lean2/src/library/basic_thms.cpp

/*
Copyright (c) 2013 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.

Author: Leonardo de Moura
*/
#include "kernel/environment.h"
#include "kernel/abstract.h"
#include "kernel/type_checker.h"
#include "library/basic_thms.h"

#include "kernel/kernel_exception.h"

namespace lean {

MK_CONSTANT(trivial,            name("Trivial"));
MK_CONSTANT(true_neq_false,     name("TrueNeFalse"));
MK_CONSTANT(false_elim_fn,      name("FalseElim"));
MK_CONSTANT(absurd_fn,          name("Absurd"));
MK_CONSTANT(em_fn,              name("EM"));
MK_CONSTANT(double_neg_fn,      name("DoubleNeg"));
MK_CONSTANT(double_neg_elim_fn, name("DoubleNegElim"));
MK_CONSTANT(mt_fn,              name("MT"));
MK_CONSTANT(contrapos_fn,       name("Contrapos"));
MK_CONSTANT(false_imp_any_fn,   name("FalseImpAny"));
MK_CONSTANT(absurd_imp_any_fn,  name("AbsurdImpAny"));
MK_CONSTANT(eq_mp_fn,           name("EqMP"));
MK_CONSTANT(not_imp1_fn,        name("NotImp1"));
MK_CONSTANT(not_imp2_fn,        name("NotImp2"));
MK_CONSTANT(conj_fn,            name("Conj"));
MK_CONSTANT(conjunct1_fn,       name("Conjunct1"));
MK_CONSTANT(conjunct2_fn,       name("Conjunct2"));
MK_CONSTANT(disj1_fn,           name("Disj1"));
MK_CONSTANT(disj2_fn,           name("Disj2"));
MK_CONSTANT(disj_cases_fn,      name("DisjCases"));
MK_CONSTANT(symm_fn,            name("Symm"));
MK_CONSTANT(trans_fn,           name("Trans"));
MK_CONSTANT(congr1_fn,          name("Congr1"));
MK_CONSTANT(congr2_fn,          name("Congr2"));
MK_CONSTANT(congr_fn,           name("Congr"));
MK_CONSTANT(eqt_elim_fn,        name("EqTElim"));
MK_CONSTANT(eqt_intro_fn,       name("EqTIntro"));
MK_CONSTANT(forall_elim_fn,     name("ForallElim"));
MK_CONSTANT(forall_intro_fn,    name("ForallIntro"));
MK_CONSTANT(exists_intro_fn,    name("ExistsIntro"));

#if 0
MK_CONSTANT(ext_fn,            name("ext"));
MK_CONSTANT(foralli_fn,        name("foralli"));
MK_CONSTANT(domain_inj_fn,     name("domain_inj"));
MK_CONSTANT(range_inj_fn,      name("range_inj"));
#endif

void import_basic_thms(environment const & env) {
    if (!env->mark_builtin_imported("basic_thms"))
        return;
    expr A  = Const("A");
    expr a  = Const("a");
    expr b  = Const("b");
    expr c  = Const("c");
    expr H  = Const("H");
    expr H1 = Const("H1");
    expr H2 = Const("H2");
    expr H3 = Const("H3");
    expr B  = Const("B");
    expr f  = Const("f");
    expr g  = Const("g");
    expr h  = Const("h");
    expr x  = Const("x");
    expr y  = Const("y");
    expr z  = Const("z");
    expr P  = Const("P");
    expr A1 = Const("A1");
    expr B1 = Const("B1");
    expr a1 = Const("a1");

    expr A_pred    = A >> Bool;
    expr q_type    = Pi({A, TypeU}, A_pred >> Bool);
    expr piABx     = Pi({x, A}, B(x));
    expr A_arrow_u = A >> TypeU;

    // Trivial : True
    env->add_theorem(trivial_name, True, Refl(Bool, True));

    // True_neq_False : Not(True = False)
    env->add_theorem(true_neq_false_name, Not(Eq(True, False)), Trivial);

    // EM : Pi (a : Bool), Or(a, Not(a))
    env->add_theorem(em_fn_name, Pi({a, Bool}, Or(a, Not(a))),
                     Fun({a, Bool}, Case(Fun({x, Bool}, Or(x, Not(x))), Trivial, Trivial, a)));

