refactor(library): more implicit_args for: and_assoc, and_comm, or_assoc, or_comm, if_pos, if_neg
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Leonardo de Moura
parent
6b7e79b62f
commit
d536475e49
12 changed files with 82 additions and 83 deletions
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@ -71,11 +71,11 @@ definition proj (a : ℕ × ℕ) : ℕ × ℕ :=
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if pr1 a ≥ pr2 a then pair (pr1 a - pr2 a) 0 else pair 0 (pr2 a - pr1 a)
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theorem proj_ge {a : ℕ × ℕ} (H : pr1 a ≥ pr2 a) : proj a = pair (pr1 a - pr2 a) 0 :=
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if_pos H _ _
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if_pos H
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theorem proj_lt {a : ℕ × ℕ} (H : pr1 a < pr2 a) : proj a = pair 0 (pr2 a - pr1 a) :=
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have H2 : ¬ pr1 a ≥ pr2 a, from lt_imp_not_ge H,
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if_neg H2 _ _
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if_neg H2
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theorem proj_le {a : ℕ × ℕ} (H : pr1 a ≤ pr2 a) : proj a = pair 0 (pr2 a - pr1 a) :=
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or_elim le_or_gt
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@ -344,7 +344,7 @@ or_elim (cases a)
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---reverse equalities, rename
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theorem cases_succ (a : ℤ) : (∃n : ℕ, a = of_nat n) ∨ (∃n : ℕ, a = - (of_nat (succ n))) :=
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or_elim (cases a)
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(assume H : (∃n : ℕ, a = of_nat n), or_intro_left _ H)
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(assume H : (∃n : ℕ, a = of_nat n), or_inl H)
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(assume H,
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obtain (n : ℕ) (H2 : a = -(of_nat n)), from H,
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discriminate
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@ -354,7 +354,7 @@ or_elim (cases a)
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a = -(of_nat n) : H2
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... = -(of_nat 0) : {H3}
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... = of_nat 0 : neg_zero,
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or_intro_left _ (exists_intro 0 H4))
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or_inl (exists_intro 0 H4))
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(take k : ℕ,
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assume H3 : n = succ k,
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have H4 : a = -(of_nat (succ k)), from subst H3 H2,
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@ -12,7 +12,7 @@ import .basic
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using nat (hiding case)
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using decidable
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using fake_simplifier
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using int
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using int eq_ops
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namespace int
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@ -121,7 +121,7 @@ discriminate
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a = a + 0 : symm (add_zero_right a)
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... = a + n : {symm H2}
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... = b : Hn,
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or_intro_right _ H3)
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or_inr H3)
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(take k : ℕ,
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assume H2 : n = succ k,
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have H3 : a + 1 + k = b, from
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@ -129,7 +129,7 @@ discriminate
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a + 1 + k = a + succ k : by simp
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... = a + n : by simp
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... = b : Hn,
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or_intro_left _ (le_intro H3))
