Used Datatype functor to define propositional logic terms.
--- a/ex/PL.ML Tue Mar 22 08:26:25 1994 +0100
+++ b/ex/PL.ML Tue Mar 22 08:28:31 1994 +0100
@@ -1,10 +1,9 @@
-(* Title: HOL/ex/prop-log.ML
+(* Title: HOL/ex/pl.ML
ID: $Id$
Author: Tobias Nipkow & Lawrence C Paulson
- Copyright 1993 TU Muenchen & University of Cambridge
+ Copyright 1994 TU Muenchen & University of Cambridge
-For ex/prop-log.thy. Inductive definition of propositional logic.
-Soundness and completeness w.r.t. truth-tables.
+Soundness and completeness of propositional logic w.r.t. truth-tables.
Prove: If H|=p then G|=p where G:Fin(H)
*)
@@ -136,7 +135,8 @@
(** The function eval **)
-val pl_ss = set_ss addsimps [pl_rec_var,pl_rec_false,pl_rec_imp];
+val pl_ss = set_ss addsimps [pl_rec_var,pl_rec_false,pl_rec_imp]
+ @ PL0.inject @ PL0.ineq;
goalw PL.thy [eval_def] "tt[false] = False";
by (simp_tac pl_ss 1);
@@ -213,8 +213,7 @@
(*This formulation is required for strong induction hypotheses*)
goal PL.thy "hyps(p,tt) |- if(tt[p], p, p->false)";
by (rtac (expand_if RS iffD2) 1);
-by(res_inst_tac[("x","p")]spec 1);
-by (rtac pl_ind 1);
+by(PL0.induct_tac "p" 1);
by (ALLGOALS (simp_tac (pl_ss addsimps [conseq_I, conseq_H])));
by (fast_tac (set_cs addIs [weaken_left_Un1, weaken_left_Un2,
weaken_right, imp_false]
@@ -250,25 +249,24 @@
(*For the case hyps(p,t)-insert(#v,Y) |- p;
we also have hyps(p,t)-{#v} <= hyps(p, t-{v}) *)
-goal PL.thy "!p.hyps(p, t-{v}) <= insert((#v)->false, hyps(p,t)-{#v})";
-by (rtac pl_ind 1);
+goal PL.thy "hyps(p, t-{v}) <= insert((#v)->false, hyps(p,t)-{#v})";
+by (PL0.induct_tac "p" 1);
by (simp_tac pl_ss 1);
by (simp_tac (pl_ss setloop (split_tac [expand_if])) 1);
-by (fast_tac (set_cs addSEs [sym RS var_neq_imp] addSDs [var_inject]) 1);
by (simp_tac pl_ss 1);
by (fast_tac set_cs 1);
-val hyps_Diff = result() RS spec;
+val hyps_Diff = result();
(*For the case hyps(p,t)-insert(#v -> false,Y) |- p;
we also have hyps(p,t)-{(#v)->false} <= hyps(p, insert(v,t)) *)
-goal PL.thy "!p.hyps(p, insert(v,t)) <= insert(#v, hyps(p,t)-{(#v)->false})";
-by (rtac pl_ind 1);
+goal PL.thy "hyps(p, insert(v,t)) <= insert(#v, hyps(p,t)-{(#v)->false})";
+by (PL0.induct_tac "p" 1);
by (simp_tac pl_ss 1);
-by (simp_tac (pl_ss setloop (split_tac [expand_if])) 1);
-by (fast_tac (set_cs addSEs [var_neq_imp, imp_inject] addSDs [var_inject]) 1);
+by (simp_tac (pl_ss addsimps [insert_subset]
+ setloop (split_tac [expand_if])) 1);
by (simp_tac pl_ss 1);
by (fast_tac set_cs 1);
-val hyps_insert = result() RS spec;
+val hyps_insert = result();
(** Two lemmas for use with weaken_left **)
@@ -282,11 +280,11 @@
(*The set hyps(p,t) is finite, and elements have the form #v or #v->false;
could probably prove the stronger hyps(p,t) : Fin(hyps(p,{}) Un hyps(p,nat))*)
-goal PL.thy "!p.hyps(p,t) : Fin(UN v:{x.True}. {#v, (#v)->false})";
-by (rtac pl_ind 1);
+goal PL.thy "hyps(p,t) : Fin(UN v:{x.True}. {#v, (#v)->false})";
+by (PL0.induct_tac "p" 1);
by (ALLGOALS (simp_tac (pl_ss setloop (split_tac [expand_if])) THEN'
fast_tac (set_cs addSIs [Fin_0I, Fin_insertI, Fin_UnI])));
-val hyps_finite = result() RS spec;
+val hyps_finite = result();
val Diff_weaken_left = subset_refl RSN (2, Diff_mono) RS weaken_left;
@@ -341,4 +339,3 @@
writeln"Reached end of file.";
-
--- a/ex/PL.thy Tue Mar 22 08:26:25 1994 +0100
+++ b/ex/PL.thy Tue Mar 22 08:28:31 1994 +0100
@@ -1,24 +1,18 @@
-(* Title: HOL/ex/prop-log
+(* Title: HOL/ex/pl.thy
ID: $Id$
Author: Tobias Nipkow
- Copyright 1991 University of Cambridge
+ Copyright 1994 TU Muenchen
Inductive definition of propositional logic.
