)
subgoal for w1' w2' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statOO
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ tr2]) apply(rule exI[of _ s2']) apply(rule exI[of _ s2'']) apply(rule exI[of _ ltr2'])
apply(rule exI[of _ trv1]) apply(rule exI[of _ sv1'']) apply(rule exI[of _ trv2]) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal by simp
subgoal by simp
subgoal unfolding lltrv1_def apply(subst f_length_ge1[OF len])
unfolding h_def t_def by simp
subgoal unfolding lltrv2_def apply(subst f_length_ge1[OF len])
unfolding h_def t_def by simp . .
qed
lemma f_not_LNil: "ltr1 ≠ [[]] ∨ ltr2 ≠ [[]] ⟹
f (w1,w2,trn, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO) ≠ [[]]"
apply(cases "llength ltr1 ≤ Suc 0 ∨ llength ltr2 ≤ Suc 0")
subgoal apply(subst f_length_1) by auto
subgoal apply(subst f_length_ge1) by auto .
lemma lvalidFromS_lltrv1:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1" "lnever isIntO ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2" "lnever isIntO ltr2"
shows "Van.lvalidFromS sv1 (lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
proof-
{fix n sv1 ltrv1
assume "∃trn w1 w2 s1 ltr1 s2 ltr2 statA sv2 statO.
n = w1 ∧
ltrv1 = lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧ lnever isIntO ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧ lnever isIntO ltr2"
hence "Van.llvalidFromS n sv1 ltrv1"
proof(coinduct rule: Van.llvalidFromS.coinduct[of "λn sv1 ltrv1.
∃trn w1 w2 s1 ltr1 s2 ltr2 statA sv2 statO.
n = w1 ∧
ltrv1 = lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧ lnever isIntO ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧ lnever isIntO ltr2"])
case (llvalidFromS n sv1 ltrv1)
then obtain trn w1 w2 s1 ltr1 s2 ltr2 statA sv2 statO
where n: "n = w1" and
ltrv1: "ltrv1 = lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1" "lnever isIntO ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2" "lnever isIntO ltr2"
by auto
have isi3: "¬ isIntO s1" using ltr1
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil llist.exhaust_sel llist.pred_inject(2))
have isi4: "¬ isIntO s2" using ltr2
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil llist.exhaust_sel llist.pred_inject(2))
show ?case proof(cases "ltr1 = [[]] ∧ ltr2 = [[]]")
case True note ltr14 = True
hence ltrv1: "ltrv1 = [[]]" unfolding ltrv1 by simp
show ?thesis unfolding ltrv1 apply(rule Van.llvalidFromS_selectLNil) by auto
next
case False hence ltr14: "ltr1 ≠ [[]] ∨ ltr2 ≠ [[]]" by auto
show ?thesis proof(cases "llength ltr1 ≤ Suc 0 ∨ llength ltr2 ≤ Suc 0")
case True note ltr14 = ltr14 True
hence ltrv1: "ltrv1 = [[sv1]]" unfolding ltrv1 by simp
show ?thesis unfolding ltrv1 apply(rule Van.llvalidFromS_selectSingl) by auto
next
case False hence current: "llength ltr1 > Suc 0" "llength ltr2 > Suc 0"
by auto
show ?thesis proof(cases trn)
case L note trn = L note current = current L
show ?thesis
proof(cases "lnever isSecO ltr1")
case True note current = current True
obtain trn' w1' w2' s1' ltr1' trv1 sv1' trv2 sv2' statOO where
ωω: "ltr1 = s1 $ ltr1'" "validTransO (s1, s1')" "Opt.lvalidFromS s1' ltr1'"
"lcompletedFromO s1' ltr1'" "lnever isIntO ltr1'" and
ω3: "ω3 Δ w1 w2 w1' w2' s1 s1' s2 statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
and trn': "trn' = (if trv1 = [] then L else R)"
and ltrv1: "ltrv1 =
lappend (llist_of trv1) (lltrv1 (trn', w1', w2', s1', ltr1', s2, ltr2, statA, sv1', sv2', statOO))"
using lltrv1_lltrv2_lnever_L[OF unw Δ r ltr1 isi4 current]
unfolding ltrv1 by blast
define ltrv1' where ltrv1': "ltrv1' ≡ lltrv1 (trn', w1', w2', s1', ltr1', s2, ltr2, statA, sv1', sv2', statOO)"
have ltrv1: "ltrv1 = lappend (llist_of trv1) ltrv1'"
unfolding ltrv1 ltrv1' ..
show ?thesis
proof(cases "trv1 = []")
case True note trv1 = True
have sv1': "sv1' = sv1"
using ω3 unfolding ω3_def by (simp add: trv1)
show ?thesis
apply(rule Van.llvalidFromS_selectDelay)
apply(rule exI[of _ "w1'"]) apply(rule exI[of _ n])
apply(rule exI[of _ sv1]) apply(rule exI[of _ "ltrv1'"])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv1 trv1 by simp
subgoal using ω3 unfolding ω3_def trv1 n by simp
subgoal apply(rule disjI1)
apply(rule exI[of _ trn'])
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1']) apply(rule exI[of _ "ltr1'"])
apply(rule exI[of _ s2]) apply(rule exI[of _ ltr2])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv2']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal ..
subgoal unfolding ltrv1' trn' trv1 sv1' using trn by simp
subgoal using ω3 unfolding ω3_def sv1' by simp
subgoal using ω3 unfolding ω3_def
by (metis Opt.reach.Step ωω(2) fst_conv r(1) snd_conv)
subgoal by fact subgoal by fact
subgoal using ω3 unfolding ω3_def
by (metis Nil_is_append_conv Van.reach_validFromS_reach last_snoc not_Cons_self2 r(4))
subgoal using ωω by simp subgoal using ωω by simp subgoal using ωω by simp
subgoal by fact subgoal by fact subgoal by fact . .
next
case False note trv1 = False
show ?thesis
apply(rule Van.llvalidFromS_selectlappend)
apply(rule exI[of _ sv1]) apply(rule exI[of _ "trv1"])
apply(rule exI[of _ sv1']) apply(rule exI[of _ "w1'"])
apply(rule exI[of _ "ltrv1'"]) apply(rule exI[of _ n])
apply(intro conjI)
subgoal .. subgoal ..
subgoal using ltrv1 .
subgoal using ω3 unfolding ω3_def
by (metis Nil_is_append_conv Van.validFromS_def Van.validS_append1 hd_append2)
subgoal by fact
subgoal using ω3 unfolding ω3_def
by (metis Van.validFromS_def Van.validS_validTrans list.sel(1) not_Cons_self2 snoc_eq_iff_butlast trv1)
subgoal apply(rule disjI1)
apply(rule exI[of _ trn'])
apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ s1']) apply(rule exI[of _ "ltr1'"])
apply(rule exI[of _ s2]) apply(rule exI[of _ ltr2])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv2']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal ..
subgoal using trv1 unfolding ltrv1' trn' by auto
subgoal using ω3 unfolding ω3_def by simp
subgoal using ω3 unfolding ω3_def
by (metis Opt.reach.Step ωω(2) fst_conv r(1) snd_conv)
subgoal by fact
subgoal using ω3 unfolding ω3_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω3 unfolding ω3_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal using ωω by auto
subgoal using ωω by auto
subgoal using ωω
using llist_all_lappend_llist_of ltr1(3) by blast
subgoal using ωω using ltr2(1) by fastforce
subgoal by fact
subgoal by fact . .
qed
next
case False note current = current False
obtain w1' w2' tr1 s1' s1'' ltr1' trv1 sv1'' trv2 sv2'' statOO where
χχ: "χχ s1 ltr1 tr1 s1' s1'' ltr1'" and
χ3': "χ3' Δ w1 w2 w1' w2' s1 tr1 s1' s1'' s2 statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
and ltrv1: "ltrv1 =
lappend (llist_of trv1) (lltrv1 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO))"
using lltrv1_lltrv2_not_lnever_L[OF unw Δ r ltr1 isi4 current]
unfolding ltrv1 by blast
define ltrv1' where ltrv1': "ltrv1' ≡ lltrv1 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO)"
have ltrv1: "ltrv1 = lappend (llist_of trv1) ltrv1'"
unfolding ltrv1 ltrv1' ..
show ?thesis apply(rule Van.llvalidFromS_selectlappend)
apply(rule exI[of _ sv1]) apply(rule exI[of _ "trv1"])
apply(rule exI[of _ sv1'']) apply(rule exI[of _ "w1'"])
apply(rule exI[of _ "ltrv1'"]) apply(rule exI[of _ "w1"])
apply(intro conjI)
subgoal unfolding n .. subgoal ..
subgoal using ltrv1 .
subgoal using χ3' unfolding χ3'_def
by (metis Nil_is_append_conv Van.validFromS_def Van.validS_append1 hd_append2)
subgoal using χ3' unfolding χ3'_def by simp
subgoal using χ3' unfolding χ3'_def
by (metis Van.validFromS Van.validS_validTrans Simple_Transition_System.validFromS_def
append_is_Nil_conv not_Cons_self2)
subgoal apply(rule disjI1)
apply(rule exI[of _ R])
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1'']) apply(rule exI[of _ "s1'' $ ltr1'"])
apply(rule exI[of _ s2]) apply(rule exI[of _ ltr2])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv2'']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal ..
subgoal unfolding ltrv1' ..
subgoal using χ3' unfolding χ3'_def by simp
subgoal using χ3' unfolding χ3'_def
by (metis Simple_Transition_System.reach_validFromS_reach χχ χχ_def
append_is_Nil_conv last_snoc not_Cons_self2 r(1))
subgoal by fact
subgoal using χ3' unfolding χ3'_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using χ3' unfolding χ3'_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr1(3) by blast
subgoal using χχ unfolding χχ_def using ltr2(1) by fastforce
subgoal by fact
subgoal by fact . .
qed
next
case R note trn = R note current = current R
show ?thesis
proof(cases "lnever isSecO ltr2")
case True note current = current True
obtain trn' w1' w2' s2' ltr2' trv1 sv1' trv2 sv2' statOO where
ωω: "ltr2 = s2 $ ltr2'" "validTransO (s2, s2')" "Opt.lvalidFromS s2' ltr2'"
"lcompletedFromO s2' ltr2'" "lnever isIntO ltr2'" and
ω4: "ω4 Δ w1 w2 w1' w2' s1 s2 s2' statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
and trn': "trn' = (if trv2 = [] then R else L)"
and ltrv1: "ltrv1 =
lappend (llist_of trv1) (lltrv1 (trn', w1', w2', s1, ltr1, s2', ltr2', statA, sv1', sv2', statOO))"
using lltrv1_lltrv2_lnever_R[OF unw Δ r ltr2(1,2) isi3 ltr2(3) current]
unfolding ltrv1 by blast
define ltrv1' where ltrv1': "ltrv1' ≡ lltrv1 (trn', w1', w2', s1, ltr1, s2', ltr2', statA, sv1', sv2', statOO)"
have ltrv1: "ltrv1 = lappend (llist_of trv1) ltrv1'"
unfolding ltrv1 ltrv1' ..
show ?thesis
proof(cases "trv1 = []")
case True note trv1 = True
have sv1': "sv1' = sv1"
using ω4 unfolding ω4_def by (simp add: trv1)
show ?thesis
apply(rule Van.llvalidFromS_selectDelay)
apply(rule exI[of _ "w1'"]) apply(rule exI[of _ n])
apply(rule exI[of _ sv1]) apply(rule exI[of _ "ltrv1'"])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv1 trv1 by simp
subgoal using ω4 unfolding ω4_def trv1 n by simp
subgoal apply(rule disjI1)
apply(rule exI[of _ trn'])
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1]) apply(rule exI[of _ "ltr1"])
apply(rule exI[of _ s2']) apply(rule exI[of _ ltr2'])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv2']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal ..
subgoal unfolding ltrv1' trn' trv1 sv1' using trn by simp
subgoal using ω4 unfolding ω4_def sv1' by simp
subgoal by fact
subgoal using ω4 unfolding ω4_def
by (metis Opt.reach.Step ωω(2) fst_conv r(2) snd_conv)
subgoal by fact
subgoal using ω4 unfolding ω4_def
by (metis Nil_is_append_conv Van.reach_validFromS_reach last_snoc not_Cons_self2 r(4))
subgoal by fact subgoal by fact subgoal by fact
subgoal using ωω by simp subgoal using ωω by simp subgoal using ωω by simp . .
next
case False note trv1 = False
show ?thesis
apply(rule Van.llvalidFromS_selectlappend)
apply(rule exI[of _ sv1]) apply(rule exI[of _ "trv1"])
apply(rule exI[of _ sv1']) apply(rule exI[of _ "w1'"])
apply(rule exI[of _ "ltrv1'"]) apply(rule exI[of _ n])
apply(intro conjI)
subgoal .. subgoal ..
subgoal using ltrv1 .
subgoal using ω4 unfolding ω4_def
by (metis Nil_is_append_conv Van.validFromS_def Van.validS_append1 hd_append2)
subgoal by fact
subgoal using ω4 unfolding ω4_def
by (metis Van.validFromS_def Van.validS_validTrans list.sel(1) not_Cons_self2 snoc_eq_iff_butlast trv1)
subgoal apply(rule disjI1)
apply(rule exI[of _ trn'])
apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ s1]) apply(rule exI[of _ "ltr1"])
apply(rule exI[of _ s2']) apply(rule exI[of _ ltr2'])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv2']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal ..
subgoal using trv1 unfolding ltrv1' trn' by auto
subgoal using ω4 unfolding ω4_def by simp
subgoal by fact
subgoal using ω4 unfolding ω4_def
by (metis Opt.reach.Step ωω(2) fst_conv r(2) snd_conv)
subgoal using ω4 unfolding ω4_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω4 unfolding ω4_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal by fact subgoal by fact subgoal by fact
subgoal using ωω by auto
subgoal using ωω by auto
subgoal using ωω
using llist_all_lappend_llist_of ltr1(3) by blast . .
qed
next
case False note current = current False
obtain w1' w2' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statOO where
χχ: "χχ s2 ltr2 tr2 s2' s2'' ltr2'" and
χ4': "χ4' Δ w1 w2 w1' w2' s1 s2 tr2 s2' s2'' statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO "
and ltrv1: "ltrv1 =
lappend (llist_of trv1) (lltrv1 (L, w1', w2', s1, ltr1, s2'', s2'' $ ltr2', statA, sv1'', sv2'', statOO))"
using lltrv1_lltrv2_not_lnever_R[OF unw Δ r ltr2(1,2) isi3 ltr2(3) current]
unfolding ltrv1 by blast
define ltrv1' where ltrv1': "ltrv1' ≡ lltrv1 (L, w1', w2', s1, ltr1, s2'', s2'' $ ltr2', statA, sv1'', sv2'', statOO)"
have ltrv1: "ltrv1 = lappend (llist_of trv1) ltrv1'"
unfolding ltrv1 ltrv1' ..
show ?thesis
proof(cases "trv1 = []")
case True note trv1 = True
hence sv1'': "sv1'' = sv1"
by (metis χ4'_def Simple_Transition_System.validFromS_Cons_iff χ4' append.simps(1))
have "w1' < w1" using trv1 χ4' unfolding χ4'_def by auto
show ?thesis
apply(rule Van.llvalidFromS_selectDelay)
apply(rule exI[of _ "w1'"]) apply(rule exI[of _ n])
apply(rule exI[of _ sv1]) apply(rule exI[of _ "ltrv1"])
apply(intro conjI)
subgoal ..
subgoal .. subgoal .. subgoal unfolding n by fact
subgoal apply(rule disjI1)
apply(rule exI[of _ L])
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1]) apply(rule exI[of _ ltr1])
apply(rule exI[of _ s2'']) apply(rule exI[of _ "s2'' $ ltr2'"])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv2'']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal ..
subgoal unfolding ltrv1 ltrv1' trv1 sv1'' by simp
subgoal using χ4' unfolding χ4'_def sv1'' by simp
subgoal by fact
subgoal using χχ unfolding χχ_def
by (metis Opt.reach_validFromS_reach Nil_is_append_conv last_snoc not_Cons_self2 r(2))
subgoal by fact
subgoal using χ4' r(4) unfolding χ4'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal by fact subgoal by fact subgoal by fact
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr2(3) by blast . .
next
case False note trv1 = False
show ?thesis
apply(rule Van.llvalidFromS_selectlappend)
apply(rule exI[of _ sv1]) apply(rule exI[of _ "trv1"])
apply(rule exI[of _ sv1''])
apply(rule exI[of _ "w1'"])
apply(rule exI[of _ "ltrv1'"])
apply(rule exI[of _ n])
apply(intro conjI)
subgoal .. subgoal ..
subgoal using ltrv1 .
subgoal using χ4' unfolding χ4'_def
by (metis Nil_is_append_conv Van.validFromS_def Van.validS_append1 hd_append2)
subgoal using trv1 .
subgoal using χ4' unfolding χ4'_def
by (metis Simple_Transition_System.validFromS_def Van.validS_validTrans list.sel(1)
not_Cons_self2 snoc_eq_iff_butlast trv1)
subgoal apply(rule disjI1)
apply(rule exI[of _ L])
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1]) apply(rule exI[of _ ltr1])
apply(rule exI[of _ s2'']) apply(rule exI[of _ "s2'' $ ltr2'"])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv2'']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal unfolding ltrv1' ..
subgoal using χ4' unfolding χ4'_def by simp
subgoal by fact
subgoal using χ4' unfolding χ4'_def
by (metis Simple_Transition_System.reach_validFromS_reach χχ χχ_def
append_is_Nil_conv last_snoc not_Cons_self2 r(2))
subgoal using χ4' unfolding χ4'_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using χ4' unfolding χ4'_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal by fact
subgoal by fact
subgoal by fact
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr2(3) by blast . .
qed
qed
qed
qed
qed
qed
}
thus ?thesis apply-apply(rule Van.llvalidFromS_imp_lvalidFromS)
using assms by blast
qed
lemma lvalidFromS_lltrv2:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1" "lnever isIntO ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2" "lnever isIntO ltr2"
shows "Van.lvalidFromS sv2 (lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
proof-
{fix n sv2 ltrv2
assume "∃trn w1 w2 s1 ltr1 s2 ltr2 statA sv1 statO.
n = w2 ∧
ltrv2 = lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧ lnever isIntO ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧ lnever isIntO ltr2"
hence "Van.llvalidFromS n sv2 ltrv2"
proof(coinduct rule: Van.llvalidFromS.coinduct[of "λn sv2 ltrv2.
∃trn w1 w2 s1 ltr1 s2 ltr2 statA sv1 statO.
n = w2 ∧
ltrv2 = lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧ lnever isIntO ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧ lnever isIntO ltr2"])
case (llvalidFromS n sv2 ltrv2)
then obtain trn w1 w2 s1 ltr1 s2 ltr2 statA sv1 statO
where n: "n = w2" and
ltrv2: "ltrv2 = lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1" "lnever isIntO ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2" "lnever isIntO ltr2"
by auto
have isi3: "¬ isIntO s1" using ltr1
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil llist.exhaust_sel llist.pred_inject(2))
have isi4: "¬ isIntO s2" using ltr2
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil llist.exhaust_sel llist.pred_inject(2))
show ?case proof(cases "ltr1 = [[]] ∧ ltr2 = [[]]")
case True note ltr14 = True
hence ltrv2: "ltrv2 = [[]]" unfolding ltrv2 by simp
show ?thesis unfolding ltrv2 apply(rule Van.llvalidFromS_selectLNil) by auto
next
case False hence ltr14: "ltr1 ≠ [[]] ∨ ltr2 ≠ [[]]" by auto
show ?thesis proof(cases "llength ltr1 ≤ Suc 0 ∨ llength ltr2 ≤ Suc 0")
case True note ltr14 = ltr14 True
hence ltrv2: "ltrv2 = [[sv2]]" unfolding ltrv2 by simp
show ?thesis unfolding ltrv2 apply(rule Van.llvalidFromS_selectSingl) by auto
next
case False hence current: "llength ltr1 > Suc 0" "llength ltr2 > Suc 0"
by auto
show ?thesis proof(cases trn)
case L note trn = L note current = current L
show ?thesis
proof(cases "lnever isSecO ltr1")
case True note current = current True
obtain trn' w1' w2' s1' ltr1' trv1 sv1' trv2 sv2' statOO where
ωω: "ltr1 = s1 $ ltr1'" "validTransO (s1, s1')" "Opt.lvalidFromS s1' ltr1'"
"lcompletedFromO s1' ltr1'" "lnever isIntO ltr1'" and
ω3: "ω3 Δ w1 w2 w1' w2' s1 s1' s2 statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
and trn': "trn' = (if trv1 = [] then L else R)"
and ltrv2: "ltrv2 =
lappend (llist_of trv2) (lltrv2(trn', w1', w2', s1', ltr1', s2, ltr2, statA, sv1', sv2', statOO))"
using lltrv1_lltrv2_lnever_L[OF unw Δ r ltr1 isi4 current]
unfolding ltrv2 by blast
define ltrv2' where ltrv2': "ltrv2' ≡ lltrv2 (trn', w1', w2', s1', ltr1', s2, ltr2, statA, sv1', sv2', statOO)"
have ltrv2: "ltrv2 = lappend (llist_of trv2) ltrv2'"
unfolding ltrv2 ltrv2' ..
show ?thesis
proof(cases "trv2 = []")
case True note trv2 = True
have sv2': "sv2' = sv2"
using ω3 unfolding ω3_def by (simp add: trv2)
show ?thesis
apply(rule Van.llvalidFromS_selectDelay)
apply(rule exI[of _ "w2'"]) apply(rule exI[of _ n])
apply(rule exI[of _ sv2]) apply(rule exI[of _ "ltrv2'"])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv2 trv2 by simp
subgoal using ω3 unfolding ω3_def trv2 n by simp
subgoal apply(rule disjI1)
apply(rule exI[of _ trn'])
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1']) apply(rule exI[of _ "ltr1'"])
apply(rule exI[of _ s2]) apply(rule exI[of _ ltr2])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv1']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal ..
subgoal unfolding ltrv2' trn' trv2 sv2' using trn by simp
subgoal using ω3 unfolding ω3_def sv2' by simp
subgoal using ω3 unfolding ω3_def
by (metis Opt.reach.Step ωω(2) fst_conv r(1) snd_conv)
subgoal by fact
subgoal using ω3 unfolding ω3_def
by (metis Nil_is_append_conv Van.reach_validFromS_reach last_snoc not_Cons_self2 r(3))
subgoal by fact
subgoal using ωω by simp subgoal using ωω by simp subgoal using ωω by simp
subgoal by fact subgoal by fact subgoal by fact . .
next
case False note trv2 = False
show ?thesis
apply(rule Van.llvalidFromS_selectlappend)
apply(rule exI[of _ sv2]) apply(rule exI[of _ "trv2"])
apply(rule exI[of _ sv2']) apply(rule exI[of _ "w2'"])
apply(rule exI[of _ "ltrv2'"]) apply(rule exI[of _ n])
apply(intro conjI)
subgoal .. subgoal ..
subgoal using ltrv2 .
subgoal using ω3 unfolding ω3_def
by (metis Nil_is_append_conv Van.validFromS_def Van.validS_append1 hd_append2)
subgoal by fact
subgoal using ω3 unfolding ω3_def
by (metis Van.validFromS_def Van.validS_validTrans append_is_Nil_conv list.sel(1) not_Cons_self2 trv2)
subgoal apply(rule disjI1)
apply(rule exI[of _ trn'])
apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ s1']) apply(rule exI[of _ "ltr1'"])
apply(rule exI[of _ s2]) apply(rule exI[of _ ltr2])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv1']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal ..
subgoal using trv2 unfolding ltrv2' trn' by auto
subgoal using ω3 unfolding ω3_def by simp
subgoal using ω3 unfolding ω3_def
by (metis Opt.reach.Step ωω(2) fst_conv r(1) snd_conv)
subgoal by fact
subgoal using ω3 unfolding ω3_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω3 unfolding ω3_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal using ωω by auto
subgoal using ωω by auto
subgoal using ωω
using llist_all_lappend_llist_of ltr1(3) by blast
subgoal using ωω using ltr2(1) by fastforce
subgoal by fact
subgoal by fact . .
qed
next
case False note current = current False
obtain w1' w2' tr1 s1' s1'' ltr1' trv1 sv1'' trv2 sv2'' statOO where
χχ: "χχ s1 ltr1 tr1 s1' s1'' ltr1'" and
χ3': "χ3' Δ w1 w2 w1' w2' s1 tr1 s1' s1'' s2 statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
and ltrv2: "ltrv2 =
lappend (llist_of trv2) (lltrv2 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO))"
using lltrv1_lltrv2_not_lnever_L[OF unw Δ r ltr1 isi4 current]
unfolding ltrv2 by blast
define ltrv2' where ltrv2': "ltrv2' ≡ lltrv2 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO)"
have ltrv2: "ltrv2 = lappend (llist_of trv2) ltrv2'"
unfolding ltrv2 ltrv2' ..
show ?thesis
proof(cases "trv2 = []")
case True note trv2 = True
hence sv2'': "sv2'' = sv2"
by (metis χ3'_def Simple_Transition_System.validFromS_Cons_iff χ3' append.simps(1))
have "w2' < w2" using trv2 χ3' unfolding χ3'_def by auto
show ?thesis
apply(rule Van.llvalidFromS_selectDelay)
apply(rule exI[of _ "w2'"]) apply(rule exI[of _ n])
apply(rule exI[of _ sv2]) apply(rule exI[of _ "ltrv2"])
apply(intro conjI)
subgoal ..
