Theory Abstract-Rewriting.SN_Order_Carrier
section ‹Carriers of Strongly Normalizing Orders›
theory SN_Order_Carrier
imports
SN_Orders
HOL.Rat
begin
text ‹
This theory shows that standard semirings can be used
in combination with polynomials, e.g. the naturals, integers,
and arbitrary Archemedean fields by using delta-orders.
It also contains the arctic integers and arctic delta-orders where
0 is -infty, 1 is zero, + is max and * is plus.
›
subsection ‹The standard semiring over the naturals›
instantiation nat :: large_ordered_semiring_1
begin
instance by (intro_classes, auto)
end
definition nat_mono :: "nat ⇒ bool" where "nat_mono x ≡ x ≠ 0"
interpretation nat_SN: SN_strict_mono_ordered_semiring_1 1 "(>) :: nat ⇒ nat ⇒ bool" nat_mono
by (unfold_locales, insert SN_nat_gt, auto simp: nat_mono_def)
interpretation nat_poly: poly_order_carrier 1 "(>) :: nat ⇒ nat ⇒ bool" True discrete
proof (unfold_locales)
fix x y :: nat
assume ge: "x ≥ y"
obtain k where k: "x - y = k" by auto
show "∃ k. x = ((+) 1 ^^ k) y"
proof (rule exI[of _ k])
from ge k have "x = k + y" by simp
also have "… = ((+) 1 ^^ k) y"
by (induct k, auto)
finally show "x = ((+) 1 ^^ k) y" .
qed
qed (auto simp: field_simps power_strict_mono)
subsection ‹The standard semiring over the Archimedean fields using delta-orderings›
definition delta_gt :: "'a :: floor_ceiling ⇒ 'a ⇒ 'a ⇒ bool" where
"delta_gt δ ≡ (λ x y. x - y ≥ δ)"
lemma non_inf_delta_gt: assumes delta: "δ > 0"
shows "non_inf {(a,b) . delta_gt δ a b}" (is "non_inf ?r")
proof
let ?gt = "delta_gt δ"
fix a :: 'a and f
assume "⋀ i. (f i, f (Suc i)) ∈ ?r"
hence gt: "⋀ i. ?gt (f i) (f (Suc i))" by simp
{
fix i
have "f i ≤ f 0 - δ * of_nat i"
proof (induct i)
case (Suc i)
thus ?case using gt[of i, unfolded delta_gt_def] by (auto simp: field_simps)
qed simp
} note fi = this
{
fix r :: 'a
have "of_nat (nat (ceiling r)) ≥ r"
by (metis ceiling_le_zero le_of_int_ceiling less_le_not_le nat_0_iff not_less of_nat_0 of_nat_nat)
} note ceil_elim = this
define i where "i = nat (ceiling ((f 0 - a) / δ))"
from fi[of i] have "f i - f 0 ≤ - δ * of_nat (nat (ceiling ((f 0 - a) / δ)))" unfolding i_def by simp
also have "… ≤ - δ * ((f 0 - a) / δ)" using ceil_elim[of "(f 0 - a) / δ"] delta
by (metis le_imp_neg_le minus_mult_commute mult_le_cancel_left_pos)
also have "… = - f 0 + a" using delta by auto
also have "… < - f 0 + a + δ" using delta by auto
finally have "¬ ?gt (f i) a" unfolding delta_gt_def by arith
thus "∃ i. (f i, a) ∉ ?r" by blast
qed
lemma delta_gt_SN: assumes dpos: "δ > 0" shows "SN {(x,y). 0 ≤ y ∧ delta_gt δ x y}"
proof -
from non_inf_imp_SN_bound[OF non_inf_delta_gt[OF dpos], of "- δ"]
show ?thesis unfolding delta_gt_def by auto
qed
definition delta_mono :: "'a :: floor_ceiling ⇒ bool" where "delta_mono x ≡ x ≥ 1"
subclass (in floor_ceiling) large_ordered_semiring_1
proof
fix x :: 'a
from ex_le_of_int[of x] obtain z where x: "x ≤ of_int z" by auto
have "z ≤ int (nat z)" by auto
with x have "x ≤ of_int (int (nat z))"
by (metis (full_types) le_cases of_int_0_le_iff of_int_of_nat_eq of_nat_0_le_iff of_nat_nat order_trans)
also have "… = of_nat (nat z)" unfolding of_int_of_nat_eq ..
