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 = {* Attrib.thms >> (fn thms => K (METHOD (fn facts => (DEPTH_SOLVE (HEADGOAL (ares_tac (facts @ thms))))))) *} "" method_setup depth_solve1 = {* Attrib.thms >> (fn thms => K (METHOD (fn facts => (DEPTH_SOLVE_1 (HEADGOAL (ares_tac (facts @ thms))))))) *} "" method_setup strip_asms = {* Attrib.thms >> (fn thms => K (METHOD (fn facts => REPEAT (resolve_tac [@{thm strip_b}, @{thm strip_s}] 1 THEN DEPTH_SOLVE_1 (ares_tac (facts @ thms) 1))))) *} "" subsection {* Simple types *} schematic_lemma "A:* |- A->A : ?T" by (depth_solve rules) schematic_lemma "A:* |- Lam a:A. a : ?T" by (depth_solve rules) schematic_lemma "A:* B:* b:B |- Lam x:A. b : ?T" by (depth_solve rules) schematic_lemma "A:* b:A |- (Lam a:A. a)^b: ?T" by (depth_solve rules) schematic_lemma "A:* B:* c:A b:B |- (Lam x:A. b)^ c: ?T" by (depth_solve rules) schematic_lemma "A:* B:* |- Lam a:A. Lam b:B. a : ?T" by (depth_solve rules) subsection {* Second-order types *} schematic_lemma (in L2) "|- Lam A:*. Lam a:A. a : ?T" by (depth_solve rules) schematic_lemma (in L2) "A:* |- (Lam B:*.Lam b:B. b)^A : ?T" by (depth_solve rules) schematic_lemma (in L2) "A:* b:A |- (Lam B:*.Lam b:B. b) ^ A ^ b: ?T" by (depth_solve rules) schematic_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 *} schematic_lemma (in Lomega) "|- Lam A:*.A->A : ?T" by (depth_solve rules) schematic_lemma (in Lomega) "B:* |- (Lam A:*.A->A) ^ B : ?T" by (depth_solve rules) schematic_lemma (in Lomega) "B:* b:B |- (Lam y:B. b): ?T" by (depth_solve rules) schematic_lemma (in Lomega) "A:* F:*->* |- F^(F^A): ?T" by (depth_solve rules) schematic_lemma (in Lomega) "A:* |- Lam F:*->*.F^(F^A): ?T" by (depth_solve rules) subsection {* LP *} schematic_lemma (in LP) "A:* |- A -> * : ?T" by (depth_solve rules) schematic_lemma (in LP) "A:* P:A->* a:A |- P^a: ?T" by (depth_solve rules) schematic_lemma (in LP) "A:* P:A->A->* a:A |- Pi a:A. P^a^a: ?T" by (depth_solve rules) schematic_lemma (in LP) "A:* P:A->* Q:A->* |- Pi a:A. P^a -> Q^a: ?T" by (depth_solve rules) schematic_lemma (in LP) "A:* P:A->* |- Pi a:A. P^a -> P^a: ?T" by (depth_solve rules) schematic_lemma (in LP) "A:* P:A->* |- Lam a:A. Lam x:P^a. x: ?T" by (depth_solve rules) schematic_lemma (in LP) "A:* P:A->* Q:* |- (Pi a:A. P^a->Q) -> (Pi a:A. P^a) -> Q : ?T" by (depth_solve rules) schematic_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 *} schematic_lemma (in L2) "A:* B:* |- Pi C:*.(A->B->C)->C : ?T" by (depth_solve rules) schematic_lemma (in Lomega2) "|- Lam A:*.Lam B:*.Pi C:*.(A->B->C)->C : ?T" by (depth_solve rules) schematic_lemma (in Lomega2) "|- Lam A:*.Lam B:*.Lam x:A. Lam y:B. x : ?T" by (depth_solve rules) schematic_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 *} schematic_lemma (in LP2) "A:* P:A->* |- Lam a:A. P^a->(Pi A:*.A) : ?T" by (depth_solve rules) schematic_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) schematic_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 *} schematic_lemma (in LPomega) "A:* |- Lam P:A->A->*.Lam a:A. P^a^a : ?T" by (depth_solve rules) schematic_lemma (in LPomega) "|- Lam A:*.Lam P:A->A->*.Lam a:A. P^a^a : ?T" by (depth_solve rules) subsection {* Constructions *} schematic_lemma (in CC) "|- Lam A:*.Lam P:A->*.Lam a:A. P^a->Pi P:*.P: ?T" by (depth_solve rules) schematic_lemma (in CC) "|- Lam A:*.Lam P:A->*.Pi a:A. P^a: ?T" by (depth_solve rules) schematic_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 *} schematic_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) schematic_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) schematic_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