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or_inl (le_intro H3))
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-- ### interaction with neg and sub
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@ -376,7 +376,7 @@ int_by_cases a
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int_by_cases_succ b
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(take m : ℕ,
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show -n ≤ m ∨ -n > m, from
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or_intro_left _ (neg_le_pos n m))
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or_inl (neg_le_pos n m))
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(take m : ℕ,
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show -n ≤ -succ m ∨ -n > -succ m, from
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or_imp_or le_or_gt
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@ -389,7 +389,7 @@ theorem trichotomy_alt (a b : ℤ) : (a < b ∨ a = b) ∨ a > b :=
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or_imp_or_left (le_or_gt a b) (assume H : a ≤ b, le_imp_lt_or_eq H)
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theorem trichotomy (a b : ℤ) : a < b ∨ a = b ∨ a > b :=
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iff_elim_left (or_assoc _ _ _) (trichotomy_alt a b)
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iff_elim_left or_assoc (trichotomy_alt a b)
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theorem le_total (a b : ℤ) : a ≤ b ∨ b ≤ a :=
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or_imp_or_right (le_or_gt a b) (assume H : b < a, lt_imp_le H)
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@ -583,13 +583,13 @@ theorem or_elim3 {a b c d : Prop} (H : a ∨ b ∨ c) (Ha : a → d) (Hb : b →
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or_elim H Ha (assume H2,or_elim H2 Hb Hc)
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theorem sign_pos {a : ℤ} (H : a > 0) : sign a = 1 :=
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if_pos H _ _
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if_pos H
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theorem sign_negative {a : ℤ} (H : a < 0) : sign a = - 1 :=
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trans (if_neg (lt_antisym H) _ _) (if_pos H _ _)
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if_neg (lt_antisym H) ⬝ if_pos H
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theorem sign_zero : sign 0 = 0 :=
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trans (if_neg (lt_irrefl 0) _ _) (if_neg (lt_irrefl 0) _ _)
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if_neg (lt_irrefl 0) ⬝ if_neg (lt_irrefl 0)
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-- add_rewrite sign_negative sign_pos to_nat_negative to_nat_nonneg_eq sign_zero mul_to_nat
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@ -197,13 +197,13 @@ theorem mem_nil (x : T) : x ∈ nil ↔ false := iff_refl _
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theorem mem_cons (x : T) (y : T) (l : list T) : mem x (cons y l) ↔ (x = y ∨ mem x l) := iff_refl _
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theorem mem_concat_imp_or (x : T) (s t : list T) : x ∈ s ++ t → x ∈ s ∨ x ∈ t :=
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list_induction_on s (or_intro_right _)
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list_induction_on s or_inr
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(take y s,
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assume IH : x ∈ s ++ t → x ∈ s ∨ x ∈ t,
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assume H1 : x ∈ (y :: s) ++ t,
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have H2 : x = y ∨ x ∈ s ++ t, from H1,
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have H3 : x = y ∨ x ∈ s ∨ x ∈ t, from or_imp_or_right H2 IH,
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iff_elim_right (or_assoc _ _ _) H3)
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iff_elim_right or_assoc H3)
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theorem mem_or_imp_concat (x : T) (s t : list T) : x ∈ s ∨ x ∈ t → x ∈ s ++ t :=
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list_induction_on s
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@ -214,9 +214,9 @@ list_induction_on s
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or_elim H
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(assume H1,
