File Hoare.ML (Isabelle2005: October 2005)

# File Hoare.ML

    (*  Title:      HOL/IMPP/Hoare.ML
        ID:         $Id: Hoare.ML,v 1.10 2005/09/17 18:14:30 wenzelm Exp $
        Author:     David von Oheimb
        Copyright   1999 TUM

    Soundness and relative completeness of Hoare rules wrt operational semantics
    *)

    Goalw [state_not_singleton_def]
            "state_not_singleton ==> !t. (!s::state. s = t) --> False";
    by (Clarify_tac 1);
    by (case_tac "ta = t" 1);
    by  (ALLGOALS (blast_tac (HOL_cs addDs [not_sym])));
    qed "single_stateE";

    Addsimps[peek_and_def];

    section "validity";

    Goalw [triple_valid_def]
      "|=n:{P}.c.{Q} = (!Z s. P Z s --> (!s'. <c,s> -n-> s' --> Q Z s'))";
    by Auto_tac;
    qed "triple_valid_def2";

    Goal "|=0:{P}. BODY pn .{Q}";
    by (simp_tac (simpset() addsimps [triple_valid_def2]) 1);
    by (Clarsimp_tac 1);
    qed "Body_triple_valid_0";

    (* only ==> direction required *)
    Goal "|=n:{P}. the (body pn) .{Q} = |=Suc n:{P}. BODY pn .{Q}";
    by (simp_tac (simpset() addsimps [triple_valid_def2]) 1);
    by (Force_tac 1);
    qed "Body_triple_valid_Suc";

    Goalw [triple_valid_def] "|=Suc n:t --> |=n:t";
    by (induct_tac "t" 1);
    by (Simp_tac 1);
    by (fast_tac (claset() addIs [evaln_Suc]) 1);
    qed_spec_mp "triple_valid_Suc";

    Goal "||=Suc n:ts ==> ||=n:ts";
    by (fast_tac (claset() addIs [triple_valid_Suc]) 1);
    qed "triples_valid_Suc";

    section "derived rules";

    Goal "[| G|-{P'}.c.{Q'}; !Z s. P Z s --> \
    \                        (!s'. (!Z'. P' Z' s --> Q' Z' s') --> Q Z s') |] \
    \      ==> G|-{P}.c.{Q}";
    by (rtac hoare_derivs.conseq 1);
    by (Blast_tac 1);
    qed "conseq12";

    Goal "[| G|-{P'}.c.{Q}; !Z s. P Z s --> P' Z s |] ==> G|-{P}.c.{Q}";
    by (etac conseq12 1);
    by (Fast_tac 1);
    qed "conseq1";

    Goal "[| G|-{P}.c.{Q'}; !Z s. Q' Z s --> Q Z s |] ==> G|-{P}.c.{Q}";
    by (etac conseq12 1);
    by (Fast_tac 1);
    qed "conseq2";

    Goal "[| G Un (%p. {P p}.      BODY p  .{Q p})`Procs  \
    \         ||- (%p. {P p}. the (body p) .{Q p})`Procs; \
    \   pn:Procs |] ==> G|-{P pn}. BODY pn .{Q pn}";
    by (dtac hoare_derivs.Body 1);
    by (etac hoare_derivs.weaken 1);
    by (Fast_tac 1);
    qed "Body1";

    Goal "(insert ({P}. BODY pn .{Q}) G) |-{P}. the (body pn) .{Q} ==> \
    \ G|-{P}. BODY pn .{Q}";
    by (rtac Body1 1);
    by (rtac singletonI 2);
    by (Clarsimp_tac 1);
    qed "BodyN";

    Goal "[| !Z s. P Z s --> G|-{%Z s'. s'=s}.c.{%Z'. Q Z} |] ==> G|-{P}.c.{Q}";
    by (rtac hoare_derivs.conseq 1);
    by (Fast_tac 1);
    qed "escape";

    Goal "[| C ==> G|-{P}.c.{Q} |] ==> G|-{%Z s. P Z s & C}.c.{Q}";
    by (rtac hoare_derivs.conseq 1);
    by (fast_tac (claset() addDs (premises())) 1);
    qed "constant";

