512 lines
17 KiB
Text
512 lines
17 KiB
Text
/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura
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-/
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import data.bool tools.helper_tactics
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open bool eq.ops decidable helper_tactics
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namespace pos_num
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theorem succ_not_is_one (a : pos_num) : is_one (succ a) = ff :=
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pos_num.induction_on a rfl (take n iH, rfl) (take n iH, rfl)
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theorem succ_one : succ one = bit0 one
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theorem succ_bit1 (a : pos_num) : succ (bit1 a) = bit0 (succ a)
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theorem succ_bit0 (a : pos_num) : succ (bit0 a) = bit1 a
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theorem ne_of_bit0_ne_bit0 {a b : pos_num} (H₁ : bit0 a ≠ bit0 b) : a ≠ b :=
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assume H : a = b,
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absurd rfl (H ▸ H₁)
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theorem ne_of_bit1_ne_bit1 {a b : pos_num} (H₁ : bit1 a ≠ bit1 b) : a ≠ b :=
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assume H : a = b,
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absurd rfl (H ▸ H₁)
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theorem pred_succ : ∀ (a : pos_num), pred (succ a) = a
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| one := rfl
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| (bit0 a) := by rewrite succ_bit0
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| (bit1 a) :=
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calc
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pred (succ (bit1 a)) = cond (is_one (succ a)) one (bit1 (pred (succ a))) : rfl
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... = cond ff one (bit1 (pred (succ a))) : succ_not_is_one
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... = bit1 (pred (succ a)) : rfl
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... = bit1 a : pred_succ a
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section
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variables (a b : pos_num)
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theorem one_add_one : one + one = bit0 one
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theorem one_add_bit0 : one + (bit0 a) = bit1 a
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theorem one_add_bit1 : one + (bit1 a) = succ (bit1 a)
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theorem bit0_add_one : (bit0 a) + one = bit1 a
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theorem bit1_add_one : (bit1 a) + one = succ (bit1 a)
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theorem bit0_add_bit0 : (bit0 a) + (bit0 b) = bit0 (a + b)
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theorem bit0_add_bit1 : (bit0 a) + (bit1 b) = bit1 (a + b)
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theorem bit1_add_bit0 : (bit1 a) + (bit0 b) = bit1 (a + b)
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theorem bit1_add_bit1 : (bit1 a) + (bit1 b) = succ (bit1 (a + b))
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theorem one_mul : one * a = a
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end
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theorem mul_one : ∀ a, a * one = a
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| one := rfl
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| (bit1 n) :=
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calc bit1 n * one = bit0 (n * one) + one : rfl
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... = bit0 n + one : mul_one n
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... = bit1 n : bit0_add_one
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| (bit0 n) :=
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calc bit0 n * one = bit0 (n * one) : rfl
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... = bit0 n : mul_one n
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theorem decidable_eq [instance] : ∀ (a b : pos_num), decidable (a = b)
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| one one := inl rfl
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| one (bit0 b) := inr (by contradiction)
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| one (bit1 b) := inr (by contradiction)
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| (bit0 a) one := inr (by contradiction)
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| (bit0 a) (bit0 b) :=
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match decidable_eq a b with
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| inl H₁ := inl (by rewrite H₁)
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| inr H₁ := inr (by intro H; injection H; contradiction)
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end
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| (bit0 a) (bit1 b) := inr (by contradiction)
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| (bit1 a) one := inr (by contradiction)
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| (bit1 a) (bit0 b) := inr (by contradiction)
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| (bit1 a) (bit1 b) :=
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match decidable_eq a b with
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| inl H₁ := inl (by rewrite H₁)
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| inr H₁ := inr (by intro H; injection H; contradiction)
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end
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local notation a < b := (lt a b = tt)
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local notation a `≮`:50 b:50 := (lt a b = ff)
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theorem lt_one_right_eq_ff : ∀ a : pos_num, a ≮ one
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| one := rfl
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| (bit0 a) := rfl
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| (bit1 a) := rfl
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theorem lt_one_succ_eq_tt : ∀ a : pos_num, one < succ a
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| one := rfl