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or_elim H1
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(take H2 : x = y, or_intro_left _ H2)
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(take H2 : x ∈ s, or_intro_right _ (IH (or_intro_left _ H2))))
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(assume H1 : x ∈ t, or_intro_right _ (IH (or_intro_right _ H1))))
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(take H2 : x = y, or_inl H2)
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(take H2 : x ∈ s, or_inr (IH (or_inl H2))))
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(assume H1 : x ∈ t, or_inr (IH (or_inr H1))))
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theorem mem_concat (x : T) (s t : list T) : x ∈ s ++ t ↔ x ∈ s ∨ x ∈ t
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:= iff_intro (mem_concat_imp_or _ _ _) (mem_or_imp_concat _ _ _)
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@ -67,7 +67,7 @@ definition pred (n : ℕ) := nat_rec 0 (fun m x, m) n
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theorem pred_zero : pred 0 = 0
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theorem pred_succ (n : ℕ) : pred (succ n) = n
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theorem pred_succ {n : ℕ} : pred (succ n) = n
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opaque_hint (hiding pred)
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@ -75,7 +75,7 @@ theorem zero_or_succ_pred (n : ℕ) : n = 0 ∨ n = succ (pred n) :=
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induction_on n
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(or_inl (refl 0))
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(take m IH, or_inr
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(show succ m = succ (pred (succ m)), from congr_arg succ (pred_succ m⁻¹)))
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(show succ m = succ (pred (succ m)), from congr_arg succ pred_succ⁻¹))
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theorem zero_or_exists_succ (n : ℕ) : n = 0 ∨ ∃k, n = succ k :=
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or_imp_or (zero_or_succ_pred n) (assume H, H)
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@ -91,9 +91,9 @@ or_elim (zero_or_succ_pred n)
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theorem succ_inj {n m : ℕ} (H : succ n = succ m) : n = m :=
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calc
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n = pred (succ n) : pred_succ n⁻¹
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n = pred (succ n) : pred_succ⁻¹
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... = pred (succ m) : {H}
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... = m : pred_succ m
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... = m : pred_succ
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theorem succ_ne_self {n : ℕ} : succ n ≠ n :=
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induction_on n
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@ -48,12 +48,12 @@ if measure x < m then f x else default
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theorem restrict_lt_eq {dom codom : Type} (default : codom) (measure : dom → ℕ) (f : dom → codom)
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(m : ℕ) (x : dom) (H : measure x < m) :
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restrict default measure f m x = f x :=
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if_pos H _ _
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if_pos H
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theorem restrict_not_lt_eq {dom codom : Type} (default : codom) (measure : dom → ℕ)
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(f : dom → codom) (m : ℕ) (x : dom) (H : ¬ measure x < m) :
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restrict default measure f m x = default :=
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if_neg H _ _
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if_neg H
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definition rec_measure_aux {dom codom : Type} (default : codom) (measure : dom → ℕ)
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(rec_val : dom → (dom → codom) → codom) : ℕ → dom → codom :=
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@ -81,7 +81,7 @@ case_strong_induction_on m
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funext
