feat(nat): redefine le and lt in the standard library
This commit is contained in:
Floris van Doorn
parent
37995edbd7
commit
7f5caab694
13 changed files with 342 additions and 155 deletions
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@ -30,6 +30,7 @@ namespace nat
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nat.cases_on a zero (λ a₁, a₁)
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-- add is defined in init.num
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definition sub (a b : nat) : nat :=
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nat.rec_on b a (λ b₁ r, pred r)
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@ -57,49 +58,52 @@ namespace nat
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/- properties of inequality -/
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definition le_of_eq {n m : ℕ} (p : n = m) : n ≤ m := p ▸ le.refl n
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theorem le_of_eq {n m : ℕ} (p : n = m) : n ≤ m := p ▸ le.refl n
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definition le_succ (n : ℕ) : n ≤ succ n := by repeat constructor
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theorem le_succ (n : ℕ) : n ≤ succ n := by repeat constructor
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definition pred_le (n : ℕ) : pred n ≤ n := by cases n;all_goals (repeat constructor)
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theorem pred_le (n : ℕ) : pred n ≤ n := by cases n;all_goals (repeat constructor)
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definition le.trans [trans] {n m k : ℕ} (H1 : n ≤ m) (H2 : m ≤ k) : n ≤ k :=
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theorem le.trans [trans] {n m k : ℕ} (H1 : n ≤ m) (H2 : m ≤ k) : n ≤ k :=
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by induction H2 with n H2 IH;exact H1;exact le.step IH
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definition le_succ_of_le {n m : ℕ} (H : n ≤ m) : n ≤ succ m := le.trans H !le_succ
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theorem le_succ_of_le {n m : ℕ} (H : n ≤ m) : n ≤ succ m := le.trans H !le_succ
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definition le_of_succ_le {n m : ℕ} (H : succ n ≤ m) : n ≤ m := le.trans !le_succ H
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theorem le_of_succ_le {n m : ℕ} (H : succ n ≤ m) : n ≤ m := le.trans !le_succ H
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definition le_of_lt {n m : ℕ} (H : n < m) : n ≤ m := le_of_succ_le H
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theorem le_of_lt {n m : ℕ} (H : n < m) : n ≤ m := le_of_succ_le H
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definition succ_le_succ [unfold-c 3] {n m : ℕ} (H : n ≤ m) : succ n ≤ succ m :=
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theorem succ_le_succ [unfold-c 3] {n m : ℕ} (H : n ≤ m) : succ n ≤ succ m :=
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by induction H;reflexivity;exact le.step v_0
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definition pred_le_pred [unfold-c 3] {n m : ℕ} (H : n ≤ m) : pred n ≤ pred m :=
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theorem pred_le_pred [unfold-c 3] {n m : ℕ} (H : n ≤ m) : pred n ≤ pred m :=
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by induction H;reflexivity;cases b;exact v_0;exact le.step v_0
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definition le_of_succ_le_succ [unfold-c 3] {n m : ℕ} (H : succ n ≤ succ m) : n ≤ m :=
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theorem le_of_succ_le_succ [unfold-c 3] {n m : ℕ} (H : succ n ≤ succ m) : n ≤ m :=
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pred_le_pred H
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definition le_succ_of_pred_le [unfold-c 1] {n m : ℕ} (H : pred n ≤ m) : n ≤ succ m :=
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theorem le_succ_of_pred_le [unfold-c 1] {n m : ℕ} (H : pred n ≤ m) : n ≤ succ m :=
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by cases n;exact le.step H;exact succ_le_succ H
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definition not_succ_le_self {n : ℕ} : ¬succ n ≤ n :=
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theorem not_succ_le_self {n : ℕ} : ¬succ n ≤ n :=
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by induction n with n IH;all_goals intros;cases a;apply IH;exact le_of_succ_le_succ a
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definition zero_le (n : ℕ) : 0 ≤ n :=
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theorem zero_le (n : ℕ) : 0 ≤ n :=
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by induction n with n IH;apply le.refl;exact le.step IH
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definition zero_lt_succ (n : ℕ) : 0 < succ n :=
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theorem lt.step {n m : ℕ} (H : n < m) : n < succ m :=
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le.step H
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theorem zero_lt_succ (n : ℕ) : 0 < succ n :=
