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refactor(library/init/logic): move theorems to library/logic

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Leonardo de Moura 2014-12-12 13:20:27 -08:00
parent d50b7a8ba1
commit d6c8e23b03
20 changed files with 207 additions and 189 deletions
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2
library/algebra/order.lean
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@ -15,7 +15,7 @@ These might not hold constructively in some applications, but we can define addi
with both < and ≤ as needed.
-/
import logic.eq
import logic.eq logic.connectives
import data.unit data.sigma data.prod
import algebra.function algebra.binary

2
library/algebra/ring.lean
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@ -9,7 +9,7 @@ Structures with multiplicative and additive components, including semirings, rin
The development is modeled after Isabelle's library.
-/
import logic.eq data.unit data.sigma data.prod
import logic.eq logic.connectives data.unit data.sigma data.prod
import algebra.function algebra.binary algebra.group
open eq eq.ops

2
library/data/fin.lean
View file

@ -79,6 +79,6 @@ namespace fin
have aux₂ : to_nat (lift (of_nat_core p) (n - succ p)) = p, from calc
to_nat (lift (of_nat_core p) (n - succ p)) = to_nat (of_nat_core p) : to_nat.lift
... = p : to_nat.of_nat_core,
heq.to_eq (heq.trans aux₁ (heq.from_eq aux₂))
heq.to_eq (heq.trans aux₁ (heq.of_eq aux₂))
end fin

16
library/data/int/basic.lean
View file

@ -885,17 +885,15 @@ theorem mul_cancel_right {a b c : ℤ} (H1 : c ≠ 0) (H2 : a * c = b * c) : a =
or.resolve_right (mul_cancel_right_or H2) H1
theorem mul_ne_zero {a b : ℤ} (Ha : a ≠ 0) (Hb : b ≠ 0) : a * b ≠ 0 :=
not_intro
(assume H : a * b = 0,
or.elim (mul_eq_zero H)
(assume H2 : a = 0, absurd H2 Ha)
(assume H2 : b = 0, absurd H2 Hb))
(assume H : a * b = 0,
or.elim (mul_eq_zero H)
(assume H2 : a = 0, absurd H2 Ha)
(assume H2 : b = 0, absurd H2 Hb))
theorem mul_ne_zero_left {a b : ℤ} (H : a * b ≠ 0) : a ≠ 0 :=
not_intro
(assume H2 : a = 0,
have H3 : a * b = 0, by simp,
absurd H3 H)
(assume H2 : a = 0,
have H3 : a * b = 0, by simp,
absurd H3 H)
theorem mul_ne_zero_right {a b : ℤ} (H : a * b ≠ 0) : b ≠ 0 :=
mul_ne_zero_left (mul_comm a b ▸ H)

25
library/data/int/order.lean
View file

@ -221,19 +221,18 @@ exists_intro n H2
-- -- ### basic facts
theorem lt_irrefl (a : ℤ) : ¬ a < a :=
not_intro
(assume H : a < a,
obtain (n : ℕ) (Hn : a + succ n = a), from lt_elim H,
have H2 : a + succ n = a + 0, from
calc
a + succ n = a : Hn
... = a + 0 : by simp,
have H3 : succ n = 0, from add_cancel_left H2,
have H4 : succ n = 0, from of_nat_inj H3,
absurd H4 !succ_ne_zero)
(assume H : a < a,
obtain (n : ℕ) (Hn : a + succ n = a), from lt_elim H,
have H2 : a + succ n = a + 0, from
calc
a + succ n = a : Hn
... = a + 0 : by simp,
have H3 : succ n = 0, from add_cancel_left H2,
have H4 : succ n = 0, from of_nat_inj H3,
absurd H4 !succ_ne_zero)
theorem lt_imp_ne {a b : ℤ} (H : a < b) : a ≠ b :=
not_intro (assume H2 : a = b, absurd (H2 ▸ H) (lt_irrefl b))
(assume H2 : a = b, absurd (H2 ▸ H) (lt_irrefl b))
theorem lt_of_nat (n m : ℕ) : (of_nat n < of_nat m) ↔ (n < m) :=
calc
@ -284,10 +283,10 @@ theorem lt_trans {a b c : ℤ} (H1 : a < b) (H2 : b < c) : a < c :=
lt_le_trans H1 (lt_imp_le H2)
theorem le_imp_not_gt {a b : ℤ} (H : a ≤ b) : ¬ a > b :=
not_intro (assume H2 : a > b, absurd (le_lt_trans H H2) (lt_irrefl a))
(assume H2 : a > b, absurd (le_lt_trans H H2) (lt_irrefl a))
theorem lt_imp_not_ge {a b : ℤ} (H : a < b) : ¬ a ≥ b :=
not_intro (assume H2 : a ≥ b, absurd (lt_le_trans H H2) (lt_irrefl a))
(assume H2 : a ≥ b, absurd (lt_le_trans H H2) (lt_irrefl a))
theorem lt_antisym {a b : ℤ} (H : a < b) : ¬ b < a :=
le_imp_not_gt (lt_imp_le H)