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(take x,
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have H3 [fact]: ¬ measure x < 0, from not_lt_zero,
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show restrict default measure f 0 x = default, from if_neg H3 _ _),
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show restrict default measure f 0 x = default, from if_neg H3),
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show f' 0 = restrict default measure f 0, from trans H1 (symm H2))
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(take m,
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assume IH: ∀n, n ≤ m → f' n = restrict default measure f n,
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@ -93,16 +93,16 @@ case_strong_induction_on m
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have H2 [fact] : f' (succ m) x = rec_val x f, from
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calc
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f' (succ m) x = if measure x < succ m then rec_val x (f' m) else default : rfl
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... = rec_val x (f' m) : if_pos H1 _ _
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... = rec_val x (f' m) : if_pos H1
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... = rec_val x (restrict default measure f m) : {IH m le_refl}
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... = rec_val x f : symm (rec_decreasing _ _ _ (lt_succ_imp_le H1)),
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have H3 : restrict default measure f (succ m) x = rec_val x f, from
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let m' := measure x in
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calc
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restrict default measure f (succ m) x = f x : if_pos H1 _ _
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restrict default measure f (succ m) x = f x : if_pos H1
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... = f' (succ m') x : refl _
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... = if measure x < succ m' then rec_val x (f' m') else default : rfl
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... = rec_val x (f' m') : if_pos self_lt_succ _ _
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... = rec_val x (f' m') : if_pos self_lt_succ
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... = rec_val x (restrict default measure f m') : {IH m' (lt_succ_imp_le H1)}
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... = rec_val x f : (rec_decreasing _ _ _ le_refl)⁻¹,
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show f' (succ m) x = restrict default measure f (succ m) x,
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@ -111,9 +111,9 @@ case_strong_induction_on m
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have H2 : f' (succ m) x = default, from
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calc
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f' (succ m) x = if measure x < succ m then rec_val x (f' m) else default : rfl
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... = default : if_neg H1 _ _,
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... = default : if_neg H1,
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have H3 : restrict default measure f (succ m) x = default,
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from if_neg H1 _ _,
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from if_neg H1,
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show f' (succ m) x = restrict default measure f (succ m) x,
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from trans H2 (symm H3))))
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@ -130,7 +130,7 @@ let m := measure x in
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calc
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f x = f' (succ m) x : rfl
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... = if measure x < succ m then rec_val x (f' m) else default : rfl
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... = rec_val x (f' m) : if_pos self_lt_succ _ _
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... = rec_val x (f' m) : if_pos self_lt_succ
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... = rec_val x (restrict default measure f m) : {rec_measure_aux_spec _ _ _ rec_decreasing _}
