michael/lean2
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

Add more theorems.

Browse source 
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura 2013-08-07 11:23:04 -07:00
parent 5acedcddbb
commit dd21cdcc95
2 changed files with 208 additions and 88 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

202
src/kernel/basic_thms.cpp
View file

@ -11,24 +11,34 @@ Author: Leonardo de Moura
namespace lean {
MK_CONSTANT(true_neq_false, name("TrueNeFalse"));
MK_CONSTANT(truth, name("Truth"));
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(mt_fn, name("MT"));
MK_CONSTANT(contrapos_fn, name("Contrapos"));
MK_CONSTANT(conj_fn, name("Conj"));
MK_CONSTANT(symm_fn, name("Symm"));
MK_CONSTANT(trans_fn, name("Trans"));
MK_CONSTANT(xtrans_fn, name("xTrans"));
MK_CONSTANT(congr1_fn, name("Congr1"));
MK_CONSTANT(congr2_fn, name("Congr2"));
MK_CONSTANT(congr_fn, name("Congr"));
MK_CONSTANT(eq_mp_fn, name("EqMP"));
MK_CONSTANT(eqt_elim_fn, name("EqTElim"));
MK_CONSTANT(forall_elim_fn, name("ForallElim"));
MK_CONSTANT(true_neq_false, name("TrueNeFalse"));
MK_CONSTANT(truth, name("Truth"));
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(xtrans_fn, name("xTrans"));
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"));
@ -45,6 +55,7 @@ void add_basic_thms(environment & env) {
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");
@ -65,9 +76,13 @@ void add_basic_thms(environment & env) {
// True_neq_False : Not(True = False)
env.add_theorem(true_neq_false_name, Not(Eq(True, False)), Trivial);
// Truth : True := Trivial
// Truth : True
env.add_theorem(truth_name, True, 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), Truth, H, a)));
@ -81,6 +96,11 @@ void add_basic_thms(environment & env) {
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, P, Not(Not(a)), a, 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)}},
@ -92,41 +112,118 @@ void add_basic_thms(environment & env) {
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, Fun({x, Bool}, x), a, b, 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))))));
// Iff : Pi (a b : Bool) (H1 : a => b) (H2 : b => a), a = b
// Symm : Pi (A : Type u) (a b : A) (H : a = b), b = a :=
// Subst A (Fun x : A => x = a) a b (Refl A a) H
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, Fun({x, A}, Eq(x,a)), a, b, Refl(A, a), H)));
// 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)));
// Trans: Pi (A: Type u) (a b c : A) (H1 : a = b) (H2 : b = c), a = c :=
// Subst A (Fun x : A => a = x) b c H1 H2
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, Fun({x, A}, Eq(a, x)), b, c, H1, H2)));
// 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))));
// xTrans: Pi (A: Type u) (B : Type u) (a : A) (b c : B) (H1 : a = b) (H2 : b = c), a = c :=
// Subst B (Fun x : B => a = x) b c H1 H2
env.add_theorem(xtrans_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, Fun({x, B}, Eq(a, x)), b, c, H1, H2)));
// OrIntro1 : Pi (a b : Bool) (H : a), Or(a, b)
env.add_theorem(name("OrIntro1"), Pi({{a, Bool}, {b, Bool}, {H, a}}, Or(a, b)),
// 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))))));
// 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)));
// Disj2 : Pi (a b : Bool) (H : b), Or(a, b)
env.add_theorem(disj2_fn_name, Pi({{a, Bool}, {b, Bool}, {H, b}}, Or(a, b)),