subgoal .. subgoal .. subgoal unfolding n by fact
subgoal apply(rule disjI1)
apply(rule exI[of _ R])
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1'']) apply(rule exI[of _ "s1'' $ ltr1'"])
apply(rule exI[of _ s2]) apply(rule exI[of _ ltr2])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv1'']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal ..
subgoal unfolding ltrv2 ltrv2' trv2 sv2'' by simp
subgoal using χ3' unfolding χ3'_def sv2'' by simp
subgoal using χχ unfolding χχ_def
by (metis Opt.reach_validFromS_reach Nil_is_append_conv last_snoc not_Cons_self2 r(1))
subgoal by fact
subgoal using χ3' r(3) unfolding χ3'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal by fact
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr1(3) by blast
subgoal by fact subgoal by fact subgoal by fact . .
next
case False note trv2 = False
show ?thesis
apply(rule Van.llvalidFromS_selectlappend)
apply(rule exI[of _ sv2]) apply(rule exI[of _ "trv2"])
apply(rule exI[of _ sv2'']) apply(rule exI[of _ "w2'"])
apply(rule exI[of _ "ltrv2'"]) apply(rule exI[of _ "w2"])
apply(intro conjI)
subgoal unfolding n .. subgoal ..
subgoal using ltrv2 .
subgoal using χ3' unfolding χ3'_def
by (metis Nil_is_append_conv Van.validFromS_def Van.validS_append1 hd_append2)
subgoal by fact
subgoal using χ3' unfolding χ3'_def
by (metis Simple_Transition_System.validFromS_def Van.validS_validTrans append_is_Nil_conv list.sel(1) not_Cons_self2 trv2)
subgoal apply(rule disjI1)
apply(rule exI[of _ R])
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1'']) apply(rule exI[of _ "s1'' $ ltr1'"])
apply(rule exI[of _ s2]) apply(rule exI[of _ ltr2])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv1'']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal ..
subgoal unfolding ltrv2' ..
subgoal using χ3' unfolding χ3'_def by simp
subgoal using χ3' unfolding χ3'_def
by (metis Simple_Transition_System.reach_validFromS_reach χχ χχ_def
append_is_Nil_conv last_snoc not_Cons_self2 r(1))
subgoal by fact
subgoal using χ3' unfolding χ3'_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using χ3' unfolding χ3'_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr1(3) by blast
subgoal using χχ unfolding χχ_def using ltr2(1) by fastforce
subgoal by fact
subgoal by fact . .
qed
qed
next
case R note trn = R note current = current R
show ?thesis
proof(cases "lnever isSecO ltr2")
case True note current = current True
obtain trn' w1' w2' s2' ltr2' trv1 sv1' trv2 sv2' statOO where
ωω: "ltr2 = s2 $ ltr2'" "validTransO (s2, s2')" "Opt.lvalidFromS s2' ltr2'"
"lcompletedFromO s2' ltr2'" "lnever isIntO ltr2'" and
ω4: "ω4 Δ w1 w2 w1' w2' s1 s2 s2' statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
and trn': "trn' = (if trv2 = [] then R else L)"
and ltrv2: "ltrv2 =
lappend (llist_of trv2) (lltrv2 (trn', w1', w2', s1, ltr1, s2', ltr2', statA, sv1', sv2', statOO))"
using lltrv1_lltrv2_lnever_R[OF unw Δ r ltr2(1,2) isi3 ltr2(3) current]
unfolding ltrv2 by blast
define ltrv2' where ltrv2': "ltrv2' ≡ lltrv2 (trn', w1', w2', s1, ltr1, s2', ltr2', statA, sv1', sv2', statOO)"
have ltrv2: "ltrv2 = lappend (llist_of trv2) ltrv2'"
unfolding ltrv2 ltrv2' ..
show ?thesis
proof(cases "trv2 = []")
case True note trv2 = True
have sv2': "sv2' = sv2"
using ω4 unfolding ω4_def by (simp add: trv2)
show ?thesis
apply(rule Van.llvalidFromS_selectDelay)
apply(rule exI[of _ "w2'"]) apply(rule exI[of _ n])
apply(rule exI[of _ sv2]) apply(rule exI[of _ "ltrv2'"])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv2 trv2 by simp
subgoal using ω4 unfolding ω4_def trv2 n by simp
subgoal apply(rule disjI1)
apply(rule exI[of _ trn'])
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1]) apply(rule exI[of _ "ltr1"])
apply(rule exI[of _ s2']) apply(rule exI[of _ ltr2'])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv1']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal ..
subgoal unfolding ltrv2' trn' trv2 sv2' using trn by simp
subgoal using ω4 unfolding ω4_def sv2' by simp
subgoal by fact
subgoal using ω4 unfolding ω4_def
by (metis Opt.reach.Step ωω(2) fst_conv r(2) snd_conv)
subgoal using ω4 unfolding ω4_def
by (metis Nil_is_append_conv Van.reach_validFromS_reach last_snoc not_Cons_self2 r(3))
subgoal by fact subgoal by fact subgoal by fact subgoal by fact
subgoal using ωω by simp subgoal using ωω by simp subgoal using ωω by simp . .
next
case False note trv2 = False
show ?thesis
apply(rule Van.llvalidFromS_selectlappend)
apply(rule exI[of _ sv2]) apply(rule exI[of _ "trv2"])
apply(rule exI[of _ sv2']) apply(rule exI[of _ "w2'"])
apply(rule exI[of _ "ltrv2'"]) apply(rule exI[of _ n])
apply(intro conjI)
subgoal .. subgoal ..
subgoal using ltrv2 .
subgoal using ω4 unfolding ω4_def
by (metis Nil_is_append_conv Van.validFromS_def Van.validS_append1 hd_append2)
subgoal by fact
subgoal using ω4 unfolding ω4_def
by (metis Van.validFromS_def Van.validS_validTrans append_is_Nil_conv list.sel(1) not_Cons_self2 trv2)
subgoal apply(rule disjI1)
apply(rule exI[of _ trn'])
apply(rule exI[of _ w1']) apply(rule exI[of _ w2']) apply(rule exI[of _ s1]) apply(rule exI[of _ "ltr1"])
apply(rule exI[of _ s2']) apply(rule exI[of _ ltr2'])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv1']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal ..
subgoal using trv2 unfolding ltrv2' trn' by auto
subgoal using ω4 unfolding ω4_def by simp
subgoal by fact
subgoal using ω4 unfolding ω4_def
by (metis Opt.reach.Step ωω(2) fst_conv r(2) snd_conv)
subgoal using ω4 unfolding ω4_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω4 unfolding ω4_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal by fact subgoal by fact subgoal by fact
subgoal using ωω by auto
subgoal using ωω by auto
subgoal using ωω
using llist_all_lappend_llist_of ltr1(3) by blast . .
qed
next
case False note current = current False
obtain w1' w2' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statOO where
χχ: "χχ s2 ltr2 tr2 s2' s2'' ltr2'" and
χ4': "χ4' Δ w1 w2 w1' w2' s1 s2 tr2 s2' s2'' statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO "
and ltrv2: "ltrv2 =
lappend (llist_of trv2) (lltrv2 (L, w1', w2', s1, ltr1, s2'', s2'' $ ltr2', statA, sv1'', sv2'', statOO))"
using lltrv1_lltrv2_not_lnever_R[OF unw Δ r ltr2(1,2) isi3 ltr2(3) current]
unfolding ltrv2 by blast
define ltrv2' where ltrv2': "ltrv2' ≡ lltrv2 (L, w1', w2', s1, ltr1, s2'', s2'' $ ltr2', statA, sv1'', sv2'', statOO)"
have ltrv2: "ltrv2 = lappend (llist_of trv2) ltrv2'"
unfolding ltrv2 ltrv2' ..
have trv2: "trv2 ≠ []" using χ4' unfolding χ4'_def by auto
show ?thesis
apply(rule Van.llvalidFromS_selectlappend)
apply(rule exI[of _ sv2]) apply(rule exI[of _ "trv2"])
apply(rule exI[of _ sv2''])
apply(rule exI[of _ "w2'"])
apply(rule exI[of _ "ltrv2'"])
apply(rule exI[of _ n])
apply(intro conjI)
subgoal .. subgoal ..
subgoal using ltrv2 .
subgoal using χ4' unfolding χ4'_def
by (metis Nil_is_append_conv Van.validFromS_def Van.validS_append1 hd_append2)
subgoal using trv2 .
subgoal using χ4' unfolding χ4'_def
by (metis Simple_Transition_System.validFromS_def Van.validS_validTrans append_is_Nil_conv list.sel(1) not_Cons_self2)
subgoal apply(rule disjI1)
apply(rule exI[of _ L])
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1]) apply(rule exI[of _ ltr1])
apply(rule exI[of _ s2'']) apply(rule exI[of _ "s2'' $ ltr2'"])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv1'']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal unfolding ltrv2' ..
subgoal using χ4' unfolding χ4'_def by simp
subgoal by fact
subgoal using χ4' unfolding χ4'_def
by (metis Simple_Transition_System.reach_validFromS_reach χχ χχ_def
append_is_Nil_conv last_snoc not_Cons_self2 r(2))
subgoal using χ4' unfolding χ4'_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using χ4' unfolding χ4'_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal by fact
subgoal by fact
subgoal by fact
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr2(3) by blast . .
qed
qed
qed
qed
qed
}
thus ?thesis apply-apply(rule Van.llvalidFromS_imp_lvalidFromS)
using assms by blast
qed
lemma lcompletedFrom_lltrv1:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1" "lnever isIntO ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2" "lnever isIntO ltr2"
shows "Van.lcompletedFrom sv1 (lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
proof-
{fix ltrv1 assume ltrv1: "ltrv1 = lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and lfin: "lfinite ltrv1"
hence "list_of ltrv1 ≠ [] ∧ finalV (last (list_of ltrv1))"
using assms(2-) proof(induct "length (list_of ltrv1)" "w1"
arbitrary: trn w2 ltrv1 s1 ltr1 s2 ltr2 statA sv1 sv2 statO
rule: less2_induct')
case (less w1 ltrv1 trn w2 s1 ltr1 s2 ltr2 statA sv1 sv2 statO)
hence ltrv1: "ltrv1 = lltrv1 (trn, w1, w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
and lfin: "lfinite ltrv1"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "lcompletedFromO s1 ltr1" "lnever isIntO ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "lcompletedFromO s2 ltr2" "lnever isIntO ltr2"
by auto
have isi3: "¬ isIntO s1" using ltr1
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil llist.exhaust_sel llist.pred_inject(2))
have isi4: "¬ isIntO s2" using ltr2
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil llist.exhaust_sel llist.pred_inject(2))
show ?case proof(cases "ltr1 = [[]] ∧ ltr2 = [[]]")
case True note ltr14 = True
hence False using ltr1(2) ltr2(2) unfolding Opt.lcompletedFrom_def by auto
thus ?thesis by auto
next
case False hence ltr14: "ltr1 ≠ [[]] ∨ ltr2 ≠ [[]]" by auto
show ?thesis proof(cases "llength ltr1 ≤ Suc 0 ∨ llength ltr2 ≤ Suc 0")
case True note ltr14 = ltr14 True
hence ltrv1: "list_of ltrv1 = [sv1]" unfolding ltrv1 by simp
have "llength ltr1 = Suc 0 ∨ llength ltr2 = Suc 0"
using ltr14
by (metis Opt.lcompletedFrom_def
Suc_ile_eq i0_less lfinite_code(1) llength_eq_0 llist.exhaust
ltr1(2) ltr2(2) nle_le not_lnull_conv zero_enat_def)
hence "ltr1 = [[s1]] ∨ ltr2 = [[s2]]"
using Opt.lcompletedFrom_singl ltr1(1) ltr1(2) ltr2(1) ltr2(2) by blast
hence "finalO s1 ∨ finalO s2"
using Opt.lcompletedFrom_LCons ltr1(2) ltr2(2) by blast
hence "finalV sv1"
using Δ r(1) r(2) r(3) r(4) unw unwindCond_def by auto
thus ?thesis unfolding ltrv1 by auto
next
case False hence current: "llength ltr1 > Suc 0" "llength ltr2 > Suc 0" by auto
show ?thesis
proof(cases trn)
case L note current = current L
show ?thesis
proof(cases "lnever isSecO ltr1")
case True note current = current True
obtain trn' w1' w2' s1' ltr1' trv1 sv1' trv2 sv2' statOO where
ωω: "ltr1 = s1 $ ltr1'" "validTransO (s1, s1')" "Opt.lvalidFromS s1' ltr1'"
"lcompletedFromO s1' ltr1'" "lnever isIntO ltr1'" and
ω3: "ω3 Δ w1 w2 w1' w2' s1 s1' s2 statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
and trn' : "trn' = (if trv1 = [] then L else R)"
and lltrv1: "ltrv1 =
lappend (llist_of trv1) (lltrv1 (trn', w1', w2', s1', ltr1', s2, ltr2, statA, sv1', sv2', statOO))"
using lltrv1_lltrv2_lnever_L[OF unw Δ r ltr1 isi4 current]
unfolding ltrv1 by blast
define ltrv1' where ltrv1': "ltrv1' = lltrv1 (trn', w1', w2', s1', ltr1', s2, ltr2, statA, sv1', sv2', statOO)"
have lltrv1: "ltrv1 = lappend (llist_of trv1) ltrv1'"
unfolding lltrv1 ltrv1' ..
have trv1ne: "trv1 ≠ [] ∨ w1' < w1" using ω3 unfolding ω3_def by auto
have lfin': "lfinite ltrv1'"
using lfin trv1ne unfolding lltrv1 by simp
have len: "length (list_of ltrv1') < length (list_of ltrv1) ∨
length (list_of ltrv1') = length (list_of ltrv1) ∧ w1' < w1"
using trv1ne lfin lfin' by (simp add: list_of_lappend lltrv1)
have 0: "list_of ltrv1' ≠ [] ∧ finalV (last (list_of ltrv1'))"
using len proof(elim disjE conjE)
assume len: "length (list_of ltrv1') < length (list_of ltrv1)"
show ?thesis
apply(rule less(1)[OF _ ltrv1'])
subgoal by fact subgoal by fact
subgoal using ω3 unfolding ω3_def by simp
subgoal by (metis Opt.reach.Step ωω(2) fst_conv r(1) snd_conv)
subgoal by fact
subgoal using ω3 unfolding ω3_def
by (metis Van.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω3 unfolding ω3_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(4))
subgoal by fact subgoal by fact subgoal by fact subgoal by fact
subgoal by fact subgoal by fact .
next
assume len: "length (list_of ltrv1') = length (list_of ltrv1)" "w1' < w1"
show ?thesis
apply(rule less(2)[OF _ _ ltrv1'])
subgoal by fact subgoal using len by simp subgoal by fact
subgoal using ω3 unfolding ω3_def by simp
subgoal by (metis Opt.reach.Step ωω(2) fst_conv r(1) snd_conv)
subgoal by fact
subgoal using ω3 unfolding ω3_def
by (metis Van.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω3 unfolding ω3_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(4))
subgoal by fact subgoal by fact subgoal by fact subgoal by fact
subgoal by fact subgoal by fact .
qed
show ?thesis unfolding lltrv1 using 0
by (simp add: lfin' list_of_lappend)
next
case False note current = current False
obtain w1' w2' tr1 s1' s1'' ltr1' trv1 sv1'' trv2 sv2'' statOO where
χχ: "χχ s1 ltr1 tr1 s1' s1'' ltr1'" and
χ3': "χ3' Δ w1 w2 w1' w2' s1 tr1 s1' s1'' s2 statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
and lltrv1: "ltrv1 =
lappend (llist_of trv1) (lltrv1 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO))"
using lltrv1_lltrv2_not_lnever_L[OF unw Δ r ltr1 isi4 current]
unfolding ltrv1 by blast
define ltrv1' where ltrv1': "ltrv1' = lltrv1 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO)"
have lltrv1: "ltrv1 = lappend (llist_of trv1) ltrv1'"
unfolding lltrv1 ltrv1' ..
have trv1ne: "trv1 ≠ []" using χ3' unfolding χ3'_def by auto
have lfin': "lfinite ltrv1'"
using lfin trv1ne unfolding lltrv1 by simp
have len: "length (list_of ltrv1') < length (list_of ltrv1)"
using trv1ne lfin lfin' by (simp add: list_of_lappend lltrv1)
have 0: "list_of ltrv1' ≠ [] ∧ finalV (last (list_of ltrv1'))"
apply(rule less(1)[OF _ ltrv1'])
subgoal by fact subgoal by fact
subgoal using χ3' unfolding χ3'_def by simp
subgoal using χχ unfolding χχ_def
by (metis Simple_Transition_System.reach_validFromS_reach r(1) snoc_eq_iff_butlast)
subgoal by fact
subgoal using χ3' unfolding χ3'_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc r(3))
subgoal using χ3' unfolding χ3'_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(4))
subgoal using χχ unfolding χχ_def by simp
subgoal using χχ unfolding χχ_def by simp
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr1 by blast
subgoal by fact subgoal by fact subgoal by fact .
show ?thesis unfolding lltrv1 using 0
by (simp add: lfin' list_of_lappend)
qed
next
case R note current = current R
show ?thesis
proof(cases "lnever isSecO ltr2")
case True note current = current True
obtain trn' w1' w2' s2' ltr2' trv1 sv1' trv2 sv2' statOO where
ωω: "ltr2 = s2 $ ltr2'" "validTransO (s2, s2')" "Opt.lvalidFromS s2' ltr2'"
"lcompletedFromO s2' ltr2'" "lnever isIntO ltr2'" and
ω4: "ω4 Δ w1 w2 w1' w2' s1 s2 s2' statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
and trn': "trn' = (if trv2 = [] then R else L)"
and ltrv1: "ltrv1 =
lappend (llist_of trv1) (lltrv1 (trn', w1', w2', s1, ltr1, s2', ltr2', statA, sv1', sv2', statOO))"
using lltrv1_lltrv2_lnever_R[OF unw Δ r ltr2(1,2) isi3 ltr2(3) current]
unfolding ltrv1 by blast
define ltrv1' where ltrv1': "ltrv1' = lltrv1 (trn', w1', w2', s1, ltr1, s2', ltr2', statA, sv1', sv2', statOO)"
have lltrv1: "ltrv1 = lappend (llist_of trv1) ltrv1'"
unfolding ltrv1 ltrv1' ..
have trv1ne: "trv1 ≠ [] ∨ w1' < w1" using ω4 unfolding ω4_def by auto
have lfin': "lfinite ltrv1'"
using lfin trv1ne unfolding lltrv1 by simp
have len: "length (list_of ltrv1') < length (list_of ltrv1) ∨
length (list_of ltrv1') = length (list_of ltrv1) ∧ w1' < w1"
using trv1ne lfin lfin' by (simp add: list_of_lappend lltrv1)
have 0: "list_of ltrv1' ≠ [] ∧ finalV (last (list_of ltrv1'))"
using len proof(elim disjE conjE)
assume len: "length (list_of ltrv1') < length (list_of ltrv1)"
show ?thesis
apply(rule less(1)[OF _ ltrv1'])
subgoal by fact subgoal by fact
subgoal using ω4 unfolding ω4_def by simp
subgoal by fact
subgoal using r(2) ωω by (metis Opt.reach.Step fst_conv snd_conv)
subgoal using ω4 unfolding ω4_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(3))
subgoal using ω4 unfolding ω4_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(4))
subgoal by fact subgoal by fact subgoal by fact subgoal by fact
subgoal by fact subgoal by fact .
next
assume len: "length (list_of ltrv1') = length (list_of ltrv1)" "w1' < w1"
show ?thesis
apply(rule less(2)[OF _ _ ltrv1'])
subgoal by fact subgoal using len by simp subgoal by fact
subgoal using ω4 unfolding ω4_def by simp
subgoal by fact
subgoal by (metis Opt.reach.Step ωω(2) fst_conv r(2) snd_conv)
subgoal using ω4 unfolding ω4_def
by (metis Van.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω4 unfolding ω4_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(4))
subgoal by fact subgoal by fact subgoal by fact subgoal by fact
subgoal by fact subgoal by fact .
qed
show ?thesis unfolding lltrv1 using 0
by (simp add: lfin' list_of_lappend)
next
case False note current = current False
obtain w1' w2' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statOO where
χχ: "χχ s2 ltr2 tr2 s2' s2'' ltr2'" and
χ4': "χ4' Δ w1 w2 w1' w2' s1 s2 tr2 s2' s2'' statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO "
and ltrv1: "ltrv1 =
lappend (llist_of trv1) (lltrv1 (L, w1', w2', s1, ltr1, s2'', s2'' $ ltr2', statA, sv1'', sv2'', statOO))"
using lltrv1_lltrv2_not_lnever_R[OF unw Δ r ltr2(1,2) isi3 ltr2(3) current]
unfolding ltrv1 by blast
define ltrv1' where ltrv1': "ltrv1' = lltrv1 (L, w1', w2', s1, ltr1, s2'', s2'' $ ltr2', statA, sv1'', sv2'', statOO)"
have lltrv1: "ltrv1 = lappend (llist_of trv1) ltrv1'"
unfolding ltrv1 ltrv1' ..
have trv1ne: "trv1 ≠ [] ∨ w1' < w1" using χ4' unfolding χ4'_def by auto
have lfin': "lfinite ltrv1'"
using lfin trv1ne unfolding lltrv1 by simp
have len: "length (list_of ltrv1') < length (list_of ltrv1) ∨
length (list_of ltrv1') = length (list_of ltrv1) ∧ w1' < w1"
using trv1ne lfin lfin' by (simp add: list_of_lappend lltrv1)
have 0: "list_of ltrv1' ≠ [] ∧ finalV (last (list_of ltrv1'))"
using len proof(elim disjE conjE)
assume len: "length (list_of ltrv1') < length (list_of ltrv1)"
show ?thesis
apply(rule less(1)[OF _ ltrv1'])
subgoal by fact subgoal by fact
subgoal using χ4' unfolding χ4'_def by simp
subgoal by fact
subgoal using r(2) χχ unfolding χχ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using χ4' unfolding χ4'_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(3))
subgoal using χ4' unfolding χ4'_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(4))
subgoal by fact subgoal by fact subgoal by fact
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr2(3) by blast .
next
assume len: "length (list_of ltrv1') = length (list_of ltrv1)" "w1' < w1"
show ?thesis
apply(rule less(2)[OF _ _ ltrv1'])
subgoal by fact subgoal using len by simp subgoal by fact
subgoal using χ4' unfolding χ4'_def by simp
subgoal by fact
subgoal using χχ unfolding χχ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(2))
subgoal using χ4' unfolding χ4'_def
by (metis Van.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using χ4' unfolding χ4'_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc r(4))
subgoal by fact subgoal by fact subgoal by fact
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr2(3) by blast .
qed
show ?thesis unfolding lltrv1 using 0
by (simp add: lfin' list_of_lappend)
qed
qed
qed
qed
qed
}
thus ?thesis unfolding Van.lcompletedFrom_def by auto
qed
lemma lcompletedFrom_lltrv2:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1" "lnever isIntO ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2" "lnever isIntO ltr2"
shows "Van.lcompletedFrom sv2 (lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
proof-
{fix ltrv2 assume ltrv2: "ltrv2 = lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and lfin: "lfinite ltrv2"
hence "list_of ltrv2 ≠ [] ∧ finalV (last (list_of ltrv2))"
using assms(2-) proof(induct "length (list_of ltrv2)" "w2"
arbitrary: ltrv2 trn w1 s1 ltr1 s2 ltr2 statA sv1 sv2 statO
rule: less2_induct')
case (less w2 ltrv2 trn w1 s1 ltr1 s2 ltr2 statA sv1 sv2 statO)
hence ltrv2: "ltrv2 = lltrv2 (trn, w1, w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
and lfin: "lfinite ltrv2"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "lcompletedFromO s1 ltr1" "lnever isIntO ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "lcompletedFromO s2 ltr2" "lnever isIntO ltr2"
by auto
have isi3: "¬ isIntO s1" using ltr1
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil llist.exhaust_sel llist.pred_inject(2))
have isi4: "¬ isIntO s2" using ltr2
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil llist.exhaust_sel llist.pred_inject(2))
show ?case proof(cases "ltr1 = [[]] ∧ ltr2 = [[]]")
case True note ltr14 = True
hence False using ltr1(2) ltr2(2) unfolding Opt.lcompletedFrom_def by auto
thus ?thesis by auto
next
case False hence ltr14: "ltr1 ≠ [[]] ∨ ltr2 ≠ [[]]" by auto
show ?thesis proof(cases "llength ltr1 ≤ Suc 0 ∨ llength ltr2 ≤ Suc 0")
case True note ltr14 = ltr14 True
hence ltrv2: "list_of ltrv2 = [sv2]" unfolding ltrv2 by simp
have "llength ltr1 = Suc 0 ∨ llength ltr2 = Suc 0"
using ltr14
by (metis Opt.lcompletedFrom_def
Suc_ile_eq i0_less lfinite_code(1) llength_eq_0 llist.exhaust
ltr1(2) ltr2(2) nle_le not_lnull_conv zero_enat_def)
hence "ltr1 = [[s1]] ∨ ltr2 = [[s2]]"
using Opt.lcompletedFrom_singl ltr1(1) ltr1(2) ltr2(1) ltr2(2) by blast
hence "finalO s1 ∨ finalO s2"
using Opt.lcompletedFrom_LCons ltr1(2) ltr2(2) by blast
hence "finalV sv2"
using Δ r(1) r(2) r(3) r(4) unw unwindCond_def by auto
thus ?thesis unfolding ltrv2 by auto
next
case False hence current: "llength ltr1 > Suc 0" "llength ltr2 > Suc 0" by auto
show ?thesis
proof(cases trn)
case L note current = current L
show ?thesis
proof(cases "lnever isSecO ltr1")
case True note current = current True
obtain trn' w1' w2' s1' ltr1' trv1 sv1' trv2 sv2' statOO where
ωω: "ltr1 = s1 $ ltr1'" "validTransO (s1, s1')" "Opt.lvalidFromS s1' ltr1'"
"lcompletedFromO s1' ltr1'" "lnever isIntO ltr1'" and
ω3: "ω3 Δ w1 w2 w1' w2' s1 s1' s2 statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
and trn' : "trn' = (if trv1 = [] then L else R)"
and lltrv2: "ltrv2 =
lappend (llist_of trv2) (lltrv2 (trn', w1', w2', s1', ltr1', s2, ltr2, statA, sv1', sv2', statOO))"
using lltrv1_lltrv2_lnever_L[OF unw Δ r ltr1 isi4 current]
unfolding ltrv2 by blast
define ltrv2' where ltrv2': "ltrv2' = lltrv2 (trn', w1', w2', s1', ltr1', s2, ltr2, statA, sv1', sv2', statOO)"
have lltrv2: "ltrv2 = lappend (llist_of trv2) ltrv2'"
unfolding lltrv2 ltrv2' ..