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| (bit0 a) := rfl
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| (bit1 a) := rfl
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theorem lt_of_lt_bit0_bit0 {a b : pos_num} (H : bit0 a < bit0 b) : a < b := H
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theorem lt_of_lt_bit0_bit1 {a b : pos_num} (H : bit1 a < bit0 b) : a < b := H
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theorem lt_of_lt_bit1_bit1 {a b : pos_num} (H : bit1 a < bit1 b) : a < b := H
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theorem lt_of_lt_bit1_bit0 {a b : pos_num} (H : bit0 a < bit1 b) : a < succ b := H
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theorem lt_bit0_bit0_eq_lt (a b : pos_num) : lt (bit0 a) (bit0 b) = lt a b :=
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rfl
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theorem lt_bit1_bit1_eq_lt (a b : pos_num) : lt (bit1 a) (bit1 b) = lt a b :=
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rfl
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theorem lt_bit1_bit0_eq_lt (a b : pos_num) : lt (bit1 a) (bit0 b) = lt a b :=
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rfl
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theorem lt_bit0_bit1_eq_lt_succ (a b : pos_num) : lt (bit0 a) (bit1 b) = lt a (succ b) :=
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rfl
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theorem lt_irrefl : ∀ (a : pos_num), a ≮ a
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| one := rfl
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| (bit0 a) :=
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begin
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rewrite lt_bit0_bit0_eq_lt, apply lt_irrefl
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end
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| (bit1 a) :=
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begin
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rewrite lt_bit1_bit1_eq_lt, apply lt_irrefl
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end
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theorem ne_of_lt_eq_tt : ∀ {a b : pos_num}, a < b → a = b → false
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| one ⌞one⌟ H₁ (eq.refl one) := absurd H₁ ff_ne_tt
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| (bit0 a) ⌞(bit0 a)⌟ H₁ (eq.refl (bit0 a)) :=
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begin
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rewrite lt_bit0_bit0_eq_lt at H₁,
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apply ne_of_lt_eq_tt H₁ (eq.refl a)
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end
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| (bit1 a) ⌞(bit1 a)⌟ H₁ (eq.refl (bit1 a)) :=
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begin
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rewrite lt_bit1_bit1_eq_lt at H₁,
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apply ne_of_lt_eq_tt H₁ (eq.refl a)
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end
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theorem lt_base : ∀ a : pos_num, a < succ a
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| one := rfl
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| (bit0 a) :=
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begin
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rewrite [succ_bit0, lt_bit0_bit1_eq_lt_succ],
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apply lt_base
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end
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| (bit1 a) :=
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begin
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rewrite [succ_bit1, lt_bit1_bit0_eq_lt],
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apply lt_base
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end
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theorem lt_step : ∀ {a b : pos_num}, a < b → a < succ b
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| one one H := rfl
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| one (bit0 b) H := rfl
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| one (bit1 b) H := rfl
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| (bit0 a) one H := absurd H ff_ne_tt
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| (bit0 a) (bit0 b) H :=
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begin
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rewrite [succ_bit0, lt_bit0_bit1_eq_lt_succ, lt_bit0_bit0_eq_lt at H],
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apply lt_step H
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end
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| (bit0 a) (bit1 b) H :=
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begin
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rewrite [succ_bit1, lt_bit0_bit0_eq_lt, lt_bit0_bit1_eq_lt_succ at H],
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exact H
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end
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| (bit1 a) one H := absurd H ff_ne_tt
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| (bit1 a) (bit0 b) H :=
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begin
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rewrite [succ_bit0, lt_bit1_bit1_eq_lt, lt_bit1_bit0_eq_lt at H],
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exact H
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end
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| (bit1 a) (bit1 b) H :=
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begin
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rewrite [succ_bit1, lt_bit1_bit0_eq_lt, lt_bit1_bit1_eq_lt at H],
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apply lt_step H
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end
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theorem lt_of_lt_succ_succ : ∀ {a b : pos_num}, succ a < succ b → a < b
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| one one H := absurd H ff_ne_tt
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| one (bit0 b) H := rfl
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| one (bit1 b) H := rfl
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| (bit0 a) one H :=
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begin