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by induction n with n IH;apply le.refl;exact le.step IH
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definition lt.trans [trans] {n m k : ℕ} (H1 : n < m) (H2 : m < k) : n < k :=
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theorem lt.trans [trans] {n m k : ℕ} (H1 : n < m) (H2 : m < k) : n < k :=
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le.trans (le.step H1) H2
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definition lt_of_le_of_lt [trans] {n m k : ℕ} (H1 : n ≤ m) (H2 : m < k) : n < k :=
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theorem lt_of_le_of_lt [trans] {n m k : ℕ} (H1 : n ≤ m) (H2 : m < k) : n < k :=
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le.trans (succ_le_succ H1) H2
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definition lt_of_lt_of_le [trans] {n m k : ℕ} (H1 : n < m) (H2 : m ≤ k) : n < k :=
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theorem lt_of_lt_of_le [trans] {n m k : ℕ} (H1 : n < m) (H2 : m ≤ k) : n < k :=
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le.trans H1 H2
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theorem le.antisymm {n m : ℕ} (H1 : n ≤ m) (H2 : m ≤ n) : n = m :=
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@ -111,7 +115,7 @@ namespace nat
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{ exfalso, apply not_succ_le_self, exact lt.trans H1' H2'}},
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end
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definition not_succ_le_zero (n : ℕ) : ¬succ n ≤ zero :=
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theorem not_succ_le_zero (n : ℕ) : ¬succ n ≤ zero :=
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by intro H; cases H
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theorem lt.irrefl (n : ℕ) : ¬n < n := not_succ_le_self
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@ -128,7 +132,7 @@ namespace nat
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theorem lt.asymm {n m : ℕ} (H1 : n < m) (H2 : m < n) : empty :=
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le_lt_antisymm (le_of_lt H1) H2
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theorem lt.by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a = b → P) (H3 : b < a → P) : P :=
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definition lt.by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a = b → P) (H3 : b < a → P) : P :=
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begin
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revert b H1 H2 H3, induction a with a IH,
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{ intros, cases b,
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@ -142,16 +146,16 @@ namespace nat
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intro H, exact H3 (succ_le_succ H)}}
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end
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definition lt.trichotomy (a b : ℕ) : a < b ⊎ a = b ⊎ b < a :=
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lt.by_cases (λH, inl H) (λH, inr (inl H)) (λH, inr (inr H))
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theorem lt_or_ge (a b : ℕ) : (a < b) ⊎ (a ≥ b) :=
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lt.by_cases inl (λH, inr (eq.rec_on H !le.refl)) (λH, inr (le_of_lt H))
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theorem lt.trichotomy (a b : ℕ) : a < b ⊎ a = b ⊎ b < a :=
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lt.by_cases inl (λH, inr (inl H)) (λH, inr (inr H))
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definition lt_ge_by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a ≥ b → P) : P :=
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sum.rec_on (lt_or_ge a b) H1 H2
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lt.by_cases H1 (λH, H2 (le_of_eq H⁻¹)) (λH, H2 (le_of_lt H))
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definition not_lt_zero (a : ℕ) : ¬ a < zero :=
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theorem lt_or_ge (a b : ℕ) : (a < b) ⊎ (a ≥ b) :=
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lt_ge_by_cases inl inr
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theorem not_lt_zero (a : ℕ) : ¬ a < zero :=
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by intro H; cases H
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-- less-than is well-founded
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@ -198,49 +202,49 @@ namespace nat
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definition decidable_gt [instance] : decidable_rel gt := _
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definition decidable_ge [instance] : decidable_rel ge := _
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definition eq_or_lt_of_le {a b : ℕ} (H : a ≤ b) : a = b ⊎ a < b :=
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theorem eq_or_lt_of_le {a b : ℕ} (H : a ≤ b) : a = b ⊎ a < b :=