    // FalseElim : Pi (a : Bool) (H : False), a
    env->add_theorem(false_elim_fn_name, Pi({{a, Bool}, {H, False}}, a),
                     Fun({{a, Bool}, {H, False}}, Case(Fun({x, Bool}, x), Trivial, H, a)));

    // Absurd : Pi (a : Bool) (H1 : a) (H2 : Not a), False
    env->add_theorem(absurd_fn_name, Pi({{a, Bool}, {H1, a}, {H2, Not(a)}}, False),
                     Fun({{a, Bool}, {H1, a}, {H2, Not(a)}},
                         MP(a, False, H2, H1)));

    // DoubleNeg : Pi (a : Bool), Eq(Not(Not(a)), a)
    env->add_theorem(double_neg_fn_name, Pi({a, Bool}, Eq(Not(Not(a)), a)),
                     Fun({a, Bool}, Case(Fun({x, Bool}, Eq(Not(Not(x)), x)), Trivial, Trivial, a)));

    // DoubleNegElim : Pi (a : Bool) (P : Bool -> Bool) (H : P (Not (Not a))), (P a)
    env->add_theorem(double_neg_elim_fn_name, Pi({{a, Bool}, {P, Bool >> Bool}, {H, P(Not(Not(a)))}}, P(a)),
                     Fun({{a, Bool}, {P, Bool >> Bool}, {H, P(Not(Not(a)))}},
                         Subst(Bool, Not(Not(a)), a, P, H, DoubleNeg(a))));

    // ModusTollens : Pi (a b : Bool) (H1 : a => b) (H2 : Not(b)), Not(a)
    env->add_theorem(mt_fn_name, Pi({{a, Bool}, {b, Bool}, {H1, Implies(a, b)}, {H2, Not(b)}}, Not(a)),
                     Fun({{a, Bool}, {b, Bool}, {H1, Implies(a, b)}, {H2, Not(b)}},
                         Discharge(a, False, Fun({H, a},
                                                 Absurd(b, MP(a, b, H1, H), H2)))));

    // Contrapositive : Pi (a b : Bool) (H : a => b), (Not(b) => Not(a))
    env->add_theorem(contrapos_fn_name, Pi({{a, Bool}, {b, Bool}, {H, Implies(a, b)}}, Implies(Not(b), Not(a))),
                     Fun({{a, Bool}, {b, Bool}, {H, Implies(a, b)}},
                         Discharge(Not(b), Not(a), Fun({H1, Not(b)}, MT(a, b, H, H1)))));

    // FalseImpliesAny : Pi (a : Bool), False => a
    env->add_theorem(false_imp_any_fn_name, Pi({a, Bool}, Implies(False, a)),
                     Fun({a, Bool}, Case(Fun({x, Bool}, Implies(False, x)), Trivial, Trivial, a)));

    // AbsurdImpliesAny : Pi (a c : Bool) (H1 : a) (H2 : Not a), c
    env->add_theorem(absurd_imp_any_fn_name, Pi({{a, Bool}, {c, Bool}, {H1, a}, {H2, Not(a)}}, c),
                     Fun({{a, Bool}, {c, Bool}, {H1, a}, {H2, Not(a)}},
                         MP(False, c, FalseImpAny(c), Absurd(a, H1, H2))));

    // EqMP : Pi (a b: Bool) (H1 : a = b) (H2 : a), b
    env->add_theorem(eq_mp_fn_name, Pi({{a, Bool}, {b, Bool}, {H1, Eq(a, b)}, {H2, a}}, b),
                     Fun({{a, Bool}, {b, Bool}, {H1, Eq(a, b)}, {H2, a}},
                         Subst(Bool, a, b, Fun({x, Bool}, x), H2, H1)));

    // NotImp1 : Pi (a b : Bool) (H : Not(Implies(a, b))), a
    env->add_theorem(not_imp1_fn_name, Pi({{a, Bool}, {b, Bool}, {H, Not(Implies(a, b))}}, a),
                     Fun({{a, Bool}, {b, Bool}, {H, Not(Implies(a, b))}},
                         EqMP(Not(Not(a)), a,
                              DoubleNeg(a),
                              Discharge(Not(a), False,
                                        Fun({H1, Not(a)},
                                            Absurd(Implies(a, b),
                                                   Discharge(a, b,
                                                             Fun({H2, a},
                                                                 FalseElim(b, Absurd(a, H2, H1)))),
                                                   H))))));

