354 lines
No EOL
13 KiB
Text
354 lines
No EOL
13 KiB
Text
import macros
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universe U ≥ 1
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variable Bool : Type
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-- The following builtin declarations can be removed as soon as Lean supports inductive datatypes and match expressions
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builtin true : Bool
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builtin false : Bool
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builtin if {A : (Type U)} : Bool → A → A → A
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definition TypeU := (Type U)
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definition not (a : Bool) := a → false
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notation 40 ¬ _ : not
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definition or (a b : Bool) := ¬ a → b
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infixr 30 || : or
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infixr 30 \/ : or
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infixr 30 ∨ : or
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definition and (a b : Bool) := ¬ (a → ¬ b)
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definition implies (a b : Bool) := a → b
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infixr 35 && : and
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infixr 35 /\ : and
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infixr 35 ∧ : and
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-- The Lean parser has special treatment for the constant exists.
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-- It allows us to write
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-- exists x y : A, P x y and ∃ x y : A, P x y
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-- as syntax sugar for
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-- exists A (fun x : A, exists A (fun y : A, P x y))
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-- That is, it treats the exists as an extra binder such as fun and forall.
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-- It also provides an alias (Exists) that should be used when we
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-- want to treat exists as a constant.
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definition Exists (A : TypeU) (P : A → Bool) := ¬ (∀ x : A, ¬ (P x))
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definition eq {A : TypeU} (a b : A) := a == b
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infix 50 = : eq
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definition neq {A : TypeU} (a b : A) := ¬ (a == b)
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infix 50 ≠ : neq
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theorem em (a : Bool) : a ∨ ¬ a
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:= λ Hna : ¬ a, Hna
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axiom case (P : Bool → Bool) (H1 : P true) (H2 : P false) (a : Bool) : P a
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axiom refl {A : TypeU} (a : A) : a == a
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axiom subst {A : TypeU} {a b : A} {P : A → Bool} (H1 : P a) (H2 : a == b) : P b
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-- Function extensionality
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axiom funext {A : TypeU} {B : A → TypeU} {f g : ∀ x : A, B x} (H : ∀ x : A, f x == g x) : f == g
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-- Forall extensionality
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axiom allext {A : TypeU} {B C : A → TypeU} (H : ∀ x : A, B x == C x) : (∀ x : A, B x) == (∀ x : A, C x)
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-- Alias for subst where we can provide P explicitly, but keep A,a,b implicit
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definition substp {A : TypeU} {a b : A} (P : A → Bool) (H1 : P a) (H2 : a == b) : P b
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:= subst H1 H2
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theorem eta {A : TypeU} {B : A → TypeU} (f : ∀ x : A, B x) : (λ x : A, f x) == f
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:= funext (λ x : A, refl (f x))
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theorem trivial : true
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:= refl true
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theorem absurd {a : Bool} (H1 : a) (H2 : ¬ a) : false
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:= H2 H1
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theorem eqmp {a b : Bool} (H1 : a == b) (H2 : a) : b
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:= subst H2 H1
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infixl 100 <| : eqmp
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infixl 100 ◂ : eqmp
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theorem boolcomplete (a : Bool) : a == true ∨ a == false
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:= case (λ x, x == true ∨ x == false) trivial trivial a
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theorem false::elim (a : Bool) (H : false) : a
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:= case (λ x, x) trivial H a
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theorem imp::trans {a b c : Bool} (H1 : a → b) (H2 : b → c) : a → c
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:= λ Ha, H2 (H1 Ha)
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theorem imp::eq::trans {a b c : Bool} (H1 : a → b) (H2 : b == c) : a → c
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:= λ Ha, H2 ◂ (H1 Ha)
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theorem eq::imp::trans {a b c : Bool} (H1 : a == b) (H2 : b → c) : a → c
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:= λ Ha, H2 (H1 ◂ Ha)
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theorem not::not::eq (a : Bool) : (¬ ¬ a) == a
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:= case (λ x, (¬ ¬ x) == x) trivial trivial a
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theorem not::not::elim {a : Bool} (H : ¬ ¬ a) : a
