Theory Fixrec
section "Package for defining recursive functions in HOLCF"
theory Fixrec
imports Cprod Sprod Ssum Up One Tr Fix
keywords "fixrec" :: thy_defn
begin
subsection ‹Pattern-match monad›
default_sort cpo
pcpodef 'a match = "UNIV::(one ++ 'a u) set"
by simp_all
definition
fail :: "'a match" where
"fail = Abs_match (sinl⋅ONE)"
definition
succeed :: "'a → 'a match" where
"succeed = (Λ x. Abs_match (sinr⋅(up⋅x)))"
lemma matchE [case_names bottom fail succeed, cases type: match]:
"⟦p = ⊥ ⟹ Q; p = fail ⟹ Q; ⋀x. p = succeed⋅x ⟹ Q⟧ ⟹ Q"
unfolding fail_def succeed_def
apply (cases p, rename_tac r)
apply (rule_tac p=r in ssumE, simp add: Abs_match_strict)
apply (rule_tac p=x in oneE, simp, simp)
apply (rule_tac p=y in upE, simp, simp add: cont_Abs_match)
done
lemma succeed_defined [simp]: "succeed⋅x ≠ ⊥"
by (simp add: succeed_def cont_Abs_match Abs_match_bottom_iff)
lemma fail_defined [simp]: "fail ≠ ⊥"
by (simp add: fail_def Abs_match_bottom_iff)
lemma succeed_eq [simp]: "(succeed⋅x = succeed⋅y) = (x = y)"
by (simp add: succeed_def cont_Abs_match Abs_match_inject)
lemma succeed_neq_fail [simp]:
"succeed⋅x ≠ fail" "fail ≠ succeed⋅x"
by (simp_all add: succeed_def fail_def cont_Abs_match Abs_match_inject)
subsubsection ‹Run operator›
definition
run :: "'a match → 'a::pcpo" where
"run = (Λ m. sscase⋅⊥⋅(fup⋅ID)⋅(Rep_match m))"
text ‹rewrite rules for run›
lemma run_strict [simp]: "run⋅⊥ = ⊥"
unfolding run_def
by (simp add: cont_Rep_match Rep_match_strict)
lemma run_fail [simp]: "run⋅fail = ⊥"
unfolding run_def fail_def
by (simp add: cont_Rep_match Abs_match_inverse)
lemma run_succeed [simp]: "run⋅(succeed⋅x) = x"
unfolding run_def succeed_def
by (simp add: cont_Rep_match cont_Abs_match Abs_match_inverse)
subsubsection ‹Monad plus operator›
definition
mplus :: "'a match → 'a match → 'a match" where
"mplus = (Λ m1 m2. sscase⋅(Λ _. m2)⋅(Λ _. m1)⋅(Rep_match m1))"
abbreviation
mplus_syn :: "['a match, 'a match] ⇒ 'a match" (infixr ‹+++› 65) where
"m1 +++ m2 == mplus⋅m1⋅m2"
text ‹rewrite rules for mplus›
lemma mplus_strict [simp]: "⊥ +++ m = ⊥"
unfolding mplus_def
by (simp add: cont_Rep_match Rep_match_strict)
lemma mplus_fail [simp]: "fail +++ m = m"
unfolding mplus_def fail_def
by (simp add: cont_Rep_match Abs_match_inverse)
lemma mplus_succeed [simp]: "succeed⋅x +++ m = succeed⋅x"
unfolding mplus_def succeed_def
by (simp add: cont_Rep_match cont_Abs_match Abs_match_inverse)
lemma mplus_fail2 [simp]: "m +++ fail = m"
by (cases m, simp_all)
lemma mplus_assoc: "(x +++ y) +++ z = x +++ (y +++ z)"
by (cases x, simp_all)
subsection ‹Match functions for built-in types›
default_sort pcpo
definition
match_bottom :: "'a → 'c match → 'c match"
where
"match_bottom = (Λ x k. seq⋅x⋅fail)"
definition
match_Pair :: "'a::cpo × 'b::cpo → ('a → 'b → 'c match) → 'c match"
where
"match_Pair = (Λ x k. csplit⋅k⋅x)"
definition
match_spair :: "'a ⊗ 'b → ('a → 'b → 'c match) → 'c match"
where
"match_spair = (Λ x k. ssplit⋅k⋅x)"
definition
match_sinl :: "'a ⊕ 'b → ('a → 'c match) → 'c match"
where
"match_sinl = (Λ x k. sscase⋅k⋅(Λ b. fail)⋅x)"