    Goal "G|-{%Z s. P Z s & ~b s}.WHILE b DO c.{P}";
    by (rtac (hoare_derivs.Loop RS conseq2) 1);
    by  (ALLGOALS Simp_tac);
    by (rtac hoare_derivs.conseq 1);
    by (Fast_tac 1);
    qed "LoopF";

    (*
    Goal "[| G'||-ts; G' <= G |] ==> G||-ts";
    by (etac hoare_derivs.cut 1);
    by (etac hoare_derivs.asm 1);
    qed "thin";
    *)
    Goal "G'||-ts ==> !G. G' <= G --> G||-ts";
    by (etac hoare_derivs.induct 1);
    by                (ALLGOALS (EVERY'[Clarify_tac, REPEAT o smp_tac 1]));
    by (rtac                  hoare_derivs.empty 1);
    by               (eatac hoare_derivs.insert 1 1);
    by              (fast_tac (claset() addIs [hoare_derivs.asm]) 1);
    by             (fast_tac (claset() addIs [hoare_derivs.cut]) 1);
    by            (fast_tac (claset() addIs [hoare_derivs.weaken]) 1);
    by           (EVERY'[rtac hoare_derivs.conseq, strip_tac, smp_tac 2,Clarify_tac,
                         smp_tac 1,rtac exI, rtac exI, eatac conjI 1] 1);
    by          (EVERY'[rtac hoare_derivs.Body,dtac spec,etac mp,Fast_tac] 7);
    by         (ALLGOALS (resolve_tac ((funpow 5 tl) hoare_derivs.intros)
                          THEN_ALL_NEW Fast_tac));
    qed_spec_mp "thin";

    Goal "G|-{P}. the (body pn) .{Q} ==> G|-{P}. BODY pn .{Q}";
    by (rtac BodyN 1);
    by (etac thin 1);
    by Auto_tac;
    qed "weak_Body";

    Goal "G||-insert t ts ==> G|-t & G||-ts";
    by (fast_tac (claset() addIs [hoare_derivs.weaken]) 1);
    qed "derivs_insertD";

    Goal "[| finite U; \
    \ !p. G |-     {P' p}.c0 p.{Q' p}       --> G |-     {P p}.c0 p.{Q p} |] ==> \
    \     G||-(%p. {P' p}.c0 p.{Q' p}) ` U --> G||-(%p. {P p}.c0 p.{Q p}) ` U";
    by (etac finite_induct 1);
    by (ALLGOALS Clarsimp_tac);
    by (dtac derivs_insertD 1);
    by (rtac hoare_derivs.insert 1);
    by  Auto_tac;
    qed_spec_mp "finite_pointwise";

    section "soundness";

    Goalw [hoare_valids_def]
     "G|={P &> b}. c .{P} ==> \
    \ G|={P}. WHILE b DO c .{P &> (Not o b)}";
    by (full_simp_tac (simpset() addsimps [triple_valid_def2]) 1);
    by (rtac allI 1);
    by (subgoal_tac "!d s s'. <d,s> -n-> s' --> \
    \ d = WHILE b DO c --> ||=n:G --> (!Z. P Z s --> P Z s' & ~b s')" 1);
    by  (EVERY'[etac thin_rl, Fast_tac] 1);
    by (EVERY'[REPEAT o rtac allI, rtac impI] 1);
    by ((etac evaln.induct THEN_ALL_NEW Simp_tac) 1);
    by  (ALLGOALS Fast_tac);
    qed "Loop_sound_lemma";

    Goalw [hoare_valids_def]
       "[| G Un (%pn. {P pn}.      BODY pn  .{Q pn})`Procs \
    \        ||=(%pn. {P pn}. the (body pn) .{Q pn})`Procs |] ==> \
    \       G||=(%pn. {P pn}.      BODY pn  .{Q pn})`Procs";
    by (rtac allI 1);
    by (induct_tac "n" 1);
    by  (fast_tac (claset() addIs [Body_triple_valid_0]) 1);
    by (Clarsimp_tac 1);
    by (dtac triples_valid_Suc 1);
    by (mp_tac 1);
    by (asm_full_simp_tac (simpset() addsimps [ball_Un]) 1);
    by (EVERY'[dtac spec, etac impE, etac conjI, atac] 1);
    by (fast_tac (claset() addSIs [Body_triple_valid_Suc RS iffD1]) 1);
    qed "Body_sound_lemma";