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by cases H with b' H; exact sum.inl rfl; exact sum.inr (succ_le_succ H)
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definition le_of_eq_or_lt {a b : ℕ} (H : a = b ⊎ a < b) : a ≤ b :=
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theorem le_of_eq_or_lt {a b : ℕ} (H : a = b ⊎ a < b) : a ≤ b :=
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by cases H with H H; exact le_of_eq H; exact le_of_lt H
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definition eq_or_lt_of_not_lt {a b : ℕ} (hnlt : ¬ a < b) : a = b ⊎ b < a :=
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theorem eq_or_lt_of_not_lt {a b : ℕ} (hnlt : ¬ a < b) : a = b ⊎ b < a :=
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sum.rec_on (lt.trichotomy a b)
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(λ hlt, absurd hlt hnlt)
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(λ h, h)
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definition lt_succ_of_le {a b : ℕ} (h : a ≤ b) : a < succ b :=
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theorem lt_succ_of_le {a b : ℕ} (h : a ≤ b) : a < succ b :=
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succ_le_succ h
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definition lt_of_succ_le {a b : ℕ} (h : succ a ≤ b) : a < b := h
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theorem lt_of_succ_le {a b : ℕ} (h : succ a ≤ b) : a < b := h
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definition succ_le_of_lt {a b : ℕ} (h : a < b) : succ a ≤ b := h
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theorem succ_le_of_lt {a b : ℕ} (h : a < b) : succ a ≤ b := h
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definition max (a b : ℕ) : ℕ := if a < b then b else a
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definition min (a b : ℕ) : ℕ := if a < b then a else b
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definition max_self (a : ℕ) : max a a = a :=
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theorem max_self (a : ℕ) : max a a = a :=
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eq.rec_on !if_t_t rfl
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definition max_eq_right {a b : ℕ} (H : a < b) : max a b = b :=
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theorem max_eq_right {a b : ℕ} (H : a < b) : max a b = b :=
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if_pos H
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definition max_eq_left {a b : ℕ} (H : ¬ a < b) : max a b = a :=
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theorem max_eq_left {a b : ℕ} (H : ¬ a < b) : max a b = a :=
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if_neg H
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definition eq_max_right {a b : ℕ} (H : a < b) : b = max a b :=
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theorem eq_max_right {a b : ℕ} (H : a < b) : b = max a b :=
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eq.rec_on (max_eq_right H) rfl
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definition eq_max_left {a b : ℕ} (H : ¬ a < b) : a = max a b :=
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theorem eq_max_left {a b : ℕ} (H : ¬ a < b) : a = max a b :=
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eq.rec_on (max_eq_left H) rfl
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definition le_max_left (a b : ℕ) : a ≤ max a b :=
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theorem le_max_left (a b : ℕ) : a ≤ max a b :=
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by_cases
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(λ h : a < b, le_of_lt (eq.rec_on (eq_max_right h) h))
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(λ h : ¬ a < b, eq.rec_on (eq_max_left h) !le.refl)
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definition le_max_right (a b : ℕ) : b ≤ max a b :=
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theorem le_max_right (a b : ℕ) : b ≤ max a b :=
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by_cases
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(λ h : a < b, eq.rec_on (eq_max_right h) !le.refl)
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(λ h : ¬ a < b, sum.rec_on (eq_or_lt_of_not_lt h)
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@ -249,21 +253,21 @@ namespace nat
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have aux : a = max a b, from eq_max_left (lt.asymm h),
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eq.rec_on aux (le_of_lt h)))
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definition succ_sub_succ_eq_sub (a b : ℕ) : succ a - succ b = a - b :=
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theorem succ_sub_succ_eq_sub (a b : ℕ) : succ a - succ b = a - b :=
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by induction b with b IH; reflexivity; apply ap pred IH
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definition sub_eq_succ_sub_succ (a b : ℕ) : a - b = succ a - succ b :=