2
library/data/nat/basic.lean
View file

@ -4,7 +4,7 @@
-- Basic operations on the natural numbers.
import data.num algebra.binary
import logic.connectives data.num algebra.binary
open eq.ops binary

11
library/data/nat/order.lean
View file

@ -54,10 +54,9 @@ obtain (k : ℕ) (Hk : n + k = 0), from le_elim H,
add.eq_zero_left Hk
theorem not_succ_zero_le (n : ℕ) : ¬ succ n ≤ 0 :=
not_intro
(assume H : succ n ≤ 0,
have H2 : succ n = 0, from le_zero H,
absurd H2 !succ_ne_zero)
(assume H : succ n ≤ 0,
have H2 : succ n = 0, from le_zero H,
absurd H2 !succ_ne_zero)
theorem le_trans {n m k : ℕ} (H1 : n ≤ m) (H2 : m ≤ k) : n ≤ k :=
le.trans H1 H2
@ -249,7 +248,7 @@ theorem lt_imp_ne {n m : ℕ} (H : n < m) : n ≠ m :=
λ heq : n = m, absurd H (heq ▸ !lt.irrefl)
theorem lt_irrefl (n : ℕ) : ¬ n < n :=
not_intro (assume H : n < n, absurd rfl (lt_imp_ne H))
(assume H : n < n, absurd rfl (lt_imp_ne H))
theorem lt_def (n m : ℕ) : n < m ↔ succ n ≤ m :=
iff.intro
@ -287,7 +286,7 @@ le.rec_on H
(λ m (h : n < m), lt.asymm h)
theorem lt_imp_not_ge {n m : ℕ} (H : n < m) : ¬ n ≥ m :=
not_intro (assume H2 : m ≤ n, absurd (lt.of_lt_of_le H H2) !lt_irrefl)
(assume H2 : m ≤ n, absurd (lt.of_lt_of_le H H2) !lt_irrefl)
theorem lt_antisym {n m : ℕ} (H : n < m) : ¬ m < n :=
lt.asymm H

2
library/data/sigma/thms.lean
View file

@ -54,7 +54,7 @@ namespace sigma
theorem ndtrip_eq {A B : Type} {C : A → B → Type} {a₁ a₂ : A} {b₁ b₂ : B}
{c₁ : C a₁ b₁} {c₂ : C a₂ b₂} (H₁ : a₁ = a₂) (H₂ : b₁ = b₂)
(H₃ : cast (congr_arg2 C H₁ H₂) c₁ = c₂) : ⟨a₁, b₁, c₁⟩ = ⟨a₂, b₂, c₂⟩ :=
hdcongr_arg3 dtrip H₁ (heq.from_eq H₂) H₃
hdcongr_arg3 dtrip H₁ (heq.of_eq H₂) H₃
theorem ndtrip_equal {A B : Type} {C : A → B → Type} {p₁ p₂ : Σa b, C a b} :
∀(H₁ : dpr1 p₁ = dpr1 p₂) (H₂ : dpr2' p₁ = dpr2' p₂)

