22 lines
680 B
Text
22 lines
680 B
Text
import standard
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using num eq_proofs
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inductive nat : Type :=
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| zero : nat
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| succ : nat → nat
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abbreviation plus (x y : nat) : nat
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:= nat_rec x (λn r, succ r) y
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definition to_nat [coercion] [inline] (n : num) : nat
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:= num_rec zero (λn, pos_num_rec (succ zero) (λn r, plus r (plus r (succ zero))) (λn r, plus r r) n) n
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definition add (x y : nat) : nat
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:= plus x y
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variable le : nat → nat → Prop
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infixl `+`:65 := add
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infix `≤`:50 := le
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axiom add_one (n:nat) : n + (succ zero) = succ n
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axiom add_le_right {n m : nat} (H : n ≤ m) (k : nat) : n + k ≤ m + k
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theorem succ_le {n m : nat} (H : n ≤ m) : succ n ≤ succ m
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:= add_one m ▸ add_one n ▸ add_le_right H 1
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