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:= (not::not::eq a) ◂ H
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theorem mt {a b : Bool} (H1 : a → b) (H2 : ¬ b) : ¬ a
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:= λ Ha, absurd (H1 Ha) H2
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theorem contrapos {a b : Bool} (H : a → b) : ¬ b → ¬ a
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:= λ Hnb : ¬ b, mt H Hnb
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theorem absurd::elim {a : Bool} (b : Bool) (H1 : a) (H2 : ¬ a) : b
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:= false::elim b (absurd H1 H2)
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theorem not::imp::eliml {a b : Bool} (Hnab : ¬ (a → b)) : a
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:= not::not::elim
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(have ¬ ¬ a :
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λ Hna : ¬ a, absurd (have a → b : λ Ha : a, absurd::elim b Ha Hna)
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Hnab)
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theorem not::imp::elimr {a b : Bool} (H : ¬ (a → b)) : ¬ b
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:= λ Hb : b, absurd (have a → b : λ Ha : a, Hb)
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(have ¬ (a → b) : H)
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theorem resolve1 {a b : Bool} (H1 : a ∨ b) (H2 : ¬ a) : b
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:= H1 H2
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-- Recall that and is defined as ¬ (a → ¬ b)
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theorem and::intro {a b : Bool} (H1 : a) (H2 : b) : a ∧ b
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:= λ H : a → ¬ b, absurd H2 (H H1)
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theorem and::eliml {a b : Bool} (H : a ∧ b) : a
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:= not::imp::eliml H
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theorem and::elimr {a b : Bool} (H : a ∧ b) : b
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:= not::not::elim (not::imp::elimr H)
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-- Recall that or is defined as ¬ a → b
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theorem or::introl {a : Bool} (H : a) (b : Bool) : a ∨ b
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:= λ H1 : ¬ a, absurd::elim b H H1
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theorem or::intror {b : Bool} (a : Bool) (H : b) : a ∨ b
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:= λ H1 : ¬ a, H
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theorem or::elim {a b c : Bool} (H1 : a ∨ b) (H2 : a → c) (H3 : b → c) : c
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:= not::not::elim
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(λ H : ¬ c,
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absurd (have c : H3 (have b : resolve1 H1 (have ¬ a : (mt (λ Ha : a, H2 Ha) H))))
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H)
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theorem refute {a : Bool} (H : ¬ a → false) : a
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:= or::elim (em a) (λ H1 : a, H1) (λ H1 : ¬ a, false::elim a (H H1))
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theorem symm {A : TypeU} {a b : A} (H : a == b) : b == a
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:= subst (refl a) H
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theorem trans {A : TypeU} {a b c : A} (H1 : a == b) (H2 : b == c) : a == c
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:= subst H1 H2
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infixl 100 ⋈ : trans
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theorem ne::symm {A : TypeU} {a b : A} (H : a ≠ b) : b ≠ a
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:= λ H1 : b = a, H (symm H1)
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theorem eq::ne::trans {A : TypeU} {a b c : A} (H1 : a = b) (H2 : b ≠ c) : a ≠ c
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:= subst H2 (symm H1)
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theorem ne::eq::trans {A : TypeU} {a b c : A} (H1 : a ≠ b) (H2 : b = c) : a ≠ c
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:= subst H1 H2
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theorem eqt::elim {a : Bool} (H : a == true) : a
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:= (symm H) ◂ trivial
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theorem congr1 {A : TypeU} {B : A → TypeU} {f g : ∀ x : A, B x} (a : A) (H : f == g) : f a == g a
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:= substp (fun h : (∀ x : A, B x), f a == h a) (refl (f a)) H
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theorem congr2 {A : TypeU} {B : A → TypeU} {a b : A} (f : ∀ x : A, B x) (H : a == b) : f a == f b
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:= substp (fun x : A, f a == f x) (refl (f a)) H
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theorem congr {A : TypeU} {B : A → TypeU} {f g : ∀ x : A, B x} {a b : A} (H1 : f == g) (H2 : a == b) : f a == g b
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:= subst (congr2 f H2) (congr1 b H1)
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-- Recall that exists is defined as ¬ ∀ x : A, ¬ P x
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theorem exists::elim {A : TypeU} {P : A → Bool} {B : Bool} (H1 : Exists A P) (H2 : ∀ (a : A) (H : P a), B) : B
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:= refute (λ R : ¬ B,
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absurd (λ a : A, mt (λ H : P a, H2 a H) R)
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H1)
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theorem exists::intro {A : TypeU} {P : A → Bool} (a : A) (H : P a) : Exists A P
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:= λ H1 : (∀ x : A, ¬ P x),
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absurd H (H1 a)
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theorem boolext {a b : Bool} (Hab : a → b) (Hba : b → a) : a == b
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:= or::elim (boolcomplete a)
