| author | wenzelm |
| Tue, 28 Aug 2012 20:16:11 +0200 | |
| changeset 48989 | 06c0e350782c |
| parent 48564 | eaa36c0d620a |
| child 63549 | b0d31c7def86 |
| permissions | -rw-r--r-- |
| 42151 | 1 |
(* Title: HOL/HOLCF/ex/Hoare.thy |
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Author: Franz Regensburger |
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Theory for an example by C.A.R. Hoare |
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p x = if b1 x |
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then p (g x) |
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else x fi |
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q x = if b1 x orelse b2 x |
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then q (g x) |
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else x fi |
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Prove: for all b1 b2 g . |
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q o p = q |
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In order to get a nice notation we fix the functions b1,b2 and g in the |
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signature of this example |
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*) |
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theory Hoare |
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imports HOLCF |
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begin |
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axiomatization |
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modernized specifications ('definition', 'axiomatization');
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b1 :: "'a -> tr" and |
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b2 :: "'a -> tr" and |
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g :: "'a -> 'a" |
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definition |
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p :: "'a -> 'a" where |
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"p = fix$(LAM f. LAM x. If b1$x then f$(g$x) else x)" |
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more robust syntax for definition/abbreviation/notation;
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definition |
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q :: "'a -> 'a" where |
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"q = fix$(LAM f. LAM x. If b1$x orelse b2$x then f$(g$x) else x)" |
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(* --------- pure HOLCF logic, some little lemmas ------ *) |
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lemma hoare_lemma2: "b~=TT ==> b=FF | b=UU" |
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apply (rule Exh_tr [THEN disjE]) |
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apply blast+ |
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done |
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lemma hoare_lemma3: " (ALL k. b1$(iterate k$g$x) = TT) | (EX k. b1$(iterate k$g$x)~=TT)" |
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apply blast |
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done |
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lemma hoare_lemma4: "(EX k. b1$(iterate k$g$x) ~= TT) ==> |
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EX k. b1$(iterate k$g$x) = FF | b1$(iterate k$g$x) = UU" |
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apply (erule exE) |
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apply (rule exI) |
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apply (rule hoare_lemma2) |
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apply assumption |
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done |
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lemma hoare_lemma5: "[|(EX k. b1$(iterate k$g$x) ~= TT); |
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k=Least(%n. b1$(iterate n$g$x) ~= TT)|] ==> |
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b1$(iterate k$g$x)=FF | b1$(iterate k$g$x)=UU" |
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apply hypsubst |
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apply (rule hoare_lemma2) |
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apply (erule exE) |
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apply (erule LeastI) |
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done |
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lemma hoare_lemma6: "b=UU ==> b~=TT" |
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apply hypsubst |
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apply (rule dist_eq_tr) |
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done |
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lemma hoare_lemma7: "b=FF ==> b~=TT" |
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apply hypsubst |
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apply (rule dist_eq_tr) |
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done |
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lemma hoare_lemma8: "[|(EX k. b1$(iterate k$g$x) ~= TT); |
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k=Least(%n. b1$(iterate n$g$x) ~= TT)|] ==> |
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ALL m. m < k --> b1$(iterate m$g$x)=TT" |
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apply hypsubst |
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apply (erule exE) |
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apply (intro strip) |
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apply (rule_tac p = "b1$ (iterate m$g$x) " in trE) |
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prefer 2 apply (assumption) |
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apply (rule le_less_trans [THEN less_irrefl [THEN notE]]) |
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prefer 2 apply (assumption) |
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apply (rule Least_le) |
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apply (erule hoare_lemma6) |
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apply (rule le_less_trans [THEN less_irrefl [THEN notE]]) |
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prefer 2 apply (assumption) |
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apply (rule Least_le) |
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apply (erule hoare_lemma7) |
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done |
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lemma hoare_lemma28: "f$(y::'a)=(UU::tr) ==> f$UU = UU" |
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by (rule strictI) |
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(* ----- access to definitions ----- *) |
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lemma p_def3: "p$x = If b1$x then p$(g$x) else x" |
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apply (rule trans) |
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apply (rule p_def [THEN eq_reflection, THEN fix_eq3]) |
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apply simp |
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done |
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lemma q_def3: "q$x = If b1$x orelse b2$x then q$(g$x) else x" |
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apply (rule trans) |