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theorem sub_eq_succ_sub_succ (a b : ℕ) : a - b = succ a - succ b :=
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eq.rec_on (succ_sub_succ_eq_sub a b) rfl
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definition zero_sub_eq_zero (a : ℕ) : zero - a = zero :=
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theorem zero_sub_eq_zero (a : ℕ) : zero - a = zero :=
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nat.rec_on a
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rfl
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(λ a₁ (ih : zero - a₁ = zero), ap pred ih)
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definition zero_eq_zero_sub (a : ℕ) : zero = zero - a :=
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theorem zero_eq_zero_sub (a : ℕ) : zero = zero - a :=
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eq.rec_on (zero_sub_eq_zero a) rfl
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definition sub_lt {a b : ℕ} : zero < a → zero < b → a - b < a :=
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theorem sub_lt {a b : ℕ} : zero < a → zero < b → a - b < a :=
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have aux : Π {a}, zero < a → Π {b}, zero < b → a - b < a, from
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λa h₁, le.rec_on h₁
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(λb h₂, le.cases_on h₂
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@ -278,7 +282,7 @@ namespace nat
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(lt.trans (@ih b₁ bpos) (lt.base a₁)))),
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λ h₁ h₂, aux h₁ h₂
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definition sub_le (a b : ℕ) : a - b ≤ a :=
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theorem sub_le (a b : ℕ) : a - b ≤ a :=
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nat.rec_on b
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(le.refl a)
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(λ b₁ ih, le.trans !pred_le ih)
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@ -42,7 +42,7 @@ encodable.mk
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begin
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cases o with a,
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begin esimp end,
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begin esimp, rewrite [if_neg !succ_ne_zero, pred_succ, encodable.encodek] end
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begin esimp, rewrite [if_neg !succ_ne_zero, encodable.encodek] end
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end)
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section sum
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@ -462,9 +462,12 @@ assume i, mem_cons_of_mem _ i
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theorem mem_upto_of_lt : ∀ {n i : nat}, i < n → i ∈ upto n
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| 0 i h := absurd h !not_lt_zero
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| (succ n) i h := or.elim (eq_or_lt_of_le h)
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(λ ieqn : i = n, by rewrite [ieqn, upto_succ]; exact !mem_cons)
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(λ iltn : i < n, mem_upto_succ_of_mem_upto (mem_upto_of_lt iltn))
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| (succ n) i h :=
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begin
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cases h with m h',
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{ rewrite upto_succ, apply mem_cons},
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{ exact mem_upto_succ_of_mem_upto (mem_upto_of_lt h')}
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end
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/- union -/
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section union
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@ -36,7 +36,7 @@ namespace nat
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definition not_bex_succ {P : nat → Prop} {n : nat} (H₁ : ¬ bex n P) (H₂ : ¬ P n) : ¬ bex (succ n) P :=
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λ H, obtain (w : nat) (Hw : w < succ n ∧ P w), from H,
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and.rec_on Hw (λ hltsn hp, or.rec_on (eq_or_lt_of_le hltsn)
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and.rec_on Hw (λ hltsn hp, or.rec_on (eq_or_lt_of_le (le_of_succ_le_succ hltsn))
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(λ heq : w = n, absurd (eq.rec_on heq hp) H₂)
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(λ hltn : w < n, absurd (exists.intro w (and.intro hltn hp)) H₁))
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@ -47,7 +47,7 @@ namespace nat
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λ x Hlt, H x (lt.step Hlt)
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definition ball_succ_of_ball {n : nat} {P : nat → Prop} (H₁ : ball n P) (H₂ : P n) : ball (succ n) P :=