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(λ Hat : a == true, or::elim (boolcomplete b)
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(λ Hbt : b == true, trans Hat (symm Hbt))
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(λ Hbf : b == false, false::elim (a == b) (subst (Hab (eqt::elim Hat)) Hbf)))
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(λ Haf : a == false, or::elim (boolcomplete b)
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(λ Hbt : b == true, false::elim (a == b) (subst (Hba (eqt::elim Hbt)) Haf))
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(λ Hbf : b == false, trans Haf (symm Hbf)))
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definition iff::intro {a b : Bool} (Hab : a → b) (Hba : b → a) := boolext Hab Hba
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theorem eqt::intro {a : Bool} (H : a) : a == true
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:= boolext (λ H1 : a, trivial)
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(λ H2 : true, H)
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theorem or::comm (a b : Bool) : (a ∨ b) == (b ∨ a)
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:= boolext (λ H, or::elim H (λ H1, or::intror b H1) (λ H2, or::introl H2 a))
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(λ H, or::elim H (λ H1, or::intror a H1) (λ H2, or::introl H2 b))
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theorem or::assoc (a b c : Bool) : ((a ∨ b) ∨ c) == (a ∨ (b ∨ c))
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:= boolext (λ H : (a ∨ b) ∨ c,
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or::elim H (λ H1 : a ∨ b, or::elim H1 (λ Ha : a, or::introl Ha (b ∨ c)) (λ Hb : b, or::intror a (or::introl Hb c)))
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(λ Hc : c, or::intror a (or::intror b Hc)))
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(λ H : a ∨ (b ∨ c),
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or::elim H (λ Ha : a, (or::introl (or::introl Ha b) c))
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(λ H1 : b ∨ c, or::elim H1 (λ Hb : b, or::introl (or::intror a Hb) c)
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(λ Hc : c, or::intror (a ∨ b) Hc)))
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theorem or::id (a : Bool) : (a ∨ a) == a
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:= boolext (λ H, or::elim H (λ H1, H1) (λ H2, H2))
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(λ H, or::introl H a)
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theorem or::falsel (a : Bool) : (a ∨ false) == a
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:= boolext (λ H, or::elim H (λ H1, H1) (λ H2, false::elim a H2))
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(λ H, or::introl H false)
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theorem or::falser (a : Bool) : (false ∨ a) == a
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:= (or::comm false a) ⋈ (or::falsel a)
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theorem or::truel (a : Bool) : (true ∨ a) == true
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:= eqt::intro (case (λ x : Bool, true ∨ x) trivial trivial a)
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theorem or::truer (a : Bool) : (a ∨ true) == true
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:= (or::comm a true) ⋈ (or::truel a)
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theorem or::tauto (a : Bool) : (a ∨ ¬ a) == true
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:= eqt::intro (em a)
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theorem and::comm (a b : Bool) : (a ∧ b) == (b ∧ a)
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:= boolext (λ H, and::intro (and::elimr H) (and::eliml H))
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(λ H, and::intro (and::elimr H) (and::eliml H))
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theorem and::id (a : Bool) : (a ∧ a) == a
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:= boolext (λ H, and::eliml H)
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(λ H, and::intro H H)
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theorem and::assoc (a b c : Bool) : ((a ∧ b) ∧ c) == (a ∧ (b ∧ c))
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:= boolext (λ H, and::intro (and::eliml (and::eliml H)) (and::intro (and::elimr (and::eliml H)) (and::elimr H)))
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(λ H, and::intro (and::intro (and::eliml H) (and::eliml (and::elimr H))) (and::elimr (and::elimr H)))
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theorem and::truer (a : Bool) : (a ∧ true) == a
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:= boolext (λ H : a ∧ true, and::eliml H)
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(λ H : a, and::intro H trivial)
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theorem and::truel (a : Bool) : (true ∧ a) == a
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:= trans (and::comm true a) (and::truer a)
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theorem and::falsel (a : Bool) : (a ∧ false) == false
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:= boolext (λ H, and::elimr H)
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(λ H, false::elim (a ∧ false) H)
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theorem and::falser (a : Bool) : (false ∧ a) == false
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:= (and::comm false a) ⋈ (and::falsel a)
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theorem and::absurd (a : Bool) : (a ∧ ¬ a) == false
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:= boolext (λ H, absurd (and::eliml H) (and::elimr H))
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(λ H, false::elim (a ∧ ¬ a) H)
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theorem not::true : (¬ true) == false
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:= trivial
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theorem not::false : (¬ false) == true
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:= trivial
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theorem not::and (a b : Bool) : (¬ (a ∧ b)) == (¬ a ∨ ¬ b)