    Goal "G||-ts ==> G||=ts";
    by (etac hoare_derivs.induct 1);
    by              (TRYALL (eresolve_tac [Loop_sound_lemma, Body_sound_lemma]
                             THEN_ALL_NEW atac));
    by            (rewtac hoare_valids_def);
    by            (Blast_tac 1);
    by           (Blast_tac 1);
    by          (Blast_tac 1); (* asm *)
    by         (Blast_tac 1); (* cut *)
    by        (Blast_tac 1); (* weaken *)
    by       (ALLGOALS (EVERY'[REPEAT o thin_tac "?x : hoare_derivs",
                               Simp_tac, Clarify_tac, REPEAT o smp_tac 1]));
    by       (ALLGOALS (full_simp_tac (simpset() addsimps [triple_valid_def2])));
    by       (EVERY'[strip_tac, smp_tac 2, Blast_tac] 1); (* conseq *)
    by      (ALLGOALS Clarsimp_tac); (* Skip, Ass, Local *)
    by   (Force_tac 3); (* Call *)
    by  (eresolve_tac evaln_elim_cases 2); (* If *)
    by   (TRYALL Blast_tac);
    qed "hoare_sound";

    section "completeness";

    (* Both versions *)

    (*unused*)
    Goalw [MGT_def] "G|-MGT c ==> \
    \ G|-{%Z s0. !s1. <c,s0> -c-> s1 --> Z=s1}. c .{%Z s1. Z=s1}";
    by (etac conseq12 1);
    by Auto_tac;
    qed "MGT_alternI";

    (* requires com_det *)
    Goalw [MGT_def] "state_not_singleton ==> \
    \ G|-{%Z s0. !s1. <c,s0> -c-> s1 --> Z=s1}. c .{%Z s1. Z=s1} ==> G|-MGT c";
    by (etac conseq12 1);
    by Auto_tac;
    by (case_tac "? t. <c,?s> -c-> t" 1);
    by  (fast_tac (claset() addEs [com_det]) 1);
    by (Clarsimp_tac 1);
    by (dtac single_stateE 1);
    by (Blast_tac 1);
    qed "MGT_alternD";

    Goalw [MGT_def]
     "{}|-(MGT c::state triple) ==> {}|={P}.c.{Q} ==> {}|-{P}.c.{Q::state assn}";
    by (etac conseq12 1);
    by (clarsimp_tac (claset(), simpset() addsimps
      [hoare_valids_def,eval_eq,triple_valid_def2]) 1);
    qed "MGF_complete";

    AddSEs WTs_elim_cases;
    (* requires com_det, escape (i.e. hoare_derivs.conseq) *)
    Goal "state_not_singleton ==> \
    \ !pn:dom body. G|-{=}.BODY pn.{->} ==> WT c --> G|-{=}.c.{->}";
    by (induct_tac "c" 1);
    by        (ALLGOALS Clarsimp_tac);
    by        (fast_tac (claset() addIs [(thm"domI")]) 7);
    by (etac MGT_alternD 6);
    by       (rewtac MGT_def);
    by       (EVERY'[dtac bspec, etac (thm"domI")] 7);
    by       (EVERY'[rtac escape, Clarsimp_tac, res_inst_tac
                 [("P1","%Z' s. s=(setlocs Z newlocs)[Loc Arg ::= fun Z]")]
                 (hoare_derivs.Call RS conseq1), etac conseq12] 7);
    by        (ALLGOALS (etac thin_rl));
    by (rtac (hoare_derivs.Skip RS conseq2) 1);
    by (rtac (hoare_derivs.Ass RS conseq1) 2);
    by        (EVERY'[rtac escape, Clarsimp_tac, res_inst_tac
                      [("P1","%Z' s. s=(Z[Loc loc::=fun Z])")]
                      (hoare_derivs.Local RS conseq1), etac conseq12] 3);
    by         (EVERY'[etac hoare_derivs.Comp, etac conseq12] 5);
    by         ((rtac hoare_derivs.If THEN_ALL_NEW etac conseq12) 6);
    by          (EVERY'[rtac (hoare_derivs.Loop RS conseq2), etac conseq12] 8);
    by           Auto_tac;
    qed_spec_mp "MGF_lemma1";