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λ (x : nat) (Hlt : x < succ n), or.elim (eq_or_lt_of_le Hlt)
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λ (x : nat) (Hlt : x < succ n), or.elim (eq_or_lt_of_le (le_of_succ_le_succ Hlt))
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(λ heq : x = n, eq.rec_on (eq.rec_on heq rfl) H₂)
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(λ hlt : x < n, H₁ x hlt)
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@ -24,7 +24,7 @@ private lemma lbp_zero : lbp 0 :=
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private lemma lbp_succ {x : nat} : lbp x → ¬ p x → lbp (succ x) :=
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λ lx npx y yltsx,
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or.elim (eq_or_lt_of_le yltsx)
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or.elim (eq_or_lt_of_le (le_of_succ_le_succ yltsx))
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(λ yeqx, by substvars; assumption)
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(λ yltx, lx y yltx)
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@ -46,7 +46,7 @@ acc.intro x (λ (y : nat) (l : y ≺ x),
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by_cases
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(λ yeqx : y = succ x, by substvars; assumption)
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(λ ynex : y ≠ succ x,
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have ygtsx : succ x < y, from lt_of_le_and_ne (succ_lt_succ ygtx) (ne.symm ynex),
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have ygtsx : succ x < y, from lt_of_le_and_ne ygtx (ne.symm ynex),
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acc.inv h (and.intro ygtsx (and.elim_right l))))
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private lemma acc_of_px_of_gt {x y : nat} : p x → y > x → acc gtb y :=
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@ -61,7 +61,7 @@ private lemma acc_of_acc_of_lt : ∀ {x y : nat}, acc gtb x → y < x → acc gt
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assert ax : acc gtb x, from acc_of_acc_succ asx,
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by_cases
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(λ yeqx : y = x, by substvars; assumption)
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(λ ynex : y ≠ x, acc_of_acc_of_lt ax (lt_of_le_and_ne yltsx ynex))
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(λ ynex : y ≠ x, acc_of_acc_of_lt ax (lt_of_le_and_ne (le_of_lt_succ yltsx) ynex))
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parameter (ex : ∃ a, p a)
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parameter [dp : decidable_pred p]
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@ -442,7 +442,7 @@ have H3 : m * k < (succ (n div k)) * k, from
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... = n div k * k + n mod k : eq_div_mul_add_mod
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... < n div k * k + k : add_lt_add_left (!mod_lt H1)
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... = (succ (n div k)) * k : succ_mul,
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lt_of_mul_lt_mul_right H3
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le_of_lt_succ (lt_of_mul_lt_mul_right H3)
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theorem le_div_iff_mul_le {m n k : ℕ} (H : k > 0) : m ≤ n div k ↔ m * k ≤ n :=
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iff.intro !mul_le_of_le_div (!le_div_of_mul_le H)
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@ -15,12 +15,6 @@ namespace nat
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theorem le_of_lt_or_eq {m n : ℕ} (H : m < n ∨ m = n) : m ≤ n :=
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or.elim H (take H1, le_of_lt H1) (take H1, H1 ▸ !le.refl)
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theorem lt.by_cases {a b : ℕ} {P : Prop}
|
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(H1 : a < b → P) (H2 : a = b → P) (H3 : b < a → P) : P :=
|
||||
or.elim !lt.trichotomy
|
||||
(assume H, H1 H)
|
||||
(assume H, or.elim H (assume H', H2 H') (assume H', H3 H'))
|
||||
|
||||
theorem lt_or_eq_of_le {m n : ℕ} (H : m ≤ n) : m < n ∨ m = n :=
|
||||
lt.by_cases
|
||||
(assume H1 : m < n, or.inl H1)
|
||||
|
|
@ -55,15 +49,8 @@ theorem le.intro {n m k : ℕ} (h : n + k = m) : n ≤ m :=
|
|||
h ▸ le_add_right n k
|
||||
|
||||
theorem le.elim {n m : ℕ} (h : n ≤ m) : ∃k, n + k = m :=
|
||||
le.rec_on h
|
||||
(exists.intro 0 rfl)
|
||||
(λ m (h : n < m), lt.rec_on h
|
||||
(exists.intro 1 rfl)
|
||||
(λ b hlt (ih : ∃ (k : ℕ), n + k = b),
|
||||
obtain (k : ℕ) (h : n + k = b), from ih,
|
||||
exists.intro (succ k) (calc
|
||||
n + succ k = succ (n + k) : add_succ
|
||||
... = succ b : h)))
|
||||
by induction h with m h ih;existsi 0; reflexivity;
|
||||
cases ih with k H; existsi succ k; exact congr_arg succ H
|
||||
|
||||
theorem le.total {m n : ℕ} : m ≤ n ∨ n ≤ m :=
|
||||
lt.by_cases
|