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... = rec_val x f : (rec_decreasing _ _ _ le_refl)⁻¹
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@ -154,8 +154,8 @@ show lhs = rhs, from
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by_cases -- (y = 0 ∨ x < y)
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(assume H1 : y = 0 ∨ x < y,
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calc
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lhs = 0 : if_pos H1 _ _
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... = rhs : symm (if_pos H1 _ _))
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lhs = 0 : if_pos H1
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... = rhs : (if_pos H1)⁻¹)
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(assume H1 : ¬ (y = 0 ∨ x < y),
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have H2 : y ≠ 0 ∧ ¬ x < y, from sorry, -- subst (not_or _ _) H1,
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have ypos : y > 0, from ne_zero_imp_pos (and_elim_left H2),
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@ -163,9 +163,9 @@ show lhs = rhs, from
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have H4 : x - y < x, from sub_lt (lt_le_trans ypos xgey) ypos,
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have H5 : x - y < m, from lt_le_trans H4 H,
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symm (calc
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rhs = succ (restrict 0 (fun x, x) g m (x - y)) : if_neg H1 _ _
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rhs = succ (restrict 0 (fun x, x) g m (x - y)) : if_neg H1
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... = succ (g (x - y)) : {restrict_lt_eq _ _ _ _ _ H5}
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... = lhs : symm (if_neg H1 _ _)))
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... = lhs : (if_neg H1)⁻¹))
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theorem div_aux_spec (y : ℕ) (x : ℕ) :
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div_aux y x = if (y = 0 ∨ x < y) then 0 else succ (div_aux y (x - y)) :=
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@ -176,12 +176,12 @@ definition idivide (x : ℕ) (y : ℕ) : ℕ := div_aux y x
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infixl `div` := idivide -- copied from Isabelle
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theorem div_zero {x : ℕ} : x div 0 = 0 :=
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trans (div_aux_spec _ _) (if_pos (or_intro_left _ (refl _)) _ _)
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trans (div_aux_spec _ _) (if_pos (or_inl rfl))
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-- add_rewrite div_zero
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theorem div_less {x y : ℕ} (H : x < y) : x div y = 0 :=
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trans (div_aux_spec _ _) (if_pos (or_intro_right _ H) _ _)
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trans (div_aux_spec _ _) (if_pos (or_inr H))
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-- add_rewrite div_less
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@ -197,7 +197,7 @@ have H3 : ¬ (y = 0 ∨ x < y), from
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or_elim H4
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(assume H5 : y = 0, not_elim lt_irrefl (subst H5 H1))
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(assume H5 : x < y, not_elim (lt_imp_not_ge H5) H2)),
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trans (div_aux_spec _ _) (if_neg H3 _ _)
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trans (div_aux_spec _ _) (if_neg H3)
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theorem div_add_self_right {x z : ℕ} (H : z > 0) : (x + z) div z = succ (x div z) :=
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have H1 : z ≤ x + z, by simp,
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@ -230,8 +230,8 @@ show lhs = rhs, from
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by_cases -- (y = 0 ∨ x < y)
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(assume H1 : y = 0 ∨ x < y,
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calc
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lhs = x : if_pos H1 _ _
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... = rhs : symm (if_pos H1 _ _))
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lhs = x : if_pos H1