-
*)
-PL = Finite +
-types pl 1
-arities pl :: (term)term
+PL = Finite + PL0 +
consts
- false :: "'a pl"
- "->" :: "['a pl,'a pl] => 'a pl" (infixr 90)
- var :: "'a => 'a pl" ("#_")
- pl_rec :: "['a pl,'a => 'b, 'b, ['b,'b] => 'b] => 'b"
axK,axS,axDN:: "'a pl set"
ruleMP,thms :: "'a pl set => 'a pl set"
"|-" :: "['a pl set, 'a pl] => bool" (infixl 50)
"|=" :: "['a pl set, 'a pl] => bool" (infixl 50)
+ pl_rec :: "['a pl,'a => 'b, 'b, ['b,'b] => 'b] => 'b"
eval :: "['a set, 'a pl] => bool" ("_[_]" [100,0] 100)
hyps :: "['a pl, 'a set] => 'a pl set"
rules
@@ -39,17 +33,13 @@
sat_def "H |= p == (!tt. (!q:H. tt[q]) --> tt[p])"
-pl_rec_var "pl_rec(#v,f,y,z) = f(v)"
-pl_rec_false "pl_rec(false,f,y,z) = y"
-pl_rec_imp "pl_rec(p->q,f,y,g) = g(pl_rec(p,f,y,g),pl_rec(q,f,y,g))"
-
-eval_def "tt[p] == pl_rec(p, %v.v:tt, False, op -->)"
+ pl_rec_var "pl_rec(#v,f,y,z) = f(v)"
+ pl_rec_false "pl_rec(false,f,y,z) = y"
+ pl_rec_imp "pl_rec(p->q,f,y,g) = g(pl_rec(p,f,y,g),pl_rec(q,f,y,g))"
-hyps_def
- "hyps(p,tt) == pl_rec(p, %a. {if(a:tt, #a, (#a)->false)}, {}, op Un)"
+ eval_def "tt[p] == pl_rec(p, %v.v:tt, False, op -->)"
-var_inject "(#v = #w) ==> v = w"
-imp_inject "[| (p -> q) = (p' -> q'); [| p = p'; q = q' |] ==> R |] ==> R"
-var_neq_imp "(#v = (p -> q)) ==> R"
-pl_ind "[| P(false); !!v. P(#v); !!p q. P(p)-->P(q)-->P(p->q)|] ==> !t.P(t)"
+ hyps_def
+ "hyps(p,tt) == pl_rec(p, %a. {if(a:tt, #a, (#a)->false)}, {}, op Un)"
+
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/ex/PL0.ML Tue Mar 22 08:28:31 1994 +0100
@@ -0,0 +1,12 @@
+(* Title: HOL/ex/pl0.ML
+ ID: $Id$
+ Author: Tobias Nipkow
+ Copyright 1994 TU Muenchen
+
+Inductive definition of propositional logic formulae.
+*)
+
+structure PL0 = DeclaredDatatype
+(val base = PL0.thy
+ val data = "'a pl = false | var('a) | \"op->\"('a pl,'a pl)"
+);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/ex/PL0.thy Tue Mar 22 08:28:31 1994 +0100
@@ -0,0 +1,16 @@
+(* Title: HOL/ex/pl0.thy
+ ID: $Id$
+ Author: Tobias Nipkow
+ Copyright 1994 TU Muenchen
+
+Syntax of propositional logic formulae.