||||
|
|
@ -124,25 +111,6 @@ lt_of_lt_of_le H2 H3
|
|||
theorem mul_lt_mul_of_pos_right {n m k : ℕ} (H : n < m) (Hk : k > 0) : n * k < m * k :=
|
||||
!mul.comm ▸ !mul.comm ▸ mul_lt_mul_of_pos_left H Hk
|
||||
|
||||
theorem le.antisymm {n m : ℕ} (H1 : n ≤ m) (H2 : m ≤ n) : n = m :=
|
||||
obtain (k : ℕ) (Hk : n + k = m), from le.elim H1,
|
||||
obtain (l : ℕ) (Hl : m + l = n), from le.elim H2,
|
||||
have L1 : k + l = 0, from
|
||||
add.cancel_left
|
||||
(calc
|
||||
n + (k + l) = n + k + l : add.assoc
|
||||
... = m + l : by subst m
|
||||
... = n : by subst n
|
||||
... = n + 0 : add_zero),
|
||||
assert L2 : k = 0, from eq_zero_of_add_eq_zero_right L1,
|
||||
calc
|
||||
n = n + 0 : add_zero
|
||||
... = n + k : by subst k
|
||||
... = m : Hk
|
||||
|
||||
theorem zero_le (n : ℕ) : 0 ≤ n :=
|
||||
le.intro !zero_add
|
||||
|
||||
/- nat is an instance of a linearly ordered semiring -/
|
||||
|
||||
section migrate_algebra
|
||||
|
|
@ -237,17 +205,6 @@ eq_zero_of_add_eq_zero_right Hk
|
|||
theorem lt_iff_succ_le (m n : nat) : m < n ↔ succ m ≤ n :=
|
||||
iff.intro succ_le_of_lt lt_of_succ_le
|
||||
|
||||
theorem not_succ_le_zero (n : ℕ) : ¬ succ n ≤ 0 :=
|
||||
(assume H : succ n ≤ 0,
|
||||
have H2 : succ n = 0, from eq_zero_of_le_zero H,
|
||||
absurd H2 !succ_ne_zero)
|
||||
|
||||
theorem succ_le_succ {n m : ℕ} (H : n ≤ m) : succ n ≤ succ m :=
|
||||
!add_one ▸ !add_one ▸ add_le_add_right H 1
|
||||
|
||||
theorem le_of_succ_le_succ {n m : ℕ} (H : succ n ≤ succ m) : n ≤ m :=
|
||||
le_of_add_le_add_right (by rewrite -add_one at H; assumption)
|
||||
|
||||
theorem self_le_succ (n : ℕ) : n ≤ succ n :=
|
||||
le.intro !add_one
|
||||
|
||||
|
|
@ -256,14 +213,6 @@ or.elim (lt_or_eq_of_le H)
|
|||
(assume H1 : n < m, or.inl (succ_le_of_lt H1))
|
||||
(assume H1 : n = m, or.inr H1)
|
||||
|
||||
theorem le_succ_of_pred_le {n m : ℕ} : pred n ≤ m → n ≤ succ m :=
|
||||
nat.cases_on n
|
||||
(assume H : pred 0 ≤ m, !zero_le)
|
||||
(take n',
|
||||
assume H : pred (succ n') ≤ m,
|
||||
have H1 : n' ≤ m, from pred_succ n' ▸ H,
|
||||
succ_le_succ H1)
|
||||
|
||||
theorem pred_le_of_le_succ {n m : ℕ} : n ≤ succ m → pred n ≤ m :=
|
||||
nat.cases_on n
|
||||
(assume H, !pred_zero⁻¹ ▸ zero_le m)
|
||||
|
|
|
|||
|
|
@ -34,7 +34,7 @@ theorem sqrt_aux_lower : ∀ {s n : nat}, s ≤ n → sqrt_aux s n * sqrt_aux s
|
|||
| (succ s) n h := by_cases
|
||||
(λ h₁ : (succ s)*(succ s) ≤ n, by rewrite [sqrt_aux_succ_of_pos h₁]; exact h₁)
|
||||
(λ h₂ : ¬ (succ s)*(succ s) ≤ n,
|
||||
assert aux : s ≤ n, from lt.step (lt_of_succ_le h),
|
||||
assert aux : s ≤ n, from le_of_succ_le h,
|
||||
by rewrite [sqrt_aux_succ_of_neg h₂]; exact (sqrt_aux_lower aux))
|
||||
|
||||
theorem sqrt_lower (n : nat) : sqrt n * sqrt n ≤ n :=
|
||||
|
|
@ -47,7 +47,7 @@ theorem sqrt_aux_upper : ∀ {s n : nat}, n ≤ s*s + s + s → n ≤ sqrt_aux s
|
|||
by rewrite [sqrt_aux_succ_of_pos h₁]; exact h)
|
||||
(λ h₂ : ¬ (succ s)*(succ s) ≤ n,
|
||||
assert h₃ : n < (succ s) * (succ s), from lt_of_not_ge h₂,
|
||||
assert h₄ : n ≤ s * s + s + s, by rewrite [succ_mul_succ_eq at h₃]; exact h₃,
|
||||
assert h₄ : n ≤ s * s + s + s, by rewrite [succ_mul_succ_eq at h₃]; exact le_of_lt_succ h₃,
|
||||
by rewrite [sqrt_aux_succ_of_neg h₂]; exact (sqrt_aux_upper h₄))
|
||||
|
||||
theorem sqrt_upper (n : nat) : n ≤ sqrt n * sqrt n + sqrt n + sqrt n :=
|
||||
|
|
@ -68,7 +68,7 @@ theorem sqrt_aux_offset_eq {n k : nat} (h₁ : k ≤ n + n) : ∀ {s}, s ≥ n
|
|||
| (succ s) h₂ := by_cases
|
||||
(λ hl : (succ s)*(succ s) ≤ n*n + k,
|
||||
have l₁ : n*n + k ≤ n*n + n + n, from by rewrite [add.assoc]; exact (add_le_add_left h₁ (n*n)),
|
||||
assert l₂ : n*n + k < n*n + n + n + 1, from l₁,
|
||||
assert l₂ : n*n + k < n*n + n + n + 1, from lt_succ_of_le l₁,
|
||||
have l₃ : n*n + k < (succ n)*(succ n), by rewrite [-succ_mul_succ_eq at l₂]; exact l₂,
|
||||
assert l₄ : (succ s)*(succ s) < (succ n)*(succ n), from lt_of_le_of_lt hl l₃,
|
||||
have ng : ¬ succ s > (succ n), from
|
||||
|
|
@ -79,7 +79,7 @@ theorem sqrt_aux_offset_eq {n k : nat} (h₁ : k ≤ n + n) : ∀ {s}, s ≥ n
|
|||
have ssnesn : succ s ≠ succ n, from
|
||||
assume sseqsn : succ s = succ n,
|
||||
by rewrite [sseqsn at l₄]; exact (absurd l₄ !lt.irrefl),
|
||||
have sslen : succ s ≤ n, from lt_of_le_and_ne sslesn ssnesn,
|
||||
have sslen : s < n, from lt_of_succ_lt_succ (lt_of_le_and_ne sslesn ssnesn),
|
||||
assert sseqn : succ s = n, from le.antisymm sslen h₂,
|
||||
by rewrite [sqrt_aux_succ_of_pos hl]; exact sseqn)
|
||||
(λ hg : ¬ (succ s)*(succ s) ≤ n*n + k,
|
||||
|
|
@ -88,8 +88,8 @@ theorem sqrt_aux_offset_eq {n k : nat} (h₁ : k ≤ n + n) : ∀ {s}, s ≥ n
|
|||
have p : n*n ≤ n*n + k, from !le_add_right,
|
||||
have n : ¬ n*n ≤ n*n + k, by rewrite [-neqss at hg]; exact hg,
|
||||
absurd p n)
|
||||
(λ sgen : s ≥ n,
|
||||
by rewrite [sqrt_aux_succ_of_neg hg]; exact (sqrt_aux_offset_eq sgen)))
|
||||
(λ sgen : succ s > n,
|
||||
by rewrite [sqrt_aux_succ_of_neg hg]; exact (sqrt_aux_offset_eq (le_of_lt_succ sgen))))
|
||||
|
||||
theorem sqrt_offset_eq {n k : nat} : k ≤ n + n → sqrt (n*n + k) = n :=