    // NotImp2 : Pi (a b : Bool) (H : Not(Implies(a, b))), Not(b)
    env->add_theorem(not_imp2_fn_name, Pi({{a, Bool}, {b, Bool}, {H, Not(Implies(a, b))}}, Not(b)),
                     Fun({{a, Bool}, {b, Bool}, {H, Not(Implies(a, b))}},
                         Discharge(b, False,
                                   Fun({H1, b},
                                       Absurd(Implies(a, b),
                                              // a => b
                                              DoubleNegElim(b, Fun({x, Bool}, Implies(a, x)),
                                                            // a => Not(Not(b))
                                                            DoubleNegElim(a, Fun({x, Bool}, Implies(x, Not(Not(b)))),
                                                                          // Not(Not(a)) => Not(Not(b))
                                                                          Contrapos(Not(b), Not(a),
                                                                                    Discharge(Not(b), Not(a),
                                                                                              Fun({H2, Not(b)},
                                                                                                  FalseElim(Not(a), Absurd(b, H1, H2))))))),
                                              H)))));

    // Conj : Pi (a b : Bool) (H1 : a) (H2 : b), And(a, b)
    env->add_theorem(conj_fn_name, Pi({{a, Bool}, {b, Bool}, {H1, a}, {H2, b}}, And(a, b)),
                     Fun({{a, Bool}, {b, Bool}, {H1, a}, {H2, b}},
                         Discharge(Implies(a, Not(b)), False, Fun({H, Implies(a, Not(b))},
                                                                  Absurd(b, H2, MP(a, Not(b), H, H1))))));


    // Conjunct1 : Pi (a b : Bool) (H : And(a, b)), a
    env->add_theorem(conjunct1_fn_name, Pi({{a, Bool}, {b, Bool}, {H, And(a, b)}}, a),
                     Fun({{a, Bool}, {b, Bool}, {H, And(a, b)}},
                         NotImp1(a, Not(b), H)));

    // Conjunct2 : Pi (a b : Bool) (H : And(a, b)), b
    env->add_theorem(conjunct2_fn_name, Pi({{a, Bool}, {b, Bool}, {H, And(a, b)}}, b),
                     Fun({{a, Bool}, {b, Bool}, {H, And(a, b)}},
                         EqMP(Not(Not(b)), b, DoubleNeg(b), NotImp2(a, Not(b), H))));

    // Disj1 : Pi (a b : Bool) (H : a), Or(a, b)
    env->add_theorem(disj1_fn_name, Pi({{a, Bool}, {b, Bool}, {H, a}}, Or(a, b)),
                     Fun({{a, Bool}, {b, Bool}, {H, a}},
                         Discharge(Not(a), b, Fun({H1, Not(a)},
                                                  FalseElim(b, Absurd(a, H, H1))))));

    // Disj2 : Pi (b a : Bool) (H : b), Or(a, b)
    env->add_theorem(disj2_fn_name, Pi({{b, Bool}, {a, Bool}, {H, b}}, Or(a, b)),
                     Fun({{b, Bool}, {a, Bool}, {H, b}},
                         // Not(a) => b
                         DoubleNegElim(b, Fun({x, Bool}, Implies(Not(a), x)),
                                       // Not(a) => Not(Not(b))
                                       Contrapos(Not(b), a,
                                                 Discharge(Not(b), a, Fun({H1, Not(b)},
                                                                          FalseElim(a, Absurd(b, H, H1))))))));

    // DisjCases : Pi (a b c: Bool) (H1 : Or(a, b)) (H2 : a -> c) (H3 : b -> c), c */
    env->add_theorem(disj_cases_fn_name, Pi({{a, Bool}, {b, Bool}, {c, Bool}, {H1, Or(a, b)}, {H2, a >> c}, {H3, b >> c}}, c),
                     Fun({{a, Bool}, {b, Bool}, {c, Bool}, {H1, Or(a, b)}, {H2, a >> c}, {H3, b >> c}},
                         EqMP(Not(Not(c)), c, DoubleNeg(c),
                              Discharge(Not(c), False,
                                        Fun({H, Not(c)},
                                            Absurd(c,
                                                   MP(b, c, Discharge(b, c, H3),
                                                      MP(Not(a), b, H1,
                                                         // Not(a)
                                                         MT(a, c, Discharge(a, c, H2), H))),
                                                   H))))));

