src/HOL/IMP/Hoare.ML

induct -> lfp_induct

(*  Title:      HOL/IMP/Hoare.ML
    ID:         $Id$
    Author:     Tobias Nipkow
    Copyright   1995 TUM

Soundness (and part of) relative completeness of Hoare rules
wrt denotational semantics
*)

Goal "[| !s. P' s --> P s; |- {P}c{Q} |] ==> |- {P'}c{Q}";
by (etac hoare.conseq 1);
by  (atac 1);
by (Fast_tac 1);
qed "hoare_conseq1";

Goal "[| |- {P}c{Q}; !s. Q s --> Q' s |] ==> |- {P}c{Q'}";
by (rtac hoare.conseq 1);
by    (atac 2);
by   (ALLGOALS Fast_tac);
qed "hoare_conseq2";

Goalw [hoare_valid_def] "|- {P}c{Q} ==> |= {P}c{Q}";
by (etac hoare.induct 1);
     by (ALLGOALS Asm_simp_tac);
  by (Fast_tac 1);
 by (Fast_tac 1);
by (EVERY' [rtac allI, rtac allI, rtac impI] 1);
by (etac lfp_induct2 1);
 by (rtac Gamma_mono 1);
by (rewtac Gamma_def);  
by (Fast_tac 1);
qed "hoare_sound";

Goalw [wp_def] "wp SKIP Q = Q";
by (Simp_tac 1);
qed "wp_SKIP";

Goalw [wp_def] "wp (x:==a) Q = (%s. Q(s[x::=a s]))";
by (Simp_tac 1);
qed "wp_Ass";

Goalw [wp_def] "wp (c;d) Q = wp c (wp d Q)";
by (Simp_tac 1);
by (rtac ext 1);
by (Fast_tac 1);
qed "wp_Semi";

Goalw [wp_def]
 "wp (IF b THEN c ELSE d) Q = (%s. (b s --> wp c Q s) &  (~b s --> wp d Q s))";
by (Simp_tac 1);
by (rtac ext 1);
by (Fast_tac 1);
qed "wp_If";

Goalw [wp_def]
  "b s ==> wp (WHILE b DO c) Q s = wp (c;WHILE b DO c) Q s";
by (stac C_While_If 1);
by (Asm_simp_tac 1);
qed "wp_While_True";

Goalw [wp_def] "~b s ==> wp (WHILE b DO c) Q s = Q s";
by (stac C_While_If 1);
by (Asm_simp_tac 1);
qed "wp_While_False";

Addsimps [wp_SKIP,wp_Ass,wp_Semi,wp_If,wp_While_True,wp_While_False];

(*Not suitable for rewriting: LOOPS!*)
Goal "wp (WHILE b DO c) Q s = (if b s then wp (c;WHILE b DO c) Q s else Q s)";
by (Simp_tac 1);
qed "wp_While_if";

Goal "wp (WHILE b DO c) Q s = \
\  (s : gfp(%S.{s. if b s then wp c (%s. s:S) s else Q s}))";
by (Simp_tac 1);
by (rtac iffI 1);
 by (rtac weak_coinduct 1);
  by (etac CollectI 1);
 by Safe_tac;
  by (rotate_tac ~1 1);
  by (Asm_full_simp_tac 1);
 by (rotate_tac ~1 1);
 by (Asm_full_simp_tac 1);
by (asm_full_simp_tac (simpset() addsimps [wp_def,Gamma_def]) 1);
by (strip_tac 1);
by (rtac mp 1);
 by (assume_tac 2);
by (etac lfp_induct2 1);
by (fast_tac (claset() addSIs [monoI]) 1);
by (stac gfp_unfold 1);
 by (fast_tac (claset() addSIs [monoI]) 1);
by (Fast_tac 1);
qed "wp_While";

Delsimps [C_while];

AddSIs [hoare.skip, hoare.ass, hoare.semi, hoare.If];

Goal "!Q. |- {wp c Q} c {Q}";
by (induct_tac "c" 1);
    by (ALLGOALS Simp_tac);
    by (REPEAT_FIRST Fast_tac);
 by (blast_tac (claset() addIs [hoare_conseq1]) 1);
by Safe_tac;
by (rtac hoare_conseq2 1);
 by (rtac hoare.While 1);
 by (rtac hoare_conseq1 1);
  by (Fast_tac 2);
 by (safe_tac HOL_cs);
 by (ALLGOALS (EVERY'[rotate_tac ~1, Asm_full_simp_tac]));
qed_spec_mp "wp_is_pre";

Goal "|= {P}c{Q} ==> |- {P}c{Q}";
by (rtac (wp_is_pre RSN (2,hoare_conseq1)) 1);
by (rewrite_goals_tac [hoare_valid_def,wp_def]);
by (Fast_tac 1);
qed "hoare_relative_complete";