have trv2ne: "trv2 ≠ [] ∨ w2' < w2" using ω3 unfolding ω3_def by auto
have lfin': "lfinite ltrv2'"
using lfin trv2ne unfolding lltrv2 by simp
have len: "length (list_of ltrv2') < length (list_of ltrv2) ∨
length (list_of ltrv2') = length (list_of ltrv2) ∧ w2' < w2"
using trv2ne lfin lfin' by (simp add: list_of_lappend lltrv2)
have 0: "list_of ltrv2' ≠ [] ∧ finalV (last (list_of ltrv2'))"
using len proof(elim disjE conjE)
assume len: "length (list_of ltrv2') < length (list_of ltrv2)"
show ?thesis
apply(rule less(1)[OF _ ltrv2'])
subgoal by fact subgoal by fact
subgoal using ω3 unfolding ω3_def by simp
subgoal by (metis Opt.reach.Step ωω(2) fst_conv r(1) snd_conv)
subgoal by fact
subgoal using ω3 unfolding ω3_def
by (metis Van.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω3 unfolding ω3_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(4))
subgoal by fact subgoal by fact subgoal by fact subgoal by fact
subgoal by fact subgoal by fact .
next
assume len: "length (list_of ltrv2') = length (list_of ltrv2)" "w2' < w2"
show ?thesis
apply(rule less(2)[OF _ _ ltrv2'])
subgoal by fact subgoal using len by simp subgoal by fact
subgoal using ω3 unfolding ω3_def by simp
subgoal by (metis Opt.reach.Step ωω(2) fst_conv r(1) snd_conv)
subgoal by fact
subgoal using ω3 unfolding ω3_def
by (metis Van.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω3 unfolding ω3_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(4))
subgoal by fact subgoal by fact subgoal by fact subgoal by fact
subgoal by fact subgoal by fact .
qed
show ?thesis unfolding lltrv2 using 0
by (simp add: lfin' list_of_lappend)
next
case False note current = current False
obtain w1' w2' tr1 s1' s1'' ltr1' trv1 sv1'' trv2 sv2'' statOO where
χχ: "χχ s1 ltr1 tr1 s1' s1'' ltr1'" and
χ3': "χ3' Δ w1 w2 w1' w2' s1 tr1 s1' s1'' s2 statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
and lltrv2: "ltrv2 =
lappend (llist_of trv2) (lltrv2 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO))"
using lltrv1_lltrv2_not_lnever_L[OF unw Δ r ltr1 isi4 current]
unfolding ltrv2 by blast
define ltrv2' where ltrv2': "ltrv2' = lltrv2 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO)"
have lltrv2: "ltrv2 = lappend (llist_of trv2) ltrv2'"
unfolding lltrv2 ltrv2' ..
have trv2ne: "trv2 ≠ [] ∨ w2' < w2" using χ3' unfolding χ3'_def by auto
have lfin': "lfinite ltrv2'"
using lfin trv2ne unfolding lltrv2 by simp
have len: "length (list_of ltrv2') < length (list_of ltrv2) ∨
length (list_of ltrv2') = length (list_of ltrv2) ∧ w2' < w2"
using trv2ne lfin lfin' by (simp add: list_of_lappend lltrv2)
have 0: "list_of ltrv2' ≠ [] ∧ finalV (last (list_of ltrv2'))"
using len proof(elim disjE conjE)
assume len: "length (list_of ltrv2') < length (list_of ltrv2)"
show ?thesis
apply(rule less(1)[OF _ ltrv2'])
subgoal by fact subgoal by fact
subgoal using χ3' unfolding χ3'_def by simp
subgoal using χχ unfolding χχ_def
by (metis Simple_Transition_System.reach_validFromS_reach r(1) snoc_eq_iff_butlast)
subgoal by fact
subgoal using χ3' unfolding χ3'_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc r(3))
subgoal using χ3' unfolding χ3'_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(4))
subgoal using χχ unfolding χχ_def by simp
subgoal using χχ unfolding χχ_def by simp
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr1 by blast
subgoal by fact subgoal by fact subgoal by fact .
next
assume len: "length (list_of ltrv2') = length (list_of ltrv2)" "w2' < w2"
show ?thesis
apply(rule less(2)[OF _ _ ltrv2'])
subgoal by fact subgoal using len by simp subgoal by fact
subgoal using χ3' unfolding χ3'_def by simp
subgoal using χχ unfolding χχ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(1))
subgoal by fact
subgoal using χ3' unfolding χ3'_def
by (metis Van.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using χ3' unfolding χ3'_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(4))
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr1(3) by blast
subgoal by fact subgoal by fact subgoal by fact .
qed
show ?thesis unfolding lltrv2 using 0
by (simp add: lfin' list_of_lappend)
qed
next
case R note current = current R
show ?thesis
proof(cases "lnever isSecO ltr2")
case True note current = current True
obtain trn' w1' w2' s2' ltr2' trv1 sv1' trv2 sv2' statOO where
ωω: "ltr2 = s2 $ ltr2'" "validTransO (s2, s2')" "Opt.lvalidFromS s2' ltr2'"
"lcompletedFromO s2' ltr2'" "lnever isIntO ltr2'" and
ω4: "ω4 Δ w1 w2 w1' w2' s1 s2 s2' statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
and trn': "trn' = (if trv2 = [] then R else L)"
and ltrv2: "ltrv2 =
lappend (llist_of trv2) (lltrv2 (trn', w1', w2', s1, ltr1, s2', ltr2', statA, sv1', sv2', statOO))"
using lltrv1_lltrv2_lnever_R[OF unw Δ r ltr2(1,2) isi3 ltr2(3) current]
unfolding ltrv2 by blast
define ltrv2' where ltrv2': "ltrv2' = lltrv2 (trn', w1', w2', s1, ltr1, s2', ltr2', statA, sv1', sv2', statOO)"
have lltrv2: "ltrv2 = lappend (llist_of trv2) ltrv2'"
unfolding ltrv2 ltrv2' ..
have trv2ne: "trv2 ≠ [] ∨ w2' < w2" using ω4 unfolding ω4_def by auto
have lfin': "lfinite ltrv2'"
using lfin trv2ne unfolding lltrv2 by simp
have len: "length (list_of ltrv2') < length (list_of ltrv2) ∨
length (list_of ltrv2') = length (list_of ltrv2) ∧ w2' < w2"
using trv2ne lfin lfin' by (simp add: list_of_lappend lltrv2)
have 0: "list_of ltrv2' ≠ [] ∧ finalV (last (list_of ltrv2'))"
using len proof(elim disjE conjE)
assume len: "length (list_of ltrv2') < length (list_of ltrv2)"
show ?thesis
apply(rule less(1)[OF _ ltrv2'])
subgoal by fact subgoal by fact
subgoal using ω4 unfolding ω4_def by simp
subgoal by fact
subgoal using r(2) ωω by (metis Opt.reach.Step fst_conv snd_conv)
subgoal using ω4 unfolding ω4_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(3))
subgoal using ω4 unfolding ω4_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(4))
subgoal by fact subgoal by fact subgoal by fact subgoal by fact
subgoal by fact subgoal by fact .
next
assume len: "length (list_of ltrv2') = length (list_of ltrv2)" "w2' < w2"
show ?thesis
apply(rule less(2)[OF _ _ ltrv2'])
subgoal by fact subgoal using len by simp subgoal by fact
subgoal using ω4 unfolding ω4_def by simp
subgoal by fact
subgoal by (metis Opt.reach.Step ωω(2) fst_conv r(2) snd_conv)
subgoal using ω4 unfolding ω4_def
by (metis Van.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω4 unfolding ω4_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(4))
subgoal by fact subgoal by fact subgoal by fact subgoal by fact
subgoal by fact subgoal by fact .
qed
show ?thesis unfolding lltrv2 using 0
by (simp add: lfin' list_of_lappend)
next
case False note current = current False
obtain w1' w2' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statOO where
χχ: "χχ s2 ltr2 tr2 s2' s2'' ltr2'" and
χ4': "χ4' Δ w1 w2 w1' w2' s1 s2 tr2 s2' s2'' statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO "
and ltrv2: "ltrv2 =
lappend (llist_of trv2) (lltrv2 (L, w1', w2', s1, ltr1, s2'', s2'' $ ltr2', statA, sv1'', sv2'', statOO))"
using lltrv1_lltrv2_not_lnever_R[OF unw Δ r ltr2(1,2) isi3 ltr2(3) current]
unfolding ltrv2 by blast
define ltrv2' where ltrv2': "ltrv2' = lltrv2 (L, w1', w2', s1, ltr1, s2'', s2'' $ ltr2', statA, sv1'', sv2'', statOO)"
have lltrv2: "ltrv2 = lappend (llist_of trv2) ltrv2'"
unfolding ltrv2 ltrv2' ..
have trv2ne: "trv2 ≠ []" using χ4' unfolding χ4'_def by auto
have lfin': "lfinite ltrv2'"
using lfin trv2ne unfolding lltrv2 by simp
have len: "length (list_of ltrv2') < length (list_of ltrv2)"
using trv2ne lfin lfin' by (simp add: list_of_lappend lltrv2)
have 0: "list_of ltrv2' ≠ [] ∧ finalV (last (list_of ltrv2'))"
apply(rule less(1)[OF _ ltrv2'])
subgoal by fact subgoal by fact
subgoal using χ4' unfolding χ4'_def by simp
subgoal by fact
subgoal using r(2) χχ unfolding χχ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using χ4' unfolding χ4'_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2 r(3))
subgoal using χ4' unfolding χ4'_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc r(4))
subgoal by fact subgoal by fact subgoal by fact
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr2(3) by blast .
show ?thesis unfolding lltrv2 using 0
by (simp add: lfin' list_of_lappend)
qed
qed
qed
qed
qed
}
thus ?thesis unfolding Van.lcompletedFrom_def by auto
qed
lemma lS_lltrv1_ltr1:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1" "lnever isIntO ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2" "lnever isIntO ltr2"
shows "Van.lS (lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)) = Opt.lS ltr1"
proof-
have cltrv1: "Van.lcompletedFrom sv1 (lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
using lcompletedFrom_lltrv1[OF assms] .
{fix trn nL nR ltrv1 ltr1
assume "∃w1 w2 s1 s2 ltr2 statA sv1 sv2 statO.
nL = w1 ∧ nR = w2 ∧
ltrv1 = lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧ lnever isIntO ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧ lnever isIntO ltr2"
hence "TwoFuncPred.sameFM1 isSecV isSecO getSecV getSecO trn nL nR ltrv1 ltr1"
proof(coinduct rule: TwoFuncPred.sameFM1.coinduct[of "λtrn nL nR ltrv1 ltr1.
∃w1 w2 s1 s2 ltr2 statA sv1 sv2 statO.
nL = w1 ∧ nR = w2 ∧
ltrv1 = lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧ lnever isIntO ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧ lnever isIntO ltr2",
where pred = isSecV and pred' = isSecO and func = getSecV and func' = getSecO])
case (2 trn nL nR ltrv1 ltr1)
then obtain w1 w2 sv1 s1 s2 ltr2 statA sv2 statO
where nL: "nL = w1" and nR: "nR = w2"
and ltrv1: "ltrv1 = lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1" "lnever isIntO ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2" "lnever isIntO ltr2"
by auto
have isi3: "¬ isIntO s1" using ltr1
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil llist.exhaust_sel llist.pred_inject(2))
have isi4: "¬ isIntO s2" using ltr2
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil llist.exhaust_sel llist.pred_inject(2))
show ?case proof(cases "ltr1 = [[]] ∧ ltr2 = [[]]")
case True note ltr14 = True
hence ltrv1: "ltrv1 = [[]]" unfolding ltrv1 by simp
show ?thesis using ltr14 unfolding ltrv1 apply-apply(rule TwoFuncPred.sameFM1_selectLNil) by auto
next
case False hence ltr14: "ltr1 ≠ [[]] ∨ ltr2 ≠ [[]]" by auto
show ?thesis proof(cases "llength ltr1 ≤ Suc 0 ∨ llength ltr2 ≤ Suc 0")
case True note ltr14 = ltr14 True
hence ltrv1: "ltrv1 = [[sv1]]" unfolding ltrv1 by simp
have "llength ltr1 = Suc 0 ∨ llength ltr2 = Suc 0"
by (metis Opt.lcompletedFrom_def Suc_ile_eq True
lfinite_LNil llength_LNil llist_eq_cong ltr1(2)
ltr2(2) nle_le order_le_imp_less_or_eq zero_enat_def zero_order(3))
hence "finalO s1 ∨ finalO s2"
using Opt.lcompletedFrom_singl ltr1(1) ltr1(2) ltr2(1) ltr2(2) by blast
hence fs1: "finalO s1"
using Δ r(1) r(2) r(3) r(4) unw unwindCond_def by auto
hence ltr1: "ltr1 = [[s1]]"
by (metis Opt.final_def Opt.lcompletedFrom_def
Opt.lvalidFromS_Cons_iff lfinite_code(1) llist.exhaust ltr1(1) ltr1(2))
have fsv1: "finalV sv1"
using Δ fs1 r(1) r(2) r(3) r(4) unw unwindCond_final by blast
have isv13: "¬ isSecV sv1 ∧ ¬ isSecO s1"
using fsv1 fs1 Opt.final_not_isSec Van.final_not_isSec by blast
show ?thesis unfolding ltrv1 ltr1 apply(rule TwoFuncPred.sameFM1_selectSingl)
using isv13 by auto
next
case False hence current: "llength ltr1 > Suc 0" "llength ltr2 > Suc 0"
by auto
show ?thesis proof(cases trn)
case L note trn = L[simp] note current = current L
show ?thesis
proof(cases "lnever isSecO ltr1")
case True note current = current True
obtain trn' w1' w2' s1' ltr1' trv1 sv1' trv2 sv2' statOO where
ωω: "ltr1 = s1 $ ltr1'" "validTransO (s1, s1')" "Opt.lvalidFromS s1' ltr1'"
"lcompletedFromO s1' ltr1'" "lnever isIntO ltr1'" and
ω3: "ω3 Δ w1 w2 w1' w2' s1 s1' s2 statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
and trn': "trn' = (if trv1 = [] then L else R)"
and lltrv1: "ltrv1 =
lappend (llist_of trv1) (lltrv1 (trn', w1', w2', s1', ltr1', s2, ltr2, statA, sv1', sv2', statOO))"
using lltrv1_lltrv2_lnever_L[OF unw Δ r ltr1 isi4 current]
unfolding ltrv1 by blast
define ltrv1' where ltrv1': "ltrv1' ≡ lltrv1 (trn', w1', w2', s1', ltr1', s2, ltr2, statA, sv1', sv2', statOO)"
have ltrv1: "ltrv1 = lappend (llist_of trv1) ltrv1'"
unfolding lltrv1 ltrv1' ..
have nis1: "¬ isSecO s1" using True ωω(1) by force
show ?thesis
proof(cases "trv1 = []")
case True note trv1 = True
hence "w1' < w1" using ω3 unfolding ω3_def by auto
have [simp]: "trn' = trn" by (simp add: trv1 trn')
show ?thesis
apply(rule TwoFuncPred.sameFM1_selectDelayL)
apply(rule exI[of _ "w1'"]) apply(rule exI[of _ "w1"])
apply(rule exI[of _ "trv1"]) apply(rule exI[of _ "[s1]"])
apply(rule exI[of _ "w2'"])
apply(rule exI[of _ "ltrv1'"]) apply(rule exI[of _ "ltr1'"])
apply(rule exI[of _ "w2"])
apply(intro conjI)
subgoal by fact
subgoal unfolding nL .. subgoal unfolding nR ..
subgoal unfolding ltrv1 trv1 by simp
subgoal unfolding ωω(1) by simp
subgoal by fact subgoal unfolding trv1 using ω3_def nis1 by simp
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1']) apply(rule exI[of _ s2])
apply(rule exI[of _ ltr2]) apply(rule exI[of _ statA])
apply(rule exI[of _ sv1']) apply(rule exI[of _ sv2'])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv1' by simp
subgoal using ω3 unfolding ω3_def by simp
subgoal using ω3 unfolding ω3_def
by (metis Opt.reach.Step ωω(2) fst_conv r(1) snd_conv)
subgoal by fact
subgoal using ω3 unfolding ω3_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω3 unfolding ω3_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal using ωω by auto
subgoal using ωω by auto
subgoal using ωω
using llist_all_lappend_llist_of ltr1(3) by blast
subgoal using ωω using ltr2(1) by fastforce
subgoal by fact
subgoal by fact . .
next
case False note trv1 = False
show ?thesis
apply(rule TwoFuncPred.sameFM1_selectlappend)
apply(rule exI[of _ "trv1"]) apply(rule exI[of _ "[s1]"])
apply(rule exI[of _ trn']) apply(rule exI[of _ "w1'"])
apply(rule exI[of _ "w2'"])
apply(rule exI[of _ "ltrv1'"]) apply(rule exI[of _ "ltr1'"])
apply(rule exI[of _ trn])
apply(rule exI[of _ "w1"])
apply(rule exI[of _ "w2"])
apply(intro conjI)
subgoal ..
subgoal unfolding nL .. subgoal unfolding nR ..
subgoal using ltrv1 .
subgoal unfolding ωω(1) by simp
subgoal by fact
subgoal using ω3 unfolding ω3_def by simp
subgoal using ltr1(3) ω3 unfolding ω3_def
by (metis Opt.S.map_filter Opt.S.simps(4) Van.S.map_filter Van.S.eq_Nil_iff(2) append_Nil
butlast_snoc filter.simps(2) nis1)
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1']) apply(rule exI[of _ s2])
apply(rule exI[of _ ltr2]) apply(rule exI[of _ statA])
apply(rule exI[of _ sv1']) apply(rule exI[of _ sv2'])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv1' ..
subgoal using ω3 unfolding ω3_def by simp
subgoal using ω3 unfolding ω3_def
by (metis Opt.reach.Step ωω(2) fst_conv r(1) snd_conv)
subgoal by fact
subgoal using ω3 unfolding ω3_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω3 unfolding ω3_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal using ωω by auto
subgoal using ωω by auto
subgoal using ωω
using llist_all_lappend_llist_of ltr1(3) by blast
subgoal using ωω using ltr2(1) by fastforce
subgoal by fact
subgoal by fact . .
qed
next
case False note current = current False
obtain w1' w2' tr1 s1' s1'' ltr1' trv1 sv1'' trv2 sv2'' statOO where
χχ: "χχ s1 ltr1 tr1 s1' s1'' ltr1'" and
χ3': "χ3' Δ w1 w2 w1' w2' s1 tr1 s1' s1'' s2 statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
and lltrv1: "ltrv1 =
lappend (llist_of trv1) (lltrv1 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO))"
using lltrv1_lltrv2_not_lnever_L[OF unw Δ r ltr1 isi4 current]
unfolding ltrv1 by blast
define ltrv1' where ltrv1': "ltrv1' ≡ lltrv1 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO)"
have ltrv1: "ltrv1 = lappend (llist_of trv1) ltrv1'"
unfolding lltrv1 ltrv1' ..
show ?thesis apply(rule TwoFuncPred.sameFM1_selectlappend)
apply(rule exI[of _ "trv1"]) apply(rule exI[of _ "tr1 ## s1'"])
apply(rule exI[of _ R])
apply(rule exI[of _ "w1'"]) apply(rule exI[of _ "w2'"])
apply(rule exI[of _ "ltrv1'"]) apply(rule exI[of _ "s1'' $ ltr1'"])
apply(rule exI[of _ trn])
apply(rule exI[of _ "w1"]) apply(rule exI[of _ "w2"])
apply(intro conjI)
subgoal .. subgoal unfolding nL .. subgoal unfolding nR ..
subgoal using ltrv1 .
subgoal using χχ unfolding χχ_def by simp
subgoal using χ3' unfolding χ3'_def by simp
subgoal by simp
subgoal using χ3' unfolding χ3'_def
by (simp add: Opt.S.map_filter Van.S.map_filter)
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1''])
apply(rule exI[of _ s2]) apply(rule exI[of _ ltr2])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv1'']) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv1' ..
subgoal using χ3' unfolding χ3'_def by simp
subgoal using χ3' unfolding χ3'_def
by (metis Simple_Transition_System.reach_validFromS_reach χχ χχ_def
append_is_Nil_conv last_snoc not_Cons_self2 r(1))
subgoal by fact
subgoal using χ3' unfolding χ3'_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using χ3' unfolding χ3'_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr1(3) by blast
subgoal using χχ unfolding χχ_def using ltr2(1) by fastforce
subgoal by fact
subgoal by fact . .
qed
next
case R note trn = R[simp] note current = current R
show ?thesis
proof(cases "lnever isSecO ltr2")
case True note current = current True
obtain trn' w1' w2' s2' ltr2' trv1 sv1' trv2 sv2' statOO where
ωω: "ltr2 = s2 $ ltr2'" "validTransO (s2, s2')" "Opt.lvalidFromS s2' ltr2'"
"lcompletedFromO s2' ltr2'" "lnever isIntO ltr2'" and
ω4: "ω4 Δ w1 w2 w1' w2' s1 s2 s2' statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
and trn': "trn' = (if trv2 = [] then R else L)"
and ltrv1: "ltrv1 =
lappend (llist_of trv1) (lltrv1 (trn', w1', w2', s1, ltr1, s2', ltr2', statA, sv1', sv2', statOO))"
using lltrv1_lltrv2_lnever_R[OF unw Δ r ltr2(1,2) isi3 ltr2(3) current]
unfolding ltrv1 by blast
define ltrv1' where ltrv1': "ltrv1' ≡ lltrv1 (trn', w1', w2', s1, ltr1, s2', ltr2', statA, sv1', sv2', statOO)"
have ltrv1: "ltrv1 = lappend (llist_of trv1) ltrv1'"
unfolding ltrv1 ltrv1' ..
have nev1: "never isSecV trv1" using ω4 unfolding ω4_def by auto
show ?thesis
proof(cases "trv2 = []")
case True note trv2 = True
have [simp]: "trn' = trn" using R trv2 trn' by auto
have "w2' < w2" using ω4 trv2 unfolding ω4_def by auto
show ?thesis
apply(rule TwoFuncPred.sameFM1_selectDelayR)
apply(rule exI[of _ "w2'"]) apply(rule exI[of _ nR])
apply(rule exI[of _ trv1]) apply(rule exI[of _ "[]"])
apply(rule exI[of _ "w1'"])
apply(rule exI[of _ "ltrv1'"]) apply(rule exI[of _ "ltr1"])
apply(rule exI[of _ nL])
apply(intro conjI)
subgoal by simp subgoal .. subgoal ..
subgoal by fact subgoal by simp
subgoal unfolding nR by fact
subgoal using nev1 by (simp add: never_Nil_filter)
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1]) apply(rule exI[of _ s2'])
apply(rule exI[of _ ltr2']) apply(rule exI[of _ statA])
apply(rule exI[of _ sv1']) apply(rule exI[of _ sv2'])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv1' by simp
subgoal using ω4 unfolding ω4_def by simp
subgoal by fact
subgoal using ω4 unfolding ω4_def
by (metis Opt.reach.Step ωω(2) fst_conv r(2) snd_conv)
subgoal using ω4 unfolding ω4_def
by (metis Van.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω4 unfolding ω4_def
by (metis Van.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal by fact subgoal by fact subgoal by fact
subgoal using ωω by auto
subgoal using ωω by auto
subgoal using ωω by auto . .
next
case False note trv2 = False
have [simp]: "trn' = L" using R trv2 trn' by auto
show ?thesis
apply(rule TwoFuncPred.sameFM1_selectRL)
apply(rule exI[of _ trv1]) apply(rule exI[of _ "[]"])
apply(rule exI[of _ "w1'"]) apply(rule exI[of _ "w2'"])
apply(rule exI[of _ "ltrv1'"]) apply(rule exI[of _ "ltr1"])
apply(rule exI[of _ "w1"]) apply(rule exI[of _ "w2"])
apply(intro conjI)
subgoal by fact
subgoal unfolding nL .. subgoal unfolding nR ..
subgoal unfolding ltrv1 ..
subgoal unfolding ωω(1) by simp
subgoal using ω4 unfolding ω4_def by (simp add: never_Nil_filter)
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1]) apply(rule exI[of _ s2'])
apply(rule exI[of _ ltr2']) apply(rule exI[of _ statA])
apply(rule exI[of _ sv1']) apply(rule exI[of _ sv2'])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv1' by simp
subgoal using ω4 unfolding ω4_def by simp
subgoal by fact
subgoal using ω4 unfolding ω4_def
by (metis Opt.reach.Step ωω(2) fst_conv r(2) snd_conv)
subgoal using ω4 unfolding ω4_def
by (metis Van.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω4 unfolding ω4_def
by (metis Van.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal by fact subgoal by fact subgoal by fact
subgoal using ωω by auto
subgoal using ωω by auto
subgoal using ωω by auto . .
qed
next
case False note current = current False
obtain w1' w2' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statOO where
χχ: "χχ s2 ltr2 tr2 s2' s2'' ltr2'" and
χ4': "χ4' Δ w1 w2 w1' w2' s1 s2 tr2 s2' s2'' statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO "
and ltrv1: "ltrv1 =
lappend (llist_of trv1) (lltrv1 (L, w1', w2', s1, ltr1, s2'', s2'' $ ltr2', statA, sv1'', sv2'', statOO))"
using lltrv1_lltrv2_not_lnever_R[OF unw Δ r ltr2(1,2) isi3 ltr2(3) current]
unfolding ltrv1 by blast
define ltrv1' where ltrv1': "ltrv1' ≡ lltrv1 (L, w1', w2', s1, ltr1, s2'', s2'' $ ltr2', statA, sv1'', sv2'', statOO)"
have ltrv1: "ltrv1 = lappend (llist_of trv1) ltrv1'"
unfolding ltrv1 ltrv1' ..
show ?thesis
apply(rule TwoFuncPred.sameFM1_selectRL)
apply(rule exI[of _ trv1]) apply(rule exI[of _ "[]"])
apply(rule exI[of _ "w1'"]) apply(rule exI[of _ "w2'"])
apply(rule exI[of _ "ltrv1'"]) apply(rule exI[of _ "ltr1"])
apply(rule exI[of _ "w1"]) apply(rule exI[of _ "w2"])
apply(intro conjI)
subgoal by fact
subgoal unfolding nL .. subgoal unfolding nR ..
subgoal unfolding ltrv1 .. subgoal by simp
subgoal using χ4' unfolding χ4'_def by (simp add: never_Nil_filter)
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1]) apply(rule exI[of _ s2''])
apply(rule exI[of _ "s2'' $ ltr2'"]) apply(rule exI[of _ statA])
apply(rule exI[of _ sv1'']) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv1' by simp
subgoal using χ4' unfolding χ4'_def by simp
subgoal by fact
subgoal using r(2) χχ unfolding χχ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(3) χ4' unfolding χ4'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using r(4) χ4' unfolding χ4'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal by fact subgoal by fact subgoal by fact
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr2(3) by blast . .
qed
qed
qed
qed
qed
}
thus ?thesis unfolding Van.lS[OF cltrv1] Opt.lS[OF ltr1(2)]
apply- apply(rule TwoFuncPred.sameFM1_lmap_lfilter)
using assms by blast
qed
lemma lS_lltrv2_ltr2:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1" "lnever isIntO ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2" "lnever isIntO ltr2"
shows "Van.lS (lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)) = Opt.lS ltr2"
proof-
have cltrv2: "Van.lcompletedFrom sv2 (lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
using lcompletedFrom_lltrv2[OF assms] .
