feat(library/data/num): prove many theorems for pos_num.lt and pos_num.le
This commit is contained in:
Leonardo de Moura
parent
f60fc5183a
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cc2d988aea
2 changed files with 482 additions and 16 deletions
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@ -5,24 +5,42 @@ Released under Apache 2.0 license as described in the file LICENSE.
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Module: data.num
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Author: Leonardo de Moura
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-/
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import logic.eq
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open bool
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import data.bool
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open bool eq.ops decidable
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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 pred.succ (a : pos_num) : pred (succ a) = a :=
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pos_num.rec_on a
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rfl
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(take (n : pos_num) (iH : pred (succ n) = n),
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theorem succ_one_eq_bit0_one : succ one = bit0 one :=
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rfl
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theorem succ_bit1_eq_bit0_succ (a : pos_num) : succ (bit1 a) = bit0 (succ a) :=
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rfl
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theorem succ_bit0_eq_bit1 (a : pos_num) : succ (bit0 a) = bit1 a :=
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rfl
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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_bit0_eq_cond (a : pos_num) : pred (bit0 a) = cond (is_one a) one (bit1 (pred a)) :=
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rfl
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theorem pred_succ : ∀ (a : pos_num), pred (succ a) = a
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| pred_succ one := rfl
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| pred_succ (bit0 a) := by rewrite succ_bit0_eq_bit1
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| pred_succ (bit1 a) :=
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calc
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pred (succ (bit1 n)) = cond (is_one (succ n)) one (bit1 (pred (succ n))) : rfl
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... = cond ff one (bit1 (pred (succ n))) : succ_not_is_one
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... = bit1 (pred (succ n)) : rfl
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... = bit1 n : iH)
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(take (n : pos_num) (iH : pred (succ n) = n), rfl)
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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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@ -69,4 +87,456 @@ namespace pos_num
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calc bit0 n * one = bit0 (n * one) : rfl
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... = bit0 n : iH)
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theorem decidable_eq [instance] : ∀ (a b : pos_num), decidable (a = b)
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| decidable_eq one one := inl rfl
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| decidable_eq one (bit0 b) := inr (λ H, pos_num.no_confusion H)
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| decidable_eq one (bit1 b) := inr (λ H, pos_num.no_confusion H)
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| decidable_eq (bit0 a) one := inr (λ H, pos_num.no_confusion H)
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| decidable_eq (bit0 a) (bit0 b) :=
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match decidable_eq a b with
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| inl H₁ := inl (eq.rec_on H₁ rfl)
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| inr H₁ := inr (λ H, pos_num.no_confusion H (λ H₂, absurd H₂ H₁))
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end
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| decidable_eq (bit0 a) (bit1 b) := inr (λ H, pos_num.no_confusion H)
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| decidable_eq (bit1 a) one := inr (λ H, pos_num.no_confusion H)
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| decidable_eq (bit1 a) (bit0 b) := inr (λ H, pos_num.no_confusion H)
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| decidable_eq (bit1 a) (bit1 b) :=
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match decidable_eq a b with
