--- a/src/HOL/Induct/Com.ML Mon Jan 01 11:52:04 2001 +0100
+++ b/src/HOL/Induct/Com.ML Tue Jan 02 10:27:10 2001 +0100
@@ -30,29 +30,15 @@
Unify.search_bound := 60;
(*Command execution is functional (deterministic) provided evaluation is*)
-Goal "Function ev ==> Function(exec ev)";
-by (simp_tac (simpset() addsimps [Function_def, Unique_def]) 1);
+Goal "univalent ev ==> univalent(exec ev)";
+by (simp_tac (simpset() addsimps [univalent_def]) 1);
by (REPEAT (rtac allI 1));
by (rtac impI 1);
by (etac exec.induct 1);
by (Blast_tac 3);
by (Blast_tac 1);
-by (rewrite_goals_tac [Function_def, Unique_def]);
+by (rewrite_goals_tac [univalent_def]);
by (REPEAT (blast_tac (claset() addEs [exec_WHILE_case]) 1));
-qed "Function_exec";
-
-
-Goalw [assign_def] "(s[v/x])x = v";
-by (Simp_tac 1);
-qed "assign_same";
+qed "univalent_exec";
-Goalw [assign_def] "y~=x ==> (s[v/x])y = s y";
-by (Asm_simp_tac 1);
-qed "assign_different";
-
-Goalw [assign_def] "s[s x/x] = s";
-by (rtac ext 1);
-by (Simp_tac 1);
-qed "assign_triv";
-
-Addsimps [assign_same, assign_different, assign_triv];
+Addsimps [fun_upd_same, fun_upd_other];
--- a/src/HOL/Induct/Com.thy Mon Jan 01 11:52:04 2001 +0100
+++ b/src/HOL/Induct/Com.thy Tue Jan 02 10:27:10 2001 +0100
@@ -6,7 +6,7 @@
Example of Mutual Induction via Iteratived Inductive Definitions: Commands
*)
-Com = Datatype +
+Com = Main +
types loc
state = "loc => nat"
@@ -14,23 +14,22 @@
arities loc :: term
-(*To avoid a mutually recursive datatype declaration, expressions and commands
- are combined to form a single datatype.*)
datatype
exp = N nat
| X loc
| Op n2n2n exp exp
- | valOf exp exp ("VALOF _ RESULTIS _" 60)
- | SKIP
+ | valOf com exp ("VALOF _ RESULTIS _" 60)
+and
+ com = SKIP
| ":=" loc exp (infixl 60)
- | Semi exp exp ("_;;_" [60, 60] 60)
- | Cond exp exp exp ("IF _ THEN _ ELSE _" 60)
- | While exp exp ("WHILE _ DO _" 60)
+ | Semi com com ("_;;_" [60, 60] 60)
+ | Cond exp com com ("IF _ THEN _ ELSE _" 60)
+ | While exp com ("WHILE _ DO _" 60)
(** Execution of commands **)
-consts exec :: "((exp*state) * (nat*state)) set => ((exp*state)*state)set"
+consts exec :: "((exp*state) * (nat*state)) set => ((com*state)*state)set"
"@exec" :: "((exp*state) * (nat*state)) set =>
- [exp*state,state] => bool" ("_/ -[_]-> _" [50,0,50] 50)
+ [com*state,state] => bool" ("_/ -[_]-> _" [50,0,50] 50)
translations "csig -[eval]-> s" == "(csig,s) : exec eval"
@@ -40,16 +39,12 @@
translations
"esig -|[eval]-> ns" => "(esig,ns) : eval"
-constdefs assign :: [state,nat,loc] => state ("_[_'/_]" [95,0,0] 95)
- "s[m/x] == (%y. if y=x then m else s y)"
-
-
(*Command execution. Natural numbers represent Booleans: 0=True, 1=False*)
inductive "exec eval"
intrs
Skip "(SKIP,s) -[eval]-> s"
- Assign "(e,s) -|[eval]-> (v,s') ==> (x := e, s) -[eval]-> s'[v/x]"
+ Assign "(e,s) -|[eval]-> (v,s') ==> (x := e, s) -[eval]-> s'(x:=v)"
Semi "[| (c0,s) -[eval]-> s2; (c1,s2) -[eval]-> s1 |]
==> (c0 ;; c1, s) -[eval]-> s1"
@@ -65,11 +60,4 @@
WhileTrue "[| (e,s) -|[eval]-> (0,s1);
(c,s1) -[eval]-> s2; (WHILE e DO c, s2) -[eval]-> s3 |]
==> (WHILE e DO c, s) -[eval]-> s3"
-
-constdefs
- Unique :: "['a, 'b, ('a*'b) set] => bool"
- "Unique x y r == ALL y'. (x,y'): r --> y = y'"
-
- Function :: "('a*'b) set => bool"
- "Function r == ALL x y. (x,y): r --> Unique x y r"
end