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... = rhs : (if_pos H1)⁻¹)
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(assume H1 : ¬ (y = 0 ∨ x < y),
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have H2 : y ≠ 0 ∧ ¬ x < y, from sorry, -- subst (not_or _ _) H1,
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have ypos : y > 0, from ne_zero_imp_pos (and_elim_left H2),
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@ -239,9 +239,9 @@ show lhs = rhs, from
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have H4 : x - y < x, from sub_lt (lt_le_trans ypos xgey) ypos,
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have H5 : x - y < m, from lt_le_trans H4 H,
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symm (calc
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rhs = restrict 0 (fun x, x) g m (x - y) : if_neg H1 _ _
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rhs = restrict 0 (fun x, x) g m (x - y) : if_neg H1
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... = g (x - y) : restrict_lt_eq _ _ _ _ _ H5
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... = lhs : symm (if_neg H1 _ _)))
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... = lhs : (if_neg H1)⁻¹))
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theorem mod_aux_spec (y : ℕ) (x : ℕ) :
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mod_aux y x = if (y = 0 ∨ x < y) then x else mod_aux y (x - y) :=
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@ -252,12 +252,12 @@ definition modulo (x : ℕ) (y : ℕ) : ℕ := mod_aux y x
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infixl `mod` := modulo
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theorem mod_zero {x : ℕ} : x mod 0 = x :=
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trans (mod_aux_spec _ _) (if_pos (or_intro_left _ (refl _)) _ _)
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trans (mod_aux_spec _ _) (if_pos (or_inl rfl))
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-- add_rewrite mod_zero
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theorem mod_lt_eq {x y : ℕ} (H : x < y) : x mod y = x :=
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trans (mod_aux_spec _ _) (if_pos (or_intro_right _ H) _ _)
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trans (mod_aux_spec _ _) (if_pos (or_inr H))
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-- add_rewrite mod_lt_eq
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@ -273,7 +273,7 @@ have H3 : ¬ (y = 0 ∨ x < y), from
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or_elim H4
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(assume H5 : y = 0, not_elim lt_irrefl (H5 ▸ H1))
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(assume H5 : x < y, not_elim (lt_imp_not_ge H5) H2)),
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trans (mod_aux_spec _ _) (if_neg H3 _ _)
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(mod_aux_spec _ _) ⬝ (if_neg H3)
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-- need more of these, add as rewrite rules
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theorem mod_add_self_right {x z : ℕ} (H : z > 0) : (x + z) mod z = x mod z :=
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@ -598,17 +598,17 @@ show lhs = rhs, from
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by_cases -- (y = 0)
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(assume H1 : y = 0,
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calc
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lhs = x : if_pos H1 _ _
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... = rhs : (if_pos H1 _ _)⁻¹)
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lhs = x : if_pos H1
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... = rhs : (if_pos H1)⁻¹)
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(assume H1 : y ≠ 0,
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have ypos : y > 0, from ne_zero_imp_pos H1,
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have H2 : gcd_aux_measure p' = x mod y, from pr2_pair _ _,
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have H3 : gcd_aux_measure p' < gcd_aux_measure p, from H2⁻¹ ▸ (mod_lt ypos),
|
||||
have H4: gcd_aux_measure p' < m, from lt_le_trans H3 H,
|
||||
symm (calc
|
||||
rhs = restrict 0 gcd_aux_measure g m p' : if_neg H1 _ _
|
||||
rhs = restrict 0 gcd_aux_measure g m p' : if_neg H1
|
||||
... = g p' : restrict_lt_eq _ _ _ _ _ H4
|