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| inl H₁ := inl (eq.rec_on H₁ rfl)
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| inr H₁ := inr (λ H, pos_num.no_confusion H (λ H₂, absurd H₂ H₁))
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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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| lt_one_right_eq_ff one := rfl
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| lt_one_right_eq_ff (bit0 a) := rfl
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| lt_one_right_eq_ff (bit1 a) := rfl
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theorem lt_one_succ_eq_tt : ∀ a : pos_num, one < succ a
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| lt_one_succ_eq_tt one := rfl
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| lt_one_succ_eq_tt (bit0 a) := rfl
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| lt_one_succ_eq_tt (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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| lt_irrefl one := rfl
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| lt_irrefl (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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| lt_irrefl (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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| @ne_of_lt_eq_tt one ⌞one⌟ H₁ (eq.refl one) := absurd H₁ ff_ne_tt
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| @ne_of_lt_eq_tt (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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| @ne_of_lt_eq_tt (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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| lt_base one := rfl
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| lt_base (bit0 a) :=
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begin
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rewrite [succ_bit0_eq_bit1, lt_bit0_bit1_eq_lt_succ],
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apply lt_base
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end
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| lt_base (bit1 a) :=
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begin
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rewrite [succ_bit1_eq_bit0_succ, 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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| @lt_step one one H := rfl
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| @lt_step one (bit0 b) H := rfl
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| @lt_step one (bit1 b) H := rfl
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| @lt_step (bit0 a) one H := absurd H ff_ne_tt
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| @lt_step (bit0 a) (bit0 b) H :=
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begin
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rewrite [succ_bit0_eq_bit1, 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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| @lt_step (bit0 a) (bit1 b) H :=
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begin
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rewrite [succ_bit1_eq_bit0_succ, 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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| @lt_step (bit1 a) one H := absurd H ff_ne_tt
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| @lt_step (bit1 a) (bit0 b) H :=
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begin
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rewrite [succ_bit0_eq_bit1, 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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| @lt_step (bit1 a) (bit1 b) H :=
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begin
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rewrite [succ_bit1_eq_bit0_succ, 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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| @lt_of_lt_succ_succ one one H := absurd H ff_ne_tt
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| @lt_of_lt_succ_succ one (bit0 b) H := rfl
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| @lt_of_lt_succ_succ one (bit1 b) H := rfl
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| @lt_of_lt_succ_succ (bit0 a) one H :=
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begin
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rewrite [succ_bit0_eq_bit1 at H, succ_one_eq_bit0_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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| @lt_of_lt_succ_succ (bit0 a) (bit0 b) H := by exact H
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| @lt_of_lt_succ_succ (bit0 a) (bit1 b) H := by exact H
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| @lt_of_lt_succ_succ (bit1 a) one H :=