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apply (rule q_def [THEN eq_reflection, THEN fix_eq3]) |
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apply simp |
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done |
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(** --------- proofs about iterations of p and q ---------- **) |
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lemma hoare_lemma9: "(ALL m. m< Suc k --> b1$(iterate m$g$x)=TT) --> |
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p$(iterate k$g$x)=p$x" |
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apply (induct_tac k) |
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apply (simp (no_asm)) |
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apply (simp (no_asm)) |
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apply (intro strip) |
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apply (rule_tac s = "p$ (iterate n$g$x) " in trans) |
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apply (rule trans) |
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apply (rule_tac [2] p_def3 [symmetric]) |
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apply (rule_tac s = "TT" and t = "b1$ (iterate n$g$x) " in ssubst) |
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apply (rule mp) |
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apply (erule spec) |
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apply (simp (no_asm) add: less_Suc_eq) |
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apply simp |
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apply (erule mp) |
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apply (intro strip) |
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apply (rule mp) |
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apply (erule spec) |
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apply (erule less_trans) |
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apply simp |
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done |
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lemma hoare_lemma24: "(ALL m. m< Suc k --> b1$(iterate m$g$x)=TT) --> |
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q$(iterate k$g$x)=q$x" |
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apply (induct_tac k) |
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apply (simp (no_asm)) |
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apply (simp (no_asm) add: less_Suc_eq) |
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apply (intro strip) |
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apply (rule_tac s = "q$ (iterate n$g$x) " in trans) |
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apply (rule trans) |
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apply (rule_tac [2] q_def3 [symmetric]) |
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apply (rule_tac s = "TT" and t = "b1$ (iterate n$g$x) " in ssubst) |
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apply blast |
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apply simp |
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apply (erule mp) |
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apply (intro strip) |
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apply (fast dest!: less_Suc_eq [THEN iffD1]) |
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done |
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(* -------- results about p for case (EX k. b1$(iterate k$g$x)~=TT) ------- *) |
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lemma hoare_lemma10: |
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"EX k. b1$(iterate k$g$x) ~= TT |
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==> Suc k = (LEAST n. b1$(iterate n$g$x) ~= TT) ==> p$(iterate k$g$x) = p$x" |
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by (rule hoare_lemma8 [THEN hoare_lemma9 [THEN mp]]) |
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lemma hoare_lemma11: "(EX n. b1$(iterate n$g$x) ~= TT) ==> |
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k=(LEAST n. b1$(iterate n$g$x) ~= TT) & b1$(iterate k$g$x)=FF |
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--> p$x = iterate k$g$x" |
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apply (case_tac "k") |
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apply hypsubst |
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apply (simp (no_asm)) |
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apply (intro strip) |
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apply (erule conjE) |
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apply (rule trans) |
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apply (rule p_def3) |
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apply simp |
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apply hypsubst |
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apply (intro strip) |
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apply (erule conjE) |
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apply (rule trans) |
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apply (erule hoare_lemma10 [symmetric]) |
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apply assumption |
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apply (rule trans) |
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apply (rule p_def3) |
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apply (rule_tac s = "TT" and t = "b1$ (iterate nat$g$x) " in ssubst) |
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apply (rule hoare_lemma8 [THEN spec, THEN mp]) |
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apply assumption |
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apply assumption |
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apply (simp (no_asm)) |
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apply (simp (no_asm)) |
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apply (rule trans) |
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apply (rule p_def3) |
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apply (simp (no_asm) del: iterate_Suc add: iterate_Suc [symmetric]) |
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apply (erule_tac s = "FF" in ssubst) |
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apply simp |
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done |
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lemma hoare_lemma12: "(EX n. b1$(iterate n$g$x) ~= TT) ==> |
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k=Least(%n. b1$(iterate n$g$x)~=TT) & b1$(iterate k$g$x)=UU |
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--> p$x = UU" |
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apply (case_tac "k") |
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apply hypsubst |
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apply (simp (no_asm)) |
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apply (intro strip) |
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apply (erule conjE) |
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apply (rule trans) |
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apply (rule p_def3) |
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apply simp |
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apply hypsubst |