finally
show "∃ y. x ≤ of_nat y" by blast
qed (auto simp: mult_right_mono mult_left_mono mult_right_mono_neg max_def)
lemma delta_interpretation: assumes dpos: "δ > 0" and default: "δ ≤ def"
shows "SN_strict_mono_ordered_semiring_1 def (delta_gt δ) delta_mono"
proof -
from dpos default have defz: "0 ≤ def" by auto
show ?thesis
proof (unfold_locales)
show "SN {(x,y). y ≥ 0 ∧ delta_gt δ x y}" by (rule delta_gt_SN[OF dpos])
next
fix x y z :: 'a
assume "delta_mono x" and yz: "delta_gt δ y z"
hence x: "1 ≤ x" unfolding delta_mono_def by simp
have "∃ d > 0. delta_gt δ = (λ x y. d ≤ x - y)"
by (rule exI[of _ δ], auto simp: dpos delta_gt_def)
from this obtain d where d: "0 < d" and rat: "delta_gt δ = (λ x y. d ≤ x - y)" by auto
from yz have yzd: "d ≤ y - z" by (simp add: rat)
show "delta_gt δ (x * y) (x * z)"
proof (simp only: rat)
let ?p = "(x - 1) * (y - z)"
from x have x1: "0 ≤ x - 1" by auto
from yzd d have yz0: "0 ≤ y - z" by auto
have "0 ≤ ?p"
by (rule mult_nonneg_nonneg[OF x1 yz0])
have "x * y - x * z = x * (y - z)" using right_diff_distrib[of x y z] by auto
also have "… = ((x - 1) + 1) * (y - z)" by auto
also have "… = ?p + 1 * ( y - z)" by (rule ring_distribs(2))
also have "… = ?p + (y - z)" by simp
also have "… ≥ (0 + d)" using yzd ‹0 ≤ ?p› by auto
finally
show "d ≤ x * y - x * z" by auto
qed
qed (insert dpos, auto simp: delta_gt_def default defz)
qed
lemma delta_poly: assumes dpos: "δ > 0" and default: "δ ≤ def"
shows "poly_order_carrier def (delta_gt δ) (1 ≤ δ) False"
proof -
from delta_interpretation[OF dpos default]
interpret SN_strict_mono_ordered_semiring_1 "def" "delta_gt δ" delta_mono .
interpret poly_order_carrier "def" "delta_gt δ" False False
proof(unfold_locales)
fix y z x :: 'a
assume gt: "delta_gt δ y z" and ge: "x ≥ 1"
from ge have ge: "x ≥ 0" and m: "delta_mono x" unfolding delta_mono_def by auto
show "delta_gt δ (y * x) (z * x)"
using mono[OF m gt ge] by (auto simp: field_simps)
next
fix x y :: 'a and n :: nat
assume False thus "delta_gt δ (x ^ n) (y ^ n)" ..
next
fix x y :: 'a
assume False
thus "∃ k. x = ((+) 1 ^^ k) y" by simp
qed
show ?thesis
proof(unfold_locales)
fix x y :: 'a and n :: nat
assume one: "1 ≤ δ" and gt: "delta_gt δ x y" and y: "y ≥ 0" and n: "1 ≤ n"
then obtain p where n: "n = Suc p" and x: "x ≥ 1" and y2: "0 ≤ y" and xy: "x ≥ y" by (cases n, auto simp: delta_gt_def)
show "delta_gt δ (x ^ n) (y ^ n)"
proof (simp only: n, induct p, simp add: gt)
case (Suc p)
from times_gt_mono[OF this x]
have one: "delta_gt δ (x ^ Suc (Suc p)) (x * y ^ Suc p)" by (auto simp: field_simps)
also have "… ≥ y * y ^ Suc p"
by (rule times_left_mono[OF _ xy], auto simp: zero_le_power[OF y2, of "Suc p", simplified])
finally show ?case by auto
qed
next
fix x y :: 'a
assume False
thus "∃ k. x = ((+) 1 ^^ k) y" by simp
qed (rule times_gt_mono, auto)
qed
lemma delta_minimal_delta: assumes "⋀ x y. (x,y) ∈ set xys ⟹ x > y"
shows "∃ δ > 0. ∀ x y. (x,y) ∈ set xys ⟶ delta_gt δ x y"
using assms
proof (induct xys)
case Nil
show ?case by (rule exI[of _ 1], auto)
next
case (Cons xy xys)
show ?case
proof (cases xy)
case (Pair x y)
with Cons have "x > y" by auto
then obtain d1 where "d1 = x - y" and d1pos: "d1 > 0" and "d1 ≤ x - y" by auto
hence xy: "delta_gt d1 x y" unfolding delta_gt_def by auto
from Cons obtain d2 where d2pos: "d2 > 0" and xys: "∀ x y. (x, y) ∈ set xys ⟶ delta_gt d2 x y" by auto
obtain d where d: "d = min d1 d2" by auto
with d1pos d2pos xy have dpos: "d > 0" and "delta_gt d x y" unfolding delta_gt_def by auto
with xys d Pair have "∀ x y. (x,y) ∈ set (xy # xys) ⟶ delta_gt d x y" unfolding delta_gt_def by force
with dpos show ?thesis by auto
qed
qed
interpretation weak_delta_SN: weak_SN_strict_mono_ordered_semiring_1 "(>)" 1 delta_mono
proof
fix xysp :: "('a × 'a) list"
assume orient: "∀ x y. (x,y) ∈ set xysp ⟶ x > y"
obtain xys where xsy: "xys = (1,0) # xysp" by auto
with orient have "⋀ x y. (x,y) ∈ set xys ⟹ x > y" by auto
with delta_minimal_delta have "∃ δ > 0. ∀ x y. (x,y) ∈ set xys ⟶ delta_gt δ x y" by auto
then obtain δ where dpos: "δ > 0" and orient: "⋀ x y. (x,y) ∈ set xys ⟹ delta_gt δ x y" by auto
from orient have orient1: "∀ x y. (x,y) ∈ set xysp ⟶ delta_gt δ x y" and orient2: "delta_gt δ 1 0" unfolding xsy by auto
from orient2 have oned: "δ ≤ 1" unfolding delta_gt_def by auto
show " ∃gt. SN_strict_mono_ordered_semiring_1 1 gt delta_mono ∧ (∀x y. (x, y) ∈ set xysp ⟶ gt x y)"
by (intro exI conjI, rule delta_interpretation[OF dpos oned], rule orient1)
qed
subsection ‹The standard semiring over the integers›
definition int_mono :: "int ⇒ bool" where "int_mono x ≡ x ≥ 1"
instantiation int :: large_ordered_semiring_1
begin
instance
proof
fix y :: int
show "∃ x. of_nat x ≥ y"
by (rule exI[of _ "nat y"], simp)
qed (auto simp: mult_right_mono mult_left_mono mult_right_mono_neg)
end
lemma non_inf_int_gt: "non_inf {(a,b :: int) . a > b}" (is "non_inf ?r")
by (rule non_inf_image[OF non_inf_delta_gt, of 1 _ rat_of_int], auto simp: delta_gt_def)
interpretation int_SN: SN_strict_mono_ordered_semiring_1 1 "(>) :: int ⇒ int ⇒ bool" int_mono
proof (unfold_locales)
have [simp]: "⋀ x :: int . (-1 < x) = (0 ≤ x)" by auto
show "SN {(x,y). y ≥ 0 ∧ (y :: int) < x}"
using non_inf_imp_SN_bound[OF non_inf_int_gt, of "-1"] by auto
qed (auto simp: mult_strict_left_mono int_mono_def)
interpretation int_poly: poly_order_carrier 1 "(>) :: int ⇒ int ⇒ bool" True discrete
proof (unfold_locales)
fix x y :: int
assume ge: "x ≥ y"
then obtain k where k: "x - y = k" and kp: "0 ≤ k" by auto
then obtain nk where nk: "nk = nat k" and k: "x - y = int nk" by auto
show "∃ k. x = ((+) 1 ^^ k) y"
proof (rule exI[of _ nk])
from k have "x = int nk + y" by simp
also have "… = ((+) 1 ^^ nk) y"
by (induct nk, auto)
finally show "x = ((+) 1 ^^ nk) y" .
qed
qed (auto simp: field_simps power_strict_mono)
subsection ‹The arctic semiring over the integers›
text ‹plus is interpreted as max, times is interpreted as plus, 0 is -infinity, 1 is 0›
datatype arctic = MinInfty | Num_arc int
instantiation arctic :: ord
begin
fun less_eq_arctic :: "arctic ⇒ arctic ⇒ bool" where
"less_eq_arctic MinInfty x = True"
| "less_eq_arctic (Num_arc _) MinInfty = False"
| "less_eq_arctic (Num_arc y) (Num_arc x) = (y ≤ x)"
fun less_arctic :: "arctic ⇒ arctic ⇒ bool" where
"less_arctic MinInfty x = True"
| "less_arctic (Num_arc _) MinInfty = False"
| "less_arctic (Num_arc y) (Num_arc x) = (y < x)"
instance ..
end
instantiation arctic :: ordered_semiring_1
begin
fun plus_arctic :: "arctic ⇒ arctic ⇒ arctic" where
"plus_arctic MinInfty y = y"
| "plus_arctic x MinInfty = x"
| "plus_arctic (Num_arc x) (Num_arc y) = (Num_arc (max x y))"
fun times_arctic :: "arctic ⇒ arctic ⇒ arctic" where
"times_arctic MinInfty y = MinInfty"
| "times_arctic x MinInfty = MinInfty"
| "times_arctic (Num_arc x) (Num_arc y) = (Num_arc (x + y))"
definition zero_arctic :: arctic where
"zero_arctic = MinInfty"
definition one_arctic :: arctic where
"one_arctic = Num_arc 0"
instance
proof
fix x y z :: arctic
show "x + y = y + x"
by (cases x, cases y, auto, cases y, auto)
show "(x + y) + z = x + (y + z)"
by (cases x, auto, cases y, auto, cases z, auto)
show "(x * y) * z = x * (y * z)"
by (cases x, auto, cases y, auto, cases z, auto)
show "x * 0 = 0"
by (cases x, auto simp: zero_arctic_def)
show "x * (y + z) = x * y + x * z"
by (cases x, auto, cases y, auto, cases z, auto)
show "(x + y) * z = x * z + y * z"
by (cases x, auto, cases y, cases z, auto, cases z, auto)
show "1 * x = x"
by (cases x, simp_all add: one_arctic_def)
show "x * 1 = x"
by (cases x, simp_all add: one_arctic_def)
show "0 + x = x"
by (simp add: zero_arctic_def)
show "0 * x = 0"
by (simp add: zero_arctic_def)
show "(0 :: arctic) ≠ 1"
by (simp add: zero_arctic_def one_arctic_def)
show "x + 0 = x" by (cases x, auto simp: zero_arctic_def)
show "x ≥ x"
by (cases x, auto)
show "(1 :: arctic) ≥ 0"
by (simp add: zero_arctic_def one_arctic_def)
show "max x y = max y x" unfolding max_def
by (cases x, (cases y, auto)+)
show "max x y ≥ x" unfolding max_def
by (cases x, (cases y, auto)+)
assume ge: "x ≥ y"
from ge show "x + z ≥ y + z"
by (cases x, cases y, cases z, auto, cases y, cases z, auto, cases z, auto)
from ge show "x * z ≥ y * z"
by (cases x, cases y, cases z, auto, cases y, cases z, auto, cases z, auto)
from ge show "max x y = x" unfolding max_def
by (cases x, (cases y, auto)+)
from ge show "max z x ≥ max z y" unfolding max_def
by (cases z, cases x, auto, cases x, (cases y, auto)+)
next
fix x y z :: arctic
assume "x ≥ y" and "y ≥ z"
thus "x ≥ z"
by (cases x, cases y, auto, cases y, cases z, auto, cases z, auto)
next
fix x y z :: arctic
assume "y ≥ z"
thus "x * y ≥ x * z"
by (cases x, cases y, cases z, auto, cases y, cases z, auto, cases z, auto)
next
fix x y z :: arctic
show "x ≥ y ⟹ 0 ≥ z ⟹ y * z ≥ x * z"
by (cases z, cases x, auto simp: zero_arctic_def)
qed
end
fun get_arctic_num :: "arctic ⇒ int"
where "get_arctic_num (Num_arc n) = n"