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begin
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rewrite [succ_bit1_eq_bit0_succ at H, succ_one_eq_bit0_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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| @lt_of_lt_succ_succ (bit1 a) (bit0 b) H :=
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begin
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rewrite [succ_bit1_eq_bit0_succ at H, succ_bit0_eq_bit1 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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| @lt_of_lt_succ_succ (bit1 a) (bit1 b) H :=
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begin
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rewrite [lt_bit1_bit1_eq_lt, *succ_bit1_eq_bit0_succ 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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| @lt_succ_succ one one H := absurd H ff_ne_tt
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| @lt_succ_succ one (bit0 b) H :=
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begin
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rewrite [succ_bit0_eq_bit1, succ_one_eq_bit0_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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| @lt_succ_succ one (bit1 b) H :=
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begin
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rewrite [succ_one_eq_bit0_one, succ_bit1_eq_bit0_succ, lt_bit0_bit0_eq_lt],
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apply lt_one_succ_eq_tt
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end
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| @lt_succ_succ (bit0 a) one H := absurd H ff_ne_tt
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| @lt_succ_succ (bit0 a) (bit0 b) H := by exact H
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| @lt_succ_succ (bit0 a) (bit1 b) H := by exact H
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| @lt_succ_succ (bit1 a) one H := absurd H ff_ne_tt
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| @lt_succ_succ (bit1 a) (bit0 b) H :=
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begin
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rewrite [succ_bit1_eq_bit0_succ, succ_bit0_eq_bit1, 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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| @lt_succ_succ (bit1 a) (bit1 b) H :=
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begin
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rewrite [lt_bit1_bit1_eq_lt at H, *succ_bit1_eq_bit0_succ, 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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| @lt_of_lt_succ one one H := absurd_of_eq_ff_of_eq_tt !lt_one_right_eq_ff H
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| @lt_of_lt_succ one (bit0 b) H := rfl
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| @lt_of_lt_succ one (bit1 b) H := rfl
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| @lt_of_lt_succ (bit0 a) one H := absurd_of_eq_ff_of_eq_tt !lt_one_right_eq_ff H
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| @lt_of_lt_succ (bit0 a) (bit0 b) H := by exact H
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| @lt_of_lt_succ (bit0 a) (bit1 b) H :=
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begin
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rewrite [succ_bit0_eq_bit1 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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| @lt_of_lt_succ (bit1 a) one H := absurd_of_eq_ff_of_eq_tt !lt_one_right_eq_ff H
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| @lt_of_lt_succ (bit1 a) (bit0 b) H :=
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begin
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rewrite [lt_bit1_bit0_eq_lt, succ_bit1_eq_bit0_succ 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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| @lt_of_lt_succ (bit1 a) (bit1 b) H :=
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begin
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rewrite [succ_bit1_eq_bit0_succ 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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| @lt_of_lt_succ_of_ne one one H₁ H₂ := absurd rfl H₂
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| @lt_of_lt_succ_of_ne one (bit0 b) H₁ H₂ := rfl
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| @lt_of_lt_succ_of_ne one (bit1 b) H₁ H₂ := rfl
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| @lt_of_lt_succ_of_ne (bit0 a) one H₁ H₂ :=
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begin