||||
... = lhs : (if_neg H1 _ _)⁻¹))
|
||||
... = lhs : (if_neg H1)⁻¹))
|
||||
|
||||
theorem gcd_aux_spec (p : ℕ × ℕ) : gcd_aux p =
|
||||
let x := pr1 p, y := pr2 p in
|
||||
|
|
@ -625,12 +625,12 @@ calc
|
|||
... = if y = 0 then x else gcd y (x mod y) : rfl
|
||||
|
||||
theorem gcd_zero (x : ℕ) : gcd x 0 = x :=
|
||||
(gcd_def x 0) ⬝ (if_pos rfl _ _)
|
||||
(gcd_def x 0) ⬝ (if_pos rfl)
|
||||
|
||||
-- add_rewrite gcd_zero
|
||||
|
||||
theorem gcd_pos (m : ℕ) {n : ℕ} (H : n > 0) : gcd m n = gcd n (m mod n) :=
|
||||
(gcd_def m n) ⬝ (if_neg (pos_imp_ne_zero H) _ _)
|
||||
gcd_def m n ⬝ if_neg (pos_imp_ne_zero H)
|
||||
|
||||
theorem gcd_zero_left (x : ℕ) : gcd 0 x = x :=
|
||||
case x (by simp) (take x, (gcd_def _ _) ⬝ (by simp))
|
||||
|
|
@ -650,7 +650,7 @@ have aux : ∀m, P m n, from
|
|||
aux m
|
||||
|
||||
theorem gcd_succ (m n : ℕ) : gcd m (succ n) = gcd (succ n) (m mod succ n) :=
|
||||
gcd_def _ _ ⬝ if_neg succ_ne_zero _ _
|
||||
gcd_def _ _ ⬝ if_neg succ_ne_zero
|
||||
|
||||
theorem gcd_one (n : ℕ) : gcd n 1 = 1 := sorry
|
||||
-- (by simp) (gcd_succ n 0)
|
||||
|
|
|
|||
|
|
@ -173,7 +173,7 @@ obtain (k : ℕ) (H2 : succ n + k = m), from (le_elim H),
|
|||
theorem le_pred_self {n : ℕ} : pred n ≤ n :=
|
||||
case n
|
||||
(pred_zero⁻¹ ▸ le_refl)
|
||||
(take k : ℕ, (pred_succ k)⁻¹ ▸ self_le_succ)
|
||||
(take k : ℕ, pred_succ⁻¹ ▸ self_le_succ)
|
||||
|
||||
theorem pred_le {n m : ℕ} (H : n ≤ m) : pred n ≤ pred m :=
|
||||
discriminate
|
||||
|
|
@ -189,8 +189,8 @@ discriminate
|
|||
have H2 : pred n + l = pred m,
|
||||
from calc
|
||||
pred n + l = pred (succ k) + l : {Hn}
|
||||
... = k + l : {pred_succ k}
|
||||
... = pred (succ (k + l)) : (pred_succ (k + l))⁻¹
|
||||
... = k + l : {pred_succ}
|
||||
... = pred (succ (k + l)) : pred_succ⁻¹
|
||||
... = pred (succ k + l) : {add_succ_left⁻¹}
|
||||
... = pred (n + l) : {Hn⁻¹}
|
||||
... = pred m : {Hl},
|
||||
|
|
@ -205,7 +205,7 @@ discriminate
|
|||
have H2 : pred n = k,
|
||||
from calc
|
||||
pred n = pred (succ k) : {Hn}
|
||||
... = k : pred_succ k,
|
||||
... = k : pred_succ,
|
||||
have H3 : k ≤ m, from subst H2 H,
|
||||
have H4 : succ k ≤ m ∨ k = m, from le_imp_succ_le_or_eq H3,
|
||||
show n ≤ m ∨ n = succ m, from
|
||||
|
|
@ -412,7 +412,7 @@ theorem trichotomy_alt {n m : ℕ} : (n < m ∨ n = m) ∨ n > m :=
|
|||
or_imp_or_left le_or_gt (assume H : n ≤ m, le_imp_lt_or_eq H)
|
||||
|
||||
theorem trichotomy {n m : ℕ} : n < m ∨ n = m ∨ n > m :=
|
||||
iff_elim_left (or_assoc _ _ _) trichotomy_alt
|
||||
iff_elim_left or_assoc trichotomy_alt
|
||||
|
||||
theorem le_total {n m : ℕ} : n ≤ m ∨ m ≤ n :=
|
||||
or_imp_or_right le_or_gt (assume H : m < n, lt_imp_le H)
|
||||
|
|
|
|||
|
|
@ -43,7 +43,7 @@ induction_on m
|
|||
(calc
|
||||
succ n - 1 = pred (succ n - 0) : sub_succ_right
|
||||
... = pred (succ n) : {sub_zero_right}
|
||||
... = n : pred_succ _
|
||||
... = n : pred_succ
|
||||
... = n - 0 : sub_zero_right⁻¹)
|
||||
(take k : nat,
|
||||
assume IH : succ n - succ k = n - k,
|
||||
|
|
@ -139,7 +139,7 @@ induction_on n
|
|||
(take k : nat,
|
||||
assume IH : pred k * m = k * m - m,
|
||||
calc
|
||||
pred (succ k) * m = k * m : {pred_succ _}
|
||||
pred (succ k) * m = k * m : {pred_succ}
|
||||
... = k * m + m - m : sub_add_left⁻¹
|
||||
... = succ k * m - m : {mul_succ_left⁻¹})
|
||||
|
||||
|
|
|
|||
|
|
@ -4,7 +4,6 @@
|
|||
|
||||
import general_notation .eq
|
||||
|
||||
|
||||
-- and
|
||||
-- ---
|
||||
|
||||
|
|
@ -145,10 +144,10 @@ iff_intro (λ Ha, subst H Ha) (λ Hb, subst (symm H) Hb)
|
|||
-- comm and assoc for and / or
|
||||
-- ---------------------------
|
||||
|
||||
theorem and_comm (a b : Prop) : a ∧ b ↔ b ∧ a :=
|
||||
theorem and_comm {a b : Prop} : a ∧ b ↔ b ∧ a :=
|
||||
iff_intro (λH, and_swap H) (λH, and_swap H)
|
||||
|
||||
theorem and_assoc (a b c : Prop) : (a ∧ b) ∧ c ↔ a ∧ (b ∧ c) :=
|
||||
theorem and_assoc {a b c : Prop} : (a ∧ b) ∧ c ↔ a ∧ (b ∧ c) :=
|
||||
iff_intro
|
||||
(assume H, and_intro
|
||||
(and_elim_left (and_elim_left H))
|
||||
|
|
@ -157,10 +156,10 @@ iff_intro
|
|||
(and_intro (and_elim_left H) (and_elim_left (and_elim_right H)))
|
||||
(and_elim_right (and_elim_right H)))
|
||||
|
||||
theorem or_comm (a b : Prop) : a ∨ b ↔ b ∨ a :=
|
||||
theorem or_comm {a b : Prop} : a ∨ b ↔ b ∨ a :=
|
||||
iff_intro (λH, or_swap H) (λH, or_swap H)
|
||||
|
||||
theorem or_assoc (a b c : Prop) : (a ∨ b) ∨ c ↔ a ∨ (b ∨ c) :=
|
||||
theorem or_assoc {a b c : Prop} : (a ∨ b) ∨ c ↔ a ∨ (b ∨ c) :=
|
||||