--- a/src/HOL/Induct/Exp.ML Mon Jan 01 11:52:04 2001 +0100
+++ b/src/HOL/Induct/Exp.ML Tue Jan 02 10:27:10 2001 +0100
@@ -19,8 +19,8 @@
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);
+Goal "(X x, s(x:=n)) -|-> (n, s(x:=n))";
+by (rtac (fun_upd_same RS subst) 1 THEN resolve_tac eval.intrs 1);
qed "var_assign_eval";
AddSIs [var_assign_eval];
@@ -66,33 +66,32 @@
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 \
+Goal "(c,s) -[eval Int {((e,s),(n,t)). ALL nt'. (e,s) -|-> nt' --> (n,t)=nt'}]-> 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 (Blast_tac 1);
+ by (Force_tac 1);
+ 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*)
+ by (ALLGOALS Blast_tac);
qed "com_Unique";
(*Expression evaluation is functional, or deterministic*)
-Goalw [Function_def] "Function eval";
-by (strip_tac 1);
+Goalw [univalent_def] "univalent eval";
+by (EVERY1[rtac allI, rtac allI, rtac impI]);
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 Full_simp_tac);
by (ALLGOALS Blast_tac);
-qed "Function_eval";
+qed "univalent_eval";
Goal "(e,s) -|-> (v,s') ==> (e = N n) --> (v=n & s'=s)";
@@ -215,11 +214,12 @@
(** Equivalence of VALOF x:=e RESULTIS x and e **)
+Delsimps [fun_upd_apply];
Goal "(e',s) -|-> (v,s'') ==> \
\ (e' = VALOF x:=e RESULTIS X x) --> \
-\ (EX s'. (e, s) -|-> (v,s') & (s'' = s'[v/x]))";
+\ (EX s'. (e, s) -|-> (v,s') & (s'' = s'(x:=v)))";
by (etac eval_induct 1);
-by (ALLGOALS Asm_simp_tac);
+ by (ALLGOALS Asm_simp_tac);
by (thin_tac "?PP-->?QQ" 1);
by (Clarify_tac 1);
by (Simp_tac 1);
@@ -228,6 +228,6 @@
Goal "(e,s) -|-> (v,s') ==> \
-\ (VALOF x:=e RESULTIS X x, s) -|-> (v,s'[v/x])";
+\ (VALOF x:=e RESULTIS X x, s) -|-> (v,s'(x:=v))";
by (Blast_tac 1);
qed "valof_assign2";
--- a/src/HOL/Induct/PropLog.ML Mon Jan 01 11:52:04 2001 +0100
+++ b/src/HOL/Induct/PropLog.ML Tue Jan 02 10:27:10 2001 +0100
@@ -56,21 +56,6 @@
(* [| H |- p->false; H |- p; q: pl |] ==> H |- q *)
bind_thm ("thms_notE", (thms.MP RS thms_falseE));
-(** The function eval **)
-
-Goalw [eval_def] "tt[[false]] = False";
-by (Simp_tac 1);
-qed "eval_false";
-
-Goalw [eval_def] "tt[[#v]] = (v:tt)";
-by (Simp_tac 1);
-qed "eval_var";
-
-Goalw [eval_def] "tt[[p->q]] = (tt[[p]]-->tt[[q]])";
-by (Simp_tac 1);
-qed "eval_imp";
-
-Addsimps [eval_false, eval_var, eval_imp];
(*Soundness of the rules wrt truth-table semantics*)
Goalw [sat_def] "H |- p ==> H |= p";
--- a/src/HOL/Induct/PropLog.thy Mon Jan 01 11:52:04 2001 +0100
+++ b/src/HOL/Induct/PropLog.thy Tue Jan 02 10:27:10 2001 +0100
@@ -14,7 +14,6 @@
thms :: 'a pl set => 'a pl set
"|-" :: ['a pl set, 'a pl] => bool (infixl 50)
"|=" :: ['a pl set, 'a pl] => bool (infixl 50)
- eval2 :: ['a pl, 'a set] => bool
eval :: ['a set, 'a pl] => bool ("_[[_]]" [100,0] 100)
hyps :: ['a pl, 'a set] => 'a pl set
@@ -31,17 +30,16 @@
defs
sat_def "H |= p == (!tt. (!q:H. tt[[q]]) --> tt[[p]])"
- eval_def "tt[[p]] == eval2 p tt"
primrec
- "eval2(false) = (%x. False)"
- "eval2(#v) = (%tt. v:tt)"
- "eval2(p->q) = (%tt. eval2 p tt-->eval2 q tt)"
+ "tt[[false]] = False"
+ "tt[[#v]] = (v:tt)"
+eval_imp "tt[[p->q]] = (tt[[p]] --> tt[[q]])"
primrec
- "hyps(false) = (%tt.{})"
- "hyps(#v) = (%tt.{if v:tt then #v else #v->false})"
- "hyps(p->q) = (%tt. hyps p tt Un hyps q tt)"
+ "hyps false tt = {}"
+ "hyps (#v) tt = {if v:tt then #v else #v->false}"
+ "hyps (p->q) tt = hyps p tt Un hyps q tt"
end