--- a/src/HOL/TLA/IntLemmas.ML Mon Feb 08 13:02:42 1999 +0100
+++ b/src/HOL/TLA/IntLemmas.ML Mon Feb 08 13:02:56 1999 +0100
@@ -1,7 +1,7 @@
(*
File: IntLemmas.ML
Author: Stephan Merz
- Copyright: 1997 University of Munich
+ Copyright: 1998 University of Munich
Lemmas and tactics for "intensional" logics.
@@ -12,32 +12,29 @@
qed_goal "substW" Intensional.thy
- "[| x .= y; w |= (P::[('v::world) => 'a, 'w::world] => bool)(x) |] ==> w |= P(y)"
+ "[| |- x = y; w |= P(x) |] ==> w |= P(y)"
(fn [prem1,prem2] => [rtac (rewrite_rule ([prem1] RL [inteq_reflection]) prem2) 1]);
(* Lift HOL rules to intensional reasoning *)
-qed_goal "reflW" Intensional.thy "x .= x"
- (fn _ => [ rtac intI 1,
- rewrite_goals_tac intensional_rews,
- rtac refl 1 ]);
+qed_goal "reflW" Intensional.thy "|- x = x"
+ (fn _ => [Simp_tac 1]);
-
-qed_goal "symW" Intensional.thy "s .= t ==> t .= s"
+qed_goal "symW" Intensional.thy "|- s = t ==> |- t = s"
(fn prems => [ cut_facts_tac prems 1,
rtac intI 1, dtac intD 1,
rewrite_goals_tac intensional_rews,
etac sym 1 ]);
-qed_goal "not_symW" Intensional.thy "s .~= t ==> t .~= s"
+qed_goal "not_symW" Intensional.thy "|- s ~= t ==> |- t ~= s"
(fn prems => [ cut_facts_tac prems 1,
rtac intI 1, dtac intD 1,
rewrite_goals_tac intensional_rews,
etac not_sym 1 ]);
qed_goal "transW" Intensional.thy
- "[| r .= s; s .= t |] ==> r .= t"
+ "[| |- r = s; |- s = t |] ==> |- r = t"
(fn prems => [ cut_facts_tac prems 1,
rtac intI 1, REPEAT (dtac intD 1),
rewrite_goals_tac intensional_rews,
@@ -45,36 +42,35 @@
atac 1 ]);
qed_goal "box_equalsW" Intensional.thy
- "[| a .= b; a .= c; b .= d |] ==> c .= d"
+ "[| |- a = b; |- a = c; |- b = d |] ==> |- c = d"
(fn prems => [ (rtac transW 1),
(rtac transW 1),
(rtac symW 1),
(REPEAT (resolve_tac prems 1)) ]);
+(* NB: Antecedent is a standard HOL (non-intensional) formula. *)
qed_goal "fun_congW" Intensional.thy
- "(f::('a => 'b)) = g ==> f[x] .= g[x]"
+ "f = g ==> |- f<x> = g<x>"
(fn prems => [ cut_facts_tac prems 1,
rtac intI 1,
rewrite_goals_tac intensional_rews,
etac fun_cong 1 ]);
qed_goal "fun_cong2W" Intensional.thy
- "(f::(['a,'b] => 'c)) = g ==> f[x,y] .= g[x,y]"
+ "f = g ==> |- f<x,y> = g<x,y>"
(fn prems => [ cut_facts_tac prems 1,
rtac intI 1,
- rewrite_goals_tac intensional_rews,
- asm_full_simp_tac HOL_ss 1 ]);
+ Asm_full_simp_tac 1 ]);
qed_goal "fun_cong3W" Intensional.thy
- "(f::(['a,'b,'c] => 'd)) = g ==> f[x,y,z] .= g[x,y,z]"
+ "f = g ==> |- f<x,y,z> = g<x,y,z>"
(fn prems => [ cut_facts_tac prems 1,
rtac intI 1,
- rewrite_goals_tac intensional_rews,
- asm_full_simp_tac HOL_ss 1 ]);
+ Asm_full_simp_tac 1 ]);
-qed_goal "arg_congW" Intensional.thy "x .= y ==> (f::'a=>'b)[x] .= f[y]"
+qed_goal "arg_congW" Intensional.thy "|- x = y ==> |- f<x> = f<y>"
(fn prems => [ cut_facts_tac prems 1,
rtac intI 1,
dtac intD 1,
@@ -82,7 +78,7 @@
etac arg_cong 1 ]);
qed_goal "arg_cong2W" Intensional.thy
- "[| u .= v; x .= y |] ==> (f::['a,'b]=>'c)[u,x] .= f[v,y]"
+ "[| |- u = v; |- x = y |] ==> |- f<u,x> = f<v,y>"
