(* $Id: Example.thy,v 1.3 2006/06/22 16:48:25 ballarin Exp $ *) header {* Lambda Cube Examples *} theory Example 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
lemma
A: * |- A -> A: *
lemma
A: * |- Lam a:A. a: A -> A
lemma
A: * B: * b: B |- Lam x:A. b: A -> B
lemma
A: * b: A |- (Lam a:A. a) ^ b: A
lemma
A: * B: * c: A b: B |- (Lam x:A. b) ^ c: B
lemma
A: * B: * |- Lam a:A. Lam b:B. a: A -> B -> A
lemma
Lam A:*. Lam a:A. a: Pi x:*. x -> x
lemma
A: * |- (Lam B:*. Lam b:B. b) ^ A: A -> A
lemma
A: * b: A |- (Lam B:*. Lam b:B. b) ^ A ^ b: A
lemma
Lam B:*. Lam a:Pi A:*. A. a ^ ((Pi A:*. A) -> B) ^ a: Pi x:*. (Pi A:*. A) -> x
lemma
Lam A:*. A -> A: * -> *
lemma
B: * |- (Lam A:*. A -> A) ^ B: *
lemma
B: * b: B |- Lam y:B. b: B -> B
lemma
A: * F: * -> * |- F ^ (F ^ A): *
lemma
A: * |- Lam F:* -> *. F ^ (F ^ A): (* -> *) -> *
lemma
A: * |- A -> *: []
lemma
A: * P: A -> * a: A |- P ^ a: *
lemma
A: * P: A -> A -> * a: A |- Pi a:A. P ^ a ^ a: *
lemma
A: * P: A -> * Q: A -> * |- Pi a:A. P ^ a -> Q ^ a: *
lemma
A: * P: A -> * |- Pi a:A. P ^ a -> P ^ a: *
lemma
A: * P: A -> * |- Lam a:A. Lam x:P ^ a. x: Pi x:A. P ^ x -> P ^ x
lemma
A: * P: A -> * Q: * |- (Pi a:A. P ^ a -> Q) -> (Pi a:A. P ^ a) -> Q: *
lemma
A: * P: A -> * Q: * a0.0: A
|- Lam x:Pi a:A. P ^ a -> Q. Lam y:Pi a:A. P ^ a. x ^ a0.0 ^ (y ^ a0.0):
(Pi a:A. P ^ a -> Q) -> (Pi a:A. P ^ a) -> Q
lemma
A: * B: * |- Pi C:*. (A -> B -> C) -> C: *
lemma
Lam A:*. Lam B:*. Pi C:*. (A -> B -> C) -> C: * -> * -> *
lemma
Lam A:*. Lam B:*. Lam x:A. Lam y:B. x: Pi x:*. Pi xa:*. x -> xa -> x
lemma
A: * B: *
|- Lam x:A -> B. Lam xa:B -> Pi P:*. P. Lam xb:A. xa ^ (x ^ xb):
(A -> B) -> (B -> Pi P:*. P) -> A -> Pi P:*. P
lemma
A: * P: A -> * |- Lam a:A. P ^ a -> Pi A:*. A: A -> *
lemma
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:
*
lemma
A: * P: A -> A -> *
|- Lam x:Pi a:A. Pi b:A. P ^ a ^ b -> P ^ b ^ a -> Pi P:*. P.
Lam xa:A. Lam xb:P ^ xa ^ xa. x ^ xa ^ xa ^ xb ^ xb:
(Pi a:A. Pi b:A. P ^ a ^ b -> P ^ b ^ a -> Pi P:*. P) ->
Pi a:A. P ^ a ^ a -> Pi P:*. P
lemma
A: * |- Lam P:A -> A -> *. Lam a:A. P ^ a ^ a: (A -> A -> *) -> A -> *
lemma
Lam A:*. Lam P:A -> A -> *. Lam a:A. P ^ a ^ a: Pi x:*. (x -> x -> *) -> x -> *
lemma
Lam A:*. Lam P:A -> *. Lam a:A. P ^ a -> Pi P:*. P: Pi x:*. (x -> *) -> x -> *
lemma
Lam A:*. Lam P:A -> *. Pi a:A. P ^ a: Pi x:*. (x -> *) -> *
lemma
A: * P: A -> * a: A |- Lam x:Pi a:A. P ^ a. x ^ a: (Pi a:A. P ^ a) -> P ^ a
lemma
A: * c: A f: A -> A
|- Lam a:A. Pi P:A -> *. P ^ c -> (Pi x:A. P ^ x -> P ^ (f ^ x)) -> P ^ a:
A -> *
lemma
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:
Pi x:*. x -> (x -> x) -> x -> *
lemma
A: * a: A b: A
|- Lam x:Pi P:A -> *. P ^ a -> P ^ b.
x ^ (Lam x:A. Pi Q:A -> *. Q ^ x -> Q ^ a) ^
(Lam xa:A -> *. Lam xb:xa ^ a. xb):
(Pi P:A -> *. P ^ a -> P ^ b) -> Pi P:A -> *. P ^ b -> P ^ a