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rewrite [succ_bit0 at H, succ_one at H, lt_bit1_bit0_eq_lt at H],
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apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff a) H
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end
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| (bit0 a) (bit0 b) H := by exact H
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| (bit0 a) (bit1 b) H := by exact H
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| (bit1 a) one H :=
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begin
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rewrite [succ_bit1 at H, succ_one at H, lt_bit0_bit0_eq_lt at H],
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apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff (succ a)) H
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end
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| (bit1 a) (bit0 b) H :=
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begin
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rewrite [succ_bit1 at H, succ_bit0 at H, lt_bit0_bit1_eq_lt_succ at H],
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rewrite lt_bit1_bit0_eq_lt,
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apply lt_of_lt_succ_succ H
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end
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| (bit1 a) (bit1 b) H :=
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begin
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rewrite [lt_bit1_bit1_eq_lt, *succ_bit1 at H, lt_bit0_bit0_eq_lt at H],
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apply lt_of_lt_succ_succ H
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end
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theorem lt_succ_succ : ∀ {a b : pos_num}, a < b → succ a < succ b
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| one one H := absurd H ff_ne_tt
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| one (bit0 b) H :=
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begin
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rewrite [succ_bit0, succ_one, lt_bit0_bit1_eq_lt_succ],
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apply lt_one_succ_eq_tt
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end
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| one (bit1 b) H :=
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begin
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rewrite [succ_one, succ_bit1, lt_bit0_bit0_eq_lt],
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apply lt_one_succ_eq_tt
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end
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| (bit0 a) one H := absurd H ff_ne_tt
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| (bit0 a) (bit0 b) H := by exact H
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| (bit0 a) (bit1 b) H := by exact H
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| (bit1 a) one H := absurd H ff_ne_tt
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| (bit1 a) (bit0 b) H :=
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begin
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rewrite [succ_bit1, succ_bit0, lt_bit0_bit1_eq_lt_succ, lt_bit1_bit0_eq_lt at H],
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apply lt_succ_succ H
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end
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| (bit1 a) (bit1 b) H :=
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begin
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rewrite [lt_bit1_bit1_eq_lt at H, *succ_bit1, lt_bit0_bit0_eq_lt],
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apply lt_succ_succ H
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end
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theorem lt_of_lt_succ : ∀ {a b : pos_num}, succ a < b → a < b
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| one one H := absurd_of_eq_ff_of_eq_tt !lt_one_right_eq_ff H
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| one (bit0 b) H := rfl
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| one (bit1 b) H := rfl
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| (bit0 a) one H := absurd_of_eq_ff_of_eq_tt !lt_one_right_eq_ff H
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| (bit0 a) (bit0 b) H := by exact H
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| (bit0 a) (bit1 b) H :=
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begin
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rewrite [succ_bit0 at H, lt_bit1_bit1_eq_lt at H, lt_bit0_bit1_eq_lt_succ],
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apply lt_step H
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end
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| (bit1 a) one H := absurd_of_eq_ff_of_eq_tt !lt_one_right_eq_ff H
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| (bit1 a) (bit0 b) H :=
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begin
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rewrite [lt_bit1_bit0_eq_lt, succ_bit1 at H, lt_bit0_bit0_eq_lt at H],
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apply lt_of_lt_succ H
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end
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| (bit1 a) (bit1 b) H :=
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begin
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rewrite [succ_bit1 at H, lt_bit0_bit1_eq_lt_succ at H, lt_bit1_bit1_eq_lt],
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apply lt_of_lt_succ_succ H
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end
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theorem lt_of_lt_succ_of_ne : ∀ {a b : pos_num}, a < succ b → a ≠ b → a < b
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| one one H₁ H₂ := absurd rfl H₂
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| one (bit0 b) H₁ H₂ := rfl
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| one (bit1 b) H₁ H₂ := rfl
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| (bit0 a) one H₁ H₂ :=
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begin
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rewrite [succ_one at H₁, lt_bit0_bit0_eq_lt at H₁],
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apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H₁
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end
