Theory Example

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theory Example
imports Cube
begin

(* $Id: Example.thy,v 1.1 2005/09/17 10:50:57 wenzelm 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

Simple types

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

Second-order types

lemma

  PROP L2 ==> Lam A:*. Lam a:A. a: Pi x:*. x -> x

lemma

  PROP L2 ==> A: *  |- (Lam B:*. Lam b:B. b) ^ A: A -> A

lemma

  PROP L2 ==> A: * b: A  |- (Lam B:*. Lam b:B. b) ^ A ^ b: A

lemma

  PROP L2
  ==> Lam B:*. Lam a:Pi A:*. A. a ^ ((Pi A:*. A) -> B) ^ a:
      Pi x:*. (Pi A:*. A) -> x

Weakly higher-order propositional logic

lemma

  PROP Lomega ==> Lam A:*. A -> A: * -> *

lemma

  PROP Lomega ==> B: *  |- (Lam A:*. A -> A) ^ B: *

lemma

  PROP Lomega ==> B: * b: B  |- Lam y:B. b: B -> B

lemma

  PROP Lomega ==> A: * F: * -> *  |- F ^ (F ^ A): *

lemma

  PROP Lomega ==> A: *  |- Lam F:* -> *. F ^ (F ^ A): (* -> *) -> *

LP

lemma

  PROP LP ==> A: *  |- A -> *: []

lemma

  PROP LP ==> A: * P: A -> * a: A  |- P ^ a: *

lemma

  PROP LP ==> A: * P: A -> A -> * a: A  |- Pi a:A. P ^ a ^ a: *

lemma

  PROP LP ==> A: * P: A -> * Q: A -> *  |- Pi a:A. P ^ a -> Q ^ a: *

lemma

  PROP LP ==> A: * P: A -> *  |- Pi a:A. P ^ a -> P ^ a: *

lemma

  PROP LP ==> A: * P: A -> *  |- Lam a:A. Lam x:P ^ a. x: Pi x:A. P ^ x -> P ^ x

lemma

  PROP LP
  ==> A: * P: A -> * Q: *  |- (Pi a:A. P ^ a -> Q) -> (Pi a:A. P ^ a) -> Q: *

lemma

  PROP LP
  ==> 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

Omega-order types

lemma

  PROP L2 ==> A: * B: *  |- Pi C:*. (A -> B -> C) -> C: *

lemma

  PROP Lomega2 ==> Lam A:*. Lam B:*. Pi C:*. (A -> B -> C) -> C: * -> * -> *

lemma

  PROP Lomega2
  ==> Lam A:*. Lam B:*. Lam x:A. Lam y:B. x: Pi x:*. Pi xa:*. x -> xa -> x

lemma

  PROP Lomega2
  ==> 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

Second-order Predicate Logic

lemma

  PROP LP2 ==> A: * P: A -> *  |- Lam a:A. P ^ a -> Pi A:*. A: A -> *

lemma

  PROP 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:
         *

lemma

  PROP LP2
  ==> 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

LPomega

lemma

  PROP LPomega
  ==> A: *  |- Lam P:A -> A -> *. Lam a:A. P ^ a ^ a: (A -> A -> *) -> A -> *

lemma

  PROP LPomega
  ==> Lam A:*. Lam P:A -> A -> *. Lam a:A. P ^ a ^ a:
      Pi x:*. (x -> x -> *) -> x -> *

Constructions

lemma

  PROP CC
  ==> Lam A:*. Lam P:A -> *. Lam a:A. P ^ a -> Pi P:*. P:
      Pi x:*. (x -> *) -> x -> *

lemma

  PROP CC ==> Lam A:*. Lam P:A -> *. Pi a:A. P ^ a: Pi x:*. (x -> *) -> *

lemma

  PROP CC
  ==> A: * P: A -> * a: A  |- Lam x:Pi a:A. P ^ a. x ^ a: (Pi a:A. P ^ a) -> P ^ a

Some random examples

lemma

  PROP 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:
         A -> *

lemma

  PROP 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:
      Pi x:*. x -> (x -> x) -> x -> *

lemma

  PROP LP2
  ==> 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