prelude
definition Prop := Sort.{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

attribute [elab_as_eliminator]
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
constant 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