5
library/data/sum.lean
View file

@ -7,6 +7,7 @@ Authors: Leonardo de Moura, Jeremy Avigad
The sum type, aka disjoint union.
-/
import logic.connectives
open inhabited decidable eq.ops
namespace sum
@ -48,13 +49,13 @@ namespace sum
(assume Hne : a₁ ≠ a₂, decidable.inr (mt inl_inj Hne)))
(take b₂,
have H₃ : (inl B a₁ = inr A b₂) ↔ false,
from iff.intro inl_neq_inr (assume H₄, false_elim H₄),
from iff.intro inl_neq_inr (assume H₄, !false.rec H₄),
show decidable (inl B a₁ = inr A b₂), from decidable_iff_equiv _ (iff.symm H₃)))
(take b₁, show decidable (inr A b₁ = s₂), from
rec_on s₂
(take a₂,
have H₃ : (inr A b₁ = inl B a₂) ↔ false,
from iff.intro (assume H₄, inl_neq_inr (H₄⁻¹)) (assume H₄, false_elim H₄),
from iff.intro (assume H₄, inl_neq_inr (H₄⁻¹)) (assume H₄, !false.rec H₄),
show decidable (inr A b₁ = inl B a₂), from decidable_iff_equiv _ (iff.symm H₃))
(take b₂, show decidable (inr A b₁ = inr A b₂), from
decidable.rec_on (H₂ b₁ b₂)