Fun({{a, Bool}, {b, 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, Fun({x, A}, Eq(x,a)), a, b, 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, Fun({x, A}, Eq(a, x)), b, c, H1, H2)));
// xTrans: Pi (A: Type u) (B : Type u) (a : A) (b c : B) (H1 : a = b) (H2 : b = c), a = c
env.add_theorem(xtrans_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, Fun({x, B}, Eq(a, x)), b, c, 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), Truth)));
// 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}, Truth)),
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)));
@ -149,39 +246,22 @@ void add_basic_thms(environment & env) {
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 :=
// Subst piABx (Fun h : piABx => f a = h a) f g (Refl piABx f) H
// 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, Fun({h, piABx}, Eq(f(a), h(a))), f, g, Refl(piABx, f), H)));
// Congr2 : Pi (A : Type u) (B : A -> Type u) (f : Pi (x : A) B x) (a b : A) (H : a = b), f a = f b :=
// Subst A (Fun x : A => f a = f x) a b (Refl A a) H
// Congr2 : Pi (A : Type u) (B : A -> Type u) (f : Pi (x : A) B x) (a b : A) (H : a = b), f a = f b
env.add_theorem(congr2_fn_name, Pi({{A, TypeU}, {B, A_arrow_u}, {f, piABx}, {a, A}, {b, A}, {H, Eq(a, b)}}, Eq(f(a), f(b))),
Fun({{A, TypeU}, {B, A_arrow_u}, {f, piABx}, {a, A}, {b, A}, {H, Eq(a, b)}},
Subst(A, Fun({x, A}, Eq(f(a), f(x))), a, b, Refl(A, 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 :=
// xTrans (B a) (B b) (f a) (f b) (g b) (congr2 A B f g b H1) (congr1 A B f a b H2)
// 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)}},
xTrans(B(a), B(b), f(a), f(b), g(b),
Congr2(A, B, f, a, b, H2), Congr1(A, B, f, g, b, H1))));
// EqMP : Pi (a b: Bool) (H1 : a = b) (H2 : a), b :=
// Subst Bool (Fun x : Bool => x) a b H2 H1
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, Fun({x, Bool}, x), a, b, H2, H1)));
// EqTElim : Pi (a : Bool) (H : a = True), a := EqMP(True, a, Symm(Bool, a, True, H), Truth)
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), Truth)));
// 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}}
// 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)),

94
src/kernel/basic_thms.h
View file

@ -9,78 +9,118 @@ Author: Leonardo de Moura
namespace lean {
expr mk_absurd_fn();
bool is_absurd_fn(expr const & e);
/** \brief (Theorem) a : Bool, H1 : a, H2 : Not(a) |- Absurd(a, H1, H2) : False */
inline expr Absurd(expr const & a, expr const & H1, expr const & H2) { return mk_app(mk_absurd_fn(), a, H1, H2); }
expr mk_true_ne_false();
/** \brief (Theorem) TrueNeFalse : Not(True = False) */
#define TrueNeFalse mk_true_ne_false();
expr mk_truth();
/** \brief (Theorem) Truth : True */
#define Truth mk_truth()
expr mk_em_fn();
/** \brief (Theorem) a : Bool |- EM(a) : Or(a, Not(a)) */
inline expr EM(expr const & a) { return mk_app(mk_em_fn(), a); }
expr mk_false_elim_fn();
bool is_false_elim_fn(expr const & e);
/** \brief (Theorem) a : Bool, H : False |- FalseElim(a, H) : a */
inline expr FalseElim(expr const & a, expr const & H) { return mk_app(mk_false_elim_fn(), a, H); }
expr mk_absurd_fn();
/** \brief (Theorem) a : Bool, H1 : a, H2 : Not(a) |- Absurd(a, H1, H2) : False */
inline expr Absurd(expr const & a, expr const & H1, expr const & H2) { return mk_app(mk_absurd_fn(), a, H1, H2); }