iff_intro
|
||||
(assume H, or_elim H
|
||||
(assume H1, or_elim H1
|
||||
|
|
|
|||
|
|
@ -14,24 +14,24 @@ using relation
|
|||
|
||||
theorem or_right_comm (a b c : Prop) : (a ∨ b) ∨ c ↔ (a ∨ c) ∨ b :=
|
||||
calc
|
||||
(a ∨ b) ∨ c ↔ a ∨ (b ∨ c) : or_assoc _ _ _
|
||||
... ↔ a ∨ (c ∨ b) : {or_comm b c}
|
||||
... ↔ (a ∨ c) ∨ b : iff_symm (or_assoc _ _ _)
|
||||
(a ∨ b) ∨ c ↔ a ∨ (b ∨ c) : or_assoc
|
||||
... ↔ a ∨ (c ∨ b) : {or_comm}
|
||||
... ↔ (a ∨ c) ∨ b : iff_symm or_assoc
|
||||
|
||||
theorem or_left_comm (a b c : Prop) : a ∨ (b ∨ c)↔ b ∨ (a ∨ c) :=
|
||||
calc
|
||||
a ∨ (b ∨ c) ↔ (a ∨ b) ∨ c : iff_symm (or_assoc _ _ _)
|
||||
... ↔ (b ∨ a) ∨ c : {or_comm a b}
|
||||
... ↔ b ∨ (a ∨ c) : or_assoc _ _ _
|
||||
a ∨ (b ∨ c) ↔ (a ∨ b) ∨ c : iff_symm or_assoc
|
||||
... ↔ (b ∨ a) ∨ c : {or_comm}
|
||||
... ↔ b ∨ (a ∨ c) : or_assoc
|
||||
|
||||
theorem and_right_comm (a b c : Prop) : (a ∧ b) ∧ c ↔ (a ∧ c) ∧ b :=
|
||||
calc
|
||||
(a ∧ b) ∧ c ↔ a ∧ (b ∧ c) : and_assoc _ _ _
|
||||
... ↔ a ∧ (c ∧ b) : {and_comm b c}
|
||||
... ↔ (a ∧ c) ∧ b : iff_symm (and_assoc _ _ _)
|
||||
(a ∧ b) ∧ c ↔ a ∧ (b ∧ c) : and_assoc
|
||||
... ↔ a ∧ (c ∧ b) : {and_comm}
|
||||
... ↔ (a ∧ c) ∧ b : iff_symm and_assoc
|
||||
|
||||
theorem and_left_comm (a b c : Prop) : a ∧ (b ∧ c)↔ b ∧ (a ∧ c) :=
|
||||
calc
|
||||
a ∧ (b ∧ c) ↔ (a ∧ b) ∧ c : iff_symm (and_assoc _ _ _)
|
||||
... ↔ (b ∧ a) ∧ c : {and_comm a b}
|
||||
... ↔ b ∧ (a ∧ c) : and_assoc _ _ _
|
||||
a ∧ (b ∧ c) ↔ (a ∧ b) ∧ c : iff_symm and_assoc
|
||||
... ↔ (b ∧ a) ∧ c : {and_comm}
|
||||
... ↔ b ∧ (a ∧ c) : and_assoc
|
||||
|
|
|
|||
|
|
@ -12,13 +12,13 @@ rec_on H (assume Hc, t) (assume Hnc, e)
|
|||
|
||||
notation `if` c `then` t `else` e:45 := ite c t e
|
||||
|
||||
theorem if_pos {c : Prop} {H : decidable c} (Hc : c) {A : Type} (t e : A) : (if c then t else e) = t :=
|
||||
theorem if_pos {c : Prop} {H : decidable c} (Hc : c) {A : Type} {t e : A} : (if c then t else e) = t :=
|
||||
decidable_rec
|
||||
(assume Hc : c, refl (@ite c (inl Hc) A t e))
|
||||
(assume Hnc : ¬c, absurd Hc Hnc)
|
||||
H
|
||||
|
||||
theorem if_neg {c : Prop} {H : decidable c} (Hnc : ¬c) {A : Type} (t e : A) : (if c then t else e) = e :=
|
||||
theorem if_neg {c : Prop} {H : decidable c} (Hnc : ¬c) {A : Type} {t e : A} : (if c then t else e) = e :=
|
||||
decidable_rec
|
||||
(assume Hc : c, absurd Hc Hnc)
|
||||
(assume Hnc : ¬c, refl (@ite c (inr Hnc) A t e))
|
||||
|
|
@ -31,20 +31,20 @@ decidable_rec
|
|||
H
|
||||
|
||||
theorem if_true {A : Type} (t e : A) : (if true then t else e) = t :=
|
||||
if_pos trivial t e
|
||||
if_pos trivial
|
||||
|
||||
theorem if_false {A : Type} (t e : A) : (if false then t else e) = e :=
|
||||
if_neg not_false_trivial t e
|
||||
if_neg not_false_trivial
|
||||
|
||||
theorem if_cond_congr {c₁ c₂ : Prop} {H₁ : decidable c₁} {H₂ : decidable c₂} (Heq : c₁ ↔ c₂) {A : Type} (t e : A)
|
||||
: (if c₁ then t else e) = (if c₂ then t else e) :=
|
||||
rec_on H₁
|
||||
(assume Hc₁ : c₁, rec_on H₂
|
||||
(assume Hc₂ : c₂, (if_pos Hc₁ t e) ⬝ (if_pos Hc₂ t e)⁻¹)
|
||||
(assume Hc₂ : c₂, if_pos Hc₁ ⬝ (if_pos Hc₂)⁻¹)
|
||||
(assume Hnc₂ : ¬c₂, absurd (iff_elim_left Heq Hc₁) Hnc₂))
|
||||
(assume Hnc₁ : ¬c₁, rec_on H₂
|
||||
(assume Hc₂ : c₂, absurd (iff_elim_right Heq Hc₂) Hnc₁)
|
||||
(assume Hnc₂ : ¬c₂, (if_neg Hnc₁ t e) ⬝ (if_neg Hnc₂ t e)⁻¹))
|
||||
(assume Hnc₂ : ¬c₂, if_neg Hnc₁ ⬝ (if_neg Hnc₂)⁻¹))
|
||||
|
||||
theorem if_congr_aux {c₁ c₂ : Prop} {H₁ : decidable c₁} {H₂ : decidable c₂} {A : Type} {t₁ t₂ e₁ e₂ : A}
|
||||
(Hc : c₁ ↔ c₂) (Ht : t₁ = t₂) (He : e₁ = e₂) :
|
||||
|
|
|
|||
|
|
@ -14,4 +14,4 @@ theorem gcd_def (x y : ℕ) : gcd x y = @ite (y = 0) (decidable_eq (pr2 (pair x
|
|||
sorry
|
||||
|
||||
theorem gcd_succ (m n : ℕ) : gcd m (succ n) = gcd (succ n) (m mod succ n) :=
|
||||
trans (gcd_def _ _) (if_neg succ_ne_zero _ _)
|
||||
trans (gcd_def _ _) (if_neg succ_ne_zero)
|
||||
|
|
|
|||
|
|
@ -813,7 +813,7 @@ theorem trichotomy_alt (n m : ℕ) : (n < m ∨ n = m) ∨ m < n
|
|||
:= or_imp_or (le_or_lt n m) (assume H : n ≤ m, le_imp_lt_or_eq H) (assume H : m < n, H)
|
||||
|
||||
theorem trichotomy (n m : ℕ) : n < m ∨ n = m ∨ m < n
|
||||
:= iff_elim_left (or_assoc _ _ _) (trichotomy_alt n m)
|
||||
:= iff_elim_left or_assoc (trichotomy_alt n m)
|
||||
|
||||
theorem le_total (n m : ℕ) : n ≤ m ∨ m ≤ n
|
||||
:= or_imp_or (le_or_lt n m) (assume H : n ≤ m, H) (assume H : m < n, lt_imp_le H)
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue