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theory InductiveInvariant(* ID: $Id: InductiveInvariant.thy,v 1.6 2006/11/17 01:20:34 wenzelm Exp $ Author: Sava Krsti\'{c} and John Matthews *) header {* Some of the results in Inductive Invariants for Nested Recursion *} theory InductiveInvariant imports Main begin text {* A formalization of some of the results in \emph{Inductive Invariants for Nested Recursion}, by Sava Krsti\'{c} and John Matthews. Appears in the proceedings of TPHOLs 2003, LNCS vol. 2758, pp. 253-269. *} text "S is an inductive invariant of the functional F with respect to the wellfounded relation r." definition indinv :: "('a * 'a) set => ('a => 'b => bool) => (('a => 'b) => ('a => 'b)) => bool" where "indinv r S F = (∀f x. (∀y. (y,x) : r --> S y (f y)) --> S x (F f x))" text "S is an inductive invariant of the functional F on set D with respect to the wellfounded relation r." definition indinv_on :: "('a * 'a) set => 'a set => ('a => 'b => bool) => (('a => 'b) => ('a => 'b)) => bool" where "indinv_on r D S F = (∀f. ∀x∈D. (∀y∈D. (y,x) ∈ r --> S y (f y)) --> S x (F f x))" text "The key theorem, corresponding to theorem 1 of the paper. All other results in this theory are proved using instances of this theorem, and theorems derived from this theorem." theorem indinv_wfrec: assumes wf: "wf r" and inv: "indinv r S F" shows "S x (wfrec r F x)" using wf proof (induct x) fix x assume IHYP: "!!y. (y,x) ∈ r ==> S y (wfrec r F y)" then have "!!y. (y,x) ∈ r ==> S y (cut (wfrec r F) r x y)" by (simp add: tfl_cut_apply) with inv have "S x (F (cut (wfrec r F) r x) x)" by (unfold indinv_def, blast) thus "S x (wfrec r F x)" using wf by (simp add: wfrec) qed theorem indinv_on_wfrec: assumes WF: "wf r" and INV: "indinv_on r D S F" and D: "x∈D" shows "S x (wfrec r F x)" apply (insert INV D indinv_wfrec [OF WF, of "% x y. x∈D --> S x y"]) by (simp add: indinv_on_def indinv_def) theorem ind_fixpoint_on_lemma: assumes WF: "wf r" and INV: "∀f. ∀x∈D. (∀y∈D. (y,x) ∈ r --> S y (wfrec r F y) & f y = wfrec r F y) --> S x (wfrec r F x) & F f x = wfrec r F x" and D: "x∈D" shows "F (wfrec r F) x = wfrec r F x & S x (wfrec r F x)" proof (rule indinv_on_wfrec [OF WF _ D, of "% a b. F (wfrec r F) a = b & wfrec r F a = b & S a b" F, simplified]) show "indinv_on r D (%a b. F (wfrec r F) a = b & wfrec r F a = b & S a b) F" proof (unfold indinv_on_def, clarify) fix f x assume A1: "∀y∈D. (y, x) ∈ r --> F (wfrec r F) y = f y & wfrec r F y = f y & S y (f y)" assume D': "x∈D" from A1 INV [THEN spec, of f, THEN bspec, OF D'] have "S x (wfrec r F x)" and "F f x = wfrec r F x" by auto moreover from A1 have "∀y∈D. (y, x) ∈ r --> S y (wfrec r F y)" by auto with D' INV [THEN spec, of "wfrec r F", simplified] have "F (wfrec r F) x = wfrec r F x" by blast ultimately show "F (wfrec r F) x = F f x & wfrec r F x = F f x & S x (F f x)" by auto qed qed theorem ind_fixpoint_lemma: assumes WF: "wf r" and INV: "∀f x. (∀y. (y,x) ∈ r --> S y (wfrec r F y) & f y = wfrec r F y) --> S x (wfrec r F x) & F f x = wfrec r F x" shows "F (wfrec r F) x = wfrec r F x & S x (wfrec r F x)" apply (rule ind_fixpoint_on_lemma [OF WF _ UNIV_I, simplified]) by (rule INV) theorem tfl_indinv_wfrec: "[| f == wfrec r F; wf r; indinv r S F |] ==> S x (f x)" by (simp add: indinv_wfrec) theorem tfl_indinv_on_wfrec: "[| f == wfrec r F; wf r; indinv_on r D S F; x∈D |] ==> S x (f x)" by (simp add: indinv_on_wfrec) end
theorem indinv_wfrec:
[| wf r; indinv r S F |] ==> S x (wfrec r F x)
theorem indinv_on_wfrec:
[| wf r; indinv_on r D S F; x ∈ D |] ==> S x (wfrec r F x)
theorem ind_fixpoint_on_lemma:
[| wf r;
∀f. ∀x∈D. (∀y∈D. (y, x) ∈ r --> S y (wfrec r F y) ∧ f y = wfrec r F y) -->
S x (wfrec r F x) ∧ F f x = wfrec r F x;
x ∈ D |]
==> F (wfrec r F) x = wfrec r F x ∧ S x (wfrec r F x)
theorem ind_fixpoint_lemma:
[| wf r;
∀f x. (∀y. (y, x) ∈ r --> S y (wfrec r F y) ∧ f y = wfrec r F y) -->
S x (wfrec r F x) ∧ F f x = wfrec r F x |]
==> F (wfrec r F) x = wfrec r F x ∧ S x (wfrec r F x)
theorem tfl_indinv_wfrec:
[| f == wfrec r F; wf r; indinv r S F |] ==> S x (f x)
theorem tfl_indinv_on_wfrec:
[| f == wfrec r F; wf r; indinv_on r D S F; x ∈ D |] ==> S x (f x)