expr mk_double_neg_fn();
bool is_double_neg_fn(expr const & e);
/** \brief (Theorem) a : Bool |- DoubleNeg(a) : Neg(Neg(a)) = a */
inline expr DoubleNeg(expr const & a) { return mk_app(mk_double_neg_fn(), a); }
expr mk_double_neg_elim_fn();
/** \brief (Theorem) a : Bool, P : Bool -> Bool, H : P (Not (Not a)) |- DoubleNegElim(a, P, H) : P a */
inline expr DoubleNegElim(expr const & a, expr const & P, expr const & H) { return mk_app(mk_double_neg_elim_fn(), a, P, H); }
expr mk_mt_fn();
bool is_mt_fn(expr const & e);
/** \brief (Theorem) a b : Bool, H1 : a => b, H2 : Not(b) |- MT(a, b, H1, H2) : Not(a) */
inline expr MT(expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app(mk_mt_fn(), a, b, H1, H2); }
expr mk_contrapos_fn();
bool is_contrapos_fn(expr const & e);
/** \brief (Theorem) a b : Bool, H : a => b |- Contrapos(a, b, H): Neg(b) => Neg(a) */
inline expr Contrapos(expr const & a, expr const & b, expr const & H) { return mk_app(mk_contrapos_fn(), a, b, H); }
expr mk_false_imp_any_fn();
/** \brief (Theorem) a : Bool, H : False |- a */
inline expr FalseImpAny(expr const & a, expr const & H) { return mk_app(mk_false_imp_any_fn(), a, H); }
expr mk_eq_mp_fn();
/** \brief (Theorem) a b : Bool, H1 : a = b, H2 : a |- EqMP(a, b, H1, H2) : b */
inline expr EqMP(expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app(mk_eq_mp_fn(), a, b, H1, H2); }
expr mk_not_imp1_fn();
/** \brief (Theorem) a b : Bool, H : Not(Implies(a, b)) |- NotImp1(a, b, H) : a */
inline expr NotImp1(expr const & a, expr const & b, expr const & H) { return mk_app(mk_not_imp1_fn(), a, b, H); }
expr mk_not_imp2_fn();
/** \brief (Theorem) a b : Bool, H : Not(Implies(a, b)) |- NotImp2(a, b, H) : Not(b) */
inline expr NotImp2(expr const & a, expr const & b, expr const & H) { return mk_app(mk_not_imp2_fn(), a, b, H); }
expr mk_conj_fn();
/** \brief (Theorem) a b : Bool, H1 : a, H2 : b |- Conj(a, b, H1, H2) : And(a, b) */
inline expr Conj(expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app(mk_conj_fn(), a, b, H1, H2); }
expr mk_conjunct1_fn();
/** \brief (Theorem) a b : Bool, H : And(a, b) |- Conjunct1(a, b, H) : a */
inline expr Conjunct1(expr const & a, expr const & b, expr const & H) { return mk_app(mk_conjunct1_fn(), a, b, H); }
expr mk_conjunct2_fn();
/** \brief (Theorem) a b : Bool, H : And(a, b) |- Conjunct2(a, b, H) : b */
inline expr Conjunct2(expr const & a, expr const & b, expr const & H) { return mk_app(mk_conjunct2_fn(), a, b, H); }
expr mk_disj1_fn();
/** \brief (Theorem) a b : Bool, H : a |- Disj1(a, b, H) : Or(a, b) */
inline expr Disj1(expr const & a, expr const & b, expr const & H) { return mk_app(mk_disj1_fn(), a, b, H); }
expr mk_disj2_fn();
/** \brief (Theorem) a b : Bool, H : b |- Disj2(a, b, H) : Or(a, b) */
inline expr Disj2(expr const & a, expr const & b, expr const & H) { return mk_app(mk_disj2_fn(), a, b, H); }
expr mk_disj_cases_fn();
/** \brief (Theorem) a b c : Bool, H1 : Or(a,b), H2 : a -> c, H3 : b -> c |- DisjCases(a, b, c, H1, H2, H3) : c */
inline expr DisjCases(expr const & a, expr const & b, expr const & c, expr const & H1, expr const & H2, expr const & H3) { return mk_app({mk_disj_cases_fn(), a, b, c, H1, H2, H3}); }
expr mk_symm_fn();
bool is_symm_fn(expr const & e);