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| (bit0 a) (bit0 b) H₁ H₂ :=
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begin
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rewrite [lt_bit0_bit0_eq_lt, succ_bit0 at H₁, lt_bit0_bit1_eq_lt_succ at H₁],
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apply lt_of_lt_succ_of_ne H₁ (ne_of_bit0_ne_bit0 H₂)
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end
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| (bit0 a) (bit1 b) H₁ H₂ :=
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begin
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rewrite [succ_bit1 at H₁, lt_bit0_bit0_eq_lt at H₁, lt_bit0_bit1_eq_lt_succ],
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exact H₁
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end
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| (bit1 a) one H₁ H₂ :=
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begin
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rewrite [succ_one at H₁, lt_bit1_bit0_eq_lt at H₁],
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apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H₁
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end
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| (bit1 a) (bit0 b) H₁ H₂ :=
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begin
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rewrite [succ_bit0 at H₁, lt_bit1_bit1_eq_lt at H₁, lt_bit1_bit0_eq_lt],
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exact H₁
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end
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| (bit1 a) (bit1 b) H₁ H₂ :=
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begin
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rewrite [succ_bit1 at H₁, lt_bit1_bit0_eq_lt at H₁, lt_bit1_bit1_eq_lt],
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apply lt_of_lt_succ_of_ne H₁ (ne_of_bit1_ne_bit1 H₂)
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end
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theorem lt_trans : ∀ {a b c : pos_num}, a < b → b < c → a < c
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| one b (bit0 c) H₁ H₂ := rfl
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| one b (bit1 c) H₁ H₂ := rfl
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| a (bit0 b) one H₁ H₂ := absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H₂
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| a (bit1 b) one H₁ H₂ := absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H₂
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| (bit0 a) (bit0 b) (bit0 c) H₁ H₂ :=
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begin
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rewrite lt_bit0_bit0_eq_lt at *, apply lt_trans H₁ H₂
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end
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| (bit0 a) (bit0 b) (bit1 c) H₁ H₂ :=
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begin
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rewrite [lt_bit0_bit1_eq_lt_succ at *, lt_bit0_bit0_eq_lt at H₁],
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apply lt_trans H₁ H₂
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end
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| (bit0 a) (bit1 b) (bit0 c) H₁ H₂ :=
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begin
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rewrite [lt_bit0_bit1_eq_lt_succ at H₁, lt_bit1_bit0_eq_lt at H₂, lt_bit0_bit0_eq_lt],
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apply @by_cases (a = b),
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begin
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intro H, rewrite -H at H₂, exact H₂
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end,
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begin
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intro H,
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apply lt_trans (lt_of_lt_succ_of_ne H₁ H) H₂
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end
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end
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| (bit0 a) (bit1 b) (bit1 c) H₁ H₂ :=
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begin
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rewrite [lt_bit0_bit1_eq_lt_succ at *, lt_bit1_bit1_eq_lt at H₂],
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apply lt_trans H₁ (lt_succ_succ H₂)
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end
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| (bit1 a) (bit0 b) (bit0 c) H₁ H₂ :=
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begin
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rewrite [lt_bit0_bit0_eq_lt at H₂, lt_bit1_bit0_eq_lt at *],
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apply lt_trans H₁ H₂
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end
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| (bit1 a) (bit0 b) (bit1 c) H₁ H₂ :=
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begin
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rewrite [lt_bit1_bit0_eq_lt at H₁, lt_bit0_bit1_eq_lt_succ at H₂, lt_bit1_bit1_eq_lt],
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apply @by_cases (b = c),
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begin
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intro H, rewrite H at H₁, exact H₁
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end,
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begin
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intro H,
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apply lt_trans H₁ (lt_of_lt_succ_of_ne H₂ H)
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end
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end
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| (bit1 a) (bit1 b) (bit0 c) H₁ H₂ :=
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begin
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rewrite [lt_bit1_bit1_eq_lt at H₁, lt_bit1_bit0_eq_lt at H₂, lt_bit1_bit0_eq_lt],
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apply lt_trans H₁ H₂
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end
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| (bit1 a) (bit1 b) (bit1 c) H₁ H₂ :=
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begin
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rewrite lt_bit1_bit1_eq_lt at *,