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rewrite [succ_one_eq_bit0_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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| @lt_of_lt_succ_of_ne (bit0 a) (bit0 b) H₁ H₂ :=
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begin
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rewrite [lt_bit0_bit0_eq_lt, succ_bit0_eq_bit1 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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| @lt_of_lt_succ_of_ne (bit0 a) (bit1 b) H₁ H₂ :=
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begin
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rewrite [succ_bit1_eq_bit0_succ 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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| @lt_of_lt_succ_of_ne (bit1 a) one H₁ H₂ :=
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begin
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rewrite [succ_one_eq_bit0_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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| @lt_of_lt_succ_of_ne (bit1 a) (bit0 b) H₁ H₂ :=
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begin
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rewrite [succ_bit0_eq_bit1 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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| @lt_of_lt_succ_of_ne (bit1 a) (bit1 b) H₁ H₂ :=
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begin
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rewrite [succ_bit1_eq_bit0_succ 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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| @lt_trans one b (bit0 c) H₁ H₂ := rfl
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| @lt_trans one b (bit1 c) H₁ H₂ := rfl
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| @lt_trans a (bit0 b) one H₁ H₂ := absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H₂
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| @lt_trans a (bit1 b) one H₁ H₂ := absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H₂
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| @lt_trans (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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| @lt_trans (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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| @lt_trans (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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| @lt_trans (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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| @lt_trans (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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| @lt_trans (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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| @lt_trans (bit1 a) (bit1 b) (bit0 c) H₁ H₂ :=
|
||||
begin
|
||||
rewrite [lt_bit1_bit1_eq_lt at H₁, lt_bit1_bit0_eq_lt at H₂, lt_bit1_bit0_eq_lt],
|
||||
apply (lt_trans H₁ H₂)
|
||||
end
|
||||
| @lt_trans (bit1 a) (bit1 b) (bit1 c) H₁ H₂ :=
|
||||
begin
|
||||
rewrite lt_bit1_bit1_eq_lt at *,
|
||||
apply (lt_trans H₁ H₂)
|
||||
end
|
||||
|
||||
theorem lt_antisymm : ∀ {a b : pos_num}, a < b → b ≮ a
|
||||
| @lt_antisymm one one H := rfl
|
||||
| @lt_antisymm one (bit0 b) H := rfl
|
||||
| @lt_antisymm one (bit1 b) H := rfl
|
||||
| @lt_antisymm (bit0 a) one H := absurd H ff_ne_tt
|
||||
| @lt_antisymm (bit0 a) (bit0 b) H :=
|
||||
begin
|
||||
rewrite lt_bit0_bit0_eq_lt at *,
|
||||
apply (lt_antisymm H)
|
||||
end
|
||||
| @lt_antisymm (bit0 a) (bit1 b) H :=
|
||||
begin
|
||||
rewrite lt_bit1_bit0_eq_lt,
|
||||
rewrite lt_bit0_bit1_eq_lt_succ at H,
|
||||
have H₁ : succ b ≮ a, from lt_antisymm H,
|
||||
apply eq_ff_of_ne_tt,
|
||||
intro H₂,
|
||||
apply (@by_cases (succ b = a)),
|
||||
show succ b = a → false,
|
||||
begin
|
||||
intro Hp,
|
||||
rewrite -Hp at H,
|
||||
apply (absurd_of_eq_ff_of_eq_tt (lt_irrefl (succ b)) H)
|
||||
end,
|
||||
show succ b ≠ a → false,
|
||||
begin
|
||||
intro Hn,
|
||||
have H₃ : succ b < succ a, from lt_succ_succ H₂,
|
||||
have H₄ : succ b < a, from lt_of_lt_succ_of_ne H₃ Hn,
|
||||
apply (absurd_of_eq_ff_of_eq_tt H₁ H₄)
|
||||
end,
|
||||
end
|
||||
| @lt_antisymm (bit1 a) one H := absurd H ff_ne_tt
|
||||
| @lt_antisymm (bit1 a) (bit0 b) H :=
|
||||
begin
|
||||
rewrite lt_bit0_bit1_eq_lt_succ,
|
||||
rewrite lt_bit1_bit0_eq_lt at H,
|
||||
have H₁ : lt b a = ff, from lt_antisymm H,
|
||||
apply eq_ff_of_ne_tt,
|
||||
intro H₂,
|
||||
apply (@by_cases (b = a)),
|
||||
show b = a → false,
|
||||
begin
|
||||
intro Hp,
|
||||
rewrite -Hp at H,
|
||||
apply (absurd_of_eq_ff_of_eq_tt (lt_irrefl b) H)
|
||||
end,
|
||||
show b ≠ a → false,
|
||||
begin
|
||||
intro Hn,
|
||||
have H₃ : b < a, from lt_of_lt_succ_of_ne H₂ Hn,
|
||||
apply (absurd_of_eq_ff_of_eq_tt H₁ H₃)
|
||||
end,
|
||||
end
|
||||
