definition Prop := Type.{0}

definition false : Prop := ∀x : Prop, x
check false

theorem false_elim (C : Prop) (H : false) : C
:= H C

definition eq {A : Type} (a b : A)
:= ∀ P : A → Prop, P a → P b

check eq

infix `=`:50 := eq

theorem refl {A : Type} (a : A) : a = a
:= λ P H, H

definition true : Prop
:= false = false

theorem trivial : true
:= refl false

theorem subst {A : Type} {P : A -> Prop} {a b : A} (H1 : a = b) (H2 : P a) : P b
:= H1 _ H2

theorem symm {A : Type} {a b : A} (H : a = b) : b = a
:= subst H (refl a)

theorem trans {A : Type} {a b c : A} (H1 : a = b) (H2 : b = c) : a = c
:= subst H2 H1

inductive nat : Type :=
zero : nat,
succ : nat → nat
namespace nat end nat open nat

print "using strict implicit arguments"
definition symmetric {A : Type} (R : A → A → Prop) := ∀ ⦃a b⦄, R a b → R b a

check symmetric
variable p : nat → nat → Prop
check symmetric p
axiom H1 : symmetric p
axiom H2 : p zero (succ zero)
check H1
check H1 H2

print "------------"
print "using implicit arguments"
definition symmetric2 {A : Type} (R : A → A → Prop) := ∀ {a b}, R a b → R b a
check symmetric2
check symmetric2 p
axiom H3 : symmetric2 p
axiom H4 : p zero (succ zero)
check H3
check H3 H4

print "-----------------"
print "using strict implicit arguments (ASCII notation)"
definition symmetric3 {A : Type} (R : A → A → Prop) := ∀ {{a b}}, R a b → R b a

check symmetric3
check symmetric3 p
axiom H5 : symmetric3 p
axiom H6 : p zero (succ zero)
check H5
check H5 H6
