--- a/src/Cube/ex/ROOT.ML Sat Sep 17 12:50:57 2005 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,4 +0,0 @@
-
-(* $Id$ *)
-
-time_use_thy "ex";
--- a/src/Cube/ex/ex.thy Sat Sep 17 12:50:57 2005 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,240 +0,0 @@
-
-(* $Id$ *)
-
-header {* Lambda Cube Examples *}
-
-theory ex
-imports Cube
-begin
-
-text {*
- Examples taken from:
-
- H. Barendregt. Introduction to Generalised Type Systems.
- J. Functional Programming.
-*}
-
-method_setup depth_solve = {*
- Method.thms_args (fn thms => Method.METHOD (fn facts =>
- (DEPTH_SOLVE (HEADGOAL (ares_tac (facts @ thms))))))
-*} ""
-
-method_setup depth_solve1 = {*
- Method.thms_args (fn thms => Method.METHOD (fn facts =>
- (DEPTH_SOLVE_1 (HEADGOAL (ares_tac (facts @ thms))))))
-*} ""
-
-method_setup strip_asms = {*
- let val strip_b = thm "strip_b" and strip_s = thm "strip_s" in
- Method.thms_args (fn thms => Method.METHOD (fn facts =>
- REPEAT (resolve_tac [strip_b, strip_s] 1 THEN DEPTH_SOLVE_1 (ares_tac (facts @ thms) 1))))
- end
-*} ""
-
-
-subsection {* Simple types *}
-
-lemma "A:* |- A->A : ?T"
- by (depth_solve rules)
-
-lemma "A:* |- Lam a:A. a : ?T"
- by (depth_solve rules)
-
-lemma "A:* B:* b:B |- Lam x:A. b : ?T"
- by (depth_solve rules)
-
-lemma "A:* b:A |- (Lam a:A. a)^b: ?T"
- by (depth_solve rules)
-
-lemma "A:* B:* c:A b:B |- (Lam x:A. b)^ c: ?T"
- by (depth_solve rules)
-
-lemma "A:* B:* |- Lam a:A. Lam b:B. a : ?T"
- by (depth_solve rules)
-
-
-subsection {* Second-order types *}
-
-lemma (in L2) "|- Lam A:*. Lam a:A. a : ?T"
- by (depth_solve rules)
-
-lemma (in L2) "A:* |- (Lam B:*.Lam b:B. b)^A : ?T"
- by (depth_solve rules)
-
-lemma (in L2) "A:* b:A |- (Lam B:*.Lam b:B. b) ^ A ^ b: ?T"
- by (depth_solve rules)
-
-lemma (in L2) "|- Lam B:*.Lam a:(Pi A:*.A).a ^ ((Pi A:*.A)->B) ^ a: ?T"
- by (depth_solve rules)
-
-
-subsection {* Weakly higher-order propositional logic *}
-
-lemma (in Lomega) "|- Lam A:*.A->A : ?T"
- by (depth_solve rules)
-
-lemma (in Lomega) "B:* |- (Lam A:*.A->A) ^ B : ?T"
- by (depth_solve rules)
-
-lemma (in Lomega) "B:* b:B |- (Lam y:B. b): ?T"
- by (depth_solve rules)
-
-lemma (in Lomega) "A:* F:*->* |- F^(F^A): ?T"
- by (depth_solve rules)
-
-lemma (in Lomega) "A:* |- Lam F:*->*.F^(F^A): ?T"
- by (depth_solve rules)
-
-
-subsection {* LP *}
-
-lemma (in LP) "A:* |- A -> * : ?T"
- by (depth_solve rules)
-
-lemma (in LP) "A:* P:A->* a:A |- P^a: ?T"
- by (depth_solve rules)
-
-lemma (in LP) "A:* P:A->A->* a:A |- Pi a:A. P^a^a: ?T"
- by (depth_solve rules)
-
-lemma (in LP) "A:* P:A->* Q:A->* |- Pi a:A. P^a -> Q^a: ?T"
- by (depth_solve rules)
-
-lemma (in LP) "A:* P:A->* |- Pi a:A. P^a -> P^a: ?T"
- by (depth_solve rules)
-
-lemma (in LP) "A:* P:A->* |- Lam a:A. Lam x:P^a. x: ?T"
- by (depth_solve rules)
-
-lemma (in LP) "A:* P:A->* Q:* |- (Pi a:A. P^a->Q) -> (Pi a:A. P^a) -> Q : ?T"
- by (depth_solve rules)
-
-lemma (in LP) "A:* P:A->* Q:* a0:A |-
- Lam x:Pi a:A. P^a->Q. Lam y:Pi a:A. P^a. x^a0^(y^a0): ?T"
- by (depth_solve rules)
-
-
-subsection {* Omega-order types *}
-