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apply lt_trans H₁ H₂
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end
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theorem lt_antisymm : ∀ {a b : pos_num}, a < b → b ≮ a
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| one one H := rfl
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| one (bit0 b) H := rfl
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| one (bit1 b) H := rfl
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| (bit0 a) one H := absurd H ff_ne_tt
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| (bit0 a) (bit0 b) H :=
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begin
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rewrite lt_bit0_bit0_eq_lt at *,
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apply lt_antisymm H
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end
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| (bit0 a) (bit1 b) H :=
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begin
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rewrite lt_bit1_bit0_eq_lt,
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rewrite lt_bit0_bit1_eq_lt_succ at H,
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have H₁ : succ b ≮ a, from lt_antisymm H,
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apply eq_ff_of_ne_tt,
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intro H₂,
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apply @by_cases (succ b = a),
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show succ b = a → false,
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begin
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intro Hp,
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rewrite -Hp at H,
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apply absurd_of_eq_ff_of_eq_tt (lt_irrefl (succ b)) H
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end,
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show succ b ≠ a → false,
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begin
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intro Hn,
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have H₃ : succ b < succ a, from lt_succ_succ H₂,
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have H₄ : succ b < a, from lt_of_lt_succ_of_ne H₃ Hn,
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apply absurd_of_eq_ff_of_eq_tt H₁ H₄
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end,
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end
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| (bit1 a) one H := absurd H ff_ne_tt
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| (bit1 a) (bit0 b) H :=
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begin
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rewrite lt_bit0_bit1_eq_lt_succ,
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rewrite lt_bit1_bit0_eq_lt at H,
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have H₁ : lt b a = ff, from lt_antisymm H,
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apply eq_ff_of_ne_tt,
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intro H₂,
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apply @by_cases (b = a),
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show b = a → false,
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begin
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intro Hp,
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rewrite -Hp at H,
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apply absurd_of_eq_ff_of_eq_tt (lt_irrefl b) H
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end,
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show b ≠ a → false,
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begin
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intro Hn,
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have H₃ : b < a, from lt_of_lt_succ_of_ne H₂ Hn,
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apply absurd_of_eq_ff_of_eq_tt H₁ H₃
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end,
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end
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| (bit1 a) (bit1 b) H :=
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begin
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rewrite lt_bit1_bit1_eq_lt at *,
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apply lt_antisymm H
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end
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local notation a ≤ b := (le a b = tt)
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theorem le_refl : ∀ a : pos_num, a ≤ a :=
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lt_base
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theorem le_eq_lt_succ {a b : pos_num} : le a b = lt a (succ b) :=
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rfl
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theorem not_lt_of_le : ∀ {a b : pos_num}, a ≤ b → b < a → false
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| one one H₁ H₂ := absurd H₂ ff_ne_tt
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| one (bit0 b) H₁ H₂ := absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H₂
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| one (bit1 b) H₁ H₂ := absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H₂
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| (bit0 a) one H₁ H₂ :=
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begin
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rewrite [le_eq_lt_succ at H₁, succ_one at H₁, lt_bit0_bit0_eq_lt at H₁],
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apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H₁
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end
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| (bit0 a) (bit0 b) H₁ H₂ :=
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begin
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rewrite [le_eq_lt_succ at H₁, succ_bit0 at H₁, lt_bit0_bit1_eq_lt_succ at H₁],
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rewrite [lt_bit0_bit0_eq_lt at H₂],
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apply not_lt_of_le H₁ H₂
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end
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| (bit0 a) (bit1 b) H₁ H₂ :=
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begin