/** \brief (Theorem) A : Type u, a b : A, H : a = b |- Symm(A, a, b, H) : b = a */
inline expr Symm(expr const & A, expr const & a, expr const & b, expr const & H) { return mk_app(mk_symm_fn(), A, a, b, H); }
expr mk_trans_fn();
bool is_trans_fn(expr const & e);
/** \brief (Theorem) A : Type u, a b c : A, H1 : a = b, H2 : b = c |- Trans(A, a, b, c, H1, H2) : a = c */
inline expr Trans(expr const & A, expr const & a, expr const & b, expr const & c, expr const & H1, expr const & H2) { return mk_app({mk_trans_fn(), A, a, b, c, H1, H2}); }
expr mk_xtrans_fn();
bool is_xtrans_fn(expr const & e);
/** \brief (Theorem) A : Type u, B : Type u, a : A, b c : B, H1 : a = b, H2 : b = c |- xTrans(A, B, a, b, c, H1, H2) : a = c */
inline expr xTrans(expr const & A, expr const & B, expr const & a, expr const & b, expr const & c, expr const & H1, expr const & H2) { return mk_app({mk_xtrans_fn(), A, B, a, b, c, H1, H2}); }
expr mk_eqt_elim_fn();
/** \brief (Theorem) a : Bool, H : a = True |- EqTElim(a, H) : a */
inline expr EqTElim(expr const & a, expr const & H) { return mk_app(mk_eqt_elim_fn(), a, H); }
expr mk_eqt_intro_fn();
/** \brief (Theorem) a : Bool, H : a |- EqTIntro(a, H) : a = True */
inline expr EqTIntro(expr const & a, expr const & H) { return mk_app(mk_eqt_intro_fn(), a, H); }
expr mk_congr1_fn();
bool is_congr1_fn(expr const & e);
/** \brief (Theorem) A : Type u, B : A -> Type u, f g : (Pi x : A, B x), a : A, H : f = g |- Congr2(A, B, f, g, a, H) : f a = g a */
inline expr Congr1(expr const & A, expr const & B, expr const & f, expr const & g, expr const & a, expr const & H) { return mk_app({mk_congr1_fn(), A, B, f, g, a, H}); }
expr mk_congr2_fn();
bool is_congr2_fn(expr const & e);
/** \brief (Theorem) A : Type u, B : A -> Type u, f : (Pi x : A, B x), a b : A, H : a = b |- Congr1(A, B, f, a, b, H) : f a = f b */
inline expr Congr2(expr const & A, expr const & B, expr const & f, expr const & a, expr const & b, expr const & H) { return mk_app({mk_congr2_fn(), A, B, f, a, b, H}); }
expr mk_congr_fn();
bool is_congr_fn(expr const & e);
/** \brief (Theorem) A : Type u, B : A -> Type u, f g : (Pi x : A, B x), a b : A, H1 : f = g, H2 : a = b |- Congr(A, B, f, g, a, b, H1, H2) : f a = g b */
inline expr Congr(expr const & A, expr const & B, expr const & f, expr const & g, expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app({mk_congr_fn(), A, B, f, g, a, b, H1, H2}); }
expr mk_eq_mp_fn();
bool is_eq_mp_fn(expr const & e);
/** \brief (Theorem) a : Bool, b : Bool, H1 : a = b, H2 : a |- EqMP(a, b, H1, H2) : b */
inline expr EqMP(expr const & a, expr const & b, expr const & H1, expr const & H2) { return mk_app(mk_eq_mp_fn(), a, b, H1, H2); }
expr mk_truth();
bool is_truth(expr const & e);
/** \brief (Theorem) Truth : True */
#define Truth mk_truth()
expr mk_eqt_elim_fn();
bool is_eqt_elim(expr const & e);
// \brief (Theorem) a : Bool, H : a = True |- EqT(a, H) : a
inline expr EqTElim(expr const & a, expr const & H) { return mk_app(mk_eqt_elim_fn(), a, H); }
expr mk_forall_elim_fn();
bool is_forall_elim_fn(expr const & e);
// \brief (Theorem) A : Type u, P : A -> Bool, H : (Forall A P), a : A |- Forallelim(A, P, H, a) : P a
inline expr ForallElim(expr const & A, expr const & P, expr const & H, expr const & a) { return mk_app(mk_forall_elim_fn(), A, P, H, a); }