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apply (simp (no_asm)) |
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apply (intro strip) |
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apply (erule conjE) |
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apply (rule trans) |
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apply (rule hoare_lemma10 [symmetric]) |
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apply assumption |
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apply assumption |
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apply (rule trans) |
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apply (rule p_def3) |
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apply (rule_tac s = "TT" and t = "b1$ (iterate nat$g$x) " in ssubst) |
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apply (rule hoare_lemma8 [THEN spec, THEN mp]) |
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apply assumption |
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apply assumption |
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apply (simp (no_asm)) |
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apply (simp) |
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apply (rule trans) |
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apply (rule p_def3) |
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apply simp |
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done |
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(* -------- results about p for case (ALL k. b1$(iterate k$g$x)=TT) ------- *) |
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lemma fernpass_lemma: "(ALL k. b1$(iterate k$g$x)=TT) ==> ALL k. p$(iterate k$g$x) = UU" |
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apply (rule p_def [THEN eq_reflection, THEN def_fix_ind]) |
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apply simp |
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apply simp |
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apply (simp (no_asm)) |
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apply (rule allI) |
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apply (rule_tac s = "TT" and t = "b1$ (iterate k$g$x) " in ssubst) |
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apply (erule spec) |
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apply (simp) |
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apply (rule iterate_Suc [THEN subst]) |
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apply (erule spec) |
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done |
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lemma hoare_lemma16: "(ALL k. b1$(iterate k$g$x)=TT) ==> p$x = UU" |
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apply (rule_tac F1 = "g" and t = "x" in iterate_0 [THEN subst]) |
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apply (erule fernpass_lemma [THEN spec]) |
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done |
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(* -------- results about q for case (ALL k. b1$(iterate k$g$x)=TT) ------- *) |
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lemma hoare_lemma17: "(ALL k. b1$(iterate k$g$x)=TT) ==> ALL k. q$(iterate k$g$x) = UU" |
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apply (rule q_def [THEN eq_reflection, THEN def_fix_ind]) |
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apply simp |
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apply simp |
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apply (rule allI) |
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apply (simp (no_asm)) |
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apply (rule_tac s = "TT" and t = "b1$ (iterate k$g$x) " in ssubst) |
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apply (erule spec) |
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apply (simp) |
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apply (rule iterate_Suc [THEN subst]) |
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apply (erule spec) |
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done |
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lemma hoare_lemma18: "(ALL k. b1$(iterate k$g$x)=TT) ==> q$x = UU" |
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apply (rule_tac F1 = "g" and t = "x" in iterate_0 [THEN subst]) |
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apply (erule hoare_lemma17 [THEN spec]) |
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done |
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lemma hoare_lemma19: |
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"(ALL k. (b1::'a->tr)$(iterate k$g$x)=TT) ==> b1$(UU::'a) = UU | (ALL y. b1$(y::'a)=TT)" |
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apply (rule flat_codom) |
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apply (rule_tac t = "x" in iterate_0 [THEN subst]) |
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apply (erule spec) |
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done |
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lemma hoare_lemma20: "(ALL y. b1$(y::'a)=TT) ==> ALL k. q$(iterate k$g$(x::'a)) = UU" |
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apply (rule q_def [THEN eq_reflection, THEN def_fix_ind]) |
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apply simp |
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apply simp |
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apply (rule allI) |
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apply (simp (no_asm)) |
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apply (rule_tac s = "TT" and t = "b1$ (iterate k$g$ (x::'a))" in ssubst) |
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apply (erule spec) |
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apply (simp) |
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apply (rule iterate_Suc [THEN subst]) |
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apply (erule spec) |
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done |
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lemma hoare_lemma21: "(ALL y. b1$(y::'a)=TT) ==> q$(x::'a) = UU" |
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apply (rule_tac F1 = "g" and t = "x" in iterate_0 [THEN subst]) |
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apply (erule hoare_lemma20 [THEN spec]) |
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done |
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lemma hoare_lemma22: "b1$(UU::'a)=UU ==> q$(UU::'a) = UU" |
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apply (subst q_def3) |
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apply simp |
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done |
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(* -------- results about q for case (EX k. b1$(iterate k$g$x) ~= TT) ------- *) |
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lemma hoare_lemma25: "EX k. b1$(iterate k$g$x) ~= TT |
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==> Suc k = (LEAST n. b1$(iterate n$g$x) ~= TT) ==> q$(iterate k$g$x) = q$x" |
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by (rule hoare_lemma8 [THEN hoare_lemma24 [THEN mp]]) |