161
library/init/logic.lean
View file

@ -10,12 +10,6 @@ import init.datatypes init.reserved_notation
-- implication
-- -----------
definition imp (a b : Prop) : Prop := a → b
-- make c explicit and rename to false.elim
theorem false_elim {c : Prop} (H : false) : c :=
false.rec c H
definition trivial := true.intro
definition not (a : Prop) := a → false
@ -24,28 +18,12 @@ prefix `¬` := not
definition absurd {a : Prop} {b : Type} (H1 : a) (H2 : ¬a) : b :=
false.rec b (H2 H1)
theorem mt {a b : Prop} (H1 : a → b) (H2 : ¬b) : ¬a :=
assume Ha : a, absurd (H1 Ha) H2
-- not
-- ---
theorem not_false : ¬false :=
assume H : false, H
theorem not_not_intro {a : Prop} (Ha : a) : ¬¬a :=
assume Hna : ¬a, absurd Ha Hna
theorem not_intro {a : Prop} (H : a → false) : ¬a := H
theorem not_elim {a : Prop} (H1 : ¬a) (H2 : a) : false := H1 H2
theorem not_implies_left {a b : Prop} (H : ¬(a → b)) : ¬¬a :=
assume Hna : ¬a, absurd (assume Ha : a, absurd Ha Hna) H
theorem not_implies_right {a b : Prop} (H : ¬(a → b)) : ¬b :=
assume Hb : b, absurd (assume Ha : a, Hb) H
-- eq
-- --
@ -60,15 +38,6 @@ namespace eq
variables {A : Type}
variables {a b c a': A}
definition drec_on {B : Πa' : A, a = a' → Type} (H₁ : a = a') (H₂ : B a (refl a)) : B a' H₁ :=
eq.rec (λH₁ : a = a, show B a H₁, from H₂) H₁ H₁
theorem id_refl (H₁ : a = a) : H₁ = (eq.refl a) :=
rfl
theorem irrel (H₁ H₂ : a = b) : H₁ = H₂ :=
!proof_irrel
theorem subst {P : A → Prop} (H₁ : a = b) (H₂ : P a) : P b :=
rec H₂ H₁
@ -154,37 +123,31 @@ namespace heq
definition elim {A : Type} {a : A} {P : A → Type} {b : A} (H₁ : a == b) (H₂ : P a) : P b :=
eq.rec_on (to_eq H₁) H₂
theorem drec_on {C : Π {B : Type} (b : B), a == b → Type} (H₁ : a == b) (H₂ : C a (refl a)) : C b H₁ :=
rec (λ H₁ : a == a, show C a H₁, from H₂) H₁ H₁
theorem subst {P : ∀T : Type, T → Prop} (H₁ : a == b) (H₂ : P A a) : P B b :=
rec_on H₁ H₂
theorem symm (H : a == b) : b == a :=
rec_on H (refl a)
definition type_eq (H : a == b) : A = B :=
heq.rec_on H (eq.refl A)
theorem from_eq (H : a = a') : a == a' :=
theorem of_eq (H : a = a') : a == a' :=
eq.subst H (refl a)
theorem trans (H₁ : a == b) (H₂ : b == c) : a == c :=
subst H₂ H₁
theorem trans_left (H₁ : a == b) (H₂ : b = b') : a == b' :=
trans H₁ (from_eq H₂)
theorem of_heq_of_eq (H₁ : a == b) (H₂ : b = b') : a == b' :=
trans H₁ (of_eq H₂)
theorem trans_right (H₁ : a = a') (H₂ : a' == b) : a == b :=
trans (from_eq H₁) H₂
theorem of_eq_of_heq (H₁ : a = a') (H₂ : a' == b) : a == b :=
trans (of_eq H₁) H₂
theorem true_elim {a : Prop} (H : a == true) : a :=
eq.true_elim (heq.to_eq H)
end heq
calc_trans heq.trans
calc_trans heq.trans_left
calc_trans heq.trans_right
calc_trans heq.of_heq_of_eq
calc_trans heq.of_eq_of_heq
calc_symm heq.symm
-- and
@ -199,23 +162,11 @@ namespace and
theorem elim (H₁ : a ∧ b) (H₂ : a → b → c) : c :=
rec H₂ H₁
theorem swap (H : a ∧ b) : b ∧ a :=
intro (elim_right H) (elim_left H)
definition not_left (b : Prop) (Hna : ¬a) : ¬(a ∧ b) :=
assume H : a ∧ b, absurd (elim_left H) Hna
definition not_right (a : Prop) {b : Prop} (Hnb : ¬b) : ¬(a ∧ b) :=
assume H : a ∧ b, absurd (elim_right H) Hnb
theorem imp_and (H₁ : a ∧ b) (H₂ : a → c) (H₃ : b → d) : c ∧ d :=
elim H₁ (assume Ha : a, assume Hb : b, intro (H₂ Ha) (H₃ Hb))
theorem imp_left (H₁ : a ∧ c) (H : a → b) : b ∧ c :=
elim H₁ (assume Ha : a, assume Hc : c, intro (H Ha) Hc)
theorem imp_right (H₁ : c ∧ a) (H : a → b) : c ∧ b :=
elim H₁ (assume Hc : c, assume Ha : a, intro Hc (H Ha))
end and
-- or
@ -233,45 +184,12 @@ namespace or
theorem elim (H₁ : a ∨ b) (H₂ : a → c) (H₃ : b → c) : c :=
rec H₂ H₃ H₁
theorem elim3 (H : a ∨ b ∨ c) (Ha : a → d) (Hb : b → d) (Hc : c → d) : d :=
elim H Ha (assume H₂, elim H₂ Hb Hc)
theorem resolve_right (H₁ : a ∨ b) (H₂ : ¬a) : b :=
elim H₁ (assume Ha, absurd Ha H₂) (assume Hb, Hb)
theorem resolve_left (H₁ : a ∨ b) (H₂ : ¬b) : a :=
elim H₁ (assume Ha, Ha) (assume Hb, absurd Hb H₂)
theorem swap (H : a ∨ b) : b ∨ a :=
elim H (assume Ha, inr Ha) (assume Hb, inl Hb)
definition not_intro (Hna : ¬a) (Hnb : ¬b) : ¬(a ∨ b) :=
assume H : a ∨ b, or.rec_on H
(assume Ha, absurd Ha Hna)
(assume Hb, absurd Hb Hnb)
theorem imp_or (H₁ : a ∨ b) (H₂ : a → c) (H₃ : b → d) : c ∨ d :=
elim H₁
(assume Ha : a, inl (H₂ Ha))
(assume Hb : b, inr (H₃ Hb))
theorem imp_or_left (H₁ : a ∨ c) (H : a → b) : b ∨ c :=
elim H₁
(assume H₂ : a, inl (H H₂))
(assume H₂ : c, inr H₂)
theorem imp_or_right (H₁ : c ∨ a) (H : a → b) : c ∨ b :=
elim H₁