    // Symm : Pi (A : Type u) (a b : A) (H : a = b), b = a
    env->add_theorem(symm_fn_name, Pi({{A, TypeU}, {a, A}, {b, A}, {H, Eq(a, b)}}, Eq(b, a)),
                     Fun({{A, TypeU}, {a, A}, {b, A}, {H, Eq(a, b)}},
                         Subst(A, a, b, Fun({x, A}, Eq(x, a)), Refl(A, a), H)));

    // Trans: Pi (A: Type u) (a b c : A) (H1 : a = b) (H2 : b = c), a = c
    env->add_theorem(trans_fn_name, Pi({{A, TypeU}, {a, A}, {b, A}, {c, A}, {H1, Eq(a, b)}, {H2, Eq(b, c)}}, Eq(a, c)),
                     Fun({{A, TypeU}, {a, A}, {b, A}, {c, A}, {H1, Eq(a, b)}, {H2, Eq(b, c)}},
                         Subst(A, b, c, Fun({x, A}, Eq(a, x)), H1, H2)));

    // EqTElim : Pi (a : Bool) (H : a = True), a
    env->add_theorem(eqt_elim_fn_name, Pi({{a, Bool}, {H, Eq(a, True)}}, a),
                     Fun({{a, Bool}, {H, Eq(a, True)}},
                         EqMP(True, a, Symm(Bool, a, True, H), Trivial)));

    // EqTIntro : Pi (a : Bool) (H : a), a = True
    env->add_theorem(eqt_intro_fn_name, Pi({{a, Bool}, {H, a}}, Eq(a, True)),
                     Fun({{a, Bool}, {H, a}},
                         ImpAntisym(a, True,
                                    Discharge(a, True, Fun({H1, a}, Trivial)),
                                    Discharge(True, a, Fun({H2, True}, H)))));


    env->add_theorem(name("OrIdempotent"), Pi({a, Bool}, Eq(Or(a, a), a)),
                     Fun({a, Bool}, Case(Fun({x, Bool}, Eq(Or(x, x), x)), Trivial, Trivial, a)));

    env->add_theorem(name("OrComm"), Pi({{a, Bool}, {b, Bool}}, Eq(Or(a, b), Or(b, a))),
                     Fun({{a, Bool}, {b, Bool}},
                         Case(Fun({x, Bool}, Eq(Or(x, b), Or(b, x))),
                              Case(Fun({y, Bool}, Eq(Or(True, y),  Or(y, True))),  Trivial, Trivial, b),
                              Case(Fun({y, Bool}, Eq(Or(False, y), Or(y, False))), Trivial, Trivial, b),
                              a)));

    env->add_theorem(name("OrAssoc"), Pi({{a, Bool}, {b, Bool}, {c, Bool}}, Eq(Or(Or(a, b), c), Or(a, Or(b, c)))),
                     Fun({{a, Bool}, {b, Bool}, {c, Bool}},
                         Case(Fun({x, Bool}, Eq(Or(Or(x, b), c), Or(x, Or(b, c)))),
                              Case(Fun({y, Bool}, Eq(Or(Or(True, y), c), Or(True, Or(y, c)))),
                                   Case(Fun({z, Bool}, Eq(Or(Or(True, True), z), Or(True, Or(True, z)))), Trivial, Trivial, c),
                                   Case(Fun({z, Bool}, Eq(Or(Or(True, False), z), Or(True, Or(False, z)))), Trivial, Trivial, c), b),
                              Case(Fun({y, Bool}, Eq(Or(Or(False, y), c), Or(False, Or(y, c)))),
                                   Case(Fun({z, Bool}, Eq(Or(Or(False, True), z), Or(False, Or(True, z)))), Trivial, Trivial, c),
                                   Case(Fun({z, Bool}, Eq(Or(Or(False, False), z), Or(False, Or(False, z)))), Trivial, Trivial, c), b), a)));

