31 lines
839 B
Text
31 lines
839 B
Text
import logic
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namespace nat
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variable nat : Type.{1}
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variable add : nat → nat → nat
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variable le : nat → nat → Prop
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variable one : nat
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infixl `+`:65 := add
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infix `≤`:50 := le
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axiom add_assoc (a b c : nat) : (a + b) + c = a + (b + c)
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axiom add_le_left {a b : nat} (H : a ≤ b) (c : nat) : c + a ≤ c + b
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end nat
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namespace int
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variable int : Type.{1}
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variable add : int → int → int
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variable le : int → int → Prop
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variable one : int
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infixl `+`:65 := add
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infix `≤`:50 := le
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axiom add_assoc (a b c : int) : (a + b) + c = a + (b + c)
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axiom add_le_left {a b : int} (H : a ≤ b) (c : int) : c + a ≤ c + b
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abbreviation lt (a b : int) := a + one ≤ b
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infix `<`:50 := lt
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end int
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open int
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open nat
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theorem add_lt_left {a b : int} (H : a < b) (c : int) : c + a < c + b :=
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subst (symm (add_assoc c a one)) (add_le_left H c)
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