More tidying and removal of "\!\!... from Goal commands
(* Title: HOL/Induct/Exp
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1997 University of Cambridge
Example of Mutual Induction via Iteratived Inductive Definitions: Expressions
*)
open Exp;
AddSIs [eval.N, eval.X];
AddIs [eval.Op, eval.valOf];
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)"
];
AddSEs eval_elim_cases;
Goal "(X x, s[n/x]) -|-> (n, s[n/x])";
by (rtac (assign_same RS subst) 1 THEN resolve_tac eval.intrs 1);
qed "var_assign_eval";
AddSIs [var_assign_eval];
(** Make the induction rule look nicer -- though eta_contract makes the new
version look worse than it is...**)
Goal "{((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; \
\ (c,s) -[eval]-> 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);
by (forward_tac [impOfSubs (Int_lower1 RS exec_mono)] 1);
by (auto_tac (claset() addIs prems,simpset() addsimps [split_lemma]));
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 "(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 (Clarify_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*)
Goalw [Function_def] "Function eval";
by (strip_tac 1);
by (split_all_tac 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 "(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));
AddSEs [eval_N_E];
(*This theorem says that "WHILE TRUE DO c" cannot terminate*)
Goal "(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 "(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_simp_tac);
by (ALLGOALS Blast_tac);
bind_thm ("while_if1", refl RSN (2, result() RS mp));
Goal "(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_simp_tac);
by (ALLGOALS Blast_tac);
bind_thm ("while_if2", refl RSN (2, result() RS mp));
Goal "((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 (IF e THEN c1 ELSE c2);;c
and IF e THEN (c1;;c) ELSE (c2;;c) **)
Goal "(c',s) -[eval]-> t ==> \
\ (c' = (IF e THEN c1 ELSE c2);;c) --> \
\ (IF e THEN (c1;;c) ELSE (c2;;c), s) -[eval]-> t";
by (etac exec.induct 1);
by (ALLGOALS Asm_simp_tac);
by (Blast_tac 1);
bind_thm ("if_semi1", refl RSN (2, result() RS mp));
Goal "(c',s) -[eval]-> t ==> \
\ (c' = IF e THEN (c1;;c) ELSE (c2;;c)) --> \
\ ((IF e THEN c1 ELSE c2);;c, s) -[eval]-> t";
by (etac exec.induct 1);
by (ALLGOALS Asm_simp_tac);
by (ALLGOALS Blast_tac);
bind_thm ("if_semi2", refl RSN (2, result() RS mp));
Goal "(((IF e THEN c1 ELSE c2);;c, s) -[eval]-> t) = \
\ ((IF e THEN (c1;;c) ELSE (c2;;c), s) -[eval]-> t)";
by (blast_tac (claset() addIs [if_semi1, if_semi2]) 1);
qed "if_semi";
(** Equivalence of VALOF c1 RESULTIS (VALOF c2 RESULTIS e)
and VALOF c1;;c2 RESULTIS e
**)
Goal "(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_simp_tac);
by (Blast_tac 1);
bind_thm ("valof_valof1", refl RSN (2, result() RS mp));
Goal "(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_simp_tac);
by (Blast_tac 1);
bind_thm ("valof_valof2", refl RSN (2, result() RS mp));
Goal "((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";
(** Equivalence of VALOF SKIP RESULTIS e and e **)
Goal "(e',s) -|-> (v,s') ==> \
\ (e' = VALOF SKIP RESULTIS e) --> \
\ (e, s) -|-> (v,s')";
by (etac eval_induct 1);
by (ALLGOALS Asm_simp_tac);
by (Blast_tac 1);
bind_thm ("valof_skip1", refl RSN (2, result() RS mp));
Goal "(e,s) -|-> (v,s') ==> (VALOF SKIP RESULTIS e, s) -|-> (v,s')";
by (Blast_tac 1);
qed "valof_skip2";
Goal "((VALOF SKIP RESULTIS e, s) -|-> (v,s')) = ((e, s) -|-> (v,s'))";
by (blast_tac (claset() addIs [valof_skip1, valof_skip2]) 1);
qed "valof_skip";
(** Equivalence of VALOF x:=e RESULTIS x and e **)
Goal "(e',s) -|-> (v,s'') ==> \
\ (e' = VALOF x:=e RESULTIS X x) --> \
\ (EX s'. (e, s) -|-> (v,s') & (s'' = s'[v/x]))";
by (etac eval_induct 1);
by (ALLGOALS Asm_simp_tac);
by (thin_tac "?PP-->?QQ" 1);
by (Clarify_tac 1);
by (Simp_tac 1);
by (Blast_tac 1);
bind_thm ("valof_assign1", refl RSN (2, result() RS mp));
Goal "(e,s) -|-> (v,s') ==> \
\ (VALOF x:=e RESULTIS X x, s) -|-> (v,s'[v/x])";
by (Blast_tac 1);
qed "valof_assign2";