{fix trn nL nR ltrv2 ltr2
assume "∃w1 w2 s1 s2 ltr1 statA sv1 sv2 statO.
nL = w1 ∧ nR = w2 ∧
ltrv2 = lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧ lnever isIntO ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧ lnever isIntO ltr2"
hence "TwoFuncPred.sameFM2 isSecV isSecO getSecV getSecO trn nL nR ltrv2 ltr2"
proof(coinduct rule: TwoFuncPred.sameFM2.coinduct[of "λtrn nL nR ltrv2 ltr2.
∃w1 w2 s1 s2 ltr1 statA sv1 sv2 statO.
nL = w1 ∧ nR = w2 ∧
ltrv2 = lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧ lnever isIntO ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧ lnever isIntO ltr2",
where pred = isSecV and pred' = isSecO and func = getSecV and func' = getSecO])
case (2 trn nL nR ltrv2 ltr2)
then obtain w1 w2 sv1 s1 s2 ltr1 statA sv2 statO
where nL: "nL = w1" and nR: "nR = w2"
and ltrv2: "ltrv2 = lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1" "lnever isIntO ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2" "lnever isIntO ltr2"
by auto
have isi3: "¬ isIntO s1" using ltr1
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil llist.exhaust_sel llist.pred_inject(2))
have isi4: "¬ isIntO s2" using ltr2
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil llist.exhaust_sel llist.pred_inject(2))
show ?case proof(cases "ltr1 = [[]] ∧ ltr2 = [[]]")
case True note ltr14 = True
hence ltrv2: "ltrv2 = [[]]" unfolding ltrv2 by simp
show ?thesis using ltr14 unfolding ltrv2 apply-apply(rule TwoFuncPred.sameFM2_selectLNil) by auto
next
case False hence ltr14: "ltr1 ≠ [[]] ∨ ltr2 ≠ [[]]" by auto
show ?thesis proof(cases "llength ltr1 ≤ Suc 0 ∨ llength ltr2 ≤ Suc 0")
case True note ltr14 = ltr14 True
hence ltrv2: "ltrv2 = [[sv2]]" unfolding ltrv2 by simp
have "llength ltr1 = Suc 0 ∨ llength ltr2 = Suc 0"
by (metis Opt.lcompletedFrom_def Suc_ile_eq True
lfinite_LNil llength_LNil llist_eq_cong ltr1(2)
ltr2(2) nle_le order_le_imp_less_or_eq zero_enat_def zero_order(3))
hence "finalO s1 ∨ finalO s2"
using Opt.lcompletedFrom_singl ltr1(1) ltr1(2) ltr2(1) ltr2(2) by blast
hence fs2: "finalO s2"
using Δ r(1) r(2) r(3) r(4) unw unwindCond_def by auto
hence ltr2: "ltr2 = [[s2]]"
by (metis Opt.final_def Opt.lcompletedFrom_def
Opt.lvalidFromS_Cons_iff lfinite_code(1) llist.exhaust ltr2(1) ltr2(2))
have fsv2: "finalV sv2"
using Δ fs2 r(1) r(2) r(3) r(4) unw unwindCond_final by blast
have isv24: "¬ isSecV sv2 ∧ ¬ isSecO s2"
using fsv2 fs2 Opt.final_not_isSec Van.final_not_isSec by blast
show ?thesis unfolding ltrv2 ltr2 apply(rule TwoFuncPred.sameFM2_selectSingl)
using isv24 by auto
next
case False hence current: "llength ltr1 > Suc 0" "llength ltr2 > Suc 0"
by auto
show ?thesis proof(cases trn)
case L note trn = L[simp] note current = current L
show ?thesis
proof(cases "lnever isSecO ltr1")
case True note current = current True
obtain trn' w1' w2' s1' ltr1' trv1 sv1' trv2 sv2' statOO where
ωω: "ltr1 = s1 $ ltr1'" "validTransO (s1, s1')" "Opt.lvalidFromS s1' ltr1'"
"lcompletedFromO s1' ltr1'" "lnever isIntO ltr1'" and
ω3: "ω3 Δ w1 w2 w1' w2' s1 s1' s2 statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
and trn': "trn' = (if trv1 = [] then L else R)"
and lltrv2: "ltrv2 =
lappend (llist_of trv2) (lltrv2 (trn', w1', w2', s1', ltr1', s2, ltr2, statA, sv1', sv2', statOO))"
using lltrv1_lltrv2_lnever_L[OF unw Δ r ltr1 isi4 current]
unfolding ltrv2 by blast
define ltrv2' where ltrv2': "ltrv2' ≡ lltrv2 (trn', w1', w2', s1', ltr1', s2, ltr2, statA, sv1', sv2', statOO)"
have ltrv2: "ltrv2 = lappend (llist_of trv2) ltrv2'"
unfolding lltrv2 ltrv2' ..
have nev2: "never isSecV trv2" using ω3 unfolding ω3_def by auto
show ?thesis
proof(cases "trv1 = []")
case True note trv1 = True
have [simp]: "trn' = trn" using L trv1 trn' by auto
have "w1' < w1" using ω3 trv1 unfolding ω3_def by auto
show ?thesis
apply(rule TwoFuncPred.sameFM2_selectDelayL)
apply(rule exI[of _ "w1'"]) apply(rule exI[of _ nL])
apply(rule exI[of _ trv2]) apply(rule exI[of _ "[]"])
apply(rule exI[of _ "w2'"])
apply(rule exI[of _ "ltrv2'"]) apply(rule exI[of _ "ltr2"])
apply(rule exI[of _ nR])
apply(intro conjI)
subgoal by simp subgoal .. subgoal ..
subgoal by fact subgoal by simp
subgoal unfolding nL by fact
subgoal using nev2 by (simp add: never_Nil_filter)
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1']) apply(rule exI[of _ s2])
apply(rule exI[of _ ltr1']) apply(rule exI[of _ statA])
apply(rule exI[of _ sv1']) apply(rule exI[of _ sv2'])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv2' by simp
subgoal using ω3 unfolding ω3_def by simp
subgoal using ω3 unfolding ω3_def
by (metis Opt.reach.Step ωω(2) fst_conv r(1) snd_conv)
subgoal by fact
subgoal using ω3 unfolding ω3_def
by (metis Van.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω3 unfolding ω3_def
by (metis Van.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal using ωω by auto
subgoal using ωω by auto
subgoal using ωω by auto
subgoal by fact subgoal by fact subgoal by fact . .
next
case False note trv1 = False
have [simp]: "trn' = R" using L trv1 trn' by auto
show ?thesis
apply(rule TwoFuncPred.sameFM2_selectLR)
apply(rule exI[of _ trv2]) apply(rule exI[of _ "[]"])
apply(rule exI[of _ "w1'"]) apply(rule exI[of _ "w2'"])
apply(rule exI[of _ "ltrv2'"]) apply(rule exI[of _ "ltr2"])
apply(rule exI[of _ "w1"]) apply(rule exI[of _ "w2"])
apply(intro conjI)
subgoal by fact
subgoal unfolding nL .. subgoal unfolding nR ..
subgoal unfolding ltrv2 ..
subgoal unfolding ωω(1) by simp
subgoal using ω3 unfolding ω3_def by (simp add: never_Nil_filter)
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1']) apply(rule exI[of _ s2])
apply(rule exI[of _ ltr1']) apply(rule exI[of _ statA])
apply(rule exI[of _ sv1']) apply(rule exI[of _ sv2'])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv2' by simp
subgoal using ω3 unfolding ω3_def by simp
subgoal using ω3 unfolding ω3_def
by (metis Opt.reach.Step ωω(2) fst_conv r(1) snd_conv)
subgoal by fact
subgoal using ω3 unfolding ω3_def
by (metis Van.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω3 unfolding ω3_def
by (metis Van.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal using ωω by auto
subgoal using ωω by auto
subgoal using ωω by auto
subgoal by fact subgoal by fact subgoal by fact . .
qed
next
case False note current = current False
obtain w1' w2' tr1 s1' s1'' ltr1' trv1 sv1'' trv2 sv2'' statOO where
χχ: "χχ s1 ltr1 tr1 s1' s1'' ltr1'" and
χ3': "χ3' Δ w1 w2 w1' w2' s1 tr1 s1' s1'' s2 statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
and lltrv2: "ltrv2 =
lappend (llist_of trv2) (lltrv2 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO))"
using lltrv1_lltrv2_not_lnever_L[OF unw Δ r ltr1 isi4 current]
unfolding ltrv2 by blast
define ltrv2' where ltrv2': "ltrv2' ≡ lltrv2 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO)"
have ltrv2: "ltrv2 = lappend (llist_of trv2) ltrv2'"
unfolding lltrv2 ltrv2' ..
show ?thesis
apply(rule TwoFuncPred.sameFM2_selectLR)
apply(rule exI[of _ trv2]) apply(rule exI[of _ "[]"])
apply(rule exI[of _ "w1'"]) apply(rule exI[of _ "w2'"])
apply(rule exI[of _ "ltrv2'"]) apply(rule exI[of _ "ltr2"])
apply(rule exI[of _ "w1"]) apply(rule exI[of _ "w2"])
apply(intro conjI)
subgoal by fact
subgoal unfolding nL .. subgoal unfolding nR ..
subgoal unfolding ltrv2 .. subgoal by simp
subgoal using χ3' unfolding χ3'_def by (simp add: never_Nil_filter)
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1'']) apply(rule exI[of _ s2])
apply(rule exI[of _ "s1'' $ ltr1'"]) apply(rule exI[of _ statA])
apply(rule exI[of _ sv1'']) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv2' by simp
subgoal using χ3' unfolding χ3'_def by simp
subgoal using r(1) χχ unfolding χχ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal by fact
subgoal using r(3) χ3' unfolding χ3'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using r(4) χ3' unfolding χ3'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr1(3) by blast
subgoal by fact subgoal by fact subgoal by fact . .
qed
next
case R note trn = R[simp] note current = current R
show ?thesis
proof(cases "lnever isSecO ltr2")
case True note current = current True
obtain trn' w1' w2' s2' ltr2' trv1 sv1' trv2 sv2' statOO where
ωω: "ltr2 = s2 $ ltr2'" "validTransO (s2, s2')" "Opt.lvalidFromS s2' ltr2'"
"lcompletedFromO s2' ltr2'" "lnever isIntO ltr2'" and
ω4: "ω4 Δ w1 w2 w1' w2' s1 s2 s2' statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
and trn': "trn' = (if trv2 = [] then R else L)"
and ltrv2: "ltrv2 =
lappend (llist_of trv2) (lltrv2 (trn', w1', w2', s1, ltr1, s2', ltr2', statA, sv1', sv2', statOO))"
using lltrv1_lltrv2_lnever_R[OF unw Δ r ltr2(1,2) isi3 ltr2(3) current]
unfolding ltrv2 by blast
define ltrv2' where ltrv2': "ltrv2' ≡ lltrv2 (trn', w1', w2', s1, ltr1, s2', ltr2', statA, sv1', sv2', statOO)"
have ltrv2: "ltrv2 = lappend (llist_of trv2) ltrv2'"
unfolding ltrv2 ltrv2' ..
have nis2: "¬ isSecO s2" using True ωω(1) by force
show ?thesis
proof(cases "trv2 = []")
case True note trv2 = True
hence "w2' < w2" using ω4 unfolding ω4_def by auto
have [simp]: "trn' = trn" by (simp add: trv2 trn')
show ?thesis
apply(rule TwoFuncPred.sameFM2_selectDelayR)
apply(rule exI[of _ "w2'"]) apply(rule exI[of _ "w2"])
apply(rule exI[of _ "trv2"]) apply(rule exI[of _ "[s2]"])
apply(rule exI[of _ "w1'"])
apply(rule exI[of _ "ltrv2'"]) apply(rule exI[of _ "ltr2'"])
apply(rule exI[of _ "w1"])
apply(intro conjI)
subgoal by fact
subgoal unfolding nL .. subgoal unfolding nR ..
subgoal unfolding ltrv2 trv2 by simp
subgoal unfolding ωω(1) by simp
subgoal by fact subgoal unfolding trv2 using ω4_def nis2 by simp
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1]) apply(rule exI[of _ s2'])
apply(rule exI[of _ ltr1]) apply(rule exI[of _ statA])
apply(rule exI[of _ sv1']) apply(rule exI[of _ sv2'])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv2' by simp
subgoal using ω4 unfolding ω4_def by simp
subgoal by fact
subgoal using ω4 unfolding ω4_def
by (metis Opt.reach.Step ωω(2) fst_conv r(2) snd_conv)
subgoal using ω4 unfolding ω4_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω4 unfolding ω4_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal by fact subgoal by fact subgoal by fact
subgoal using ωω by auto
subgoal using ωω by auto
subgoal using ωω
using llist_all_lappend_llist_of ltr1(3) by blast . .
next
case False note trv2 = False
show ?thesis
apply(rule TwoFuncPred.sameFM2_selectlappend)
apply(rule exI[of _ "trv2"]) apply(rule exI[of _ "[s2]"])
apply(rule exI[of _ trn']) apply(rule exI[of _ "w1'"])
apply(rule exI[of _ "w2'"])
apply(rule exI[of _ "ltrv2'"]) apply(rule exI[of _ "ltr2'"])
apply(rule exI[of _ trn])
apply(rule exI[of _ "w1"])
apply(rule exI[of _ "w2"])
apply(intro conjI)
subgoal ..
subgoal unfolding nL .. subgoal unfolding nR ..
subgoal using ltrv2 .
subgoal unfolding ωω(1) by simp
subgoal by fact
subgoal using ω4 unfolding ω4_def by simp
subgoal using ltr1(3) ω4 unfolding ω4_def
by (simp add: never_Nil_filter nis2)
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1]) apply(rule exI[of _ s2'])
apply(rule exI[of _ ltr1]) apply(rule exI[of _ statA])
apply(rule exI[of _ sv1']) apply(rule exI[of _ sv2'])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv2' ..
subgoal using ω4 unfolding ω4_def by simp
subgoal by fact
subgoal using ω4 unfolding ω4_def
by (metis Opt.reach.Step ωω(2) fst_conv r(2) snd_conv)
subgoal using ω4 unfolding ω4_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω4 unfolding ω4_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal by fact subgoal by fact subgoal by fact
subgoal using ωω by auto
subgoal using ωω by auto
subgoal using ωω
using llist_all_lappend_llist_of ltr1(3) by blast . .
qed
next
case False note current = current False
obtain w1' w2' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statOO where
χχ: "χχ s2 ltr2 tr2 s2' s2'' ltr2'" and
χ4': "χ4' Δ w1 w2 w1' w2' s1 s2 tr2 s2' s2'' statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO "
and ltrv2: "ltrv2 =
lappend (llist_of trv2) (lltrv2 (L, w1', w2', s1, ltr1, s2'', s2'' $ ltr2', statA, sv1'', sv2'', statOO))"
using lltrv1_lltrv2_not_lnever_R[OF unw Δ r ltr2(1,2) isi3 ltr2(3) current]
unfolding ltrv2 by blast
define ltrv2' where ltrv2': "ltrv2' ≡ lltrv2 (L, w1', w2', s1, ltr1, s2'', s2'' $ ltr2', statA, sv1'', sv2'', statOO)"
have ltrv2: "ltrv2 = lappend (llist_of trv2) ltrv2'"
unfolding ltrv2 ltrv2' ..
show ?thesis
apply(rule TwoFuncPred.sameFM2_selectlappend)
apply(rule exI[of _ "trv2"]) apply(rule exI[of _ "tr2 ## s2'"])
apply(rule exI[of _ L])
apply(rule exI[of _ "w1'"]) apply(rule exI[of _ "w2'"])
apply(rule exI[of _ "ltrv2'"]) apply(rule exI[of _ "s2'' $ ltr2'"])
apply(rule exI[of _ trn])
apply(rule exI[of _ "w1"]) apply(rule exI[of _ "w2"])
apply(intro conjI)
subgoal .. subgoal unfolding nL .. subgoal unfolding nR ..
subgoal using ltrv2 .
subgoal using χχ unfolding χχ_def by simp
subgoal using χ4' unfolding χ4'_def by simp
subgoal by simp
subgoal using χ4' unfolding χ4'_def
by (simp add: Opt.S.map_filter Van.S.map_filter)
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1])
apply(rule exI[of _ s2'']) apply(rule exI[of _ ltr1])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv1'']) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv2' ..
subgoal using χ4' unfolding χ4'_def by simp
subgoal by fact
subgoal using χ4' unfolding χ4'_def
by (metis Simple_Transition_System.reach_validFromS_reach χχ χχ_def
append_is_Nil_conv last_snoc not_Cons_self2 r(2))
subgoal using χ4' unfolding χ4'_def
by (metis Van.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using χ4' unfolding χ4'_def
by (metis Van.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal by fact subgoal by fact subgoal by fact
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr2(3) by blast . .
qed
qed
qed
qed
qed
}
thus ?thesis unfolding Van.lS[OF cltrv2] Opt.lS[OF ltr2(2)]
apply- apply(rule TwoFuncPred.sameFM2_lmap_lfilter)
using assms by blast
qed
lemma lA_lltrv1_lltrv2:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1" "lnever isIntO ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2" "lnever isIntO ltr2"
shows "Van.lA (lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)) =
Van.lA (lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
proof-
have cltrv1: "Van.lcompletedFrom sv1 (lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
using lcompletedFrom_lltrv1[OF assms] .
have cltrv2: "Van.lcompletedFrom sv2 (lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
using lcompletedFrom_lltrv2[OF assms] .
{fix nL nR ltrv1 ltrv2
assume "∃trn w1 w2 s1 ltr1 s2 ltr2 statA sv1 sv2 statO.
nL = w1 ∧ nR = w2 ∧
ltrv1 = lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
ltrv2 = lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧ lnever isIntO ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧ lnever isIntO ltr2"
hence "TwoFuncPred.sameFM isIntV isIntV getActV getActV nL nR ltrv1 ltrv2"
proof(coinduct rule: TwoFuncPred.sameFM.coinduct[of "λnL nR ltrv1 ltrv2.
∃ trn w1 w2 s1 ltr1 s2 ltr2 statA sv1 sv2 statO.
nL = w1 ∧ nR = w2 ∧
ltrv1 = lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
ltrv2 = lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧ lnever isIntO ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧ lnever isIntO ltr2"])
case (2 nL nR ltrv1 ltrv2)
then obtain trn w1 w2 s1 ltr1 s2 ltr2 statA sv1 sv2 statO
where nL: "nL = w1" and nR: "nR = w2"
and ltrv1: "ltrv1 = lltrv1 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and ltrv2: "ltrv2 = lltrv2 (trn,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1" "lnever isIntO ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2" "lnever isIntO ltr2"
by auto
have isi3: "¬ isIntO s1" using ltr1
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil llist.exhaust_sel llist.pred_inject(2))
have isi4: "¬ isIntO s2" using ltr2
by (metis Opt.lcompletedFrom_def Opt.lvalidFromS_def lfinite_LNil llist.exhaust_sel llist.pred_inject(2))
show ?case proof(cases "ltr1 = [[]] ∧ ltr2 = [[]]")
case True note ltr14 = True
hence ltrv1: "ltrv1 = [[]]" unfolding ltrv1 by simp
show ?thesis using ltr14 unfolding ltrv1 ltrv2 apply-apply(rule TwoFuncPred.sameFM_selectLNil) by auto
next
case False hence ltr14: "ltr1 ≠ [[]] ∨ ltr2 ≠ [[]]" by auto
show ?thesis proof(cases "llength ltr1 ≤ Suc 0 ∨ llength ltr2 ≤ Suc 0")
case True note ltr14 = ltr14 True
hence ltrv1: "ltrv1 = [[sv1]]" and ltrv2: "ltrv2 = [[sv2]]" unfolding ltrv1 ltrv2 by auto
have "llength ltr1 = Suc 0 ∨ llength ltr2 = Suc 0"
by (metis Opt.lcompletedFrom_def Suc_ile_eq True
lfinite_LNil llength_LNil llist_eq_cong ltr1(2)
ltr2(2) nle_le order_le_imp_less_or_eq zero_enat_def zero_order(3))
hence "finalO s1 ∨ finalO s2"
using Opt.lcompletedFrom_singl ltr1(1) ltr1(2) ltr2(1) ltr2(2) by blast
hence fs1: "finalO s1 ∧ finalO s2"
using Δ r(1) r(2) r(3) r(4) unw unwindCond_def by auto
have fsv12: "finalV sv1 ∧ finalV sv2"
using Δ fs1 r(1) r(2) r(3) r(4) unw unwindCond_final by blast
have isv12: "¬ isIntV sv1 ∧ ¬ isIntV sv2"
using fsv12 Van.final_not_isInt by blast
show ?thesis unfolding ltrv1 ltrv2 apply(rule TwoFuncPred.sameFM_selectSingl)
using isv12 by auto
next
case False hence current: "llength ltr1 > Suc 0" "llength ltr2 > Suc 0"
by auto
show ?thesis proof(cases trn)
case L note current = current L
show ?thesis
proof(cases "lnever isSecO ltr1")
case True note current = current True
obtain trn' w1' w2' s1' ltr1' trv1 sv1' trv2 sv2' statOO where
ωω: "ltr1 = s1 $ ltr1'" "validTransO (s1, s1')" "Opt.lvalidFromS s1' ltr1'"
"lcompletedFromO s1' ltr1'" "lnever isIntO ltr1'" and
ω3: "ω3 Δ w1 w2 w1' w2' s1 s1' s2 statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
and trn': "trn' = (if trv1 = [] then L else R)"
and lltrv1: "ltrv1 =
lappend (llist_of trv1) (lltrv1 (trn', w1', w2', s1', ltr1', s2, ltr2, statA, sv1', sv2', statOO))"
and lltrv2: "ltrv2 =
lappend (llist_of trv2) (lltrv2 (trn', w1', w2', s1', ltr1', s2, ltr2, statA, sv1', sv2', statOO))"
using lltrv1_lltrv2_lnever_L[OF unw Δ r ltr1 isi4 current]
unfolding ltrv1 ltrv2 by blast
define ltrv1' where ltrv1': "ltrv1' ≡ lltrv1 (trn', w1', w2', s1', ltr1', s2, ltr2, statA, sv1', sv2', statOO)"
have lltrv1: "ltrv1 = lappend (llist_of trv1) ltrv1'"
unfolding lltrv1 ltrv1' ..
