54 lines
1.2 KiB
Text
54 lines
1.2 KiB
Text
import data.num
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using num
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variables int nat real : Type.{1}
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variable nat_add : nat → nat → nat
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variable int_add : int → int → int
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variable real_add : real → real → real
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inductive add_struct (A : Type) :=
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| mk : (A → A → A) → add_struct A
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definition add {A : Type} {S : add_struct A} (a b : A) : A :=
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add_struct_rec (λ m, m) S a b
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infixl `+`:65 := add
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definition add_nat_struct [instance] : add_struct nat := mk nat_add
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definition add_int_struct [instance] : add_struct int := mk int_add
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definition add_real_struct [instance] : add_struct real := mk real_add
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variables n m : nat
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variables i j : int
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variables x y : real
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variable num_to_nat : num → nat
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variable nat_to_int : nat → int
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variable int_to_real : int → real
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coercion num_to_nat
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coercion nat_to_int
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coercion int_to_real
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set_option pp.implicit true
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set_option pp.coercion true
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check n + m
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check i + j
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check x + y
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check i + n
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check i + x
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check n + i
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check x + i
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check n + x
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check x + n
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check x + i + n
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check n + 0
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check 0 + n
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check 0 + i
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check i + 0
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check 0 + x
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check x + 0
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variable eq {A : Type} : A → A → Prop
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infixl `=`:50 := eq
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abbreviation id (A : Type) (a : A) := a
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notation A `=` B `:` C := @eq C A B
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check nat_to_int n + nat_to_int m = (n + m) : int
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