-lemma (in L2) "A:* B:* |- Pi C:*.(A->B->C)->C : ?T"
- by (depth_solve rules)
-
-lemma (in Lomega2) "|- Lam A:*.Lam B:*.Pi C:*.(A->B->C)->C : ?T"
- by (depth_solve rules)
-
-lemma (in Lomega2) "|- Lam A:*.Lam B:*.Lam x:A. Lam y:B. x : ?T"
- by (depth_solve rules)
-
-lemma (in Lomega2) "A:* B:* |- ?p : (A->B) -> ((B->Pi P:*.P)->(A->Pi P:*.P))"
- apply (strip_asms rules)
- apply (rule lam_ss)
- apply (depth_solve1 rules)
- prefer 2
- apply (depth_solve1 rules)
- apply (rule lam_ss)
- apply (depth_solve1 rules)
- prefer 2
- apply (depth_solve1 rules)
- apply (rule lam_ss)
- apply assumption
- prefer 2
- apply (depth_solve1 rules)
- apply (erule pi_elim)
- apply assumption
- apply (erule pi_elim)
- apply assumption
- apply assumption
- done
-
-
-subsection {* Second-order Predicate Logic *}
-
-lemma (in LP2) "A:* P:A->* |- Lam a:A. P^a->(Pi A:*.A) : ?T"
- by (depth_solve rules)
-
-lemma (in LP2) "A:* P:A->A->* |-
- (Pi a:A. Pi b:A. P^a^b->P^b^a->Pi P:*.P) -> Pi a:A. P^a^a->Pi P:*.P : ?T"
- by (depth_solve rules)
-
-lemma (in LP2) "A:* P:A->A->* |-
- ?p: (Pi a:A. Pi b:A. P^a^b->P^b^a->Pi P:*.P) -> Pi a:A. P^a^a->Pi P:*.P"
- -- {* Antisymmetry implies irreflexivity: *}
- apply (strip_asms rules)
- apply (rule lam_ss)
- apply (depth_solve1 rules)
- prefer 2
- apply (depth_solve1 rules)
- apply (rule lam_ss)
- apply assumption
- prefer 2
- apply (depth_solve1 rules)
- apply (rule lam_ss)
- apply (depth_solve1 rules)
- prefer 2
- apply (depth_solve1 rules)
- apply (erule pi_elim, assumption, assumption?)+
- done
-
-
-subsection {* LPomega *}
-
-lemma (in LPomega) "A:* |- Lam P:A->A->*.Lam a:A. P^a^a : ?T"
- by (depth_solve rules)
-
-lemma (in LPomega) "|- Lam A:*.Lam P:A->A->*.Lam a:A. P^a^a : ?T"
- by (depth_solve rules)
-
-
-subsection {* Constructions *}
-
-lemma (in CC) "|- Lam A:*.Lam P:A->*.Lam a:A. P^a->Pi P:*.P: ?T"
- by (depth_solve rules)
-
-lemma (in CC) "|- Lam A:*.Lam P:A->*.Pi a:A. P^a: ?T"
- by (depth_solve rules)
-
-lemma (in CC) "A:* P:A->* a:A |- ?p : (Pi a:A. P^a)->P^a"
- apply (strip_asms rules)
- apply (rule lam_ss)
- apply (depth_solve1 rules)
- prefer 2
- apply (depth_solve1 rules)
- apply (erule pi_elim, assumption, assumption)
- done
-
-
-subsection {* Some random examples *}
-
-lemma (in LP2) "A:* c:A f:A->A |-
- Lam a:A. Pi P:A->*.P^c -> (Pi x:A. P^x->P^(f^x)) -> P^a : ?T"
- by (depth_solve rules)
-
-lemma (in CC) "Lam A:*.Lam c:A. Lam f:A->A.
- Lam a:A. Pi P:A->*.P^c -> (Pi x:A. P^x->P^(f^x)) -> P^a : ?T"
- by (depth_solve rules)
-
-lemma (in LP2)
- "A:* a:A b:A |- ?p: (Pi P:A->*.P^a->P^b) -> (Pi P:A->*.P^b->P^a)"
- -- {* Symmetry of Leibnitz equality *}
- apply (strip_asms rules)
- apply (rule lam_ss)
- apply (depth_solve1 rules)
- prefer 2
- apply (depth_solve1 rules)
- apply (erule_tac a = "Lam x:A. Pi Q:A->*.Q^x->Q^a" in pi_elim)
- apply (depth_solve1 rules)
- apply (unfold beta)
- apply (erule imp_elim)
- apply (rule lam_bs)
- apply (depth_solve1 rules)
- prefer 2
- apply (depth_solve1 rules)
- apply (rule lam_ss)
- apply (depth_solve1 rules)
- prefer 2
- apply (depth_solve1 rules)
- apply assumption
- apply assumption
- done
-
-end