36 lines
1.4 KiB
Text
36 lines
1.4 KiB
Text
import logic
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open eq_ops eq
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inductive nat : Type :=
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zero : nat,
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succ : nat → nat
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namespace nat
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definition add (x y : nat) : nat := nat.rec x (λn r, succ r) y
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infixl `+`:65 := add
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definition mul (n m : nat) := nat.rec zero (fun m x, x + n) m
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infixl `*`:75 := mul
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axiom add_one (n:nat) : n + (succ zero) = succ n
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axiom mul_zero_right (n : nat) : n * zero = zero
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axiom add_zero_right (n : nat) : n + zero = n
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axiom mul_succ_right (n m : nat) : n * succ m = n * m + n
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axiom add_assoc (n m k : nat) : (n + m) + k = n + (m + k)
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axiom add_right_comm (n m k : nat) : n + m + k = n + k + m
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axiom induction_on {P : nat → Prop} (a : nat) (H1 : P zero) (H2 : ∀ (n : nat) (IH : P n), P (succ n)) : P a
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set_option unifier.max_steps 50000
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theorem mul_add_distr_left (n m k : nat) : (n + m) * k = n * k + m * k
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:= induction_on k
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(calc
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(n + m) * zero = zero : eq.refl _
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... = n * zero + m * zero : eq.refl _)
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(take l IH,
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calc
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(n + m) * succ l = (n + m) * l + (n + m) : mul_succ_right _ _
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... = n * l + m * l + (n + m) : {IH}
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... = n * l + m * l + n + m : symm (add_assoc _ _ _)
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... = n * l + n + m * l + m : {add_right_comm _ _ _}
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... = n * l + n + (m * l + m) : add_assoc _ _ _
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... = n * succ l + (m * l + m) : {symm (mul_succ_right _ _)}
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... = n * succ l + m * succ l : {symm (mul_succ_right _ _)})
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end nat
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