(fn prems => [ cut_facts_tac prems 1,
rtac intI 1,
REPEAT (dtac intD 1),
@@ -91,7 +87,7 @@
rtac refl 1 ]);
qed_goal "arg_cong3W" Intensional.thy
- "[| r .= s; u .= v; x .= y |] ==> (f::['a,'b,'c]=>'d)[r,u,x] .= f[s,v,y]"
+ "[| |- r = s; |- u = v; |- x = y |] ==> |- f<r,u,x> = f<s,v,y>"
(fn prems => [ cut_facts_tac prems 1,
rtac intI 1,
REPEAT (dtac intD 1),
@@ -100,7 +96,7 @@
rtac refl 1 ]);
qed_goal "congW" Intensional.thy
- "[| (f::'a=>'b) = g; x .= y |] ==> f[x] .= g[y]"
+ "[| f = g; |- x = y |] ==> |- f<x> = g<y>"
(fn prems => [ rtac box_equalsW 1,
rtac reflW 3,
rtac arg_congW 1,
@@ -110,7 +106,7 @@
resolve_tac prems 1 ]);
qed_goal "cong2W" Intensional.thy
- "[| (f::['a,'b]=>'c) = g; u .= v; x .= y |] ==> f[u,x] .= g[v,y]"
+ "[| f = g; |- u = v; |- x = y |] ==> |- f<u,x> = g<v,y>"
(fn prems => [ rtac box_equalsW 1,
rtac reflW 3,
rtac arg_cong2W 1,
@@ -120,7 +116,7 @@
resolve_tac prems 1 ]);
qed_goal "cong3W" Intensional.thy
- "[| (f::['a,'b,'c]=>'d) = g; r .= s; u .= v; x .= y |] ==> (f[r,u,x]) .= (g[s,v,y])"
+ "[| f = g; |- r = s; |- u = v; |- x = y |] ==> |- f<r,u,x> = g<s,v,y>"
(fn prems => [ rtac box_equalsW 1,
rtac reflW 3,
rtac arg_cong3W 1,
@@ -133,48 +129,38 @@
(** Lifted equivalence **)
(* Note the object-level implication in the hypothesis. Meta-level implication
- would not be correct! *)
+ would be incorrect! *)
qed_goal "iffIW" Intensional.thy
- "[| A .-> B; B .-> A |] ==> A .= B"
+ "[| |- A --> B; |- B --> A |] ==> |- A = B"
(fn prems => [ cut_facts_tac prems 1,
- rtac intI 1,
- REPEAT (dtac intD 1),
- rewrite_goals_tac intensional_rews,
- (fast_tac prop_cs 1) ]);
+ rewrite_goals_tac (Valid_def::intensional_rews),
+ Blast_tac 1 ]);
qed_goal "iffD2W" Intensional.thy
- "[| (P::('w::world) form) .= Q; w |= Q |] ==> w |= P"
- (fn prems =>
- [cut_facts_tac prems 1,
- dtac intD 1,
- rewrite_goals_tac intensional_rews,
- fast_tac prop_cs 1 ]);
+ "[| |- P = Q; w |= Q |] ==> w |= P"
+ (fn prems => [ cut_facts_tac prems 1,
+ rewrite_goals_tac (Valid_def::intensional_rews),
+ Blast_tac 1 ]);
val iffD1W = symW RS iffD2W;
(** #True **)
-qed_goal "TrueIW" Intensional.thy "#True"
- (fn _ => [rtac intI 1, rewrite_goals_tac intensional_rews, rtac TrueI 1]);
-
-
-qed_goal "eqTrueIW" Intensional.thy "(P::('w::world) form) ==> P .= #True"
+qed_goal "eqTrueIW" Intensional.thy "|- P ==> |- P = #True"
(fn prems => [cut_facts_tac prems 1,
rtac intI 1,
dtac intD 1,
- rewrite_goals_tac intensional_rews,
- asm_full_simp_tac HOL_ss 1] );
+ Asm_full_simp_tac 1]);
-qed_goal "eqTrueEW" Intensional.thy "P .= #True ==> (P::('w::world) form)"
+qed_goal "eqTrueEW" Intensional.thy "|- P = #True ==> |- P"
(fn prems => [cut_facts_tac prems 1,
rtac intI 1,
dtac intD 1,
- rewrite_goals_tac intensional_rews,
- asm_full_simp_tac HOL_ss 1] );
+ Asm_full_simp_tac 1]);
(** #False **)
-qed_goal "FalseEW" Intensional.thy "#False ==> P::('w::world) form"
+qed_goal "FalseEW" Intensional.thy "|- #False ==> |- P"
(fn prems => [cut_facts_tac prems 1,
rtac intI 1,
dtac intD 1,
@@ -182,23 +168,20 @@