    (* Version: nested single recursion *)

    Goal "[| !!G ts. ts <= G ==> P G ts;\
    \ !!G pn. P (insert (mgt_call pn) G) {mgt(the(body pn))} ==> P G {mgt_call pn};\
    \         !!G c. [| wt c; !pn:U. P G {mgt_call pn} |] ==> P G {mgt c}; \
    \         !!pn. pn : U ==> wt (the (body pn)); \
    \         finite U; uG = mgt_call`U |] ==> \
    \ !G. G <= uG --> n <= card uG --> card G = card uG - n --> (!c. wt c --> P G {mgt c})";
    by (cut_facts_tac (premises()) 1);
    by (induct_tac "n" 1);
    by  (ALLGOALS Clarsimp_tac);
    by  (subgoal_tac "G = mgt_call ` U" 1);
    by   (asm_simp_tac (simpset() addsimps [card_seteq, finite_imageI]) 2);
    by  (Asm_full_simp_tac 1);
    by  (eresolve_tac (tl(tl(premises()))(*MGF_lemma1*)) 1);
    by (rtac ballI 1);
    by  (resolve_tac (premises()(*hoare_derivs.asm*)) 1);
    by  (Fast_tac 1);
    by (eresolve_tac (tl(tl(premises()))(*MGF_lemma1*)) 1);
    by (rtac ballI 1);
    by (case_tac "mgt_call pn : G" 1);
    by  (resolve_tac (premises()(*hoare_derivs.asm*)) 1);
    by  (Fast_tac 1);
    by (resolve_tac (tl(premises())(*MGT_BodyN*)) 1);
    byev[dtac spec 1, etac impE 1, etac impE 2, dtac spec 3,etac mp 3];
    by   (eresolve_tac (tl(tl(tl(premises())))) 3);
    by   (Fast_tac 1);
    by (dtac finite_subset 1);
    by (etac finite_imageI 1);
    by (Asm_simp_tac 1);
    by (arith_tac 1);
    qed_spec_mp "nesting_lemma";

    Goalw [MGT_def] "insert ({=}.BODY pn.{->}) G|-{=}. the (body pn) .{->} ==> \
    \ G|-{=}.BODY pn.{->}";
    by (rtac BodyN 1);
    by (etac conseq2 1);
    by (Force_tac 1);
    qed "MGT_BodyN";

    (* requires BodyN, com_det *)
    Goal "[| state_not_singleton; WT_bodies; WT c |] ==> {}|-MGT c";
    by (res_inst_tac [("P","%G ts. G||-ts"),("U","dom body")] nesting_lemma 1);
    by (etac hoare_derivs.asm 1);
    by (etac MGT_BodyN 1);
    by (rtac finite_dom_body 3);
    by (etac MGF_lemma1 1);
    by (assume_tac 2);
    by       (Blast_tac 1);
    by      (Clarsimp_tac 1);
    by     (eatac WT_bodiesD 1 1);
    by (rtac le_refl 3);
    by    Auto_tac;
    qed "MGF";

    (* Version: simultaneous recursion in call rule *)

    (* finiteness not really necessary here *)
    Goalw [MGT_def]     "[| G Un (%pn. {=}.      BODY pn  .{->})`Procs \
    \                         ||-(%pn. {=}. the (body pn) .{->})`Procs; \
    \ finite Procs |] ==>   G ||-(%pn. {=}.      BODY pn  .{->})`Procs";
    by (rtac hoare_derivs.Body 1);
    by (etac finite_pointwise 1);
    by (assume_tac 2);
    by (Clarify_tac 1);
    by (etac conseq2 1);
    by Auto_tac;
    qed "MGT_Body";

