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lean2/src/library/basic_thms.cpp
Leonardo de Moura 874f67c605 feat(normalizer): remove normalization rule t == t ==> true 
This normalization rule is not really a computational rule.
It is essentially encoding the reflexivity axiom as computation.
It can also be abaused. For example, with this rule,
the following definition is valid:

Theorem Th : a = a := Refl b

Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
2013-10-22 14:02:48 -07:00

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/*
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"
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(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(trans_ext_fn, name("TransExt"));
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"));
#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 & env) {
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)));
// 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)));
// TransExt: Pi (A: Type u) (B : Type u) (a : A) (b c : B) (H1 : a = b) (H2 : b = c), a = c
env.add_theorem(trans_ext_fn_name, Pi({{A, TypeU}, {B, TypeU}, {a, A}, {b, B}, {c, B}, {H1, Eq(a, b)}, {H2, Eq(b, c)}}, Eq(a, c)),
Fun({{A, TypeU}, {B, TypeU}, {a, A}, {b, B}, {c, B}, {H1, Eq(a, b)}, {H2, Eq(b, c)}},
Subst(B, b, c, Fun({x, B}, 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), 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))));
#if 0
// STOPPED HERE
// foralli : Pi (A : Type u) (P : A -> bool) (H : Pi (x : A), P x), (forall A P)
env.add_axiom(foralli_fn_name, Pi({{A, TypeU}, {P, A_pred}, {H, Pi({x, A}, P(x))}}, Forall(A, P)));
// ext : Pi (A : Type u) (B : A -> Type u) (f g : Pi (x : A) B x) (H : Pi x : A, (f x) = (g x)), f = g
env.add_axiom(ext_fn_name, Pi({{A, TypeU}, {B, A_arrow_u}, {f, piABx}, {g, piABx}, {H, Pi({x, A}, Eq(f(x), g(x)))}}, Eq(f, g)));
// domain_inj : Pi (A A1: Type u) (B : A -> Type u) (B1 : A1 -> Type u) (H : (Pi (x : A), B x) = (Pi (x : A1), B1 x)), A = A1
expr piA1B1x = Pi({x, A1}, B1(x));
expr A1_arrow_u = A1 >> TypeU;
env.add_axiom(domain_inj_fn_name, Pi({{A, TypeU}, {A1, TypeU}, {B, A_arrow_u}, {B1, A1_arrow_u}, {H, Eq(piABx, piA1B1x)}}, Eq(A, A1)));
// range_inj : Pi (A A1: Type u) (B : A -> Type u) (B1 : A1 -> Type u) (a : A) (a1 : A1) (H : (Pi (x : A), B x) = (Pi (x : A1), B1 x)), (B a) = (B1 a1)
env.add_axiom(range_inj_fn_name, Pi({{A, TypeU}, {A1, TypeU}, {B, A_arrow_u}, {B1, A1_arrow_u}, {a, A}, {a1, A1}, {H, Eq(piABx, piA1B1x)}}, Eq(B(a), B1(a1))));
#endif
}
}
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