etac FalseE 1]);
qed_goal "False_neq_TrueW" Intensional.thy
- "(#False::('w::world) form) .= #True ==> P::('w::world) form"
+ "|- #False = #True ==> |- P"
(fn [prem] => [rtac (prem RS eqTrueEW RS FalseEW) 1]);
(** Negation **)
(* Again use object-level implication *)
-qed_goal "notIW" Intensional.thy "(P .-> #False) ==> .~P"
+qed_goal "notIW" Intensional.thy "|- P --> #False ==> |- ~P"
(fn prems => [cut_facts_tac prems 1,
- rtac intI 1,
- dtac intD 1,
- rewrite_goals_tac intensional_rews,
- fast_tac prop_cs 1]);
-
+ rewrite_goals_tac (Valid_def::intensional_rews),
+ Blast_tac 1]);
qed_goal "notEWV" Intensional.thy
- "[| .~P; P::('w::world) form |] ==> R::('w::world) form"
+ "[| |- ~P; |- P |] ==> |- R"
(fn prems => [cut_facts_tac prems 1,
rtac intI 1,
REPEAT (dtac intD 1),
@@ -210,7 +193,7 @@
are allowed to be (intensional) formulas of different types! *)
qed_goal "notEW" Intensional.thy
- "[| w |= .~P; w |= P |] ==> R::('w::world) form"
+ "[| w |= ~P; w |= P |] ==> |- R"
(fn prems => [cut_facts_tac prems 1,
rtac intI 1,
rewrite_goals_tac intensional_rews,
@@ -218,14 +201,14 @@
(** Implication **)
-qed_goal "impIW" Intensional.thy "(!!w. (w |= A) ==> (w |= B)) ==> A .-> B"
+qed_goal "impIW" Intensional.thy "(!!w. (w |= A) ==> (w |= B)) ==> |- A --> B"
(fn [prem] => [ rtac intI 1,
rewrite_goals_tac intensional_rews,
rtac impI 1,
etac prem 1 ]);
-qed_goal "mpW" Intensional.thy "[| A .-> B; w |= A |] ==> w |= B"
+qed_goal "mpW" Intensional.thy "[| |- A --> B; w |= A |] ==> w |= B"
(fn prems => [ cut_facts_tac prems 1,
dtac intD 1,
rewrite_goals_tac intensional_rews,
@@ -233,124 +216,111 @@
atac 1 ]);
qed_goal "impEW" Intensional.thy
- "[| A .-> B; w |= A; w |= B ==> w |= C |] ==> w |= (C::('w::world) form)"
+ "[| |- A --> B; w |= A; w |= B ==> w |= C |] ==> w |= C"
(fn prems => [ (REPEAT (resolve_tac (prems@[mpW]) 1)) ]);
-qed_goal "rev_mpW" Intensional.thy "[| w |= P; P .-> Q |] ==> w |= Q"
+qed_goal "rev_mpW" Intensional.thy "[| w |= P; |- P --> Q |] ==> w |= Q"
(fn prems => [ (REPEAT (resolve_tac (prems@[mpW]) 1)) ]);
-qed_goal "contraposW" Intensional.thy "[| w |= .~Q; P .-> Q |] ==> w |= .~P"
- (fn [major,minor] => [rewrite_goals_tac intensional_rews,
- rtac contrapos 1,
- rtac (rewrite_rule intensional_rews major) 1,
+qed_goalw "contraposW" Intensional.thy intensional_rews
+ "[| w |= ~Q; |- P --> Q |] ==> w |= ~P"
+ (fn [major,minor] => [rtac (major RS contrapos) 1,
etac rev_mpW 1,
rtac minor 1]);
qed_goal "iffEW" Intensional.thy
- "[| (P::('w::world) form) .= Q; [| P .-> Q; Q .-> P |] ==> R::('w::world) form |] ==> R"
+ "[| |- P = Q; [| |- P --> Q; |- Q --> P |] ==> R |] ==> R"
(fn [p1,p2] => [REPEAT(ares_tac([p1 RS iffD2W, p1 RS iffD1W, p2, impIW])1)]);
(** Conjunction **)
-qed_goal "conjIW" Intensional.thy "[| w |= P; w |= Q |] ==> w |= P .& Q"
- (fn prems => [rewrite_goals_tac intensional_rews,
- REPEAT (resolve_tac ([conjI]@prems) 1)]);
+qed_goalw "conjIW" Intensional.thy intensional_rews "[| w |= P; w |= Q |] ==> w |= P & Q"