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lemma hoare_lemma26: "(EX n. b1$(iterate n$g$x)~=TT) ==> |
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k=Least(%n. b1$(iterate n$g$x) ~= TT) & b1$(iterate k$g$x) =FF |
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--> q$x = q$(iterate k$g$x)" |
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apply (case_tac "k") |
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apply hypsubst |
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apply (intro strip) |
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apply (simp (no_asm)) |
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apply hypsubst |
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apply (intro strip) |
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apply (erule conjE) |
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apply (rule trans) |
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apply (rule hoare_lemma25 [symmetric]) |
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apply assumption |
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apply assumption |
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apply (rule trans) |
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apply (rule q_def3) |
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apply (rule_tac s = "TT" and t = "b1$ (iterate nat$g$x) " in ssubst) |
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apply (rule hoare_lemma8 [THEN spec, THEN mp]) |
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apply assumption |
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apply assumption |
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apply (simp (no_asm)) |
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apply (simp (no_asm)) |
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done |
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lemma hoare_lemma27: "(EX n. b1$(iterate n$g$x) ~= TT) ==> |
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k=Least(%n. b1$(iterate n$g$x)~=TT) & b1$(iterate k$g$x)=UU |
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--> q$x = UU" |
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apply (case_tac "k") |
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apply hypsubst |
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apply (simp (no_asm)) |
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apply (intro strip) |
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apply (erule conjE) |
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apply (subst q_def3) |
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apply (simp) |
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apply hypsubst |
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apply (simp (no_asm)) |
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apply (intro strip) |
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apply (erule conjE) |
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apply (rule trans) |
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apply (rule hoare_lemma25 [symmetric]) |
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apply assumption |
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apply assumption |
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apply (rule trans) |
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apply (rule q_def3) |
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apply (rule_tac s = "TT" and t = "b1$ (iterate nat$g$x) " in ssubst) |
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apply (rule hoare_lemma8 [THEN spec, THEN mp]) |
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apply assumption |
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apply assumption |
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apply (simp (no_asm)) |
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apply (simp) |
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apply (rule trans) |
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apply (rule q_def3) |
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apply (simp) |
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done |
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(* ------- (ALL k. b1$(iterate k$g$x)=TT) ==> q o p = q ----- *) |
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lemma hoare_lemma23: "(ALL k. b1$(iterate k$g$x)=TT) ==> q$(p$x) = q$x" |
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apply (subst hoare_lemma16) |
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apply assumption |
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apply (rule hoare_lemma19 [THEN disjE]) |
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apply assumption |
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apply (simplesubst hoare_lemma18) |
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apply assumption |
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apply (simplesubst hoare_lemma22) |
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apply assumption |
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apply (rule refl) |
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apply (simplesubst hoare_lemma21) |
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apply assumption |
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apply (simplesubst hoare_lemma21) |
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apply assumption |
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apply (rule refl) |
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done |
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(* ------------ EX k. b1~(iterate k$g$x) ~= TT ==> q o p = q ----- *) |
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||
380 |
lemma hoare_lemma29: "EX k. b1$(iterate k$g$x) ~= TT ==> q$(p$x) = q$x" |
|
381 |
apply (rule hoare_lemma5 [THEN disjE]) |
|
382 |
apply assumption |
|
383 |
apply (rule refl) |
|
384 |
apply (subst hoare_lemma11 [THEN mp]) |
|
385 |
apply assumption |
|
386 |
apply (rule conjI) |
|
387 |
apply (rule refl) |
|
388 |
apply assumption |
|
389 |
apply (rule hoare_lemma26 [THEN mp, THEN subst]) |
|
390 |
apply assumption |
|
391 |
apply (rule conjI) |
|
392 |
apply (rule refl) |
|
393 |
apply assumption |
|
394 |
apply (rule refl) |
|
395 |
apply (subst hoare_lemma12 [THEN mp]) |
|
396 |
apply assumption |
|
397 |
apply (rule conjI) |
|
398 |
apply (rule refl) |
|
399 |
apply assumption |
|
400 |
apply (subst hoare_lemma22) |
|
401 |
apply (subst hoare_lemma28) |
|
402 |
apply assumption |
|
403 |
apply (rule refl) |
|
404 |
apply (rule sym) |
|
405 |
apply (subst hoare_lemma27 [THEN mp]) |
|
406 |
apply assumption |
|
407 |
apply (rule conjI) |
|
408 |
apply (rule refl) |
|
409 |
apply assumption |
|
410 |
apply (rule refl) |
|
411 |
done |
|
412 |
||
413 |
(* ------ the main proof q o p = q ------ *) |
|
414 |
||
415 |
theorem hoare_main: "q oo p = q" |
|
|
40002
c5b5f7a3a3b1
new theorem names: fun_below_iff, fun_belowI, cfun_eq_iff, cfun_eqI, cfun_below_iff, cfun_belowI
huffman
parents:
35948
diff
changeset
|
416 |
apply (rule cfun_eqI) |
| 19742 | 417 |
apply (subst cfcomp2) |
418 |
apply (rule hoare_lemma3 [THEN disjE]) |
|
419 |
apply (erule hoare_lemma23) |
|
420 |
apply (erule hoare_lemma29) |
|
421 |
done |
|
| 17291 | 422 |
|
| 244 | 423 |
end |