definition
match_sinr :: "'a ⊕ 'b → ('b → 'c match) → 'c match"
where
"match_sinr = (Λ x k. sscase⋅(Λ a. fail)⋅k⋅x)"
definition
match_up :: "'a::cpo u → ('a → 'c match) → 'c match"
where
"match_up = (Λ x k. fup⋅k⋅x)"
definition
match_ONE :: "one → 'c match → 'c match"
where
"match_ONE = (Λ ONE k. k)"
definition
match_TT :: "tr → 'c match → 'c match"
where
"match_TT = (Λ x k. If x then k else fail)"
definition
match_FF :: "tr → 'c match → 'c match"
where
"match_FF = (Λ x k. If x then fail else k)"
lemma match_bottom_simps [simp]:
"match_bottom⋅x⋅k = (if x = ⊥ then ⊥ else fail)"
by (simp add: match_bottom_def)
lemma match_Pair_simps [simp]:
"match_Pair⋅(x, y)⋅k = k⋅x⋅y"
by (simp_all add: match_Pair_def)
lemma match_spair_simps [simp]:
"⟦x ≠ ⊥; y ≠ ⊥⟧ ⟹ match_spair⋅(:x, y:)⋅k = k⋅x⋅y"
"match_spair⋅⊥⋅k = ⊥"
by (simp_all add: match_spair_def)
lemma match_sinl_simps [simp]:
"x ≠ ⊥ ⟹ match_sinl⋅(sinl⋅x)⋅k = k⋅x"
"y ≠ ⊥ ⟹ match_sinl⋅(sinr⋅y)⋅k = fail"
"match_sinl⋅⊥⋅k = ⊥"
by (simp_all add: match_sinl_def)
lemma match_sinr_simps [simp]:
"x ≠ ⊥ ⟹ match_sinr⋅(sinl⋅x)⋅k = fail"
"y ≠ ⊥ ⟹ match_sinr⋅(sinr⋅y)⋅k = k⋅y"
"match_sinr⋅⊥⋅k = ⊥"
by (simp_all add: match_sinr_def)
lemma match_up_simps [simp]:
"match_up⋅(up⋅x)⋅k = k⋅x"
"match_up⋅⊥⋅k = ⊥"
by (simp_all add: match_up_def)
lemma match_ONE_simps [simp]:
"match_ONE⋅ONE⋅k = k"
"match_ONE⋅⊥⋅k = ⊥"
by (simp_all add: match_ONE_def)
lemma match_TT_simps [simp]:
"match_TT⋅TT⋅k = k"
"match_TT⋅FF⋅k = fail"
"match_TT⋅⊥⋅k = ⊥"
by (simp_all add: match_TT_def)
lemma match_FF_simps [simp]:
"match_FF⋅FF⋅k = k"
"match_FF⋅TT⋅k = fail"
"match_FF⋅⊥⋅k = ⊥"
by (simp_all add: match_FF_def)
subsection ‹Mutual recursion›
text ‹
The following rules are used to prove unfolding theorems from
fixed-point definitions of mutually recursive functions.
›
lemma Pair_equalI: "⟦x ≡ fst p; y ≡ snd p⟧ ⟹ (x, y) ≡ p"
by simp
lemma Pair_eqD1: "(x, y) = (x', y') ⟹ x = x'"
by simp
lemma Pair_eqD2: "(x, y) = (x', y') ⟹ y = y'"
by simp
lemma def_cont_fix_eq:
"⟦f ≡ fix⋅(Abs_cfun F); cont F⟧ ⟹ f = F f"
by (simp, subst fix_eq, simp)
lemma def_cont_fix_ind:
"⟦f ≡ fix⋅(Abs_cfun F); cont F; adm P; P ⊥; ⋀x. P x ⟹ P (F x)⟧ ⟹ P f"
by (simp add: fix_ind)
text ‹lemma for proving rewrite rules›
lemma ssubst_lhs: "⟦t = s; P s = Q⟧ ⟹ P t = Q"
by simp
subsection ‹Initializing the fixrec package›
ML_file ‹Tools/holcf_library.ML›
ML_file ‹Tools/fixrec.ML›
method_setup fixrec_simp = ‹
Scan.succeed (SIMPLE_METHOD' o Fixrec.fixrec_simp_tac)
› "pattern prover for fixrec constants"
setup ‹
Fixrec.add_matchers
[ (\<^const_name>‹up›, \<^const_name>‹match_up›),
(\<^const_name>‹sinl›, \<^const_name>‹match_sinl›),
(\<^const_name>‹sinr›, \<^const_name>‹match_sinr›),
(\<^const_name>‹spair›, \<^const_name>‹match_spair›),
(\<^const_name>‹Pair›, \<^const_name>‹match_Pair›),
(\<^const_name>‹ONE›, \<^const_name>‹match_ONE›),
(\<^const_name>‹TT›, \<^const_name>‹match_TT›),
(\<^const_name>‹FF›, \<^const_name>‹match_FF›),
(\<^const_name>‹bottom›, \<^const_name>‹match_bottom›) ]
›
hide_const (open) succeed fail run
end