+ (fn prems => [REPEAT (resolve_tac ([conjI]@prems) 1)]);
-qed_goal "conjunct1W" Intensional.thy "(w |= P .& Q) ==> w |= P"
+qed_goal "conjunct1W" Intensional.thy "(w |= P & Q) ==> w |= P"
(fn prems => [cut_facts_tac prems 1,
rewrite_goals_tac intensional_rews,
etac conjunct1 1]);
-qed_goal "conjunct2W" Intensional.thy "(w |= P .& Q) ==> w |= Q"
+qed_goal "conjunct2W" Intensional.thy "(w |= P & Q) ==> w |= Q"
(fn prems => [cut_facts_tac prems 1,
rewrite_goals_tac intensional_rews,
etac conjunct2 1]);
qed_goal "conjEW" Intensional.thy
- "[| w |= P .& Q; [| w |= P; w |= Q |] ==> w |= R |] ==> w |= (R::('w::world) form)"
+ "[| w |= P & Q; [| w |= P; w |= Q |] ==> w |= R |] ==> w |= R"
(fn prems => [cut_facts_tac prems 1, resolve_tac prems 1,
etac conjunct1W 1, etac conjunct2W 1]);
(** Disjunction **)
-qed_goal "disjI1W" Intensional.thy "w |= P ==> w |= P .| Q"
- (fn [prem] => [rewrite_goals_tac intensional_rews,
- rtac disjI1 1,
- rtac prem 1]);
+qed_goalw "disjI1W" Intensional.thy intensional_rews "w |= P ==> w |= P | Q"
+ (fn [prem] => [REPEAT (resolve_tac [disjI1,prem] 1)]);
-qed_goal "disjI2W" Intensional.thy "w |= Q ==> w |= P .| Q"
- (fn [prem] => [rewrite_goals_tac intensional_rews,
- rtac disjI2 1,
- rtac prem 1]);
+qed_goalw "disjI2W" Intensional.thy intensional_rews "w |= Q ==> w |= P | Q"
+ (fn [prem] => [REPEAT (resolve_tac [disjI2,prem] 1)]);
qed_goal "disjEW" Intensional.thy
- "[| w |= P .| Q; P .-> R; Q .-> R |] ==> w |= R"
+ "[| w |= P | Q; |- P --> R; |- Q --> R |] ==> w |= R"
(fn prems => [cut_facts_tac prems 1,
REPEAT (dtac intD 1),
rewrite_goals_tac intensional_rews,
- fast_tac prop_cs 1]);
+ Blast_tac 1]);
(** Classical propositional logic **)
-qed_goal "classicalW" Intensional.thy "(.~P .-> P) ==> P::('w::world)form"
- (fn prems => [cut_facts_tac prems 1,
- rtac intI 1,
- dtac intD 1,
- rewrite_goals_tac intensional_rews,
- fast_tac prop_cs 1]);
+qed_goalw "classicalW" Intensional.thy (Valid_def::intensional_rews)
+ "!!P. |- ~P --> P ==> |- P"
+ (fn prems => [Blast_tac 1]);
-qed_goal "notnotDW" Intensional.thy ".~.~P ==> P::('w::world) form"
- (fn prems => [cut_facts_tac prems 1,
- rtac intI 1,
+qed_goal "notnotDW" Intensional.thy "!!P. |- ~~P ==> |- P"
+ (fn prems => [rtac intI 1,
dtac intD 1,
rewrite_goals_tac intensional_rews,
etac notnotD 1]);
-qed_goal "disjCIW" Intensional.thy "(w |= .~Q .-> P) ==> (w |= P.|Q)"
- (fn prems => [cut_facts_tac prems 1,
- rewrite_goals_tac intensional_rews,
- fast_tac prop_cs 1]);
+qed_goal "disjCIW" Intensional.thy "!!P Q. (w |= ~Q --> P) ==> (w |= P|Q)"
+ (fn prems => [rewrite_goals_tac intensional_rews,
+ Blast_tac 1]);
qed_goal "impCEW" Intensional.thy
- "[| P.->Q; (w |= .~P) ==> (w |= R); (w |= Q) ==> (w |= R) |] ==> w |= (R::('w::world) form)"
+ "[| |- P --> Q; (w |= ~P) ==> (w |= R); (w |= Q) ==> (w |= R) |] ==> w |= R"
(fn [a1,a2,a3] =>
[rtac (excluded_middle RS disjE) 1,
etac (rewrite_rule intensional_rews a2) 1,
rtac a3 1,
etac (a1 RS mpW) 1]);
-(* The following generates too many parse trees...