fun pos_arctic :: "arctic ⇒ bool"
where "pos_arctic MinInfty = False"
| "pos_arctic (Num_arc n) = (0 <= n)"
interpretation arctic_SN: SN_both_mono_ordered_semiring_1 1 "(>)" pos_arctic
proof
fix x y z :: arctic
assume "x ≥ y" and "y > z"
thus "x > z"
by (cases z, simp, cases y, simp, cases x, auto)
next
fix x y z :: arctic
assume "x > y" and "y ≥ z"
thus "x > z"
by (cases z, simp, cases y, simp, cases x, auto)
next
fix x y z :: arctic
assume "x > y"
thus "x ≥ y"
by (cases x, (cases y, auto)+)
next
fix x y z u :: arctic
assume "x > y" and "z > u"
thus "x + z > y + u"
by (cases y, cases u, simp, cases z, auto, cases x, auto, cases u, auto, cases z, auto, cases x, auto, cases x, auto, cases z, auto, cases x, auto)
next
fix x y z :: arctic
assume "x > y"
thus "x * z > y * z"
by (cases y, simp, cases z, simp, cases x, auto)
next
fix x :: arctic
assume "0 > x"
thus "x = 0"
by (cases x, auto simp: zero_arctic_def)
next
fix x :: arctic
show "pos_arctic 1" unfolding one_arctic_def by simp
show "x > 0" unfolding zero_arctic_def by simp
show "(1 :: arctic) ≥ 0" unfolding zero_arctic_def by simp
show "x ≥ 0" unfolding zero_arctic_def by simp
show "¬ pos_arctic 0" unfolding zero_arctic_def by simp
next
fix x y
assume "pos_arctic x"
thus "pos_arctic (x + y)" by (cases x, simp, cases y, auto)
next
fix x y
assume "pos_arctic x" and "pos_arctic y"
thus "pos_arctic (x * y)" by (cases x, simp, cases y, auto)
next
show "SN {(x,y). pos_arctic y ∧ x > y}" (is "SN ?rel")
proof - {
fix x
assume "∃ f . f 0 = x ∧ (∀ i. (f i, f (Suc i)) ∈ ?rel)"
from this obtain f where "f 0 = x" and seq: "∀ i. (f i, f (Suc i)) ∈ ?rel" by auto
from seq have steps: "∀ i. f i > f (Suc i) ∧ pos_arctic (f (Suc i)) " by auto
let ?g = "λ i. get_arctic_num (f i)"
have "∀ i. ?g (Suc i) ≥ 0 ∧ ?g i > ?g (Suc i)"
proof
fix i
from steps have i: "f i > f (Suc i) ∧ pos_arctic (f (Suc i))" by auto
from i obtain n where fi: "f i = Num_arc n" by (cases "f (Suc i)", simp, cases "f i", auto)
from i obtain m where fsi: "f (Suc i) = Num_arc m" by (cases "f (Suc i)", auto)
with i have gz: "0 ≤ m" by simp
from i fi fsi have "n > m" by auto
with fi fsi gz
show "?g (Suc i) ≥ 0 ∧ ?g i > ?g (Suc i)" by auto
qed
from this obtain g where "∀ i. g (Suc i) ≥ 0 ∧ ((>) :: int ⇒ int ⇒ bool) (g i) (g (Suc i))" by auto
hence "∃ f. f 0 = g 0 ∧ (∀ i. (f i, f (Suc i)) ∈ {(x,y). y ≥ 0 ∧ x > y})" by auto
with int_SN.SN have False unfolding SN_defs by auto
}
thus ?thesis unfolding SN_defs by auto
qed
next
fix y z x :: arctic
assume "y > z"
thus "x * y > x * z"
by (cases x, simp, cases z, simp, cases y, auto)
next
fix c d
assume "pos_arctic d"
then obtain n where d: "d = Num_arc n" and n: "0 ≤ n"
by (cases d, auto)
show "∃ e. e ≥ 0 ∧ pos_arctic e ∧ ¬ c ≥ d * e"
proof (cases c)
case MinInfty
show ?thesis
by (rule exI[of _ "Num_arc 0"],
unfold d MinInfty zero_arctic_def, simp)
next
case (Num_arc m)
show ?thesis
by (rule exI[of _ "Num_arc (abs m + 1)"], insert n,
unfold d Num_arc zero_arctic_def, simp)
qed
qed
subsection ‹The arctic semiring over an arbitrary archimedean field›
text ‹completely analogous to the integers, where one has to use delta-orderings›