    (* requires empty, insert, com_det *)
    Goal "[| state_not_singleton; WT_bodies; \
    \ F<=(%pn. {=}.the (body pn).{->})`dom body |] ==> \
    \    (%pn. {=}.     BODY pn .{->})`dom body||-F";
    by (ftac finite_subset 1);
    by (rtac (finite_dom_body RS finite_imageI) 1);
    by (rotate_tac 2 1);
    by (make_imp_tac 1);
    by (etac finite_induct 1);
    by  (ALLGOALS (clarsimp_tac (
            claset() addSIs [hoare_derivs.empty,hoare_derivs.insert],
            simpset() delsimps [range_composition])));
    by (etac MGF_lemma1 1);
    by  (fast_tac (claset() addDs [WT_bodiesD]) 2);
    by (Clarsimp_tac 1);
    by (rtac hoare_derivs.asm 1);
    by (fast_tac (claset() addIs [(thm"domI")]) 1);
    qed_spec_mp "MGF_lemma2_simult";

    (* requires Body, empty, insert, com_det *)
    Goal "[| state_not_singleton; WT_bodies; WT c |] ==> {}|-MGT c";
    by (rtac MGF_lemma1 1);
    by (assume_tac 1);
    by (assume_tac 2);
    by (Clarsimp_tac 1);
    by (subgoal_tac "{}||-(%pn. {=}. BODY pn .{->})`dom body" 1);
    by (etac hoare_derivs.weaken 1);
    by  (fast_tac (claset() addIs [(thm"domI")]) 1);
    by (rtac (finite_dom_body RSN (2,MGT_Body)) 1);
    by (Simp_tac 1);
    by (eatac MGF_lemma2_simult 1 1);
    by (rtac subset_refl 1);
    qed "MGF'";

    (* requires Body+empty+insert / BodyN, com_det *)
    bind_thm ("hoare_complete", MGF' RS MGF_complete);

    section "unused derived rules";

    Goal "G|-{%Z s. False}.c.{Q}";
    by (rtac hoare_derivs.conseq 1);
    by (Fast_tac 1);
    qed "falseE";

    Goal "G|-{P}.c.{%Z s. True}";
    by (rtac hoare_derivs.conseq 1);
    by (fast_tac (claset() addSIs [falseE]) 1);
    qed "trueI";

    Goal "[| G|-{P}.c.{Q}; G|-{P'}.c.{Q'} |] \
    \       ==> G|-{%Z s. P Z s | P' Z s}.c.{%Z s. Q Z s | Q' Z s}";
    by (rtac hoare_derivs.conseq 1);
    by (fast_tac (claset() addEs [conseq12]) 1);
    qed "disj"; (* analogue conj non-derivable *)

    Goal "(!Z s. P Z s --> Q Z s) ==> G|-{P}. SKIP .{Q}";
    by (rtac conseq12 1);
    by (rtac hoare_derivs.Skip 1);
    by (Fast_tac 1);
    qed "hoare_SkipI";

    section "useful derived rules";

    Goal "{t}|-t";
    by (rtac hoare_derivs.asm 1);
    by (rtac subset_refl 1);
    qed "single_asm";

    Goal "[| !!s'. G|-{%Z s. s'=s & P Z s}.c.{Q} |] ==> G|-{P}.c.{Q}";
    by (rtac hoare_derivs.conseq 1);
    by (Clarsimp_tac 1);
    by (cut_facts_tac (premises()) 1);
    by (Fast_tac 1);
    qed "export_s";

    Goal "[| G|-{P}. c .{Q}; !k Z s. Q Z s --> Q Z (s[Loc Y::=k]) |] ==> \
    \   G|-{%Z s. P Z (s[Loc Y::=a s])}. LOCAL Y:=a IN c .{Q}";
    by (rtac export_s 1);
    by (rtac hoare_derivs.Local 1);
    by (etac conseq2 1);
    by (etac spec 1);
    qed "weak_Local";

    (*
    Goal "!Q. G |-{%Z s. ~(? s'. <c,s> -c-> s')}. c .{Q}";
    by (induct_tac "c" 1);
    by Auto_tac;
    by (rtac conseq1 1);
    by (rtac hoare_derivs.Skip 1);
    force 1;
    by (rtac conseq1 1);
    by (rtac hoare_derivs.Ass 1);
    force 1;
    by (defer_tac 1);
    ###
    by (rtac hoare_derivs.Comp 1);
    by (dtac spec 2);
    by (dtac spec 2);
    by (assume_tac 2);
    by (etac conseq1 2);
    by (Clarsimp_tac 2);
    force 1;
    *)