+*)
+
+PL0 = HOL +
+types 'a pl
+arities pl :: (term)term
+consts
+ false :: "'a pl"
+ "->" :: "['a pl,'a pl] => 'a pl" (infixr 90)
+ var :: "'a => 'a pl" ("#_")
+end
--- a/ex/pl.ML Tue Mar 22 08:26:25 1994 +0100
+++ b/ex/pl.ML Tue Mar 22 08:28:31 1994 +0100
@@ -1,10 +1,9 @@
-(* Title: HOL/ex/prop-log.ML
+(* Title: HOL/ex/pl.ML
ID: $Id$
Author: Tobias Nipkow & Lawrence C Paulson
- Copyright 1993 TU Muenchen & University of Cambridge
+ Copyright 1994 TU Muenchen & University of Cambridge
-For ex/prop-log.thy. Inductive definition of propositional logic.
-Soundness and completeness w.r.t. truth-tables.
+Soundness and completeness of propositional logic w.r.t. truth-tables.
Prove: If H|=p then G|=p where G:Fin(H)
*)
@@ -136,7 +135,8 @@
(** The function eval **)
-val pl_ss = set_ss addsimps [pl_rec_var,pl_rec_false,pl_rec_imp];
+val pl_ss = set_ss addsimps [pl_rec_var,pl_rec_false,pl_rec_imp]
+ @ PL0.inject @ PL0.ineq;
goalw PL.thy [eval_def] "tt[false] = False";
by (simp_tac pl_ss 1);
@@ -213,8 +213,7 @@
(*This formulation is required for strong induction hypotheses*)
goal PL.thy "hyps(p,tt) |- if(tt[p], p, p->false)";
by (rtac (expand_if RS iffD2) 1);
-by(res_inst_tac[("x","p")]spec 1);
-by (rtac pl_ind 1);
+by(PL0.induct_tac "p" 1);
by (ALLGOALS (simp_tac (pl_ss addsimps [conseq_I, conseq_H])));
by (fast_tac (set_cs addIs [weaken_left_Un1, weaken_left_Un2,
weaken_right, imp_false]
@@ -250,25 +249,24 @@
(*For the case hyps(p,t)-insert(#v,Y) |- p;
we also have hyps(p,t)-{#v} <= hyps(p, t-{v}) *)
-goal PL.thy "!p.hyps(p, t-{v}) <= insert((#v)->false, hyps(p,t)-{#v})";
-by (rtac pl_ind 1);
+goal PL.thy "hyps(p, t-{v}) <= insert((#v)->false, hyps(p,t)-{#v})";
+by (PL0.induct_tac "p" 1);
by (simp_tac pl_ss 1);
by (simp_tac (pl_ss setloop (split_tac [expand_if])) 1);
-by (fast_tac (set_cs addSEs [sym RS var_neq_imp] addSDs [var_inject]) 1);
by (simp_tac pl_ss 1);
by (fast_tac set_cs 1);
-val hyps_Diff = result() RS spec;
+val hyps_Diff = result();
(*For the case hyps(p,t)-insert(#v -> false,Y) |- p;
we also have hyps(p,t)-{(#v)->false} <= hyps(p, insert(v,t)) *)
-goal PL.thy "!p.hyps(p, insert(v,t)) <= insert(#v, hyps(p,t)-{(#v)->false})";
-by (rtac pl_ind 1);
+goal PL.thy "hyps(p, insert(v,t)) <= insert(#v, hyps(p,t)-{(#v)->false})";
+by (PL0.induct_tac "p" 1);
by (simp_tac pl_ss 1);
-by (simp_tac (pl_ss setloop (split_tac [expand_if])) 1);
-by (fast_tac (set_cs addSEs [var_neq_imp, imp_inject] addSDs [var_inject]) 1);
+by (simp_tac (pl_ss addsimps [insert_subset]
+ setloop (split_tac [expand_if])) 1);
by (simp_tac pl_ss 1);
by (fast_tac set_cs 1);
-val hyps_insert = result() RS spec;
+val hyps_insert = result();
(** Two lemmas for use with weaken_left **)
@@ -282,11 +280,11 @@
(*The set hyps(p,t) is finite, and elements have the form #v or #v->false;
could probably prove the stronger hyps(p,t) : Fin(hyps(p,{}) Un hyps(p,nat))*)
-goal PL.thy "!p.hyps(p,t) : Fin(UN v:{x.True}. {#v, (#v)->false})";
-by (rtac pl_ind 1);
+goal PL.thy "hyps(p,t) : Fin(UN v:{x.True}. {#v, (#v)->false})";
+by (PL0.induct_tac "p" 1);
by (ALLGOALS (simp_tac (pl_ss setloop (split_tac [expand_if])) THEN'
fast_tac (set_cs addSIs [Fin_0I, Fin_insertI, Fin_UnI])));
-val hyps_finite = result() RS spec;
+val hyps_finite = result();
val Diff_weaken_left = subset_refl RSN (2, Diff_mono) RS weaken_left;
@@ -341,4 +339,3 @@
writeln"Reached end of file.";
-
--- a/ex/pl.thy Tue Mar 22 08:26:25 1994 +0100
+++ b/ex/pl.thy Tue Mar 22 08:28:31 1994 +0100
@@ -1,24 +1,18 @@
-(* Title: HOL/ex/prop-log
+(* Title: HOL/ex/pl.thy
ID: $Id$
Author: Tobias Nipkow
- Copyright 1991 University of Cambridge
+ Copyright 1994 TU Muenchen
Inductive definition of propositional logic.