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:= case (λ x, (¬ (x ∧ b)) == (¬ x ∨ ¬ b))
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(case (λ y, (¬ (true ∧ y)) == (¬ true ∨ ¬ y)) trivial trivial b)
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(case (λ y, (¬ (false ∧ y)) == (¬ false ∨ ¬ y)) trivial trivial b)
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a
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theorem not::and::elim {a b : Bool} (H : ¬ (a ∧ b)) : ¬ a ∨ ¬ b
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:= (not::and a b) ◂ H
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theorem not::or (a b : Bool) : (¬ (a ∨ b)) == (¬ a ∧ ¬ b)
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:= case (λ x, (¬ (x ∨ b)) == (¬ x ∧ ¬ b))
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(case (λ y, (¬ (true ∨ y)) == (¬ true ∧ ¬ y)) trivial trivial b)
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(case (λ y, (¬ (false ∨ y)) == (¬ false ∧ ¬ y)) trivial trivial b)
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a
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theorem not::or::elim {a b : Bool} (H : ¬ (a ∨ b)) : ¬ a ∧ ¬ b
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:= (not::or a b) ◂ H
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theorem not::iff (a b : Bool) : (¬ (a == b)) == ((¬ a) == b)
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:= case (λ x, (¬ (x == b)) == ((¬ x) == b))
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(case (λ y, (¬ (true == y)) == ((¬ true) == y)) trivial trivial b)
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(case (λ y, (¬ (false == y)) == ((¬ false) == y)) trivial trivial b)
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a
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theorem not::iff::elim {a b : Bool} (H : ¬ (a == b)) : (¬ a) == b
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:= (not::iff a b) ◂ H
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theorem not::implies (a b : Bool) : (¬ (a → b)) == (a ∧ ¬ b)
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:= case (λ x, (¬ (x → b)) == (x ∧ ¬ b))
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(case (λ y, (¬ (true → y)) == (true ∧ ¬ y)) trivial trivial b)
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(case (λ y, (¬ (false → y)) == (false ∧ ¬ y)) trivial trivial b)
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a
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theorem not::implies::elim {a b : Bool} (H : ¬ (a → b)) : a ∧ ¬ b
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:= (not::implies a b) ◂ H
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theorem not::congr {a b : Bool} (H : a == b) : (¬ a) == (¬ b)
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:= congr2 not H
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theorem eq::exists::intro {A : (Type U)} {P Q : A → Bool} (H : ∀ x : A, P x == Q x) : (∃ x : A, P x) == (∃ x : A, Q x)
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:= congr2 (Exists A) (funext H)
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theorem not::forall (A : (Type U)) (P : A → Bool) : (¬ (∀ x : A, P x)) == (∃ x : A, ¬ P x)
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:= calc (¬ ∀ x : A, P x) = (¬ ∀ x : A, ¬ ¬ P x) : not::congr (allext (λ x : A, symm (not::not::eq (P x))))
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... = (∃ x : A, ¬ P x) : refl (∃ x : A, ¬ P x)
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theorem not::forall::elim {A : (Type U)} {P : A → Bool} (H : ¬ (∀ x : A, P x)) : ∃ x : A, ¬ P x
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:= (not::forall A P) ◂ H
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theorem not::exists (A : (Type U)) (P : A → Bool) : (¬ ∃ x : A, P x) == (∀ x : A, ¬ P x)
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:= calc (¬ ∃ x : A, P x) = (¬ ¬ ∀ x : A, ¬ P x) : refl (¬ ∃ x : A, P x)
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... = (∀ x : A, ¬ P x) : not::not::eq (∀ x : A, ¬ P x)
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theorem not::exists::elim {A : (Type U)} {P : A → Bool} (H : ¬ ∃ x : A, P x) : ∀ x : A, ¬ P x
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:= (not::exists A P) ◂ H
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theorem exists::unfold1 {A : TypeU} {P : A → Bool} (a : A) (H : ∃ x : A, P x) : P a ∨ (∃ x : A, x ≠ a ∧ P x)
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:= exists::elim H
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(λ (w : A) (H1 : P w),
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or::elim (em (w = a))
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(λ Heq : w = a, or::introl (subst H1 Heq) (∃ x : A, x ≠ a ∧ P x))
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(λ Hne : w ≠ a, or::intror (P a) (exists::intro w (and::intro Hne H1))))
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theorem exists::unfold2 {A : TypeU} {P : A → Bool} (a : A) (H : P a ∨ (∃ x : A, x ≠ a ∧ P x)) : ∃ x : A, P x
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:= or::elim H
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(λ H1 : P a, exists::intro a H1)
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(λ H2 : (∃ x : A, x ≠ a ∧ P x),
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exists::elim H2
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(λ (w : A) (Hw : w ≠ a ∧ P w),
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exists::intro w (and::elimr Hw)))
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theorem exists::unfold {A : TypeU} (P : A → Bool) (a : A) : (∃ x : A, P x) = (P a ∨ (∃ x : A, x ≠ a ∧ P x))
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:= boolext (λ H : (∃ x : A, P x), exists::unfold1 a H)
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(λ H : (P a ∨ (∃ x : A, x ≠ a ∧ P x)), exists::unfold2 a H)
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set::opaque exists true
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set::opaque not true
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set::opaque or true
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set::opaque and true
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set::opaque implies true |