|
||||
assume h,
|
||||
|
|
|
|||
|
|
@ -6,25 +6,43 @@ Authors: Floris van Doorn, Leonardo de Moura
|
|||
prelude
|
||||
import init.wf init.tactic init.num
|
||||
|
||||
open eq.ops decidable
|
||||
open eq.ops decidable or
|
||||
|
||||
namespace nat
|
||||
notation `ℕ` := nat
|
||||
|
||||
inductive lt (a : nat) : nat → Prop :=
|
||||
| base : lt a (succ a)
|
||||
| step : Π {b}, lt a b → lt a (succ b)
|
||||
/- basic definitions on natural numbers -/
|
||||
inductive le (a : ℕ) : ℕ → Prop :=
|
||||
| refl : le a a
|
||||
| step : Π {b}, le a b → le a (succ b)
|
||||
|
||||
notation a < b := lt a b
|
||||
infix `≤` := le
|
||||
attribute le.refl [refl]
|
||||
|
||||
definition le [reducible] (a b : nat) : Prop := a < succ b
|
||||
definition lt [reducible] (n m : ℕ) := succ n ≤ m
|
||||
definition ge [reducible] (n m : ℕ) := m ≤ n
|
||||
definition gt [reducible] (n m : ℕ) := succ m ≤ n
|
||||
infix `<` := lt
|
||||
infix `≥` := ge
|
||||
infix `>` := gt
|
||||
|
||||
notation a <= b := le a b
|
||||
notation a ≤ b := le a b
|
||||
|
||||
definition pred (a : nat) : nat :=
|
||||
definition pred [unfold-c 1] (a : nat) : nat :=
|
||||
nat.cases_on a zero (λ a₁, a₁)
|
||||
|
||||
-- add is defined in init.num
|
||||
|
||||
definition sub (a b : nat) : nat :=
|
||||
nat.rec_on b a (λ b₁ r, pred r)
|
||||
|
||||
definition mul (a b : nat) : nat :=
|
||||
nat.rec_on b zero (λ b₁ r, r + a)
|
||||
|
||||
notation a - b := sub a b
|
||||
notation a * b := mul a b
|
||||
|
||||
|
||||
/- properties of ℕ -/
|
||||
|
||||
protected definition is_inhabited [instance] : inhabited nat :=
|
||||
inhabited.mk zero
|
||||
|
||||
|
|
@ -35,27 +53,242 @@ namespace nat
|
|||
| has_decidable_eq (succ x) (succ y) :=
|
||||
match has_decidable_eq x y with
|
||||
| inl xeqy := inl (by rewrite xeqy)
|
||||
| inr xney := inr (by intro h; injection h; contradiction)
|
||||
| inr xney := inr (λ h : succ x = succ y, by injection h with xeqy; exact absurd xeqy xney)
|
||||
end
|
||||
|
||||
/- properties of inequality -/
|
||||
|
||||
theorem le_of_eq {n m : ℕ} (p : n = m) : n ≤ m := p ▸ le.refl n
|
||||
|
||||
theorem le_succ (n : ℕ) : n ≤ succ n := by repeat constructor
|
||||
|
||||
theorem pred_le (n : ℕ) : pred n ≤ n := by cases n;all_goals (repeat constructor)
|
||||
|
||||
theorem le.trans [trans] {n m k : ℕ} (H1 : n ≤ m) (H2 : m ≤ k) : n ≤ k :=
|
||||
by induction H2 with n H2 IH;exact H1;exact le.step IH
|
||||
|
||||
theorem le_succ_of_le {n m : ℕ} (H : n ≤ m) : n ≤ succ m := le.trans H !le_succ
|
||||
|
||||
theorem le_of_succ_le {n m : ℕ} (H : succ n ≤ m) : n ≤ m := le.trans !le_succ H
|
||||
|
||||
theorem le_of_lt {n m : ℕ} (H : n < m) : n ≤ m := le_of_succ_le H
|
||||
|
||||
theorem succ_le_succ [unfold-c 3] {n m : ℕ} (H : n ≤ m) : succ n ≤ succ m :=
|
||||
by induction H;reflexivity;exact le.step v_0
|
||||
|
||||
theorem pred_le_pred [unfold-c 3] {n m : ℕ} (H : n ≤ m) : pred n ≤ pred m :=
|
||||
by induction H;reflexivity;cases b;exact v_0;exact le.step v_0
|
||||
|
||||
theorem le_of_succ_le_succ [unfold-c 3] {n m : ℕ} (H : succ n ≤ succ m) : n ≤ m :=
|
||||
pred_le_pred H
|
||||
|
||||
theorem le_succ_of_pred_le [unfold-c 1] {n m : ℕ} (H : pred n ≤ m) : n ≤ succ m :=
|
||||
by cases n;exact le.step H;exact succ_le_succ H
|
||||
|
||||
theorem not_succ_le_self {n : ℕ} : ¬succ n ≤ n :=
|
||||
by induction n with n IH;all_goals intros;cases a;apply IH;exact le_of_succ_le_succ a
|
||||
|
||||
theorem zero_le (n : ℕ) : 0 ≤ n :=
|
||||
by induction n with n IH;apply le.refl;exact le.step IH
|
||||
|
||||
theorem lt.step {n m : ℕ} (H : n < m) : n < succ m :=
|
||||
le.step H
|
||||
|
||||
theorem zero_lt_succ (n : ℕ) : 0 < succ n :=
|
||||
by induction n with n IH;apply le.refl;exact le.step IH
|
||||
|
||||
theorem lt.trans [trans] {n m k : ℕ} (H1 : n < m) (H2 : m < k) : n < k :=
|
||||
le.trans (le.step H1) H2
|
||||
|
||||
theorem lt_of_le_of_lt [trans] {n m k : ℕ} (H1 : n ≤ m) (H2 : m < k) : n < k :=
|
||||
le.trans (succ_le_succ H1) H2
|
||||
|
||||
theorem lt_of_lt_of_le [trans] {n m k : ℕ} (H1 : n < m) (H2 : m ≤ k) : n < k :=
|
||||
le.trans H1 H2
|
||||
|
||||
theorem le.antisymm {n m : ℕ} (H1 : n ≤ m) (H2 : m ≤ n) : n = m :=
|
||||
begin
|
||||
cases H1 with m' H1',
|
||||
{ reflexivity},
|
||||
{ cases H2 with n' H2',
|
||||
{ reflexivity},
|
||||
{ exfalso, apply not_succ_le_self, exact lt.trans H1' H2'}},
|
||||
end
|
||||
|
||||
theorem not_succ_le_zero (n : ℕ) : ¬succ n ≤ zero :=
|
||||
by intro H; cases H
|
||||
|
||||
theorem lt.irrefl (n : ℕ) : ¬n < n := not_succ_le_self
|
||||
|
||||
theorem self_lt_succ (n : ℕ) : n < succ n := !le.refl
|
||||
theorem lt.base (n : ℕ) : n < succ n := !le.refl
|
||||
|
||||
theorem le_lt_antisymm {n m : ℕ} (H1 : n ≤ m) (H2 : m < n) : false :=
|
||||
!lt.irrefl (lt_of_le_of_lt H1 H2)
|
||||
|
||||
theorem lt_le_antisymm {n m : ℕ} (H1 : n < m) (H2 : m ≤ n) : false :=
|
||||
le_lt_antisymm H2 H1
|
||||
|
||||
theorem lt.asymm {n m : ℕ} (H1 : n < m) (H2 : m < n) : false :=
|
||||
le_lt_antisymm (le_of_lt H1) H2
|
||||
|
||||