(assume H₂ : c, inl H₂)
(assume H₂ : a, inr (H H₂))
end or
theorem not_not_em {p : Prop} : ¬¬(p ∨ ¬p) :=
assume not_em : ¬(p ∨ ¬p),
have Hnp : ¬p, from
assume Hp : p, absurd (or.inl Hp) not_em,
absurd (or.inr Hnp) not_em
-- iff
-- ---
definition iff (a b : Prop) := (a → b) ∧ (b → a)
@ -280,9 +198,6 @@ notation a <-> b := iff a b
notation a ↔ b := iff a b
namespace iff
definition def : (a ↔ b) = ((a → b) ∧ (b → a)) :=
rfl
definition intro (H₁ : a → b) (H₂ : b → a) : a ↔ b :=
and.intro H₁ H₂
@ -299,8 +214,8 @@ namespace iff
definition flip_sign (H₁ : a ↔ b) : ¬a ↔ ¬b :=
intro
(assume Hna, mt (elim_right H₁) Hna)
(assume Hnb, mt (elim_left H₁) Hnb)
(assume (Hna : ¬ a) (Hb : b), absurd (elim_right H₁ Hb) Hna)
(assume (Hnb : ¬ b) (Ha : a), absurd (elim_left H₁ Ha) Hnb)
definition refl (a : Prop) : a ↔ a :=
intro (assume H, H) (assume H, H)
@ -332,40 +247,6 @@ end iff
calc_refl iff.refl
calc_trans iff.trans
-- comm and assoc for and / or
-- ---------------------------
namespace and
theorem comm : a ∧ b ↔ b ∧ a :=
iff.intro (λH, swap H) (λH, swap H)
theorem assoc : (a ∧ b) ∧ c ↔ a ∧ (b ∧ c) :=
iff.intro
(assume H, intro
(elim_left (elim_left H))
(intro (elim_right (elim_left H)) (elim_right H)))
(assume H, intro
(intro (elim_left H) (elim_left (elim_right H)))
(elim_right (elim_right H)))
end and
namespace or
theorem comm : a ∨ b ↔ b ∨ a :=
iff.intro (λH, swap H) (λH, swap H)
theorem assoc : (a ∨ b) ∨ c ↔ a ∨ (b ∨ c) :=
iff.intro
(assume H, elim H
(assume H₁, elim H₁
(assume Ha, inl Ha)
(assume Hb, inr (inl Hb)))
(assume Hc, inr (inr Hc)))
(assume H, elim H
(assume Ha, (inl (inl Ha)))
(assume H₁, elim H₁
(assume Hb, inl (inr Hb))
(assume Hc, inr Hc)))
end or
inductive Exists {A : Type} (P : A → Prop) : Prop :=
intro : ∀ (a : A), P a → Exists P
@ -377,24 +258,10 @@ notation `∃` binders `,` r:(scoped P, Exists P) := r
theorem exists_elim {A : Type} {p : A → Prop} {B : Prop} (H1 : ∃x, p x) (H2 : ∀ (a : A) (H : p a), B) : B :=
Exists.rec H2 H1
definition exists_unique {A : Type} (p : A → Prop) :=
∃x, p x ∧ ∀y, p y → y = x
notation `∃!` binders `,` r:(scoped P, exists_unique P) := r
theorem exists_unique_intro {A : Type} {p : A → Prop} (w : A) (H1 : p w) (H2 : ∀y, p y → y = w) : ∃!x, p x :=
exists_intro w (and.intro H1 H2)
theorem exists_unique_elim {A : Type} {p : A → Prop} {b : Prop}
(H2 : ∃!x, p x) (H1 : ∀x, p x → (∀y, p y → y = x) → b) : b :=
obtain w Hw, from H2,
H1 w (and.elim_left Hw) (and.elim_right Hw)
inductive decidable [class] (p : Prop) : Type :=
inl : p → decidable p,
inr : ¬p → decidable p
definition true.decidable [instance] : decidable true :=
decidable.inl trivial
@ -406,7 +273,7 @@ namespace decidable
definition rec_on_true [H : decidable p] {H1 : p → Type} {H2 : ¬p → Type} (H3 : p) (H4 : H1 H3)
: rec_on H H1 H2 :=
rec_on H (λh, H4) (λh, false.rec _ (h H3))
rec_on H (λh, H4) (λh, !false.rec (h H3))
definition rec_on_false [H : decidable p] {H1 : p → Type} {H2 : ¬p → Type} (H3 : ¬p) (H4 : H2 H3)
: rec_on H H1 H2 :=
@ -421,7 +288,7 @@ namespace decidable
theorem by_contradiction [Hp : decidable p] (H : ¬p → false) : p :=
by_cases
(assume H1 : p, H1)
(assume H1 : ¬p, false_elim (H H1))
(assume H1 : ¬p, false.rec _ (H H1))
definition decidable_iff_equiv (Hp : decidable p) (H : p ↔ q) : decidable q :=
rec_on Hp
@ -452,7 +319,7 @@ section
definition not.decidable [instance] (Hp : decidable p) : decidable (¬p) :=
rec_on Hp
(assume Hp, inr (not_not_intro Hp))
(assume Hp, inr (λ Hnp, absurd Hp Hnp))
(assume Hnp, inl Hnp)
definition implies.decidable [instance] (Hp : decidable p) (Hq : decidable q) : decidable (p → q) :=
@ -577,13 +444,13 @@ definition is_false (c : Prop) [H : decidable c] : Prop :=
if c then false else true
theorem of_is_true {c : Prop} [H₁ : decidable c] (H₂ : is_true c) : c :=
decidable.rec_on H₁ (λ Hc, Hc) (λ Hnc, false_elim (if_neg Hnc ▸ H₂))
decidable.rec_on H₁ (λ Hc, Hc) (λ Hnc, !false.rec (if_neg Hnc ▸ H₂))
theorem not_of_not_is_true {c : Prop} [H₁ : decidable c] (H₂ : ¬ is_true c) : ¬ c :=
decidable.rec_on H₁ (λ Hc, absurd true.intro (if_pos Hc ▸ H₂)) (λ Hnc, Hnc)
theorem not_of_is_false {c : Prop} [H₁ : decidable c] (H₂ : is_false c) : ¬ c :=
decidable.rec_on H₁ (λ Hc, false_elim (if_pos Hc ▸ H₂)) (λ Hnc, Hnc)
decidable.rec_on H₁ (λ Hc, !false.rec (if_pos Hc ▸ H₂)) (λ Hnc, Hnc)
theorem of_not_is_false {c : Prop} [H₁ : decidable c] (H₂ : ¬ is_false c) : c :=
decidable.rec_on H₁ (λ Hc, Hc) (λ Hnc, absurd true.intro (if_neg Hnc ▸ H₂))

2
library/logic/axioms/classical.lean
View file

@ -6,7 +6,7 @@ Module: logic.axims.classical
Author: Leonardo de Moura
-/
import logic.quantifiers logic.cast algebra.relation
import logic.connectives logic.quantifiers logic.cast algebra.relation
open eq.ops

8
library/logic/cast.lean
View file

@ -27,6 +27,12 @@ namespace heq
universe variable u
variables {A B C : Type.{u}} {a a' : A} {b b' : B} {c : C}
definition type_eq (H : a == b) : A = B :=
heq.rec_on H (eq.refl A)
theorem drec_on {C : Π {B : Type} (b : B), a == b → Type} (H₁ : a == b) (H₂ : C a (refl a)) : C b H₁ :=
rec (λ H₁ : a == a, show C a H₁, from H₂) H₁ H₁
theorem to_cast_eq (H : a == b) : cast (type_eq H) a = b :=
drec_on H !cast_eq
end heq
@ -106,7 +112,7 @@ section
theorem cast_app (H : P = P') (f : Π x, P x) (a : A) : cast (pi_eq H) f a == f a :=
have H₁ : ∀ (H : (Π x, P x) = (Π x, P x)), cast H f a == f a, from
assume H, heq.from_eq (congr_fun (cast_eq H f) a),
assume H, heq.of_eq (congr_fun (cast_eq H f) a),
have H₂ : ∀ (H : (Π x, P x) = (Π x, P' x)), cast H f a == f a, from
H ▸ H₁,
H₂ (pi_eq H)

138
library/logic/connectives.lean Normal file
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@ -0,0 +1,138 @@
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Authors: Jeremy Avigad, Leonardo de Moura
definition imp (a b : Prop) : Prop := a → b
variables {a b c d : Prop}
theorem mt (H1 : a → b) (H2 : ¬b) : ¬a :=
assume Ha : a, absurd (H1 Ha) H2
-- make c explicit and rename to false.elim
theorem false_elim {c : Prop} (H : false) : c :=
false.rec c H
-- not
-- ---
theorem not_elim (H1 : ¬a) (H2 : a) : false := H1 H2
theorem not_intro (H : a → false) : ¬a := H
theorem not_not_intro (Ha : a) : ¬¬a :=
assume Hna : ¬a, absurd Ha Hna
theorem not_implies_left (H : ¬(a → b)) : ¬¬a :=
assume Hna : ¬a, absurd (assume Ha : a, absurd Ha Hna) H
theorem not_implies_right (H : ¬(a → b)) : ¬b :=
assume Hb : b, absurd (assume Ha : a, Hb) H
theorem not_not_em : ¬¬(a ∨ ¬a) :=
assume not_em : ¬(a ∨ ¬a),
have Hnp : ¬a, from
assume Hp : a, absurd (or.inl Hp) not_em,
absurd (or.inr Hnp) not_em
-- and
-- ---
namespace and
theorem swap (H : a ∧ b) : b ∧ a :=
intro (elim_right H) (elim_left H)
theorem imp_and (H₁ : a ∧ b) (H₂ : a → c) (H₃ : b → d) : c ∧ d :=
elim H₁ (assume Ha : a, assume Hb : b, intro (H₂ Ha) (H₃ Hb))
theorem imp_left (H₁ : a ∧ c) (H : a → b) : b ∧ c :=
elim H₁ (assume Ha : a, assume Hc : c, intro (H Ha) Hc)
theorem imp_right (H₁ : c ∧ a) (H : a → b) : c ∧ b :=
elim H₁ (assume Hc : c, assume Ha : a, intro Hc (H Ha))
theorem comm : a ∧ b ↔ b ∧ a :=
iff.intro (λH, swap H) (λH, swap H)
theorem assoc : (a ∧ b) ∧ c ↔ a ∧ (b ∧ c) :=
iff.intro
(assume H, intro
(elim_left (elim_left H))
(intro (elim_right (elim_left H)) (elim_right H)))
(assume H, intro
(intro (elim_left H) (elim_left (elim_right H)))
(elim_right (elim_right H)))
end and
-- or
-- --
namespace or
theorem imp_or (H₁ : a ∨ b) (H₂ : a → c) (H₃ : b → d) : c ∨ d :=
elim H₁
(assume Ha : a, inl (H₂ Ha))
(assume Hb : b, inr (H₃ Hb))
theorem imp_or_left (H₁ : a ∨ c) (H : a → b) : b ∨ c :=
elim H₁
(assume H₂ : a, inl (H H₂))
(assume H₂ : c, inr H₂)
theorem imp_or_right (H₁ : c ∨ a) (H : a → b) : c ∨ b :=
elim H₁
(assume H₂ : c, inl H₂)
(assume H₂ : a, inr (H H₂))
theorem elim3 (H : a ∨ b ∨ c) (Ha : a → d) (Hb : b → d) (Hc : c → d) : d :=
elim H Ha (assume H₂, elim H₂ Hb Hc)
theorem resolve_right (H₁ : a ∨ b) (H₂ : ¬a) : b :=
elim H₁ (assume Ha, absurd Ha H₂) (assume Hb, Hb)
theorem resolve_left (H₁ : a ∨ b) (H₂ : ¬b) : a :=
elim H₁ (assume Ha, Ha) (assume Hb, absurd Hb H₂)
theorem swap (H : a ∨ b) : b ∨ a :=
elim H (assume Ha, inr Ha) (assume Hb, inl Hb)
theorem comm : a ∨ b ↔ b ∨ a :=
iff.intro (λH, swap H) (λH, swap H)
theorem assoc : (a ∨ b) ∨ c ↔ a ∨ (b ∨ c) :=
iff.intro
(assume H, elim H
(assume H₁, elim H₁
(assume Ha, inl Ha)
(assume Hb, inr (inl Hb)))
(assume Hc, inr (inr Hc)))
(assume H, elim H
(assume Ha, (inl (inl Ha)))
(assume H₁, elim H₁
(assume Hb, inl (inr Hb))
(assume Hc, inr Hc)))
end or
-- iff
-- ---
namespace iff
definition def : (a ↔ b) = ((a → b) ∧ (b → a)) :=
!eq.refl
end iff
-- exists_unique
-- -------------
definition exists_unique {A : Type} (p : A → Prop) :=
∃x, p x ∧ ∀y, p y → y = x
notation `∃!` binders `,` r:(scoped P, exists_unique P) := r
theorem exists_unique_intro {A : Type} {p : A → Prop} (w : A) (H1 : p w) (H2 : ∀y, p y → y = w) : ∃!x, p x :=
exists_intro w (and.intro H1 H2)
theorem exists_unique_elim {A : Type} {p : A → Prop} {b : Prop}
(H2 : ∃!x, p x) (H1 : ∀x, p x → (∀y, p y → y = x) → b) : b :=
obtain w Hw, from H2,
H1 w (and.elim_left Hw) (and.elim_right Hw)