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rewrite [le_eq_lt_succ at H₁, succ_bit1 at H₁, lt_bit0_bit0_eq_lt at H₁],
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rewrite [lt_bit1_bit0_eq_lt at H₂],
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apply not_lt_of_le H₁ H₂
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end
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| (bit1 a) one H₁ H₂ :=
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begin
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rewrite [le_eq_lt_succ at H₁, succ_one at H₁, lt_bit1_bit0_eq_lt at H₁],
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apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H₁
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end
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| (bit1 a) (bit0 b) H₁ H₂ :=
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begin
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rewrite [le_eq_lt_succ at H₁, succ_bit0 at H₁, lt_bit1_bit1_eq_lt at H₁],
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rewrite lt_bit0_bit1_eq_lt_succ at H₂,
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have H₃ : a < succ b, from lt_step H₁,
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apply @by_cases (b = a),
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begin
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intro Hba, rewrite -Hba at H₁,
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apply absurd_of_eq_ff_of_eq_tt (lt_irrefl b) H₁
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end,
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begin
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intro Hnba,
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have H₄ : b < a, from lt_of_lt_succ_of_ne H₂ Hnba,
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apply not_lt_of_le H₃ H₄
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end
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end
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| (bit1 a) (bit1 b) H₁ H₂ :=
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begin
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rewrite [le_eq_lt_succ at H₁, succ_bit1 at H₁, lt_bit1_bit0_eq_lt at H₁],
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rewrite [lt_bit1_bit1_eq_lt at H₂],
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apply not_lt_of_le H₁ H₂
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end
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theorem le_antisymm : ∀ {a b : pos_num}, a ≤ b → b ≤ a → a = b
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| one one H₁ H₂ := rfl
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| one (bit0 b) H₁ H₂ :=
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by apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff b) H₂
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| one (bit1 b) H₁ H₂ :=
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by apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff b) H₂
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| (bit0 a) one H₁ H₂ :=
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by apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff a) H₁
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| (bit0 a) (bit0 b) H₁ H₂ :=
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begin
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rewrite [le_eq_lt_succ at *, succ_bit0 at *, lt_bit0_bit1_eq_lt_succ at *],
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have H : a = b, from le_antisymm H₁ H₂,
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rewrite H
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end
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| (bit0 a) (bit1 b) H₁ H₂ :=
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begin
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rewrite [le_eq_lt_succ at *, succ_bit1 at H₁, succ_bit0 at H₂],
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rewrite [lt_bit0_bit0_eq_lt at H₁, lt_bit1_bit1_eq_lt at H₂],
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apply false.rec _ (not_lt_of_le H₁ H₂)
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end
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| (bit1 a) one H₁ H₂ :=
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by apply absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff a) H₁
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| (bit1 a) (bit0 b) H₁ H₂ :=
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begin
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rewrite [le_eq_lt_succ at *, succ_bit0 at H₁, succ_bit1 at H₂],
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rewrite [lt_bit1_bit1_eq_lt at H₁, lt_bit0_bit0_eq_lt at H₂],
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apply false.rec _ (not_lt_of_le H₂ H₁)
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end
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| (bit1 a) (bit1 b) H₁ H₂ :=
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begin
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rewrite [le_eq_lt_succ at *, succ_bit1 at *, lt_bit1_bit0_eq_lt at *],
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have H : a = b, from le_antisymm H₁ H₂,
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rewrite H
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end
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theorem le_trans {a b c : pos_num} : a ≤ b → b ≤ c → a ≤ c :=
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begin
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intro H₁ H₂,
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rewrite [le_eq_lt_succ at *],
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apply @by_cases (a = b),
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begin
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intro Hab, rewrite Hab, exact H₂
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end,
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begin
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intro Hnab,
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have Haltb : a < b, from lt_of_lt_succ_of_ne H₁ Hnab,
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apply lt_trans Haltb H₂
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end,
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end
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end pos_num
|