| @lt_antisymm (bit1 a) (bit1 b) H :=
|
||||
begin
|
||||
rewrite lt_bit1_bit1_eq_lt at *,
|
||||
apply (lt_antisymm H)
|
||||
end
|
||||
|
||||
local notation a ≤ b := (le a b = tt)
|
||||
|
||||
theorem le_refl : ∀ a : pos_num, a ≤ a :=
|
||||
lt_base
|
||||
|
||||
theorem le_eq_lt_succ {a b : pos_num} : le a b = lt a (succ b) :=
|
||||
rfl
|
||||
|
||||
theorem not_lt_of_le : ∀ {a b : pos_num}, a ≤ b → b < a → false
|
||||
| @not_lt_of_le one one H₁ H₂ := absurd H₂ ff_ne_tt
|
||||
| @not_lt_of_le one (bit0 b) H₁ H₂ := absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H₂
|
||||
| @not_lt_of_le one (bit1 b) H₁ H₂ := absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H₂
|
||||
| @not_lt_of_le (bit0 a) one H₁ H₂ :=
|
||||
begin
|
||||
rewrite [le_eq_lt_succ at H₁, succ_one_eq_bit0_one at H₁, lt_bit0_bit0_eq_lt at H₁],
|
||||
apply (absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H₁)
|
||||
end
|
||||
| @not_lt_of_le (bit0 a) (bit0 b) H₁ H₂ :=
|
||||
begin
|
||||
rewrite [le_eq_lt_succ at H₁, succ_bit0_eq_bit1 at H₁, lt_bit0_bit1_eq_lt_succ at H₁],
|
||||
rewrite [lt_bit0_bit0_eq_lt at H₂],
|
||||
apply (not_lt_of_le H₁ H₂)
|
||||
end
|
||||
| @not_lt_of_le (bit0 a) (bit1 b) H₁ H₂ :=
|
||||
begin
|
||||
rewrite [le_eq_lt_succ at H₁, succ_bit1_eq_bit0_succ at H₁, lt_bit0_bit0_eq_lt at H₁],
|
||||
rewrite [lt_bit1_bit0_eq_lt at H₂],
|
||||
apply (not_lt_of_le H₁ H₂)
|
||||
end
|
||||
| @not_lt_of_le (bit1 a) one H₁ H₂ :=
|
||||
begin
|
||||
rewrite [le_eq_lt_succ at H₁, succ_one_eq_bit0_one at H₁, lt_bit1_bit0_eq_lt at H₁],
|
||||
apply (absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff _) H₁)
|
||||
end
|
||||
| @not_lt_of_le (bit1 a) (bit0 b) H₁ H₂ :=
|
||||
begin
|
||||
rewrite [le_eq_lt_succ at H₁, succ_bit0_eq_bit1 at H₁, lt_bit1_bit1_eq_lt at H₁],
|
||||
rewrite lt_bit0_bit1_eq_lt_succ at H₂,
|
||||
have H₃ : a < succ b, from lt_step H₁,
|
||||
apply (@by_cases (b = a)),
|
||||
begin
|
||||
intro Hba, rewrite -Hba at H₁,
|
||||
apply (absurd_of_eq_ff_of_eq_tt (lt_irrefl b) H₁)
|
||||
end,
|
||||
begin
|
||||
intro Hnba,
|
||||
have H₄ : b < a, from lt_of_lt_succ_of_ne H₂ Hnba,
|
||||
apply (not_lt_of_le H₃ H₄)
|
||||
end
|
||||
end
|
||||
| @not_lt_of_le (bit1 a) (bit1 b) H₁ H₂ :=
|
||||
begin
|
||||
rewrite [le_eq_lt_succ at H₁, succ_bit1_eq_bit0_succ at H₁, lt_bit1_bit0_eq_lt at H₁],
|
||||
rewrite [lt_bit1_bit1_eq_lt at H₂],
|
||||
apply (not_lt_of_le H₁ H₂)
|
||||
end
|
||||
|
||||
theorem le_antisymm : ∀ {a b : pos_num}, a ≤ b → b ≤ a → a = b
|
||||
| @le_antisymm one one H₁ H₂ := rfl
|
||||
| @le_antisymm one (bit0 b) H₁ H₂ :=
|
||||
by apply (absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff b) H₂)
|
||||
| @le_antisymm one (bit1 b) H₁ H₂ :=
|
||||
by apply (absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff b) H₂)
|
||||
| @le_antisymm (bit0 a) one H₁ H₂ :=
|
||||
by apply (absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff a) H₁)
|
||||
| @le_antisymm (bit0 a) (bit0 b) H₁ H₂ :=
|
||||
begin
|
||||
rewrite [le_eq_lt_succ at *, succ_bit0_eq_bit1 at *, lt_bit0_bit1_eq_lt_succ at *],
|
||||
have H : a = b, from le_antisymm H₁ H₂,
|
||||
rewrite H
|
||||
end
|
||||
| @le_antisymm (bit0 a) (bit1 b) H₁ H₂ :=
|
||||
begin
|
||||
rewrite [le_eq_lt_succ at *, succ_bit1_eq_bit0_succ at H₁, succ_bit0_eq_bit1 at H₂],
|
||||
rewrite [lt_bit0_bit0_eq_lt at H₁, lt_bit1_bit1_eq_lt at H₂],
|
||||
apply (false.rec _ (not_lt_of_le H₁ H₂))
|
||||
end
|
||||
| @le_antisymm (bit1 a) one H₁ H₂ :=
|
||||
by apply (absurd_of_eq_ff_of_eq_tt (lt_one_right_eq_ff a) H₁)
|
||||
| @le_antisymm (bit1 a) (bit0 b) H₁ H₂ :=
|
||||
begin
|
||||
rewrite [le_eq_lt_succ at *, succ_bit0_eq_bit1 at H₁, succ_bit1_eq_bit0_succ at H₂],
|
||||
rewrite [lt_bit1_bit1_eq_lt at H₁, lt_bit0_bit0_eq_lt at H₂],
|
||||
apply (false.rec _ (not_lt_of_le H₂ H₁))
|
||||
end
|
||||
| @le_antisymm (bit1 a) (bit1 b) H₁ H₂ :=
|
||||
begin
|
||||
rewrite [le_eq_lt_succ at *, succ_bit1_eq_bit0_succ at *, lt_bit1_bit0_eq_lt at *],
|
||||
have H : a = b, from le_antisymm H₁ H₂,
|
||||
rewrite H
|
||||
end
|
||||
|
||||
theorem le_trans {a b c : pos_num} : a ≤ b → b ≤ c → a ≤ c :=
|
||||
begin
|
||||
intros (H₁, H₂),
|
||||
rewrite [le_eq_lt_succ at *],
|
||||
apply (@by_cases (a = b)),
|
||||
begin
|
||||
intro Hab, rewrite Hab, exact H₂
|
||||
end,
|
||||
begin
|
||||
intro Hnab,
|
||||
have Haltb : a < b, from lt_of_lt_succ_of_ne H₁ Hnab,
|
||||
apply (lt_trans Haltb H₂)
|
||||
end,
|
||||
end
|
||||
|
||||
end pos_num
|
||||
|
|
|
|||
|
|
@ -64,10 +64,6 @@ namespace pos_num
|
|||
|
||||
definition le (a b : pos_num) : bool :=
|
||||
lt a (succ b)
|
||||
|
||||
definition equal (a b : pos_num) : bool :=
|
||||
le a b && le b a
|
||||
|
||||
end pos_num
|
||||
|
||||
definition num.is_inhabited [instance] : inhabited num :=
|
||||
|
|
|
|||
Loading…
Reference in a new issue