2
library/logic/default.lean
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@ -2,5 +2,5 @@
--- Released under Apache 2.0 license as described in the file LICENSE.
--- Author: Jeremy Avigad
import logic.eq logic.cast logic.subsingleton
import logic.eq logic.connectives logic.cast logic.subsingleton
import logic.quantifiers logic.instances logic.identities

10
library/logic/eq.lean
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@ -15,6 +15,16 @@ open eq.ops
namespace eq
variables {A B : Type} {a a' a₁ a₂ a₃ a₄ : A}
theorem irrel (H₁ H₂ : a = a') : H₁ = H₂ :=
!proof_irrel
theorem id_refl (H₁ : a = a) : H₁ = (eq.refl a) :=
rfl
definition drec_on {B : Πa' : A, a = a' → Type} (H₁ : a = a') (H₂ : B a (refl a)) : B a' H₁ :=
eq.rec (λH₁ : a = a, show B a H₁, from H₂) H₁ H₁
theorem rec_on_id {B : A → Type} (H : a = a) (b : B a) : rec_on H b = b :=
rfl

2
library/logic/identities.lean
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@ -8,7 +8,7 @@
-- Useful logical identities. In the absence of propositional extensionality, some of the
-- calculations use the type class support provided by logic.instances
import logic.instances logic.quantifiers logic.cast
import logic.connectives logic.instances logic.quantifiers logic.cast
open relation decidable relation.iff_ops

2
library/logic/instances.lean
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@ -8,7 +8,7 @@ Author: Jeremy Avigad
Class instances for iff and eq.
-/
import algebra.relation
import logic.connectives algebra.relation
namespace relation

1
tests/lean/interactive/findp.input.expected.out
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@ -3,7 +3,6 @@
-- BEGINFINDP STALE
false|Prop
false.rec|Π (C : Type), false → C
false_elim|false → ?c
false.of_ne|?a ≠ ?a → false
false.rec_on|Π (C : Type), false → C
false.cases_on|Π (C : Type), false → C

2
tests/lean/run/pquot.lean
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@ -55,7 +55,7 @@ definition pquot.lift {A : Type} {R : A → A → Prop} {B : Type}
(sound : ∀ (a b : A), R a b → f a = f b)
(q : pquot R)
: B :=
pquot.rec_on q f (λ (a b : A) (H : R a b), heq.from_eq (sound a b H))
pquot.rec_on q f (λ (a b : A) (H : R a b), heq.of_eq (sound a b H))
theorem pquot.induction_on {A : Type} {R : A → A → Prop} {P : pquot R → Prop}
(q : pquot R)

1
tests/lean/run/sum_bug.lean
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@ -1,6 +1,7 @@
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Leonardo de Moura, Jeremy Avigad
import logic
open inhabited decidable
namespace play
-- TODO: take this outside the namespace when the inductive package handles it better