define ltrv2' where ltrv2': "ltrv2' ≡ lltrv2 (trn', w1', w2', s1', ltr1', s2, ltr2, statA, sv1', sv2', statOO)"
have lltrv2: "ltrv2 = lappend (llist_of trv2) ltrv2'"
unfolding lltrv2 ltrv2' ..
show ?thesis
apply(rule TwoFuncPred.sameFM_selectlappend)
apply(rule exI[of _ "trv1"]) apply(rule exI[of _ "w1'"]) apply(rule exI[of _ "w1"])
apply(rule exI[of _ "trv2"]) apply(rule exI[of _ "w2'"]) apply(rule exI[of _ "w2"])
apply(rule exI[of _ "ltrv1'"]) apply(rule exI[of _ "ltrv2'"])
apply(intro conjI)
subgoal unfolding nL .. subgoal unfolding nR ..
subgoal using lltrv1 .
subgoal using lltrv2 .
subgoal using ω3 unfolding ω3_def by simp
subgoal using ω3 unfolding ω3_def by simp
subgoal using ω3 unfolding ω3_def by (simp add: Van.A.map_filter)
subgoal apply(rule disjI1)
apply(rule exI[of _ trn']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1']) apply(rule exI[of _ ltr1'])
apply(rule exI[of _ s2]) apply(rule exI[of _ ltr2])
apply(rule exI[of _ statA])
apply(rule exI[of _ sv1']) apply(rule exI[of _ sv2'])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv1' ..
subgoal unfolding ltrv2' ..
subgoal using ω3 unfolding ω3_def by simp
subgoal using ω3 unfolding ω3_def
by (metis Opt.reach.Step ωω(2) fst_conv r(1) snd_conv)
subgoal by fact
subgoal using ω3 unfolding ω3_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω3 unfolding ω3_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal using ωω by auto
subgoal using ωω by auto
subgoal using ωω
using llist_all_lappend_llist_of ltr1(3) by blast
subgoal using ωω using ltr2(1) by fastforce
subgoal by fact
subgoal by fact . .
next
case False note current = current False
obtain w1' w2' tr1 s1' s1'' ltr1' trv1 sv1'' trv2 sv2'' statOO where
χχ: "χχ s1 ltr1 tr1 s1' s1'' ltr1'" and
χ3': "χ3' Δ w1 w2 w1' w2' s1 tr1 s1' s1'' s2 statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
and lltrv1: "ltrv1 =
lappend (llist_of trv1) (lltrv1 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO))"
and lltrv2: "ltrv2 =
lappend (llist_of trv2) (lltrv2 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO))"
using lltrv1_lltrv2_not_lnever_L[OF unw Δ r ltr1 isi4 current]
unfolding ltrv1 ltrv2 by blast
define ltrv1' where ltrv1': "ltrv1' ≡ lltrv1 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO)"
have lltrv1: "ltrv1 = lappend (llist_of trv1) ltrv1'"
unfolding lltrv1 ltrv1' ..
define ltrv2' where ltrv2': "ltrv2' ≡ lltrv2 (R,w1',w2',s1'',s1'' $ ltr1',s2,ltr2,statA,sv1'',sv2'',statOO)"
have lltrv2: "ltrv2 = lappend (llist_of trv2) ltrv2'"
unfolding lltrv2 ltrv2' ..
show ?thesis apply(rule TwoFuncPred.sameFM_selectlappend)
apply(rule exI[of _ "trv1"]) apply(rule exI[of _ "w1'"]) apply(rule exI[of _ "w1"])
apply(rule exI[of _ "trv2"]) apply(rule exI[of _ "w2'"]) apply(rule exI[of _ "w2"])
apply(rule exI[of _ "ltrv1'"]) apply(rule exI[of _ "ltrv2'"])
apply(intro conjI)
subgoal unfolding nL .. subgoal unfolding nR ..
subgoal using lltrv1 .
subgoal using lltrv2 .
subgoal using χ3' unfolding χ3'_def by auto
subgoal using χ3' unfolding χ3'_def by auto
subgoal using χ3' unfolding χ3'_def by (simp add: Van.A.map_filter)
subgoal apply(rule disjI1)
apply(rule exI[of _ R]) apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1'']) apply(rule exI[of _ "s1'' $ ltr1'"])
apply(rule exI[of _ s2]) apply(rule exI[of _ "ltr2"])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv1'']) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal ..subgoal ..
subgoal unfolding ltrv1' ..
subgoal unfolding ltrv2' ..
subgoal using χ3' unfolding χ3'_def by simp
subgoal using χ3' unfolding χ3'_def
by (metis Simple_Transition_System.reach_validFromS_reach χχ χχ_def
append_is_Nil_conv last_snoc not_Cons_self2 r(1))
subgoal by fact
subgoal using χ3' unfolding χ3'_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using χ3' unfolding χ3'_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr1(3) by blast
subgoal using χχ unfolding χχ_def using ltr2(1) by fastforce
subgoal by fact
subgoal by fact . .
qed
next
case R note current = current R
show ?thesis
proof(cases "lnever isSecO ltr2")
case True note current = current True
obtain trn' w1' w2' s2' ltr2' trv1 sv1' trv2 sv2' statOO where
ωω: "ltr2 = s2 $ ltr2'" "validTransO (s2, s2')" "Opt.lvalidFromS s2' ltr2'"
"lcompletedFromO s2' ltr2'" "lnever isIntO ltr2'" and
ω4: "ω4 Δ w1 w2 w1' w2' s1 s2 s2' statA sv1 trv1 sv1' sv2 trv2 sv2' statOO"
and trn': "trn' = (if trv2 = [] then R else L)"
and ltrv1: "ltrv1 =
lappend (llist_of trv1) (lltrv1 (trn', w1', w2', s1, ltr1, s2', ltr2', statA, sv1', sv2', statOO))"
and ltrv2: "ltrv2 =
lappend (llist_of trv2) (lltrv2 (trn', w1', w2', s1, ltr1, s2', ltr2', statA, sv1', sv2', statOO))"
using lltrv1_lltrv2_lnever_R[OF unw Δ r ltr2(1,2) isi3 ltr2(3) current]
unfolding ltrv1 ltrv2 by blast
define ltrv1' where ltrv1': "ltrv1' ≡ lltrv1 (trn', w1', w2', s1, ltr1, s2', ltr2', statA, sv1', sv2', statOO)"
have lltrv1: "ltrv1 = lappend (llist_of trv1) ltrv1'"
unfolding ltrv1 ltrv1' ..
define ltrv2' where ltrv2': "ltrv2' ≡ lltrv2 (trn', w1', w2', s1, ltr1, s2', ltr2', statA, sv1', sv2', statOO)"
have lltrv2: "ltrv2 = lappend (llist_of trv2) ltrv2'"
unfolding ltrv2 ltrv2' ..
show ?thesis
apply(rule TwoFuncPred.sameFM_selectlappend)
apply(rule exI[of _ "trv1"]) apply(rule exI[of _ "w1'"]) apply(rule exI[of _ "w1"])
apply(rule exI[of _ "trv2"]) apply(rule exI[of _ "w2'"]) apply(rule exI[of _ "w2"])
apply(rule exI[of _ "ltrv1'"]) apply(rule exI[of _ "ltrv2'"])
apply(intro conjI)
subgoal unfolding nL .. subgoal unfolding nR ..
subgoal using lltrv1 .
subgoal using lltrv2 .
subgoal using ω4 unfolding ω4_def by simp
subgoal using ω4 unfolding ω4_def by simp
subgoal using ω4 unfolding ω4_def by (simp add: Van.A.map_filter)
subgoal apply(rule disjI1)
apply(rule exI[of _ trn']) apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1]) apply(rule exI[of _ "ltr1"])
apply(rule exI[of _ s2']) apply(rule exI[of _ "ltr2'"])
apply(rule exI[of _ statA]) apply(rule exI[of _ sv1']) apply(rule exI[of _ sv2'])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv1' ..
subgoal unfolding ltrv2' ..
subgoal using ω4 unfolding ω4_def by simp
subgoal by fact
subgoal using ω4 unfolding ω4_def
by (metis Opt.reach.Step ωω(2) fst_conv r(2) snd_conv)
subgoal using ω4 unfolding ω4_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using ω4 unfolding ω4_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal by fact
subgoal by fact
subgoal by fact
subgoal by fact
subgoal using ωω by auto
subgoal using ωω by auto . .
next
case False note current = current False
obtain w1' w2' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statOO where
χχ: "χχ s2 ltr2 tr2 s2' s2'' ltr2'" and
χ4': "χ4' Δ w1 w2 w1' w2' s1 s2 tr2 s2' s2'' statA sv1 trv1 sv1'' sv2 trv2 sv2'' statOO "
and ltrv1: "ltrv1 =
lappend (llist_of trv1) (lltrv1 (L, w1', w2', s1, ltr1, s2'', s2'' $ ltr2', statA, sv1'', sv2'', statOO))"
and ltrv2: "ltrv2 =
lappend (llist_of trv2) (lltrv2 (L, w1', w2', s1, ltr1, s2'', s2'' $ ltr2', statA, sv1'', sv2'', statOO))"
using lltrv1_lltrv2_not_lnever_R[OF unw Δ r ltr2(1,2) isi3 ltr2(3) current]
unfolding ltrv1 ltrv2 by blast
define ltrv1' where ltrv1': "ltrv1' ≡ lltrv1 (L, w1', w2', s1, ltr1, s2'', s2'' $ ltr2', statA, sv1'', sv2'', statOO)"
have lltrv1: "ltrv1 = lappend (llist_of trv1) ltrv1'"
unfolding ltrv1 ltrv1' ..
define ltrv2' where ltrv2': "ltrv2' ≡ lltrv2 (L, w1', w2', s1, ltr1, s2'', s2'' $ ltr2', statA, sv1'', sv2'', statOO)"
have lltrv2: "ltrv2 = lappend (llist_of trv2) ltrv2'"
unfolding ltrv2 ltrv2' ..
show ?thesis
apply(rule TwoFuncPred.sameFM_selectlappend)
apply(rule exI[of _ "trv1"]) apply(rule exI[of _ "w1'"]) apply(rule exI[of _ "w1"])
apply(rule exI[of _ "trv2"]) apply(rule exI[of _ "w2'"]) apply(rule exI[of _ "w2"])
apply(rule exI[of _ "ltrv1'"]) apply(rule exI[of _ "ltrv2'"])
apply(intro conjI)
subgoal unfolding nL .. subgoal unfolding nR ..
subgoal using lltrv1 .
subgoal using lltrv2 .
subgoal using χ4' unfolding χ4'_def by simp
subgoal using χ4' unfolding χ4'_def by simp
subgoal using χ4' unfolding χ4'_def by (simp add: Van.A.map_filter)
subgoal apply(rule disjI1)
apply(rule exI[of _ L]) apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1]) apply(rule exI[of _ "ltr1"])
apply(rule exI[of _ s2'']) apply(rule exI[of _ "s2'' $ ltr2'"])
apply(rule exI[of _ statA])
apply(rule exI[of _ sv1'']) apply(rule exI[of _ sv2'']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrv1' ..
subgoal unfolding ltrv2' ..
subgoal using χ4' unfolding χ4'_def by simp
subgoal by fact
subgoal using χ4' unfolding χ4'_def
by (metis Simple_Transition_System.reach_validFromS_reach χχ χχ_def
append_is_Nil_conv last_snoc not_Cons_self2 r(2))
subgoal using χ4' unfolding χ4'_def
by (metis Simple_Transition_System.reach_validFromS_reach r(3) snoc_eq_iff_butlast)
subgoal using χ4' unfolding χ4'_def
by (metis Simple_Transition_System.reach_validFromS_reach r(4) snoc_eq_iff_butlast)
subgoal by fact
subgoal by fact
subgoal by fact
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def by auto
subgoal using χχ unfolding χχ_def
using llist_all_lappend_llist_of ltr2(3) by blast . .
qed
qed
qed
qed
qed
}
thus ?thesis unfolding Van.lA[OF cltrv1] Van.lA[OF cltrv2]
apply- apply(rule TwoFuncPred.sameFM_lmap_lfilter)
using assms by blast
qed
fun isN :: "('stateO,'stateV) tuple34 ⇒ bool"
where
"isN (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ⟷ ltr1 = [[]] ∨ ltr2 = [[]]"
fun H_T ::
"('stateO,'stateV)tuple34 ⇒
('stateO,'stateV)tuple12 ×
('stateO,'stateV)tuple34"
where
"H_T (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
(let (tr1,s1',s1'',ltr1',tr2,s2',s2'',ltr2') =
(SOME k. case k of (tr1,s1',s1'',ltr1',tr2,s2',s2'',ltr2') ⇒
φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2')
in let (w1',w2',trv1,sv1'',trv2,sv2'',statAA,statOO) =
(SOME k'. case k' of (w1',w2',trv1,sv1'',trv2,sv2'',statAA,statOO) ⇒
φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO)
in ((trv1,sv1'',trv2,sv2'',statAA,statOO),
(w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO))
)"
declare H_T.simps[simp del]
definition "H ≡ fst o H_T"
definition "T ≡ snd o H_T"
fun Econd where "Econd (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) = (lnever isIntO ltr1)"
fun E where "E (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) = f (L,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
definition F :: "('stateO,'stateV)tuple34 ⇒ ('stateO,'stateV)tuple12 llist"
where "F ≡ ccorec_llist isN H Econd E T"
lemma F_LNil:
"ltr1 = [[]] ∨ ltr2 = [[]] ⟹ F (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) = [[]]"
unfolding F_def apply(subst llist_ccorec(1)) by auto
lemma F_lnever:
assumes "ltr1 ≠ [[]]" "ltr2 ≠ [[]]" "lnever isIntO ltr1"
shows "F (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) = f (L, w1, w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
using assms unfolding F_def apply(subst llist_ccorec(2))
subgoal unfolding E.simps lnull_def apply(rule f_not_LNil) by auto
subgoal using assms by auto
subgoal unfolding Econd.simps by auto .
lemma F_not_lnever:
assumes "ltr1 ≠ [[]]" "ltr2 ≠ [[]]" "¬ lnever isIntO ltr1"
shows "F (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
LCons (H (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)) (F (T (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ))"
using assms unfolding F_def apply(subst llist_ccorec(2))
subgoal unfolding E.simps lnull_def apply(rule f_not_LNil) by auto
subgoal using assms by auto
subgoal unfolding Econd.simps by auto .
definition ltrv1 ::" ('stateO,'stateV)tuple34 ⇒ 'stateV llist" where
"ltrv1 tp = lconcat (lmap (λ(trv1,sv1'',trv2,sv2'',statAA,statOO). llist_of trv1) (F tp))"
definition firstHolds1 :: "('stateO,'stateV)tuple34 ⇒ nat" where
"firstHolds1 tp = firstNC (lmap (λ(trv1,sv1'',trv2,sv2'',statAA,statOO). trv1) (F tp))"
definition ltrv2 ::" ('stateO,'stateV)tuple34 ⇒ 'stateV llist" where
"ltrv2 tp = lconcat (lmap (λ(trv1,sv1'',trv2,sv2'',statAA,statOO). llist_of trv2) (F tp))"
definition firstHolds2 :: "('stateO,'stateV)tuple34 ⇒ nat" where
"firstHolds2 tp = firstNC (lmap (λ(trv1,sv1'',trv2,sv2'',statAA,statOO). trv2) (F tp))"
lemma ltrv1_ne_imp:
assumes "ltrv1 tp ≠ [[]]"
shows "∃trv1 sv1'' trv2 sv2'' statAA statOO. (trv1,sv1'',trv2,sv2'',statAA,statOO) ∈ lset (F tp) ∧
trv1 ≠ []"
using assms unfolding ltrv1_def unfolding lconcat_eq_LNil_iff by force
lemma ltrv2_ne_imp:
assumes "ltrv2 tp ≠ [[]]"
shows "∃trv1 sv1'' trv2 sv2'' statAA statOO. (trv1,sv1'',trv2,sv2'',statAA,statOO) ∈ lset (F tp) ∧
trv2 ≠ []"
using assms unfolding ltrv2_def unfolding lconcat_eq_LNil_iff by force
lemma ltrv1_LNil[simp]:
"ltr1 = [[]] ∨ ltr2 = [[]] ⟹ ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) = [[]]"
unfolding ltrv1_def F_LNil by simp
lemma ltrv2_LNil[simp]:
"ltr1 = [[]] ∨ ltr2 = [[]] ⟹ ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) = [[]]"
unfolding ltrv2_def F_LNil by simp
lemma ltrv1_lnever:
assumes "ltr1 ≠ [[]]" "ltr2 ≠ [[]]" "lnever isIntO ltr1"
shows "ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) = lltrv1 (L, w1, w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
unfolding ltrv1_def F_lnever[OF assms] lltrv1_def ..
lemma ltrv2_lnever:
assumes "ltr1 ≠ [[]]" "ltr2 ≠ [[]]" "lnever isIntO ltr1"
shows "ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) = lltrv2 (L, w1, w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
unfolding ltrv2_def F_lnever[OF assms] lltrv2_def ..
lemma H_T_not_lnever:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and l: "¬ lnever isIntO ltr1" "Opt.lA ltr1 = Opt.lA ltr2"
shows "∃ w1' w2' tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statAA statOO.
φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2' ∧
φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO ∧
H_T (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
((trv1,sv1'',trv2,sv2'',statAA,statOO),
(w1',w2', s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO))"
proof-
obtain tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2'
where φφ: "φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2'"
using isIntO_φφ[OF ltr1 ltr2 l(2,1)]
by auto
define tp where
"tp = (SOME k. case k of (tr1,s1',s1'',ltr1',tr2,s2',s2'',ltr2') ⇒
φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2')"
have 0: "case tp of (tr1,s1',s1'',ltr1',tr2,s2',s2'',ltr2') ⇒
φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2'"
using φφ unfolding tp_def apply- apply(rule someI_ex)
apply(rule exI[of _ "(tr1,s1',s1'',ltr1',tr2,s2',s2'',ltr2')"]) by auto
obtain tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2' where
tp: "tp = (tr1,s1',s1'',ltr1',tr2,s2',s2'',ltr2')" by(cases tp, auto)
have φφ: "φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2'"
using 0 unfolding tp by auto
obtain w1' w2' trv1 sv1'' trv2 sv2'' statAA statOO
where φ': "φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
using unwindCond_ex_φ'[OF unw Δ r stat, of tr1 s1' s1'' tr2 s2' s2'']
using φφ unfolding φφ_def by auto
define tp' where
"tp' = (SOME k'. case k' of (w1',w2',trv1,sv1'',trv2,sv2'',statAA,statOO) ⇒
φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO)"
have 1: "case tp' of (w1',w2',trv1,sv1'',trv2,sv2'',statAA,statOO) ⇒
φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
using φ' unfolding tp'_def apply- apply(rule someI_ex)
apply(rule exI[of _ "(w1',w2',trv1,sv1'',trv2,sv2'',statAA,statOO)"]) by auto
obtain w1' w2' trv1 sv1'' trv2 sv2'' statAA statOO where
tp': "tp' = (w1',w2',trv1,sv1'',trv2,sv2'',statAA,statOO)" by(cases tp', auto)
have φ': "φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
using 1 unfolding tp' by auto
show ?thesis
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ tr1]) apply(rule exI[of _ s1']) apply(rule exI[of _ s1'']) apply(rule exI[of _ ltr1'])
apply(rule exI[of _ tr2]) apply(rule exI[of _ s2']) apply(rule exI[of _ s2'']) apply(rule exI[of _ ltr2'])
apply(rule exI[of _ trv1]) apply(rule exI[of _ sv1'']) apply(rule exI[of _ trv2]) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statAA]) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal using φφ .
subgoal using φ' .
subgoal unfolding H_T.simps
unfolding tp_def[symmetric] tp apply simp
unfolding tp'_def[symmetric] tp' by simp .
qed
lemma ltrv1_ltrv2_not_lnever:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and l: "¬ lnever isIntO ltr1" "Opt.lA ltr1 = Opt.lA ltr2"
shows "∃ w1' w2' tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statAA statOO.
φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2' ∧
φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO ∧
ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
lappend (llist_of trv1) (ltrv1 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO)) ∧
ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) =
lappend (llist_of trv2) (ltrv2 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO))"
proof-
have ltr1NE: "ltr1 ≠ [[]]" using l(1) by auto
hence ltr2NE: "ltr2 ≠ [[]]" using l(2)
using Opt.lcompletedFrom_def ltr2(2) by blast
show ?thesis
using H_T_not_lnever[OF assms] apply(elim exE)
subgoal for w1' w2' tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statAA statOO
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ tr1]) apply(rule exI[of _ s1']) apply(rule exI[of _ s1'']) apply(rule exI[of _ ltr1'])
apply(rule exI[of _ tr2]) apply(rule exI[of _ s2']) apply(rule exI[of _ s2'']) apply(rule exI[of _ ltr2'])
apply(rule exI[of _ trv1]) apply(rule exI[of _ sv1'']) apply(rule exI[of _ trv2]) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statAA]) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal by simp
subgoal by simp
subgoal unfolding ltrv1_def apply(subst F_not_lnever[OF ltr1NE ltr2NE l(1)])
unfolding H_def T_def by simp
subgoal unfolding ltrv2_def apply(subst F_not_lnever[OF ltr1NE ltr2NE l(1)])
unfolding H_def T_def by simp . .
qed
lemma lvalidFromS_ltrv1:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and ltr14: "Opt.lA ltr1 = Opt.lA ltr2"
shows "Van.lvalidFromS sv1 (ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
proof-
{fix n1 sv1 ltrr1
assume "∃w1 w2 s1 ltr1 s2 ltr2 statA sv2 statO.
ltrr1 = ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
n1 = w1 ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
(statA = Diff ⟶ statO = Diff) ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧
Opt.lA ltr1 = Opt.lA ltr2"
hence "Van.llvalidFromS n1 sv1 ltrr1"
proof(coinduct rule: Van.llvalidFromS.coinduct[of "λn1 sv1 ltrr1.
∃w1 w2 s1 ltr1 s2 ltr2 statA sv2 statO.
ltrr1 = ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
n1 = w1 ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
(statA = Diff ⟶ statO = Diff) ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧
Opt.lA ltr1 = Opt.lA ltr2"])
case (llvalidFromS n1 sv1 ltrr1)
then obtain w1 w2 s1 ltr1 s2 ltr2 statA sv2 statO
where ltrr1: "ltrr1 = ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and n1: "n1 = w1"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and A34: "Opt.lA ltr1 = Opt.lA ltr2"
by auto
have current: "ltr1 ≠ [[]]" "ltr2 ≠ [[]]"
using ltr1(2) ltr2(2) unfolding Opt.lcompletedFrom_def by auto
show ?case proof(cases "lnever isIntO ltr1")
case True note current = current True
hence "lnever isIntO ltr2"
by (metis Opt.lA lfiltermap_LNil_never lfiltermap_lmap_lfilter ltr1(2) A34 ltr2(2))
note ln34 = True this
have ltrr1: "ltrr1 = lltrv1 (L, w1, w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
unfolding ltrr1 ltrv1_lnever[OF current] by simp
show ?thesis apply(rule Van.llvalidFromS_selectlvalidFromS)
unfolding ltrr1 apply simp
apply(rule lvalidFromS_lltrv1)
using ln34 Δ llvalidFromS ln34(2) ltr1(1) ltr1(2) ltr2(1) ltr2(2) r(1) r(2) r(4) unw by auto
next
case False note ln3 = False
hence ln4: "¬ lnever isIntO ltr2"
by (metis Opt.lA lfiltermap_LNil_never lfiltermap_lmap_lfilter ltr1(2) A34 ltr2(2))
have "ltr1 ≠ [[s1]]" using ln3 ltr1
using Opt.final_not_isInt by auto
hence "llength ltr1 > Suc 0"
by (metis Opt.lcompletedFrom_singl Suc_ile_eq current(1) enat_0_iff(1)
linorder_not_less llength_LNil llist_eq_cong ltr1(1) ltr1(2) nle_le not_less_zero)
hence "¬ finalO s1"
by (metis Opt.final_def Opt.lvalidFromS_Cons_iff current(1) eSuc_enat enat_0
linorder_neq_iff llength_LCons llength_LNil llist.exhaust_sel ltr1(1))
hence nf12: "¬ finalV sv1 ∧ ¬ finalV sv2"
using Δ r(1) r(2) r(3) r(4) unw unwindCond_def by force
obtain w1' w2' tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statAA statOO
where φφ: "φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2'"
and φ': "φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
and ltrr1: "ltrr1 =
lappend (llist_of trv1) (ltrv1 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO))"
using ltrv1_ltrv2_not_lnever[OF unw Δ r stat ltr1 ltr2 ln3 A34]
unfolding ltrr1 by blast
define ltrr1' where ltrr1': "ltrr1' = ltrv1 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO)"
have ltrr1: "ltrr1 = lappend (llist_of trv1) ltrr1'"
unfolding ltrr1 ltrr1' ..
have ne: "trv1 ≠ [] ∨ (trv1 = [] ∧ w1' < w1)"
using φ' unfolding φ'_def ltrr1 by simp
show ?thesis using ne proof(elim disjE conjE)
assume trv1: "trv1 ≠ []"
show ?thesis
apply(rule Van.llvalidFromS_selectlappend)
apply(rule exI[of _ sv1]) apply(rule exI[of _ trv1])
apply(rule exI[of _ sv1'']) apply(rule exI[of _ "w1'"])
apply(rule exI[of _ ltrr1']) apply(rule exI[of _ "w1"])
apply(intro conjI)
subgoal unfolding n1 .. subgoal ..
subgoal unfolding ltrr1 ..
subgoal using φ' unfolding φ'_def
by (metis Van.validS_append1 Van.validFromS_def append_is_Nil_conv hd_append2)
subgoal by fact
subgoal using φ' unfolding φ'_def
by (metis Simple_Transition_System.validFromS_def Van.validS_validTrans append_is_Nil_conv list.sel(1) not_Cons_self2 trv1)
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1'']) apply(rule exI[of _ "s1'' $ ltr1'"])
apply(rule exI[of _ s2'']) apply(rule exI[of _ "s2'' $ ltr2'"])
apply(rule exI[of _ statAA]) apply(rule exI[of _ sv2'']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal unfolding ltrr1' ..
subgoal ..
subgoal using φ' unfolding φ'_def by auto
subgoal using r(1) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(2) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(3) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach append_is_Nil_conv last_snoc trv1)
subgoal using r(4) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using φ' unfolding φ'_def by auto
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp . .
next
assume trv1[simp]: "trv1 = []" and MM': "w1' < w1"
hence sv1''[simp]: "sv1'' = sv1" using φ' unfolding φ'_def by simp
show ?thesis
apply(rule Van.llvalidFromS_selectDelay)
apply(rule exI[of _ "w1'"]) apply(rule exI[of _ "w1"])
apply(rule exI[of _ sv1'']) apply(rule exI[of _ ltrr1'])
apply(intro conjI)
subgoal unfolding n1 .. subgoal by simp
subgoal unfolding ltrr1 by simp subgoal by fact
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1'']) apply(rule exI[of _ "s1'' $ ltr1'"])
apply(rule exI[of _ s2'']) apply(rule exI[of _ "s2'' $ ltr2'"])
apply(rule exI[of _ statAA]) apply(rule exI[of _ sv2'']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal unfolding ltrr1' ..
subgoal ..
subgoal using φ' unfolding φ'_def by auto
subgoal using r(1) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(2) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal unfolding sv1'' by fact
subgoal using r(4) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using φ' unfolding φ'_def by auto
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp . .
qed
qed
qed
}
thus ?thesis apply-apply(rule Van.llvalidFromS_imp_lvalidFromS)
using assms by blast
qed
lemma lvalidFromS_ltrv2:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and ltr14: "Opt.lA ltr1 = Opt.lA ltr2"
shows "Van.lvalidFromS sv2 (ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
proof-
{fix n2 sv2 ltrr2
assume "∃w1 w2 s1 ltr1 s2 ltr2 statA sv1 statO.
ltrr2 = ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
n2 = w2 ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
(statA = Diff ⟶ statO = Diff) ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧
Opt.lA ltr1 = Opt.lA ltr2"
hence "Van.llvalidFromS n2 sv2 ltrr2"
proof(coinduct rule: Van.llvalidFromS.coinduct[of "λn2 sv2 ltrr2.
∃w1 w2 s1 ltr1 s2 ltr2 statA sv1 statO.
ltrr2 = ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
n2 = w2 ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
(statA = Diff ⟶ statO = Diff) ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧
Opt.lA ltr1 = Opt.lA ltr2"])
case (llvalidFromS n2 sv2 ltrr2)
then obtain w1 w2 s1 ltr1 s2 ltr2 statA sv1 statO
where ltrr2: "ltrr2 = ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and n2: "n2 = w2"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and A34: "Opt.lA ltr1 = Opt.lA ltr2"
by auto
have current: "ltr1 ≠ [[]]" "ltr2 ≠ [[]]"
using ltr1(2) ltr2(2) unfolding Opt.lcompletedFrom_def by auto
show ?case proof(cases "lnever isIntO ltr1")
case True note current = current True
hence "lnever isIntO ltr2"
by (metis Opt.lA lfiltermap_LNil_never lfiltermap_lmap_lfilter ltr1(2) A34 ltr2(2))
note ln34 = True this
have ltrr2: "ltrr2 = lltrv2 (L, w1, w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
unfolding ltrr2 ltrv2_lnever[OF current] by simp
show ?thesis apply(rule Van.llvalidFromS_selectlvalidFromS)
unfolding ltrr2 apply simp
apply(rule lvalidFromS_lltrv2)
using ln34 Δ llvalidFromS ln34(2) ltr1(1) ltr1(2) ltr2(1) ltr2(2) r(1) r(2) r(3) unw by auto
next
case False note ln3 = False
hence ln4: "¬ lnever isIntO ltr2"
by (metis Opt.lA lfiltermap_LNil_never lfiltermap_lmap_lfilter ltr1(2) A34 ltr2(2))
have "ltr2 ≠ [[s2]]" using ln4 ltr2
using Opt.final_not_isInt by auto
hence "llength ltr2 > Suc 0"
by (metis Opt.lcompletedFrom_singl Suc_ile_eq current(2) enat_0_iff(2)
linorder_not_less llength_LNil llist_eq_cong ltr2(1) ltr2(2) nle_le not_iless0)
hence "¬ finalO s2"
by (metis Opt.final_def Opt.lvalidFromS_Cons_iff current(2) eSuc_enat enat_0
linorder_neq_iff llength_LCons llength_LNil llist.exhaust_sel ltr2(1))
hence nf12: "¬ finalV sv1 ∧ ¬ finalV sv2"
using Δ r(1) r(2) r(3) r(4) unw unwindCond_def by force
obtain w1' w2' tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statAA statOO
where φφ: "φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2'"
and φ': "φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
and ltrr2: "ltrr2 =
lappend (llist_of trv2) (ltrv2 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO))"
using ltrv1_ltrv2_not_lnever[OF unw Δ r stat ltr1 ltr2 ln3 A34]
unfolding ltrr2 by blast
define ltrr2' where ltrr2': "ltrr2' = ltrv2 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO)"
have ltrr2: "ltrr2 = lappend (llist_of trv2) ltrr2'"
unfolding ltrr2 ltrr2' ..
have ne: "trv2 ≠ [] ∨ (trv2 = [] ∧ w2' < w2)"
using φ' unfolding φ'_def ltrr2 by simp
show ?thesis using ne proof(elim disjE conjE)
assume trv2: "trv2 ≠ []"
show ?thesis
apply(rule Van.llvalidFromS_selectlappend)
apply(rule exI[of _ sv2]) apply(rule exI[of _ trv2])
apply(rule exI[of _ sv2'']) apply(rule exI[of _ "w2'"])
apply(rule exI[of _ ltrr2']) apply(rule exI[of _ "w2"])
apply(intro conjI)
subgoal unfolding n2 .. subgoal ..
subgoal unfolding ltrr2 ..
subgoal using φ' unfolding φ'_def
by (metis Van.validS_append1 Van.validFromS_def append_is_Nil_conv hd_append2)
subgoal by fact
subgoal using φ' unfolding φ'_def
by (metis Simple_Transition_System.validFromS_def Van.validS_validTrans append_is_Nil_conv list.distinct(1) list.sel(1) trv2)
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1'']) apply(rule exI[of _ "s1'' $ ltr1'"])
apply(rule exI[of _ s2'']) apply(rule exI[of _ "s2'' $ ltr2'"])
apply(rule exI[of _ statAA]) apply(rule exI[of _ sv1'']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal unfolding ltrr2' ..
subgoal ..
subgoal using φ' unfolding φ'_def by auto
subgoal using r(1) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(2) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(3) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using r(4) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using φ' unfolding φ'_def by auto
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp . .
next
assume trv2[simp]: "trv2 = []" and MM': "w2' < w2"
hence sv2''[simp]: "sv2'' = sv2" using φ' unfolding φ'_def by simp
show ?thesis
apply(rule Van.llvalidFromS_selectDelay)
apply(rule exI[of _ "w2'"]) apply(rule exI[of _ "w2"])
apply(rule exI[of _ sv2'']) apply(rule exI[of _ ltrr2'])
apply(intro conjI)
subgoal unfolding n2 .. subgoal by simp
subgoal unfolding ltrr2 by simp subgoal by fact
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1'']) apply(rule exI[of _ "s1'' $ ltr1'"])
apply(rule exI[of _ s2'']) apply(rule exI[of _ "s2'' $ ltr2'"])
apply(rule exI[of _ statAA]) apply(rule exI[of _ sv1'']) apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal unfolding ltrr2' ..
subgoal ..
subgoal using φ' unfolding φ'_def by auto
subgoal using r(1) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(2) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(3) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal unfolding sv2'' by fact
subgoal using φ' unfolding φ'_def by auto
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp . .
qed
qed
qed
}
thus ?thesis apply-apply(rule Van.llvalidFromS_imp_lvalidFromS)
using assms by blast
qed
lemma lcompletedFrom_ltrv1:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and A34: "Opt.lA ltr1 = Opt.lA ltr2"
shows "Van.lcompletedFrom sv1 (ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
proof-
{fix ltrr1 assume ltrr1: "ltrr1 = ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and lfin: "lfinite ltrr1"
hence "list_of ltrr1 ≠ [] ∧ finalV (last (list_of ltrr1))"
using assms(2-) proof(induct "length (list_of ltrr1)" "w1"
arbitrary: w2 ltrr1 s1 ltr1 s2 ltr2 statA sv1 sv2 statO
rule: less2_induct')
case (less w1 ltrr1 w2 s1 ltr1 s2 ltr2 statA sv1 sv2 statO)
hence ltrr1: "ltrr1 = ltrv1 (w1,w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
and lfin: "lfinite ltrr1"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "lcompletedFromO s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "lcompletedFromO s2 ltr2"
and A34: "Opt.lA ltr1 = Opt.lA ltr2"
by auto
have current: "ltr1 ≠ [[]]" "ltr2 ≠ [[]]"
using ltr1(2) ltr2(2) unfolding Opt.lcompletedFrom_def by auto
show ?case proof(cases "lnever isIntO ltr1")
case True note ln3 = True note current = current True
hence ln4: "lnever isIntO ltr2"
by (metis Opt.lA lfiltermap_LNil_never lfiltermap_lmap_lfilter ltr1(2) A34 ltr2(2))
note ln34 = True this
have ltrr1: "ltrr1 = lltrv1 (L, w1, w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
unfolding ltrr1 ltrv1_lnever[OF current] by simp
show ?thesis
using lcompletedFrom_lltrv1[OF unw Δ r ltr1 ln3 ltr2 ln4, of L]
using lfin[unfolded ltrr1]
unfolding Van.lcompletedFrom_def ltrr1[symmetric]
using llist_of_list_of by fastforce
next
case False note ln3 = False
hence ln4: "¬ lnever isIntO ltr2"
by (metis Opt.lA lfiltermap_LNil_never lfiltermap_lmap_lfilter ltr1(2) A34 ltr2(2))
have "ltr1 ≠ [[s1]]" using ln3 ltr1
using Opt.final_not_isInt by auto
hence "llength ltr1 > Suc 0"
by (metis Opt.lcompletedFrom_singl Suc_ile_eq current(1) enat_0_iff(1)
linorder_not_less llength_LNil llist_eq_cong ltr1(1) ltr1(2) nle_le not_less_zero)
hence "¬ finalO s1"
by (metis Opt.final_def Opt.lvalidFromS_Cons_iff current(1) eSuc_enat enat_0
linorder_neq_iff llength_LCons llength_LNil llist.exhaust_sel ltr1(1))
hence nf12: "¬ finalV sv1 ∧ ¬ finalV sv2"
using Δ r(1) r(2) r(3) r(4) unw unwindCond_def by force
obtain w1' w2' tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statAA statOO
where φφ: "φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2'"
and φ': "φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
and ltrr1: "ltrr1 =
lappend (llist_of trv1) (ltrv1 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO))"
using ltrv1_ltrv2_not_lnever[OF unw Δ r stat ltr1 ltr2 ln3 A34]
unfolding ltrr1 by blast
define ltrr1' where ltrr1': "ltrr1' = ltrv1 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO)"
have ltrr1: "ltrr1 = lappend (llist_of trv1) ltrr1'"
unfolding ltrr1 ltrr1' ..
have ne: "trv1 ≠ [] ∨ (trv1 = [] ∧ w1' < w1)"
using φ' unfolding φ'_def ltrr1 by simp
have lfin': "lfinite ltrr1'"
using lfin ne unfolding ltrr1 by simp
have len: "length (list_of ltrr1') < length (list_of ltrr1) ∨
length (list_of ltrr1') = length (list_of ltrr1) ∧ w1' < w1"
using ne lfin lfin' by (simp add: list_of_lappend ltrr1)
have 0: "list_of ltrr1' ≠ [] ∧ finalV (last (list_of ltrr1'))"
using len proof(elim disjE conjE)
assume len: "length (list_of ltrr1') < length (list_of ltrr1)"
show ?thesis
apply(rule less(1)[OF _ ltrr1'])
subgoal by fact subgoal by fact
subgoal using φ' unfolding φ'_def by simp
subgoal using r(1) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(2) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(3) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using r(4) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using φ' unfolding φ'_def by auto
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp .
next
assume len: "length (list_of ltrr1') = length (list_of ltrr1)" "w1' < w1"
show ?thesis
apply(rule less(2)[OF _ _ ltrr1'])
subgoal by fact subgoal unfolding len ..
subgoal by fact
subgoal using φ' unfolding φ'_def by simp
subgoal using r(1) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(2) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(3) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using r(4) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using φ' unfolding φ'_def by auto
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp .
qed
show ?thesis unfolding ltrr1 using 0
by (simp add: lfin' list_of_lappend)
qed
qed
}
thus ?thesis unfolding Van.lcompletedFrom_def by auto
qed
lemma lcompletedFrom_ltrv2:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and A34: "Opt.lA ltr1 = Opt.lA ltr2"
shows "Van.lcompletedFrom sv2 (ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
proof-
{fix ltrr2 assume ltrr2: "ltrr2 = ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and lfin: "lfinite ltrr2"
hence "list_of ltrr2 ≠ [] ∧ finalV (last (list_of ltrr2))"
using assms(2-) proof(induct "length (list_of ltrr2)" "w2"
arbitrary: w1 ltrr2 s1 ltr1 s2 ltr2 statA sv1 sv2 statO
rule: less2_induct')
case (less w2 ltrr2 w1 s1 ltr1 s2 ltr2 statA sv1 sv2 statO)
hence ltrr2: "ltrr2 = ltrv2 (w1,w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
and lfin: "lfinite ltrr2"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "lcompletedFromO s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "lcompletedFromO s2 ltr2"
and A34: "Opt.lA ltr1 = Opt.lA ltr2"
by auto
have current: "ltr1 ≠ [[]]" "ltr2 ≠ [[]]"
using ltr1(2) ltr2(2) unfolding Opt.lcompletedFrom_def by auto
show ?case proof(cases "lnever isIntO ltr1")
case True note ln3 = True note current = current True
hence ln4: "lnever isIntO ltr2"
by (metis Opt.lA lfiltermap_LNil_never lfiltermap_lmap_lfilter ltr1(2) A34 ltr2(2))
note ln34 = True this
have ltrr2: "ltrr2 = lltrv2 (L, w1, w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
unfolding ltrr2 ltrv2_lnever[OF current] by simp
show ?thesis
using lcompletedFrom_lltrv2[OF unw Δ r ltr1 ln3 ltr2 ln4, of L]
using lfin[unfolded ltrr2]
unfolding Van.lcompletedFrom_def ltrr2[symmetric]
using llist_of_list_of by fastforce
next
case False note ln3 = False
hence ln4: "¬ lnever isIntO ltr2"
by (metis Opt.lA lfiltermap_LNil_never lfiltermap_lmap_lfilter ltr1(2) A34 ltr2(2))
have "ltr2 ≠ [[s2]]" using ln4 ltr2
using Opt.final_not_isInt by auto
hence "llength ltr2 > Suc 0"
by (metis Opt.lcompletedFrom_singl Suc_ile_eq current(2) enat_0_iff(2)
linorder_not_less llength_LNil llist_eq_cong ltr2(1) ltr2(2) nle_le not_less_zero)
hence "¬ finalO s2"
by (metis Opt.final_def Opt.lvalidFromS_Cons_iff current(2) eSuc_enat enat_0
linorder_neq_iff llength_LCons llength_LNil llist.exhaust_sel ltr2(1))
hence nf12: "¬ finalV sv1 ∧ ¬ finalV sv2"
using Δ r(1) r(2) r(3) r(4) unw unwindCond_def by force
obtain w1' w2' tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statAA statOO
where φφ: "φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2'"
and φ': "φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
and ltrr2: "ltrr2 =
lappend (llist_of trv2) (ltrv2 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO))"
using ltrv1_ltrv2_not_lnever[OF unw Δ r stat ltr1 ltr2 ln3 A34]
unfolding ltrr2 by blast
define ltrr2' where ltrr2': "ltrr2' = ltrv2 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO)"
have ltrr2: "ltrr2 = lappend (llist_of trv2) ltrr2'"
unfolding ltrr2 ltrr2' ..
have ne: "trv2 ≠ [] ∨ (trv2 = [] ∧ w2' < w2)"
using φ' unfolding φ'_def ltrr2 by simp
have lfin': "lfinite ltrr2'"
using lfin ne unfolding ltrr2 by simp
have len: "length (list_of ltrr2') < length (list_of ltrr2) ∨
length (list_of ltrr2') = length (list_of ltrr2) ∧ w2' < w2"
using ne lfin lfin' by (simp add: list_of_lappend ltrr2)
have 0: "list_of ltrr2' ≠ [] ∧ finalV (last (list_of ltrr2'))"
using len proof(elim disjE conjE)
assume len: "length (list_of ltrr2') < length (list_of ltrr2)"
show ?thesis
apply(rule less(1)[OF _ ltrr2'])
subgoal by fact subgoal by fact
subgoal using φ' unfolding φ'_def by simp
subgoal using r(1) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(2) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(3) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using r(4) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using φ' unfolding φ'_def by auto
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp .
next
assume len: "length (list_of ltrr2') = length (list_of ltrr2)" "w2' < w2"
show ?thesis
apply(rule less(2)[OF _ _ ltrr2'])
subgoal by fact subgoal unfolding len ..
subgoal by fact
subgoal using φ' unfolding φ'_def by simp
subgoal using r(1) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(2) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(3) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using r(4) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using φ' unfolding φ'_def by auto
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp
subgoal using φφ unfolding φφ_def by simp .
qed
show ?thesis unfolding ltrr2 using 0
by (simp add: lfin' list_of_lappend)
qed
qed
}
thus ?thesis unfolding Van.lcompletedFrom_def by auto
qed
lemma lS_ltrv1_ltr1:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and A34: "Opt.lA ltr1 = Opt.lA ltr2"
shows "Van.lS (ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)) = Opt.lS ltr1"
proof-
have cltrv1: "Van.lcompletedFrom sv1 (ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
using lcompletedFrom_ltrv1[OF assms] .
{fix nL nR ltrr1 ltr1
assume "∃w1 w2 s1 s2 ltr2 statA sv1 sv2 statO.
nL = w1 ∧ nR = w1 ∧
ltrr1 = ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
(statA = Diff ⟶ statO = Diff) ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧
Opt.lA ltr1 = Opt.lA ltr2"
hence "TwoFuncPred.sameFM isSecV isSecO getSecV getSecO nL nR ltrr1 ltr1"
proof(coinduct rule: TwoFuncPred.sameFM.coinduct[of "λnL nR ltrr1 ltr1.
∃ w1 w2 s1 s2 ltr2 statA sv1 sv2 statO.
nL = w1 ∧ nR = w1 ∧
ltrr1 = ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
(statA = Diff ⟶ statO = Diff) ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧
Opt.lA ltr1 = Opt.lA ltr2"])
case (2 nL nR ltrr1 ltr1)
then obtain w1 w2 s1 s2 ltr2 statA sv1 sv2 statO
where nL: "nL = w1" and nR: "nR = w1"
and ltrr1: "ltrr1 = ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and A34: "Opt.lA ltr1 = Opt.lA ltr2"
by auto
have current: "ltr1 ≠ [[]]" "ltr2 ≠ [[]]"
using ltr1(2) ltr2(2) unfolding Opt.lcompletedFrom_def by auto
show ?case proof(cases "lnever isIntO ltr1")
case True note ln3 = True note current = current True
hence ln4: "lnever isIntO ltr2"
by (metis Opt.lA lfiltermap_LNil_never lfiltermap_lmap_lfilter ltr1(2) A34 ltr2(2))
note ln34 = True this
have ltrr1: "ltrr1 = lltrv1 (L, w1, w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
unfolding ltrr1 ltrv1_lnever[OF current] by simp
have clltrv1: "Van.lcompletedFrom sv1 (lltrv1 (L,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
using lcompletedFrom_lltrv1[OF unw Δ r ltr1 ln3 ltr2 ln4] .
show ?thesis apply(rule TwoFuncPred.sameFM_selectlmap_lfilter)
unfolding ltrr1 apply simp
using lS_lltrv1_ltr1[OF unw Δ r ltr1 ln3 ltr2 ln4, of L]
unfolding Van.lS[OF clltrv1] Opt.lS[OF ltr1(2)] .
next
case False note ln3 = False
hence ln4: "¬ lnever isIntO ltr2"
by (metis Opt.lA lfiltermap_LNil_never lfiltermap_lmap_lfilter ltr1(2) A34 ltr2(2))
have "ltr1 ≠ [[s1]]" using ln3 ltr1
using Opt.final_not_isInt by auto
hence "llength ltr1 > Suc 0"
by (metis Opt.lcompletedFrom_singl Suc_ile_eq current(1) enat_0_iff(1)
linorder_not_less llength_LNil llist_eq_cong ltr1(1) ltr1(2) nle_le not_less_zero)
hence "¬ finalO s1"
by (metis Opt.final_def Opt.lvalidFromS_Cons_iff current(1) eSuc_enat enat_0
linorder_neq_iff llength_LCons llength_LNil llist.exhaust_sel ltr1(1))
hence nf12: "¬ finalV sv1 ∧ ¬ finalV sv2"
using Δ r(1) r(2) r(3) r(4) unw unwindCond_def by force
obtain w1' w2' tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statAA statOO
where φφ: "φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2'"
and φ': "φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
and ltrr1: "ltrr1 =
lappend (llist_of trv1) (ltrv1 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO))"
using ltrv1_ltrv2_not_lnever[OF unw Δ r stat ltr1 ltr2 ln3 A34]
unfolding ltrr1 by blast
define ltrr1' where ltrr1': "ltrr1' = ltrv1 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO)"
have ltrr1: "ltrr1 = lappend (llist_of trv1) ltrr1'"
unfolding ltrr1 ltrr1' ..
have ne1: "trv1 ≠ [] ∨ w1' < w1"
using φ' unfolding φ'_def ltrr1 by simp
show ?thesis
apply(rule TwoFuncPred.sameFM_selectlappend)
apply(rule exI[of _ "trv1"]) apply(rule exI[of _ "w1'"]) apply(rule exI[of _ "w1"])
apply(rule exI[of _ "tr1 ## s1'"]) apply(rule exI[of _ "w1'"]) apply(rule exI[of _ "w1"])
apply(rule exI[of _ "ltrr1'"]) apply(rule exI[of _ "s1'' $ ltr1'"])
apply(intro conjI)
subgoal unfolding nL .. subgoal unfolding nR ..
subgoal using ltrr1 .
subgoal using φφ unfolding φφ_def by simp
subgoal by fact
subgoal by simp
subgoal using φ' unfolding φ'_def unfolding Van.S.map_filter Opt.S.map_filter by simp
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1''])
apply(rule exI[of _ s2'']) apply(rule exI[of _ "s2'' $ ltr2'"])
apply(rule exI[of _ statAA])
apply(rule exI[of _ sv1'']) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrr1' ..
subgoal using φ' unfolding φ'_def by simp
subgoal using r(1) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(2) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(3) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using r(4) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using φ' unfolding φ'_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp . .
qed
qed
}
thus ?thesis apply-
unfolding Van.lS[OF cltrv1] Opt.lS[OF ltr1(2)] apply(rule TwoFuncPred.sameFM_lmap_lfilter)
using assms by blast
qed
lemma lS_ltrv2_ltr2:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and A34: "Opt.lA ltr1 = Opt.lA ltr2"
shows "Van.lS (ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)) = Opt.lS ltr2"
proof-
have cltrv2: "Van.lcompletedFrom sv2 (ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
using lcompletedFrom_ltrv2[OF assms] .
{fix nL nR ltrr2 ltr2
assume "∃w1 w2 s1 s2 ltr1 statA sv1 sv2 statO.
nL = w2 ∧ nR = w2 ∧
ltrr2 = ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
(statA = Diff ⟶ statO = Diff) ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧
Opt.lA ltr1 = Opt.lA ltr2"
hence "TwoFuncPred.sameFM isSecV isSecO getSecV getSecO nL nR ltrr2 ltr2"
proof(coinduct rule: TwoFuncPred.sameFM.coinduct[of "λnL nR ltrr2 ltr2.
∃ w1 w2 s1 s2 ltr1 statA sv1 sv2 statO.
nL = w2 ∧ nR = w2 ∧
ltrr2 = ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
(statA = Diff ⟶ statO = Diff) ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧
Opt.lA ltr1 = Opt.lA ltr2"])
case (2 nL nR ltrr2 ltr2)
then obtain w1 w2 s1 s2 ltr1 statA sv1 sv2 statO
where nL: "nL = w2" and nR: "nR = w2"
and ltrr2: "ltrr2 = ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and A34: "Opt.lA ltr1 = Opt.lA ltr2"
by auto
have current: "ltr1 ≠ [[]]" "ltr2 ≠ [[]]"
using ltr1(2) ltr2(2) unfolding Opt.lcompletedFrom_def by auto
show ?case proof(cases "lnever isIntO ltr1")
case True note ln3 = True note current = current True
hence ln4: "lnever isIntO ltr2"
by (metis Opt.lA lfiltermap_LNil_never lfiltermap_lmap_lfilter ltr1(2) A34 ltr2(2))
note ln34 = True this
have ltrr2: "ltrr2 = lltrv2 (L, w1, w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
unfolding ltrr2 ltrv2_lnever[OF current] by simp
have clltrv2: "Van.lcompletedFrom sv2 (lltrv2 (L,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
using lcompletedFrom_lltrv2[OF unw Δ r ltr1 ln3 ltr2 ln4] .
show ?thesis apply(rule TwoFuncPred.sameFM_selectlmap_lfilter)
unfolding ltrr2 apply simp
using lS_lltrv2_ltr2[OF unw Δ r ltr1 ln3 ltr2 ln4, of L]
unfolding Van.lS[OF clltrv2] Opt.lS[OF ltr2(2)] .
next
case False note ln3 = False
hence ln4: "¬ lnever isIntO ltr2"
by (metis Opt.lA lfiltermap_LNil_never lfiltermap_lmap_lfilter ltr1(2) A34 ltr2(2))
have "ltr2 ≠ [[s2]]" using ln4 ltr2
using Opt.final_not_isInt by auto
hence "llength ltr2 > Suc 0"
by (metis Opt.lcompletedFrom_singl Suc_ile_eq current(2) enat_0_iff(2)
linorder_not_less llength_LNil llist_eq_cong ltr2(1) ltr2(2) nle_le not_less_zero)
hence "¬ finalO s2"
by (metis Opt.final_def Opt.lvalidFromS_Cons_iff current(2) eSuc_enat enat_0
linorder_neq_iff llength_LCons llength_LNil llist.exhaust_sel ltr2(1))
hence nf12: "¬ finalV sv1 ∧ ¬ finalV sv2"
using Δ r(1) r(2) r(3) r(4) unw unwindCond_def by force
obtain w1' w2' tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statAA statOO
where φφ: "φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2'"
and φ': "φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
and ltrr2: "ltrr2 =
lappend (llist_of trv2) (ltrv2 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO))"
using ltrv1_ltrv2_not_lnever[OF unw Δ r stat ltr1 ltr2 ln3 A34]
unfolding ltrr2 by blast
define ltrr2' where ltrr2': "ltrr2' = ltrv2 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO)"
have ltrr1: "ltrr2 = lappend (llist_of trv2) ltrr2'"
unfolding ltrr2 ltrr2' ..
have ne2: "trv2 ≠ [] ∨ w2' < w2"
using φ' unfolding φ'_def ltrr2 by simp
show ?thesis
apply(rule TwoFuncPred.sameFM_selectlappend)
apply(rule exI[of _ "trv2"]) apply(rule exI[of _ "w2'"]) apply(rule exI[of _ "w2"])
apply(rule exI[of _ "tr2 ## s2'"]) apply(rule exI[of _ "w2'"]) apply(rule exI[of _ "w2"])
apply(rule exI[of _ "ltrr2'"]) apply(rule exI[of _ "s2'' $ ltr2'"])
apply(intro conjI)
subgoal unfolding nL .. subgoal unfolding nR ..
subgoal using ltrr1 .
subgoal using φφ unfolding φφ_def by simp
subgoal by fact
subgoal by simp
subgoal using φ' unfolding φ'_def unfolding Van.S.map_filter Opt.S.map_filter by simp
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1''])
apply(rule exI[of _ s2'']) apply(rule exI[of _ "s1'' $ ltr1'"])
apply(rule exI[of _ statAA])
apply(rule exI[of _ sv1'']) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrr2' ..
subgoal using φ' unfolding φ'_def by simp
subgoal using r(1) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(2) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(3) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using r(4) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using φ' unfolding φ'_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp . .
qed
qed
}
thus ?thesis apply-
unfolding Van.lS[OF cltrv2] Opt.lS[OF ltr2(2)] apply(rule TwoFuncPred.sameFM_lmap_lfilter)
using assms by blast
qed
lemma lA_ltrv1_ltrv2:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and A34: "Opt.lA ltr1 = Opt.lA ltr2"
shows "Van.lA (ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)) =
Van.lA (ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
proof-
have cltrv1: "Van.lcompletedFrom sv1 (ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
using lcompletedFrom_ltrv1[OF assms] .
have cltrv2: "Van.lcompletedFrom sv2 (ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
using lcompletedFrom_ltrv2[OF assms] .
{fix nL nR ltrr1 ltrr2
assume "∃w1 w2 s1 ltr1 s2 ltr2 statA sv1 sv2 statO.
nL = w1 ∧ nR = w2 ∧
ltrr1 = ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
ltrr2 = ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
(statA = Diff ⟶ statO = Diff) ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧
Opt.lA ltr1 = Opt.lA ltr2"
hence "TwoFuncPred.sameFM isIntV isIntV getActV getActV nL nR ltrr1 ltrr2"
proof(coinduct rule: TwoFuncPred.sameFM.coinduct[of "λnL nR ltrr1 ltrr2.
∃ w1 w2 s1 ltr1 s2 ltr2 statA sv1 sv2 statO.
nL = w1 ∧ nR = w2 ∧
ltrr1 = ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
ltrr2 = ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
(statA = Diff ⟶ statO = Diff) ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧
Opt.lA ltr1 = Opt.lA ltr2"])
case (2 nL nR ltrr1 ltrr2)
then obtain w1 w2 s1 ltr1 s2 ltr2 statA sv1 sv2 statO
where nL: "nL = w1" and nR: "nR = w2"
and ltrr1: "ltrr1 = ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and ltrr2: "ltrr2 = ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and A34: "Opt.lA ltr1 = Opt.lA ltr2"
by auto
have current: "ltr1 ≠ [[]]" "ltr2 ≠ [[]]"
using ltr1(2) ltr2(2) unfolding Opt.lcompletedFrom_def by auto
show ?case proof(cases "lnever isIntO ltr1")
case True note ln3 = True note current = current True
hence ln4: "lnever isIntO ltr2"
by (metis Opt.lA lfiltermap_LNil_never lfiltermap_lmap_lfilter ltr1(2) A34 ltr2(2))
note ln34 = True this
have ltrr1: "ltrr1 = lltrv1 (L, w1, w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
unfolding ltrr1 ltrv1_lnever[OF current] by simp
have ltrr2: "ltrr2 = lltrv2 (L, w1, w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
unfolding ltrr2 ltrv2_lnever[OF current] by simp
have clltrv1: "Van.lcompletedFrom sv1 (lltrv1 (L,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
using lcompletedFrom_lltrv1[OF unw Δ r ltr1 ln3 ltr2 ln4] .
have clltrv2: "Van.lcompletedFrom sv2 (lltrv2 (L,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
using lcompletedFrom_lltrv2[OF unw Δ r ltr1 ln3 ltr2 ln4] .
show ?thesis apply(rule TwoFuncPred.sameFM_selectlmap_lfilter)
unfolding ltrr1 ltrr2 apply simp
using lA_lltrv1_lltrv2[OF unw Δ r ltr1 ln3 ltr2 ln4, of L]
unfolding Van.lA[OF clltrv1] Van.lA[OF clltrv2] .
next
case False note ln3 = False
hence ln4: "¬ lnever isIntO ltr2"
by (metis Opt.lA lfiltermap_LNil_never lfiltermap_lmap_lfilter ltr1(2) A34 ltr2(2))
have "ltr1 ≠ [[s1]]" using ln3 ltr1
using Opt.final_not_isInt by auto
hence "llength ltr1 > Suc 0"
by (metis Opt.lcompletedFrom_singl Suc_ile_eq current(1) enat_0_iff(1)
linorder_not_less llength_LNil llist_eq_cong ltr1(1) ltr1(2) nle_le not_less_zero)
hence "¬ finalO s1"
by (metis Opt.final_def Opt.lvalidFromS_Cons_iff current(1) eSuc_enat enat_0
linorder_neq_iff llength_LCons llength_LNil llist.exhaust_sel ltr1(1))
hence nf12: "¬ finalV sv1 ∧ ¬ finalV sv2"
using Δ r(1) r(2) r(3) r(4) unw unwindCond_def by force
obtain w1' w2' tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statAA statOO
where φφ: "φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2'"
and φ': "φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
and ltrr1: "ltrr1 =
lappend (llist_of trv1) (ltrv1 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO))"
and ltrr2: "ltrr2 =
lappend (llist_of trv2) (ltrv2 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO))"
using ltrv1_ltrv2_not_lnever[OF unw Δ r stat ltr1 ltr2 ln3 A34]
unfolding ltrr1 ltrr2 by blast
define ltrr1' where ltrr1': "ltrr1' = ltrv1 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO)"
have ltrr1: "ltrr1 = lappend (llist_of trv1) ltrr1'"
unfolding ltrr1 ltrr1' ..
have ne1: "trv1 ≠ [] ∨ w1' < w1"
using φ' unfolding φ'_def ltrr1 by simp
define ltrr2' where ltrr2': "ltrr2' = ltrv2 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO)"
have ltrr2: "ltrr2 = lappend (llist_of trv2) ltrr2'"
unfolding ltrr2 ltrr2' ..
have ne2: "trv2 ≠ [] ∨ w2' < w2"
using φ' unfolding φ'_def ltrr1 by simp
show ?thesis
apply(rule TwoFuncPred.sameFM_selectlappend)
apply(rule exI[of _ "trv1"]) apply(rule exI[of _ "w1'"]) apply(rule exI[of _ "w1"])
apply(rule exI[of _ "trv2"]) apply(rule exI[of _ "w2'"]) apply(rule exI[of _ "w2"])
apply(rule exI[of _ "ltrr1'"]) apply(rule exI[of _ "ltrr2'"])
apply(intro conjI)
subgoal unfolding nL .. subgoal unfolding nR ..
subgoal using ltrr1 .
subgoal using ltrr2 .
subgoal by fact subgoal by fact
subgoal using φ' unfolding φ'_def unfolding Van.A.map_filter by simp
subgoal apply(rule disjI1)
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1'']) apply(rule exI[of _ "s1'' $ ltr1'"])
apply(rule exI[of _ s2'']) apply(rule exI[of _ "s2'' $ ltr2'"])
apply(rule exI[of _ statAA])
apply(rule exI[of _ sv1'']) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal .. subgoal ..
subgoal unfolding ltrr1' ..
subgoal unfolding ltrr2' ..
subgoal using φ' unfolding φ'_def by simp
subgoal using r(1) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(2) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(3) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using r(4) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using φ' unfolding φ'_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp . .
qed
qed
}
thus ?thesis apply-
unfolding Van.lA[OF cltrv1] Van.lA[OF cltrv2] apply(rule TwoFuncPred.sameFM_lmap_lfilter)
using assms by blast
qed
lemma lO_ltrv1_ltrv2:
assumes unw: "unwindCond Δ"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and A34: "Opt.lA ltr1 = Opt.lA ltr2"
and O12: "Van.lO (ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)) =
Van.lO (ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
and stO: "statO = Eq"
shows "Opt.lO ltr1 = Opt.lO ltr2"
proof-
have cltrv1: "Van.lcompletedFrom sv1 (ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
using lcompletedFrom_ltrv1[OF assms(1-12)] .
have cltrv2: "Van.lcompletedFrom sv2 (ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
using lcompletedFrom_ltrv2[OF assms(1-12)] .
{fix nL nR ltr1 ltr2
assume "∃ltrr1 ltrr2 w1 w2 s1 s2 statA sv1 sv2 statO.
ltrr1 = ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
ltrr2 = ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
(statA = Diff ⟶ statO = Diff) ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧
Opt.lA ltr1 = Opt.lA ltr2 ∧
Van.lO ltrr1 = Van.lO ltrr2 ∧
statO = Eq"
hence "TwoFuncPred.sameFM isIntO isIntO getObsO getObsO nL nR ltr1 ltr2"
proof(coinduct rule: TwoFuncPred.sameFM.coinduct[of "λnL nR ltr1 ltr2.
∃ ltrr1 ltrr2 w1 w2 s1 s2 statA sv1 sv2 statO.
ltrr1 = ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
ltrr2 = ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO) ∧
Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO ∧
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
(statA = Diff ⟶ statO = Diff) ∧
Opt.lvalidFromS s1 ltr1 ∧ Opt.lcompletedFrom s1 ltr1 ∧
Opt.lvalidFromS s2 ltr2 ∧ Opt.lcompletedFrom s2 ltr2 ∧
Opt.lA ltr1 = Opt.lA ltr2 ∧
Van.lO ltrr1 = Van.lO ltrr2 ∧
statO = Eq"])
case (2 nL nR ltr1 ltr2)
then obtain ltrr1 ltrr2 w1 w2 s1 s2 statA sv1 sv2 statO where
ltrr11: "ltrr1 = ltrv1 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and ltrr22: "ltrr2 = ltrv2 (w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO)"
and Δ: "Δ ∞ w1 w2 s1 s2 statA sv1 sv2 statO"
and r: "reachO s1" "reachO s2" "reachV sv1" "reachV sv2"
and stat: "statA = Diff ⟶ statO = Diff"
and ltr1: "Opt.lvalidFromS s1 ltr1" "Opt.lcompletedFrom s1 ltr1"
and ltr2: "Opt.lvalidFromS s2 ltr2" "Opt.lcompletedFrom s2 ltr2"
and A34: "Opt.lA ltr1 = Opt.lA ltr2"
and O12: "Van.lO ltrr1 = Van.lO ltrr2"
and stO: "statO = Eq"
by auto
have stA: "statA = Eq" using stat stO
using status.exhaust by blast
have current: "ltr1 ≠ [[]]" "ltr2 ≠ [[]]"
using ltr1(2) ltr2(2) unfolding Opt.lcompletedFrom_def by auto
show ?case proof(cases "lnever isIntO ltr1")
case True note ln3 = True note current = current True
hence ln4: "lnever isIntO ltr2"
by (metis Opt.lA lfiltermap_LNil_never lfiltermap_lmap_lfilter ltr1(2) A34 ltr2(2))
note ln34 = True this
have ltrr1: "ltrr1 = lltrv1 (L, w1, w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
unfolding ltrr11 ltrv1_lnever[OF current] by simp
have ltrr2: "ltrr2 = lltrv2 (L, w1, w2, s1, ltr1, s2, ltr2, statA, sv1, sv2, statO)"
unfolding ltrr22 ltrv2_lnever[OF current] by simp
have clltrv1: "Van.lcompletedFrom sv1 (lltrv1 (L,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
using lcompletedFrom_lltrv1[OF unw Δ r ltr1 ln3 ltr2 ln4] .
have clltrv2: "Van.lcompletedFrom sv2 (lltrv2 (L,w1,w2,s1,ltr1,s2,ltr2,statA,sv1,sv2,statO))"
using lcompletedFrom_lltrv2[OF unw Δ r ltr1 ln3 ltr2 ln4] .
show ?thesis apply(rule TwoFuncPred.sameFM_selectlmap_lfilter)
unfolding Opt.lO[OF ltr1(2)] Opt.lO[OF ltr2(2)]
by (metis ln3 ln4 lnever_LNil_lfilter')
next
case False note ln3 = False
hence ln4: "¬ lnever isIntO ltr2"
by (metis Opt.lA lfiltermap_LNil_never lfiltermap_lmap_lfilter ltr1(2) A34 ltr2(2))
have "ltr1 ≠ [[s1]]" using ln3 ltr1
using Opt.final_not_isInt by auto
hence "llength ltr1 > Suc 0"
by (metis Opt.lcompletedFrom_singl Suc_ile_eq current(1) enat_0_iff(1)
linorder_not_less llength_LNil llist_eq_cong ltr1(1) ltr1(2) nle_le not_less_zero)
hence "¬ finalO s1"
by (metis Opt.final_def Opt.lvalidFromS_Cons_iff current(1) eSuc_enat enat_0
linorder_neq_iff llength_LCons llength_LNil llist.exhaust_sel ltr1(1))
hence nf12: "¬ finalV sv1 ∧ ¬ finalV sv2"
using Δ r(1) r(2) r(3) r(4) unw unwindCond_def by force
obtain w1' w2' tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2' trv1 sv1'' trv2 sv2'' statAA statOO
where φφ: "φφ s1 ltr1 s2 ltr2 tr1 s1' s1'' ltr1' tr2 s2' s2'' ltr2'"
and φ': "φ' Δ w1 w2 w1' w2' statA s1 tr1 s1' s1'' s2 tr2 s2' s2'' statAA statO sv1 trv1 sv1'' sv2 trv2 sv2'' statOO"
and ltrr1: "ltrr1 =
lappend (llist_of trv1) (ltrv1 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO))"
and ltrr2: "ltrr2 =
lappend (llist_of trv2) (ltrv2 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO))"
using ltrv1_ltrv2_not_lnever[OF unw Δ r stat ltr1 ltr2 ln3 A34]
unfolding ltrr11 ltrr22 by blast
define ltrr1' where ltrr1': "ltrr1' = ltrv1 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO)"
have ltrr1: "ltrr1 = lappend (llist_of trv1) ltrr1'"
unfolding ltrr1 ltrr1' ..
have ne1: "trv1 ≠ [] ∨ w1' < w1"
using φ' unfolding φ'_def ltrr1 by simp
define ltrr2' where ltrr2': "ltrr2' = ltrv2 (w1',w2',s1'',s1'' $ ltr1',s2'',s2'' $ ltr2',statAA,sv1'',sv2'',statOO)"
have ltrr2: "ltrr2 = lappend (llist_of trv2) ltrr2'"
unfolding ltrr2 ltrr2' ..
have ne2: "trv2 ≠ [] ∨ w2' < w2"
using φ' unfolding φ'_def ltrr1 by simp
have ltr1_eq: "ltr1 = lappend (llist_of (tr1 ## s1')) (s1'' $ ltr1')"
and ltr2_eq: "ltr2 = lappend (llist_of (tr2 ## s2')) (s2'' $ ltr2')" using φφ unfolding φφ_def by auto
have sst: "statOO = Diff ⟷ Van.O (trv1 ## sv1'') ≠ Van.O (trv2 ## sv2'')"
"statA = Eq ⟹ statAA = Diff ⟷ Opt.O ((tr1 ## s1') ## s1'') ≠ Opt.O ((tr2 ## s2') ## s2'')"
"statO = Diff ⟹ statOO = Diff"
"statAA = Diff ⟹ statOO = Diff"
using φ' stO unfolding φ'_def by auto
have Atrv12': "Van.A (trv1 ## sv1'') = Van.A (trv2 ## sv2'')"
using φ' unfolding φ'_def by auto
have Δ': "Δ ∞ w1' w2' s1'' s2'' statAA sv1'' sv2'' statOO"
using φ' unfolding φ'_def by auto
have vltrv1: "Van.lvalidFromS sv1 ltrr1"
unfolding ltrr11 using lvalidFromS_ltrv1
using A34 Δ ltr1(1) ltr1(2) ltr2(1) ltr2(2) r(1) r(2) r(3) r(4) stat unw by blast
have cltrv1: "Van.lcompletedFrom sv1 ltrr1"
unfolding ltrr11 using lcompletedFrom_ltrv1
using A34 Δ ltr1(1) ltr1(2) ltr2(1) ltr2(2) r(1) r(2) r(3) r(4) stat unw by blast
have vltrv2: "Van.lvalidFromS sv2 ltrr2"
unfolding ltrr22 using lvalidFromS_ltrv2
using A34 Δ ltr1(1) ltr1(2) ltr2(1) ltr2(2) r(1) r(2) r(3) r(4) stat unw by blast
have cltrv2: "Van.lcompletedFrom sv2 ltrr2"
unfolding ltrr22 using lcompletedFrom_ltrv2
using A34 Δ ltr1(1) ltr1(2) ltr2(1) ltr2(2) r(1) r(2) r(3) r(4) stat unw by blast
have Oltrr1: "Van.lO ltrr1 = lmap getObsV (lfilter isIntV ltrr1)"
using Van.lO[OF cltrv1] .
have Oltrr2: "Van.lO ltrr2 = lmap getObsV (lfilter isIntV ltrr2)"
using Van.lO[OF cltrv2] .
have cltrv1': "Van.lcompletedFrom (lastt sv1 trv1) ltrr1'"
using cltrv1 unfolding ltrr1 Van.lcompletedFrom_def Van.final_def apply simp
using φ' unfolding φ'_def
by (metis Van.validS_validTrans Van.validFromS_def lappend_LNil2 last_appendR lfinite_llist_of list_of_lappend
list_of_llist_of llist_of.simps(1) snoc_eq_iff_butlast)
have cltrv2': "Van.lcompletedFrom (lastt sv2 trv2) ltrr2'"
using cltrv2 unfolding ltrr2 Van.lcompletedFrom_def Van.final_def apply simp
using φ' unfolding φ'_def
by (metis Van.validS_validTrans Van.validFromS_def lappend_LNil2 last_appendR lfinite_llist_of list_of_lappend
list_of_llist_of llist_of.simps(1) snoc_eq_iff_butlast)
have Oltrr1': "Van.lO ltrr1' = lmap getObsV (lfilter isIntV ltrr1')"
using Van.lO[OF cltrv1'] .
have Oltrr2': "Van.lO ltrr2' = lmap getObsV (lfilter isIntV ltrr2')"
using Van.lO[OF cltrv2'] .
have "Van.O (trv1 ## sv1'') = Van.O (trv2 ## sv2'') ∧
Van.lO ltrr1' = Van.lO ltrr2'"
using Atrv12' O12 Oltrr1' Oltrr2' unfolding Oltrr1 Oltrr2 unfolding ltrr1 ltrr2
unfolding Van.A.map_filter Van.O.map_filter Van.lO.lmap_lfilter apply simp
unfolding lmap_lappend_distrib apply simp apply(subst (asm) lappend_llist_of_inj)
using map_eq_imp_length_eq by auto
hence O12'': "Van.O (trv1 ## sv1'') = Van.O (trv2 ## sv2'')"
and O12': "Van.lO ltrr1' = Van.lO ltrr2'" by auto
have stOO: "statOO = Eq" using O12'' sst by(cases statOO, auto)
have O34': "Opt.O ((tr1 ## s1') ## s1'') = Opt.O ((tr2 ## s2') ## s2'')"
using stOO sst(2) sst(4) stA by blast
hence s14': "getObsO s1' = getObsO s2'"
using φφ unfolding φφ_def Opt.O.map_filter by (simp add: never_Nil_filter)
show ?thesis
apply(rule TwoFuncPred.sameFM_selectlappend)
apply(rule exI[of _ "tr1 ## s1'"]) apply(rule exI[of _ undefined]) apply(rule exI[of _ nL])
apply(rule exI[of _ "tr2 ## s2'"]) apply(rule exI[of _ undefined]) apply(rule exI[of _ nR])
apply(rule exI[of _ "s1'' $ ltr1'"]) apply(rule exI[of _ "s2'' $ ltr2'"])
apply(intro conjI)
subgoal .. subgoal ..
subgoal by fact subgoal by fact
subgoal by simp subgoal by simp
subgoal using φφ unfolding φφ_def unfolding Opt.O.map_filter
by (simp add: s14' never_Nil_filter)
subgoal apply(rule disjI1)
apply(rule exI[of _ ltrr1']) apply(rule exI[of _ ltrr2'])
apply(rule exI[of _ w1']) apply(rule exI[of _ w2'])
apply(rule exI[of _ s1'']) apply(rule exI[of _ s2''])
apply(rule exI[of _ statAA])
apply(rule exI[of _ sv1'']) apply(rule exI[of _ sv2''])
apply(rule exI[of _ statOO])
apply(intro conjI)
subgoal unfolding ltrr1' ..
subgoal unfolding ltrr2' ..
subgoal using φ' unfolding φ'_def by simp
subgoal using r(1) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(2) φφ unfolding φφ_def
by (metis Opt.reach_validFromS_reach append_is_Nil_conv last_snoc not_Cons_self2)
subgoal using r(3) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using r(4) φ' unfolding φ'_def
by (metis Van.reach_validFromS_reach snoc_eq_iff_butlast)
subgoal using φ' unfolding φ'_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal using r(2) φφ unfolding φφ_def by simp
subgoal by fact
subgoal by fact . .
qed
qed
}
thus ?thesis
unfolding Opt.lO[OF ltr1(2)] Opt.lO[OF ltr2(2)] apply(rule TwoFuncPred.sameFM_lmap_lfilter[where wL = undefined and wR = undefined])
using assms by blast
qed
end
theorem unwind_lrsecure:
assumes init: "initCond Δ" and unwind: "unwindCond Δ"
shows lrsecure
unfolding lrsecure_def2 proof clarify
fix s1 tr1 s2 tr2
assume 3: "istateO s1" "Opt.lvalidFromS s1 tr1" "lcompletedFromO s1 tr1"
and 4: "istateO s2" "Opt.lvalidFromS s2 tr2" "lcompletedFromO s2 tr2"
and A34: "Opt.lA tr1 = Opt.lA tr2" and O34: "Opt.lO tr1 ≠ Opt.lO tr2"
obtain sv1 sv2 where
isv12: "istateV sv1" "istateV sv2" and c12: "corrState sv1 s1" "corrState sv2 s2"
and Δ: "Δ ∞ ∞ ∞ s1 s2 Eq sv1 sv2 Eq"
using init 3 4 unfolding initCond_def by blast
have r: "reachV sv1" "reachV sv2" "reachO s1" "reachO s2"
by (auto simp: Van.Istate isv12 Opt.Istate 3 4)
note all = 3 4 A34 isv12 Δ unwind r
show "∃sv1 trv1 sv2 trv2.
istateV sv1 ∧ istateV sv2 ∧ corrState sv1 s1 ∧ corrState sv2 s2 ∧
Van.lvalidFromS sv1 trv1 ∧ lcompletedFromV sv1 trv1 ∧ Van.lvalidFromS sv2 trv2 ∧
lcompletedFromV sv2 trv2 ∧ Van.lS trv1 = Opt.lS tr1 ∧ Van.lS trv2 = Opt.lS tr2 ∧
Van.lA trv1 = Van.lA trv2 ∧ Van.lO trv1 ≠ Van.lO trv2"
apply(rule exI[of _ sv1])
apply(rule exI[of _ "ltrv1 Δ (∞, ∞, s1, tr1, s2, tr2, Eq, sv1, sv2, Eq)"])
apply(rule exI[of _ sv2])
apply(rule exI[of _ "ltrv2 Δ (∞, ∞, s1, tr1, s2, tr2, Eq, sv1, sv2, Eq)"])
apply(intro conjI)
subgoal by fact subgoal by fact subgoal by fact subgoal by fact
subgoal apply(rule lvalidFromS_ltrv1) using all by auto
subgoal apply(rule lcompletedFrom_ltrv1) using all by auto
subgoal apply(rule lvalidFromS_ltrv2) using all by auto
subgoal apply(rule lcompletedFrom_ltrv2) using all by auto
subgoal apply(rule lS_ltrv1_ltr1) using all by auto
subgoal apply(rule lS_ltrv2_ltr2) using all by auto
subgoal apply(rule lA_ltrv1_ltrv2) using all by auto
subgoal using O34 apply- apply(erule contrapos_nn)
apply(rule lO_ltrv1_ltrv2) using all by auto .
qed
subsection ‹ Compositional unwinding ›
text ‹ We allow networks of unwinding relations that unwind into each other,
which offer a compositional alternative to monolithic unwinding. ›
definition unwindIntoCond ::
"(enat ⇒ enat ⇒ enat ⇒ 'stateO ⇒ 'stateO ⇒ status ⇒ 'stateV ⇒ 'stateV ⇒ status ⇒ bool) ⇒
(enat ⇒ enat ⇒ enat ⇒ 'stateO ⇒ 'stateO ⇒ status ⇒ 'stateV ⇒ 'stateV ⇒ status ⇒ bool)
⇒ bool"
where
"unwindIntoCond Δ Δ' ≡ ∀w w1 w2 s1 s2 statA sv1 sv2 statO.
reachO s1 ∧ reachO s2 ∧ reachV sv1 ∧ reachV sv2 ∧
Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟶
(finalO s1 ⟷ finalO s2) ∧ (finalV sv1 ⟷ finalO s1) ∧ (finalV sv2 ⟷ finalO s2)
∧
(statA = Eq ⟶ (isIntO s1 ⟷ isIntO s2))
∧
((∃v<w. proact Δ' v w1 w2 s1 s2 statA sv1 sv2 statO)
∨
react Δ' w1 w2 s1 s2 statA sv1 sv2 statO)"
theorem distrib_unwind_lrsecure:
assumes m: "0 < m" and nxt: "⋀i. i < (m::nat) ⟹ nxt i ⊆ {0..<m}"
and init: "initCond (Δs 0)"
and step: "⋀i. i < m ⟹
unwindIntoCond (Δs i) (λw w1 w2 s1 s2 statA sv1 sv2 statO.
∃j ∈ nxt i. Δs j w w1 w2 s1 s2 statA sv1 sv2 statO)"
shows lrsecure
proof-
define Δ where D: "Δ ≡ λw w1 w2 s1 s2 statA sv1 sv2 statO. ∃i < m. Δs i w w1 w2 s1 s2 statA sv1 sv2 statO"
have i: "initCond Δ" using init m unfolding initCond_def D by meson
have c: "unwindCond Δ" unfolding unwindCond_def apply(intro allI impI allI)
apply(subst (asm) D) apply (elim exE conjE)
subgoal for w w1 w2 s1 s2 statA sv1 sv2 statO i
apply(frule step) unfolding unwindIntoCond_def
apply(erule allE[of _ w]) apply(erule allE[of _ w1]) apply(erule allE[of _ w2])
apply(erule allE[of _ s1]) apply(erule allE[of _ s2]) apply(erule allE[of _ statA])
apply(erule allE[of _ sv1]) apply(erule allE[of _ sv2]) apply(erule allE[of _ statO])
apply simp apply(elim conjE)
apply(erule disjE)
subgoal apply(rule disjI1)
subgoal apply(elim exE conjE) subgoal for v
apply(rule exI[of _ v], simp)
apply(rule proact_mono[of "λw w1 w2 s1 s2 statA sv1 sv2 statO. ∃j∈nxt i. Δs j w w1 w2 s1 s2 statA sv1 sv2 statO"])
subgoal unfolding le_fun_def D by simp (meson atLeastLessThan_iff nxt subsetD)
subgoal . . . .
subgoal apply(rule disjI2)
apply(rule match_mono[of "λw w1 w2 s1 s2 statA sv1 sv2 statO. ∃j∈nxt i. Δs j w w1 w2 s1 s2 statA sv1 sv2 statO"])
subgoal unfolding le_fun_def D by simp (meson atLeastLessThan_iff nxt subsetD)
subgoal . . . .
show ?thesis using unwind_lrsecure[OF i c] .
qed
lemma unwindIntoCond_simpleI:
assumes
"⋀w w1 w2 s1 s2 statA sv1 sv2 statO.
reachO s1 ⟹ reachO s2 ⟹ reachV sv1 ⟹ reachV sv2 ⟹
Δ w w1 w2 s1 s2 statA sv1 sv2 statO
⟹
(finalO s1 ⟷ finalO s2) ∧ (finalV sv1 ⟷ finalO s1) ∧ (finalV sv2 ⟷ finalO s2)"
and
"⋀w w1 w2 s1 s2 statA sv1 sv2 statO.
reachO s1 ⟹ reachO s2 ⟹ reachV sv1 ⟹ reachV sv2 ⟹
Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹
statA = Eq
⟹
isIntO s1 ⟷ isIntO s2"
"⋀w w1 w2 s1 s2 statA sv1 sv2 statO.
reachO s1 ⟹ reachO s2 ⟹ reachV sv1 ⟹ reachV sv2 ⟹
Δ w w1 w2 s1 s2 statA sv1 sv2 statO
⟹
react Δ' w1 w2 s1 s2 statA sv1 sv2 statO"
shows "unwindIntoCond Δ Δ'"
using assms unfolding unwindIntoCond_def by auto
lemma unwindIntoCond_simpleI2:
assumes
"⋀w w1 w2 s1 s2 statA sv1 sv2 statO.
reachO s1 ⟹ reachO s2 ⟹ reachV sv1 ⟹ reachV sv2 ⟹
Δ w w1 w2 s1 s2 statA sv1 sv2 statO
⟹
(finalO s1 ⟷ finalO s2) ∧ (finalV sv1 ⟷ finalO s1) ∧ (finalV sv2 ⟷ finalO s2)"
and
"⋀w w1 w2 s1 s2 statA sv1 sv2 statO.
reachO s1 ⟹ reachO s2 ⟹ reachV sv1 ⟹ reachV sv2 ⟹
Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹
statA = Eq
⟹
isIntO s1 ⟷ isIntO s2"
and
"⋀w w1 w2 s1 s2 statA sv1 sv2 statO.
reachO s1 ⟹ reachO s2 ⟹ reachV sv1 ⟹ reachV sv2 ⟹
Δ w w1 w2 s1 s2 statA sv1 sv2 statO
⟹
(∃v<w. proact Δ' v w1 w2 s1 s2 statA sv1 sv2 statO)"
shows "unwindIntoCond Δ Δ'"
using assms unfolding unwindIntoCond_def by auto
lemma unwindIntoCond_simpleIB:
assumes
"⋀w w1 w2 s1 s2 statA sv1 sv2 statO.
reachO s1 ⟹ reachO s2 ⟹ reachV sv1 ⟹ reachV sv2 ⟹
Δ w w1 w2 s1 s2 statA sv1 sv2 statO
⟹
(finalO s1 ⟷ finalO s2) ∧ (finalV sv1 ⟷ finalO s1) ∧ (finalV sv2 ⟷ finalO s2)"
and
"⋀w w1 w2 s1 s2 statA sv1 sv2 statO.
reachO s1 ⟹ reachO s2 ⟹ reachV sv1 ⟹ reachV sv2 ⟹
Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹
statA = Eq
⟹
isIntO s1 ⟷ isIntO s2"
and
"⋀w w1 w2 s1 s2 statA sv1 sv2 statO.
reachO s1 ⟹ reachO s2 ⟹ reachV sv1 ⟹ reachV sv2 ⟹
Δ w w1 w2 s1 s2 statA sv1 sv2 statO
⟹
(∃v<w. proact Δ' v w1 w2 s1 s2 statA sv1 sv2 statO) ∨ react Δ' w1 w2 s1 s2 statA sv1 sv2 statO"
shows "unwindIntoCond Δ Δ'"
using assms unfolding unwindIntoCond_def by auto
definition oor where
"oor Δ Δ⇩2 ≡ λw w1 w2 s1 s2 statA sv1 sv2 statO.
Δ w w1 w2 s1 s2 statA sv1 sv2 statO ∨ Δ⇩2 w w1 w2 s1 s2 statA sv1 sv2 statO"
lemma oorI1:
"Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ oor Δ Δ⇩2 w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding oor_def by simp
lemma oorI2:
"Δ⇩2 w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ oor Δ Δ⇩2 w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding oor_def by simp
definition oor3 where
"oor3 Δ Δ⇩2 Δ⇩3 ≡ λw w1 w2 s1 s2 statA sv1 sv2 statO.
Δ w w1 w2 s1 s2 statA sv1 sv2 statO ∨ Δ⇩2 w w1 w2 s1 s2 statA sv1 sv2 statO ∨
Δ⇩3 w w1 w2 s1 s2 statA sv1 sv2 statO"
lemma oor3I1:
"Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ oor3 Δ Δ⇩2 Δ⇩3 w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding oor3_def by simp
lemma oor3I2:
"Δ⇩2 w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ oor3 Δ Δ⇩2 Δ⇩3 w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding oor3_def by simp
lemma oor3I3:
"Δ⇩3 w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ oor3 Δ Δ⇩2 Δ⇩3 w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding oor3_def by simp
definition oor4 where
"oor4 Δ Δ⇩2 Δ⇩3 Δ⇩4 ≡ λw w1 w2 s1 s2 statA sv1 sv2 statO.
Δ w w1 w2 s1 s2 statA sv1 sv2 statO ∨ Δ⇩2 w w1 w2 s1 s2 statA sv1 sv2 statO ∨
Δ⇩3 w w1 w2 s1 s2 statA sv1 sv2 statO ∨ Δ⇩4 w w1 w2 s1 s2 statA sv1 sv2 statO"
lemma oor4I1:
"Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ oor4 Δ Δ⇩2 Δ⇩3 Δ⇩4 w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding oor4_def by simp
lemma oor4I2:
"Δ⇩2 w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ oor4 Δ Δ⇩2 Δ⇩3 Δ⇩4 w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding oor4_def by simp
lemma oor4I3:
"Δ⇩3 w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ oor4 Δ Δ⇩2 Δ⇩3 Δ⇩4 w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding oor4_def by simp
lemma oor4I4:
"Δ⇩4 w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ oor4 Δ Δ⇩2 Δ⇩3 Δ⇩4 w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding oor4_def by simp
definition oor5 where
"oor5 Δ Δ⇩2 Δ⇩3 Δ⇩4 Δ⇩5 ≡ λw w1 w2 s1 s2 statA sv1 sv2 statO.
Δ w w1 w2 s1 s2 statA sv1 sv2 statO ∨ Δ⇩2 w w1 w2 s1 s2 statA sv1 sv2 statO ∨
Δ⇩3 w w1 w2 s1 s2 statA sv1 sv2 statO ∨ Δ⇩4 w w1 w2 s1 s2 statA sv1 sv2 statO ∨
Δ⇩5 w w1 w2 s1 s2 statA sv1 sv2 statO"
lemma oor5I1:
"Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ oor5 Δ Δ⇩2 Δ⇩3 Δ⇩4 Δ⇩5 w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding oor5_def by simp
lemma oor5I2:
"Δ⇩2 w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ oor5 Δ Δ⇩2 Δ⇩3 Δ⇩4 Δ⇩5 w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding oor5_def by simp
lemma oor5I3:
"Δ⇩3 w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ oor5 Δ Δ⇩2 Δ⇩3 Δ⇩4 Δ⇩5 w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding oor5_def by simp
lemma oor5I4:
"Δ⇩4 w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ oor5 Δ Δ⇩2 Δ⇩3 Δ⇩4 Δ⇩5 w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding oor5_def by simp
lemma oor5I5:
"Δ⇩5 w w1 w2 s1 s2 statA sv1 sv2 statO ⟹ oor5 Δ Δ⇩2 Δ⇩3 Δ⇩4 Δ⇩5 w w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding oor5_def by simp
lemma isIntO_match1: "isIntO s1 ⟹ match1 Δ w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding match1_def by auto
lemma isIntO_match2: "isIntO s2 ⟹ match2 Δ w1 w2 s1 s2 statA sv1 sv2 statO"
unfolding match2_def by auto
lemma isIntO_match:
assumes ‹isIntO s1› and ‹isIntO s2›
and ‹match12 Δ w1 w2 s1 s2 statA sv1 sv2 statO›
shows ‹react Δ w1 w2 s1 s2 statA sv1 sv2 statO›
unfolding react_def apply (intro conjI)
subgoal
using assms(1) by (rule isIntO_match1)
subgoal
using assms(2) by (rule isIntO_match2)
subgoal
using assms(3) by assumption
.
lemma match1_1_oorI1:
"match1_1 Δ w1 w2 s1 s1' s2 statA sv1 sv2 statO ⟹
match1_1 (oor Δ Δ⇩2) w1 w2 s1 s1' s2 statA sv1 sv2 statO"
apply(rule match1_1_mono) unfolding le_fun_def oor_def by auto
lemma match1_1_oorI2:
"match1_1 Δ⇩2 w1 w2 s1 s1' s2 statA sv1 sv2 statO ⟹
match1_1 (oor Δ Δ⇩2) w1 w2 s1 s1' s2 statA sv1 sv2 statO"
apply(rule match1_1_mono) unfolding le_fun_def oor_def by auto
lemma match1_oorI1:
"match1 Δ w1 w2 s1 s2 statA sv1 sv2 statO ⟹
match1 (oor Δ Δ⇩2) w1 w2 s1 s2 statA sv1 sv2 statO"
apply(rule match1_mono) unfolding le_fun_def oor_def by auto
lemma match1_oorI2:
"match1 Δ⇩2 w1 w2 s1 s2 statA sv1 sv2 statO ⟹
match1 (oor Δ Δ⇩2) w1 w2 s1 s2 statA sv1 sv2 statO"
apply(rule match1_mono) unfolding le_fun_def oor_def by auto
lemma match2_1_oorI1:
"match2_1 Δ w1 w2 s1 s2 s2' statA sv1 sv2 statO ⟹
match2_1 (oor Δ Δ⇩2) w1 w2 s1 s2 s2' statA sv1 sv2 statO"
apply(rule match2_1_mono) unfolding le_fun_def oor_def by auto
lemma match2_1_oorI2:
"match2_1 Δ⇩2 w1 w2 s1 s2 s2' statA sv1 sv2 statO ⟹
match2_1 (oor Δ Δ⇩2) w1 w2 s1 s2 s2' statA sv1 sv2 statO"
apply(rule match2_1_mono) unfolding le_fun_def oor_def by auto
lemma match2_oorI1:
"match2 Δ w1 w2 s1 s2 statA sv1 sv2 statO ⟹
match2 (oor Δ Δ⇩2) w1 w2 s1 s2 statA sv1 sv2 statO"
apply(rule match2_mono) unfolding le_fun_def oor_def by auto
lemma match2_oorI2:
"match2 Δ⇩2 w1 w2 s1 s2 statA sv1 sv2 statO ⟹
match2 (oor Δ Δ⇩2) w1 w2 s1 s2 statA sv1 sv2 statO"
apply(rule match2_mono) unfolding le_fun_def oor_def by auto
lemma match12_oorI1:
"match12 Δ w1 w2 s1 s2 statA sv1 sv2 statO ⟹
match12 (oor Δ Δ⇩2) w1 w2 s1 s2 statA sv1 sv2 statO"
apply(rule match12_mono) unfolding le_fun_def oor_def by auto
lemma match12_oorI2:
"match12 Δ⇩2 w1 w2 s1 s2 statA sv1 sv2 statO ⟹
match12 (oor Δ Δ⇩2) w1 w2 s1 s2 statA sv1 sv2 statO"
apply(rule match12_mono) unfolding le_fun_def oor_def by auto
lemma match12_1_oorI1:
"match12_1 Δ w1 w2 s1' s2' statA' sv1 sv2 statO ⟹
match12_1 (oor Δ Δ⇩2) w1 w2 s1' s2' statA' sv1 sv2 statO"
apply(rule match12_1_mono) unfolding le_fun_def oor_def by auto
lemma match12_1_oorI2:
"match12_1 Δ⇩2 w1 w2 s1' s2' statA' sv1 sv2 statO ⟹
match12_1 (oor Δ Δ⇩2) w1 w2 s1' s2' statA' sv1 sv2 statO"
apply(rule match12_1_mono) unfolding le_fun_def oor_def by auto
lemma match12_2_oorI1:
"match12_2 Δ w1 w2 s2 s2' statA' sv1 sv2 statO ⟹
match12_2 (oor Δ Δ⇩2) w1 w2 s2 s2' statA' sv1 sv2 statO"
apply(rule match12_2_mono) unfolding le_fun_def oor_def by auto
lemma match12_2_oorI2:
"match12_2 Δ⇩2 w1 w2 s2 s2' statA' sv1 sv2 statO ⟹
match12_2 (oor Δ Δ⇩2) w1 w2 s2 s2' statA' sv1 sv2 statO"
apply(rule match12_2_mono) unfolding le_fun_def oor_def by auto
lemma match12_12_oorI1:
"match12_12 Δ w1 w2 s1' s2' statA' sv1 sv2 statO ⟹
match12_12 (oor Δ Δ⇩2) w1 w2 s1' s2' statA' sv1 sv2 statO"
apply(rule match12_12_mono) unfolding le_fun_def oor_def by auto
lemma match12_12_oorI2:
"match12_12 Δ⇩2 w1 w2 s1' s2' statA' sv1 sv2 statO ⟹
match12_12 (oor Δ Δ⇩2) w1 w2 s1' s2' statA' sv1 sv2 statO"
apply(rule match12_12_mono) unfolding le_fun_def oor_def by auto
lemma match_oorI1:
"react Δ w1 w2 s1 s2 statA sv1 sv2 statO ⟹
react (oor Δ Δ⇩2) w1 w2 s1 s2 statA sv1 sv2 statO"
apply(rule match_mono) unfolding le_fun_def oor_def by auto
lemma match_oorI2:
"react Δ⇩2 w1 w2 s1 s2 statA sv1 sv2 statO ⟹
react (oor Δ Δ⇩2) w1 w2 s1 s2 statA sv1 sv2 statO"
apply(rule match_mono) unfolding le_fun_def oor_def by auto
lemma proact_oorI1:
"proact Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹
proact (oor Δ Δ⇩2) w w1 w2 s1 s2 statA sv1 sv2 statO"
apply(rule proact_mono) unfolding le_fun_def oor_def by auto
lemma proact_oorI2:
"proact Δ⇩2 w w1 w2 s1 s2 statA sv1 sv2 statO ⟹
proact (oor Δ Δ⇩2) w w1 w2 s1 s2 statA sv1 sv2 statO"
apply(rule proact_mono) unfolding le_fun_def oor_def by auto
lemma move_1_oorI1:
"move_1 Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹
move_1 (oor Δ Δ⇩2) w w1 w2 s1 s2 statA sv1 sv2 statO"
apply(rule move_1_mono) unfolding le_fun_def oor_def by auto
lemma move_1_oorI2:
"move_1 Δ⇩2 w w1 w2 s1 s2 statA sv1 sv2 statO ⟹
move_1 (oor Δ Δ⇩2) w w1 w2 s1 s2 statA sv1 sv2 statO"
apply(rule move_1_mono) unfolding le_fun_def oor_def by auto
lemma move_2_oorI1:
"move_2 Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹
move_2 (oor Δ Δ⇩2) w w1 w2 s1 s2 statA sv1 sv2 statO"
apply(rule move_2_mono) unfolding le_fun_def oor_def by auto
lemma move_2_oorI2:
"move_2 Δ⇩2 w w1 w2 s1 s2 statA sv1 sv2 statO ⟹
move_2 (oor Δ Δ⇩2) w w1 w2 s1 s2 statA sv1 sv2 statO"
apply(rule move_2_mono) unfolding le_fun_def oor_def by auto
lemma move_12_oorI1:
"move_12 Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟹
move_12 (oor Δ Δ⇩2) w w1 w2 s1 s2 statA sv1 sv2 statO"
apply(rule move_12_mono) unfolding le_fun_def oor_def by auto
lemma move_12_oorI2:
"move_12 Δ⇩2 w w1 w2 s1 s2 statA sv1 sv2 statO ⟹
move_12 (oor Δ Δ⇩2) w w1 w2 s1 s2 statA sv1 sv2 statO"
apply(rule move_12_mono) unfolding le_fun_def oor_def by auto
end
context Relative_Security_Determ
begin
lemma match1_1_defD: "match1_1 Δ w1 w2 s1 s1' s2 statA sv1 sv2 statO ⟷
¬ finalV sv1 ∧ Δ ∞ w1 w2 s1' s2 statA (nextO sv1) sv2 statO"
unfolding match1_1_def validTrans_iff_next by simp
lemma match1_12_defD: "match1_12 Δ w1 w2 s1 s1' s2 statA sv1 sv2 statO ⟷
¬ finalV sv1 ∧ ¬ finalV sv2 ∧
Δ ∞ w1 w2 s1' s2 statA (nextO sv1) (nextO sv2) (sstatO' statO sv1 sv2)"
unfolding match1_12_def validTrans_iff_next by simp
lemmas match1_defsD = match1_def match1_1_defD match1_12_defD
lemma match2_1_defD: "match2_1 Δ w1 w2 s1 s2 s2' statA sv1 sv2 statO ⟷
¬ finalV sv2 ∧ Δ ∞ w1 w2 s1 s2' statA sv1 (nextO sv2) statO"
unfolding match2_1_def validTrans_iff_next by simp
lemma match2_12_defD: "match2_12 Δ w1 w2 s1 s2 s2' statA sv1 sv2 statO ⟷
¬ finalV sv1 ∧ ¬ finalV sv2 ∧ Δ ∞ w1 w2 s1 s2' statA (nextO sv1) (nextO sv2) (sstatO' statO sv1 sv2)"
unfolding match2_12_def validTrans_iff_next by simp
lemmas match2_defsD = match2_def match2_1_defD match2_12_defD
lemma match12_1_defD: "match12_1 Δ w1 w2 s1' s2' statA' sv1 sv2 statO ⟷
¬ finalV sv1 ∧ Δ ∞ w1 w2 s1' s2' statA' (nextO sv1) sv2 statO"
unfolding match12_1_def validTrans_iff_next by simp
lemma match12_2_defD: "match12_2 Δ w1 w2 s1' s2' statA' sv1 sv2 statO ⟷
¬ finalV sv2 ∧ Δ ∞ w1 w2 s1' s2' statA' sv1 (nextO sv2) statO"
unfolding match12_2_def validTrans_iff_next by simp
lemma match12_12_defD: "match12_12 Δ w1 w2 s1' s2' statA' sv1 sv2 statO ⟷
(let statO' = sstatO' statO sv1 sv2 in
¬ finalV sv1 ∧ ¬ finalV sv2 ∧
(statA' = Diff ⟶ statO' = Diff) ∧
Δ ∞ w1 w2 s1' s2' statA' (nextO sv1) (nextO sv2) statO')"
unfolding match12_12_def validTrans_iff_next by simp
lemmas match12_defsD = match12_def match12_1_defD match12_2_defD match12_12_defD
lemmas match_deep_defsD = match1_defsD match2_defsD match12_defsD
lemma move_1_defD: "move_1 Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟷
¬ finalV sv1 ∧ Δ w w1 w2 s1 s2 statA (nextO sv1) sv2 statO"
unfolding move_1_def validTrans_iff_next by simp
lemma move_2_defD: "move_2 Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟷
¬ finalV sv2 ∧ Δ w w1 w2 s1 s2 statA sv1 (nextO sv2) statO"
unfolding move_2_def validTrans_iff_next by simp
lemma move_12_defD: "move_12 Δ w w1 w2 s1 s2 statA sv1 sv2 statO ⟷
(let statO' = sstatO' statO sv1 sv2 in
¬ finalV sv1 ∧ ¬ finalV sv2 ∧
Δ w w1 w2 s1 s2 statA (nextO sv1) (nextO sv2) statO')"
unfolding move_12_def validTrans_iff_next by simp
lemmas proact_defsD = proact_def move_1_defD move_2_defD move_12_defD
end
end