-
*)
-PL = Finite +
-types pl 1
-arities pl :: (term)term
+PL = Finite + PL0 +
consts
- false :: "'a pl"
- "->" :: "['a pl,'a pl] => 'a pl" (infixr 90)
- var :: "'a => 'a pl" ("#_")
- pl_rec :: "['a pl,'a => 'b, 'b, ['b,'b] => 'b] => 'b"
axK,axS,axDN:: "'a pl set"
ruleMP,thms :: "'a pl set => 'a pl set"
"|-" :: "['a pl set, 'a pl] => bool" (infixl 50)
"|=" :: "['a pl set, 'a pl] => bool" (infixl 50)
+ pl_rec :: "['a pl,'a => 'b, 'b, ['b,'b] => 'b] => 'b"
eval :: "['a set, 'a pl] => bool" ("_[_]" [100,0] 100)
hyps :: "['a pl, 'a set] => 'a pl set"
rules
@@ -39,17 +33,13 @@
sat_def "H |= p == (!tt. (!q:H. tt[q]) --> tt[p])"
-pl_rec_var "pl_rec(#v,f,y,z) = f(v)"
-pl_rec_false "pl_rec(false,f,y,z) = y"
-pl_rec_imp "pl_rec(p->q,f,y,g) = g(pl_rec(p,f,y,g),pl_rec(q,f,y,g))"
-
-eval_def "tt[p] == pl_rec(p, %v.v:tt, False, op -->)"
+ pl_rec_var "pl_rec(#v,f,y,z) = f(v)"
+ pl_rec_false "pl_rec(false,f,y,z) = y"
+ pl_rec_imp "pl_rec(p->q,f,y,g) = g(pl_rec(p,f,y,g),pl_rec(q,f,y,g))"
-hyps_def
- "hyps(p,tt) == pl_rec(p, %a. {if(a:tt, #a, (#a)->false)}, {}, op Un)"
+ eval_def "tt[p] == pl_rec(p, %v.v:tt, False, op -->)"
-var_inject "(#v = #w) ==> v = w"
-imp_inject "[| (p -> q) = (p' -> q'); [| p = p'; q = q' |] ==> R |] ==> R"
-var_neq_imp "(#v = (p -> q)) ==> R"
-pl_ind "[| P(false); !!v. P(#v); !!p q. P(p)-->P(q)-->P(p->q)|] ==> !t.P(t)"
+ hyps_def
+ "hyps(p,tt) == pl_rec(p, %a. {if(a:tt, #a, (#a)->false)}, {}, op Un)"
+
end
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/ex/pl0.ML Tue Mar 22 08:28:31 1994 +0100
@@ -0,0 +1,12 @@
+(* Title: HOL/ex/pl0.ML
+ ID: $Id$
+ Author: Tobias Nipkow
+ Copyright 1994 TU Muenchen
+
+Inductive definition of propositional logic formulae.
+*)
+
+structure PL0 = DeclaredDatatype
+(val base = PL0.thy
+ val data = "'a pl = false | var('a) | \"op->\"('a pl,'a pl)"
+);
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/ex/pl0.thy Tue Mar 22 08:28:31 1994 +0100
@@ -0,0 +1,16 @@
+(* Title: HOL/ex/pl0.thy
+ ID: $Id$
+ Author: Tobias Nipkow
+ Copyright 1994 TU Muenchen
+
+Syntax of propositional logic formulae.
+*)
+
+PL0 = HOL +
+types 'a pl
+arities pl :: (term)term
+consts
+ false :: "'a pl"
+ "->" :: "['a pl,'a pl] => 'a pl" (infixr 90)
+ var :: "'a => 'a pl" ("#_")
+end