    // Congr1 : Pi (A : Type u) (B : A -> Type u) (f g: Pi (x : A) B x) (a : A) (H : f = g), f a = g a
    env->add_theorem(congr1_fn_name, Pi({{A, TypeU}, {B, A_arrow_u}, {f, piABx}, {g, piABx}, {a, A}, {H, Eq(f, g)}}, Eq(f(a), g(a))),
                     Fun({{A, TypeU}, {B, A_arrow_u}, {f, piABx}, {g, piABx}, {a, A}, {H, Eq(f, g)}},
                         Subst(piABx, f, g, Fun({h, piABx}, Eq(f(a), h(a))), Refl(B(a), f(a)), H)));

    // Congr2 : Pi (A : Type u) (B : A -> Type u)  (a b : A) (f : Pi (x : A) B x) (H : a = b), f a = f b
    env->add_theorem(congr2_fn_name, Pi({{A, TypeU}, {B, A_arrow_u}, {a, A}, {b, A}, {f, piABx}, {H, Eq(a, b)}}, Eq(f(a), f(b))),
                     Fun({{A, TypeU}, {B, A_arrow_u}, {a, A}, {b, A}, {f, piABx}, {H, Eq(a, b)}},
                         Subst(A, a, b, Fun({x, A}, Eq(f(a), f(x))), Refl(B(a), f(a)), H)));

    // Congr : Pi (A : Type u) (B : A -> Type u) (f g : Pi (x : A) B x) (a b : A) (H1 : f = g) (H2 : a = b), f a = g b
    env->add_theorem(congr_fn_name, Pi({{A, TypeU}, {B, A_arrow_u}, {f, piABx}, {g, piABx}, {a, A}, {b, A}, {H1, Eq(f, g)}, {H2, Eq(a, b)}}, Eq(f(a), g(b))),
                     Fun({{A, TypeU}, {B, A_arrow_u}, {f, piABx}, {g, piABx}, {a, A}, {b, A}, {H1, Eq(f, g)}, {H2, Eq(a, b)}},
                         TransExt(B(a), B(b), B(b), f(a), f(b), g(b),
                                  Congr2(A, B, a, b, f, H2), Congr1(A, B, f, g, b, H1))));


    // ForallElim : Pi (A : Type u) (P : A -> bool) (H : (forall A P)) (a : A), P a
    env->add_theorem(forall_elim_fn_name, Pi({{A, TypeU}, {P, A_pred}, {H, mk_forall(A, P)}, {a, A}}, P(a)),
                     Fun({{A, TypeU}, {P, A_pred}, {H, mk_forall(A, P)}, {a, A}},
                         EqTElim(P(a), Congr1(A, Fun({x, A}, Bool), P, Fun({x, A}, True), a, H))));

    // ForallIntro : Pi (A : Type u) (P : A -> bool) (H : Pi (x : A), P x), (forall A P)
    env->add_theorem(forall_intro_fn_name, Pi({{A, TypeU}, {P, A_pred}, {H, Pi({x, A}, P(x))}}, Forall(A, P)),
                     Fun({{A, TypeU}, {P, A_pred}, {H, Pi({x, A}, P(x))}},
                         Trans(A_pred, P, Fun({x, A}, P(x)), Fun({x, A}, True),
                               Symm(A_pred, Fun({x, A}, P(x)), P, Eta(A, Fun({x, A}, Bool), P)), // P == fun x : A, P x
                               Abst(A, Fun({x, A}, Bool), Fun({x, A}, P(x)), Fun({x, A}, True), Fun({x, A}, EqTIntro(P(x), H(x)))))));

    // ExistsIntro : Pi (A : Type u) (P : A -> bool) (a : A) (H : P a), exists A P
    env->add_theorem(exists_intro_fn_name, Pi({{A, TypeU}, {P, A_pred}, {a, A}, {H, P(a)}}, mk_exists(A, P)),
                     Fun({{A, TypeU}, {P, A_pred}, {a, A}, {H, P(a)}},
                         Discharge(mk_forall(A, Fun({x, A}, Not(P(x)))), False,
                                   Fun({H2, mk_forall(A, Fun({x, A}, Not(P(x))))},
                                       Absurd(P(a), H, ForallElim(A, Fun({x, A}, Not(P(x))), H2, a))))));
}
}