src/HOL/Induct/Exp.ML
 author paulson Thu, 08 May 1997 12:22:01 +0200 changeset 3144 04b0d8941365 parent 3120 c58423c20740 child 3145 809a2c9902f7 permissions -rw-r--r--
New proofs about WHILE and VALOF
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(*  Title:      HOL/Induct/Exp
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

Example of Mutual Induction via Iteratived Inductive Definitions: Expressions
*)

open Exp;

val eval_elim_cases = map (eval.mk_cases exp.simps)
["(N(n),sigma) -|-> (n',s')", "(X(x),sigma) -|-> (n,s')",
"(Op f a1 a2,sigma)  -|-> (n,s')",
"(VALOF c RESULTIS e, s) -|-> (n, s1)"
];

(** Make the induction rule look nicer -- though eta_contract makes the new
version look worse than it is...**)

goal thy "{((e,s),(n,s')). P e s n s'} = \
\         Collect (split (%v. split (split P v)))";
by (rtac Collect_cong 1);
by (split_all_tac 1);
by (Simp_tac 1);
val split_lemma = result();

(*New induction rule.  Note the form of the VALOF induction hypothesis*)
val major::prems = goal thy
"[| (e,s) -|-> (n,s');                                         \
\     !!n s. P (N n) s n s;                                      \
\     !!s x. P (X x) s (s x) s;                                  \
\     !!e0 e1 f n0 n1 s s0 s1.                                   \
\        [| (e0,s) -|-> (n0,s0); P e0 s n0 s0;                   \
\           (e1,s0) -|-> (n1,s1); P e1 s0 n1 s1                  \
\        |] ==> P (Op f e0 e1) s (f n0 n1) s1;                   \
\     !!c e n s s0 s1.                                           \
\        [| (c,s) -[eval Int {((e,s),(n,s')). P e s n s'}]-> s0; \
\           (e,s0) -|-> (n,s1); P e s0 n s1 |]                   \
\        ==> P (VALOF c RESULTIS e) s n s1                       \
\  |] ==> P e s n s'";
by (rtac (major RS eval.induct) 1);
by (blast_tac (!claset addIs prems) 1);
by (blast_tac (!claset addIs prems) 1);
by (blast_tac (!claset addIs prems) 1);
qed "eval_induct";

(*Lemma for Function_eval.  The major premise is that (c,s) executes to s1
using eval restricted to its functional part.  Note that the execution
(c,s) -[eval]-> s2 can use unrestricted eval!  The reason is that
the execution (c,s) -[eval Int {...}]-> s1 assures us that execution is
functional on the argument (c,s).
*)
goal thy
"!!x. (c,s) -[eval Int {((e,s),(n,s')). Unique (e,s) (n,s') eval}]-> s1 \
\         ==> (ALL s2. (c,s) -[eval]-> s2 --> s2=s1)";
by (etac exec.induct 1);
by (ALLGOALS Full_simp_tac);
by (Blast_tac 3);
by (Blast_tac 1);
by (rewtac Unique_def);
by (Blast_tac 1);
by (Blast_tac 1);
by (Blast_tac 1);
by (blast_tac (!claset addEs [exec_WHILE_case]) 1);
by (thin_tac "(?c,s2) -[?ev]-> s3" 1);
by (Step_tac 1);
by (etac exec_WHILE_case 1);
by (ALLGOALS Fast_tac);         (*Blast_tac: proof fails*)
qed "com_Unique";

(*Expression evaluation is functional, or deterministic*)
goal thy "Function eval";
by (simp_tac (!simpset addsimps [Function_def]) 1);
by (REPEAT (rtac allI 1));
by (rtac impI 1);
by (etac eval_induct 1);
by (dtac com_Unique 4);
by (ALLGOALS (full_simp_tac (!simpset addsimps [Unique_def])));
by (ALLGOALS Blast_tac);
qed "Function_eval";

goal thy "!!x. (e,s) -|-> (v,s') ==> (e = N n) --> (v=n & s'=s)";
by (etac eval.induct 1);
by (ALLGOALS Asm_simp_tac);
val lemma = result();
bind_thm ("eval_N_E", refl RSN (2, lemma RS mp));

(*This theorem says that "WHILE TRUE DO c" cannot terminate*)
goal thy "!!x. (c', s) -[eval]-> t ==> (c' = WHILE (N 0) DO c) --> False";
by (etac exec.induct 1);
by (Auto_tac());
bind_thm ("while_true_E", refl RSN (2, result() RS mp));

(** Equivalence of IF e THEN c;;(WHILE e DO c) ELSE SKIP  and  WHILE e DO c **)

goal thy "!!x. (c',s) -[eval]-> t ==> \
\              (c' = WHILE e DO c) --> \
\              (IF e THEN c;;c' ELSE SKIP, s) -[eval]-> t";
by (etac exec.induct 1);
by (ALLGOALS Asm_full_simp_tac);
by (ALLGOALS Blast_tac);
bind_thm ("while_if1", refl RSN (2, result() RS mp));

goal thy "!!x. (c',s) -[eval]-> t ==> \
\              (c' = IF e THEN c;;(WHILE e DO c) ELSE SKIP) --> \
\              (WHILE e DO c, s) -[eval]-> t";
by (etac exec.induct 1);
by (ALLGOALS Asm_full_simp_tac);
by (ALLGOALS Blast_tac);
bind_thm ("while_if2", refl RSN (2, result() RS mp));

goal thy "((IF e THEN c;;(WHILE e DO c) ELSE SKIP, s) -[eval]-> t)  =  \
\         ((WHILE e DO c, s) -[eval]-> t)";
by (blast_tac (!claset addIs [while_if1, while_if2]) 1);
qed "while_if";

(** Equivalence of  VALOF c1 RESULTIS (VALOF c2 RESULTIS e)
and  VALOF c1;;c2 RESULTIS e
**)

goal thy "!!x. (e',s) -|-> (v,s') ==> \
\              (e' = VALOF c1 RESULTIS (VALOF c2 RESULTIS e)) --> \
\              (VALOF c1;;c2 RESULTIS e, s) -|-> (v,s')";
by (etac eval.induct 1);
by (ALLGOALS Asm_full_simp_tac);
(*The destruction rule below replaces (c,s)-[eval Int ...]->t by
(c,s)-[eval]->t.  The new induction rule might include this assumption*)
by (blast_tac (!claset addSDs [impOfSubs (Int_lower1 RS exec_mono)]) 1);
bind_thm ("valof_valof1", refl RSN (2, result() RS mp));

goal thy "!!x. (e',s) -|-> (v,s') ==> \
\              (e' = VALOF c1;;c2 RESULTIS e) --> \
\              (VALOF c1 RESULTIS (VALOF c2 RESULTIS e), s) -|-> (v,s')";
by (etac eval.induct 1);
by (ALLGOALS Asm_full_simp_tac);
by (blast_tac (!claset addSDs [impOfSubs (Int_lower1 RS exec_mono)]) 1);
bind_thm ("valof_valof2", refl RSN (2, result() RS mp));

goal thy "((VALOF c1 RESULTIS (VALOF c2 RESULTIS e), s) -|-> (v,s'))  =  \
\         ((VALOF c1;;c2 RESULTIS e, s) -|-> (v,s'))";
by (blast_tac (!claset addIs [valof_valof1, valof_valof2]) 1);
qed "valof_valof";

```