-
-qed_goal "iffCEW" Intensional.thy
- "[| P .= Q; \
+qed_goalw "iffCEW" Intensional.thy intensional_rews
+ "[| |- P = Q; \
\ [| (w |= P); (w |= Q) |] ==> (w |= R); \
-\ [| (w |= .~P); (w |= .~Q) |] ==> (w |= R) \
-\ |] ==> w |= (R::('w::world) form)"
-
-*)
+\ [| (w |= ~P); (w |= ~Q) |] ==> (w |= R) \
+\ |] ==> w |= R"
+ (fn [a1,a2,a3] =>
+ [rtac iffCE 1,
+ etac a2 2, atac 2,
+ etac a3 2, atac 2,
+ rtac (int_unlift a1) 1]);
qed_goal "case_split_thmW" Intensional.thy
- "[| P .-> Q; .~P .-> Q |] ==> Q::('w::world) form"
- (fn prems => [cut_facts_tac prems 1,
- rtac intI 1,
- REPEAT (dtac intD 1),
- rewrite_goals_tac intensional_rews,
- fast_tac prop_cs 1]);
+ "!!P. [| |- P --> Q; |- ~P --> Q |] ==> |- Q"
+ (fn _ => [rewrite_goals_tac (Valid_def::intensional_rews),
+ Blast_tac 1]);
fun case_tacW a = res_inst_tac [("P",a)] case_split_thmW;
(** Rigid quantifiers **)
-qed_goal "allIW" Intensional.thy "(!!x. P(x)) ==> RALL x. P(x)"
+qed_goal "allIW" Intensional.thy "(!!x. |- P x) ==> |- ! x. P(x)"
(fn [prem] => [rtac intI 1,
rewrite_goals_tac intensional_rews,
rtac allI 1,
- rtac (prem RS intE) 1]);
+ rtac (prem RS intD) 1]);
-qed_goal "specW" Intensional.thy "(RALL x. P(x)) ==> P(x)"
+qed_goal "specW" Intensional.thy "|- ! x. P x ==> |- P x"
(fn prems => [cut_facts_tac prems 1,
rtac intI 1,
dtac intD 1,
@@ -359,24 +329,24 @@
qed_goal "allEW" Intensional.thy
- "[| RALL x. P(x); P(x) ==> R |] ==> R::('w::world) form"
+ "[| |- ! x. P x; |- P x ==> |- R |] ==> |- R"
(fn major::prems=>
[ (REPEAT (resolve_tac (prems @ [major RS specW]) 1)) ]);
qed_goal "all_dupEW" Intensional.thy
- "[| RALL x. P(x); [| P(x); RALL x. P(x) |] ==> R |] ==> R::('w::world) form"
+ "[| |- ! x. P x; [| |- P x; |- ! x. P x |] ==> |- R |] ==> |- R"
(fn prems =>
[ (REPEAT (resolve_tac (prems @ (prems RL [specW])) 1)) ]);
-qed_goal "exIW" Intensional.thy "P(x) ==> REX x. P(x)"
+qed_goal "exIW" Intensional.thy "|- P x ==> |- ? x. P x"
(fn [prem] => [rtac intI 1,
rewrite_goals_tac intensional_rews,
rtac exI 1,
rtac (prem RS intD) 1]);
qed_goal "exEW" Intensional.thy
- "[| w |= REX x. P(x); !!x. P(x) .-> Q |] ==> w |= Q"
+ "[| w |= ? x. P x; !!x. |- P x --> Q |] ==> w |= Q"
(fn [major,minor] => [rtac exE 1,
rtac (rewrite_rule intensional_rews major) 1,
etac rev_mpW 1,
@@ -385,8 +355,7 @@
(** Classical quantifier reasoning **)
qed_goal "exCIW" Intensional.thy
- "(w |= (RALL x. .~P(x)) .-> P(a)) ==> w |= REX x. P(x)"
- (fn prems => [cut_facts_tac prems 1,
- rewrite_goals_tac intensional_rews,
- fast_tac HOL_cs 1]);
+ "!!P. w |= (! x. ~P x) --> P a ==> w |= ? x. P x"
+ (fn prems => [rewrite_goals_tac intensional_rews,
+ Blast_tac 1]);