definition lt.by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a = b → P) (H3 : b < a → P) : P :=
|
||||
begin
|
||||
revert b H1 H2 H3, induction a with a IH,
|
||||
{ intros, cases b,
|
||||
exact H2 rfl,
|
||||
exact H1 !zero_lt_succ},
|
||||
{ intros, cases b with b,
|
||||
exact H3 !zero_lt_succ,
|
||||
{ apply IH,
|
||||
intro H, exact H1 (succ_le_succ H),
|
||||
intro H, exact H2 (congr rfl H),
|
||||
intro H, exact H3 (succ_le_succ H)}}
|
||||
end
|
||||
|
||||
theorem lt.trichotomy (a b : ℕ) : a < b ∨ a = b ∨ b < a :=
|
||||
lt.by_cases (λH, inl H) (λH, inr (inl H)) (λH, inr (inr H))
|
||||
|
||||
definition lt_ge_by_cases {a b : ℕ} {P : Type} (H1 : a < b → P) (H2 : a ≥ b → P) : P :=
|
||||
lt.by_cases H1 (λH, H2 (le_of_eq H⁻¹)) (λH, H2 (le_of_lt H))
|
||||
|
||||
theorem lt_or_ge (a b : ℕ) : (a < b) ∨ (a ≥ b) :=
|
||||
lt_ge_by_cases inl inr
|
||||
|
||||
definition not_lt_zero (a : ℕ) : ¬ a < zero :=
|
||||
by intro H; cases H
|
||||
|
||||
-- less-than is well-founded
|
||||
theorem lt.wf [instance] : well_founded lt :=
|
||||
well_founded.intro (λn, nat.rec_on n
|
||||
(acc.intro zero (λ (y : nat) (hlt : y < zero),
|
||||
have aux : ∀ {n₁}, y < n₁ → zero = n₁ → acc lt y, from
|
||||
λ n₁ hlt, nat.lt.cases_on hlt
|
||||
(by contradiction)
|
||||
(by contradiction),
|
||||
aux hlt rfl))
|
||||
(λ (n : nat) (ih : acc lt n),
|
||||
acc.intro (succ n) (λ (m : nat) (hlt : m < succ n),
|
||||
have aux : ∀ {n₁} (hlt : m < n₁), succ n = n₁ → acc lt m, from
|
||||
λ n₁ hlt, nat.lt.cases_on hlt
|
||||
(λ (sn_eq_sm : succ n = succ m),
|
||||
by injection sn_eq_sm with neqm; rewrite neqm at ih; exact ih)
|
||||
(λ b (hlt : m < b) (sn_eq_sb : succ n = succ b),
|
||||
by injection sn_eq_sb with neqb; rewrite neqb at ih; exact acc.inv ih hlt),
|
||||
aux hlt rfl)))
|
||||
definition lt.wf [instance] : well_founded lt :=
|
||||
begin
|
||||
constructor, intro n, induction n with n IH,
|
||||
{ constructor, intros n H, exfalso, exact !not_lt_zero H},
|
||||
{ constructor, intros m H,
|
||||
assert aux : ∀ {n₁} (hlt : m < n₁), succ n = n₁ → acc lt m,
|
||||
{ intros n₁ hlt, induction hlt,
|
||||
{ intro p, injection p with q, exact q ▸ IH},
|
||||
{ intro p, injection p with q, exact (acc.inv (q ▸ IH) a)}},
|
||||
apply aux H rfl},
|
||||
end
|
||||
|
||||
definition measure {A : Type} (f : A → ℕ) : A → A → Prop :=
|
||||
inv_image lt f
|
||||
|
||||
definition measure.wf {A : Type} (f : A → ℕ) : well_founded (measure f) :=
|
||||
inv_image.wf f lt.wf
|
||||
|
||||
theorem succ_lt_succ {a b : ℕ} (H : a < b) : succ a < succ b :=
|
||||
succ_le_succ H
|
||||
|
||||
theorem lt_of_succ_lt {a b : ℕ} (H : succ a < b) : a < b :=
|
||||
le_of_succ_le H
|
||||
|
||||
theorem lt_of_succ_lt_succ {a b : ℕ} (H : succ a < succ b) : a < b :=
|
||||
le_of_succ_le_succ H
|
||||
|
||||
definition decidable_le [instance] : decidable_rel le :=
|
||||
begin
|
||||
intros n, induction n with n IH,
|
||||
{ intro m, left, apply zero_le},
|
||||
{ intro m, cases m with m,
|
||||
{ right, apply not_succ_le_zero},
|
||||
{ let H := IH m, clear IH,
|
||||
cases H with H H,
|
||||
left, exact succ_le_succ H,
|
||||
right, intro H2, exact H (le_of_succ_le_succ H2)}}
|
||||
end
|
||||
|
||||
definition decidable_lt [instance] : decidable_rel lt := _
|
||||
definition decidable_gt [instance] : decidable_rel gt := _
|
||||
definition decidable_ge [instance] : decidable_rel ge := _
|
||||
|
||||
theorem eq_or_lt_of_le {a b : ℕ} (H : a ≤ b) : a = b ∨ a < b :=
|
||||
by cases H with b' H; exact inl rfl; exact inr (succ_le_succ H)
|
||||
|
||||
theorem le_of_eq_or_lt {a b : ℕ} (H : a = b ∨ a < b) : a ≤ b :=
|
||||
by cases H with H H; exact le_of_eq H; exact le_of_lt H
|
||||
|
||||
theorem eq_or_lt_of_not_lt {a b : ℕ} (hnlt : ¬ a < b) : a = b ∨ b < a :=
|
||||
or.rec_on (lt.trichotomy a b)
|
||||
(λ hlt, absurd hlt hnlt)
|
||||
(λ h, h)
|
||||
|
||||
theorem lt_succ_of_le {a b : ℕ} (h : a ≤ b) : a < succ b :=
|
||||
succ_le_succ h
|
||||
|
||||
theorem lt_of_succ_le {a b : ℕ} (h : succ a ≤ b) : a < b := h
|
||||
|
||||
theorem succ_le_of_lt {a b : ℕ} (h : a < b) : succ a ≤ b := h
|
||||
|
||||
definition max (a b : ℕ) : ℕ := if a < b then b else a
|
||||
definition min (a b : ℕ) : ℕ := if a < b then a else b
|
||||
|
||||
theorem max_self (a : ℕ) : max a a = a :=
|
||||
eq.rec_on !if_t_t rfl
|
||||
|
||||
theorem max_eq_right {a b : ℕ} (H : a < b) : max a b = b :=
|
||||
if_pos H
|
||||
|
||||
theorem max_eq_left {a b : ℕ} (H : ¬ a < b) : max a b = a :=
|
||||
if_neg H
|
||||
|
||||
theorem eq_max_right {a b : ℕ} (H : a < b) : b = max a b :=
|
||||
eq.rec_on (max_eq_right H) rfl
|
||||
|
||||
theorem eq_max_left {a b : ℕ} (H : ¬ a < b) : a = max a b :=
|
||||
eq.rec_on (max_eq_left H) rfl
|
||||
|
||||
theorem le_max_left (a b : ℕ) : a ≤ max a b :=
|
||||
by_cases
|
||||
(λ h : a < b, le_of_lt (eq.rec_on (eq_max_right h) h))
|
||||
(λ h : ¬ a < b, eq.rec_on (eq_max_left h) !le.refl)
|
||||
|
||||
theorem le_max_right (a b : ℕ) : b ≤ max a b :=
|
||||
by_cases
|
||||
(λ h : a < b, eq.rec_on (eq_max_right h) !le.refl)
|
||||
(λ h : ¬ a < b, or.rec_on (eq_or_lt_of_not_lt h)
|
||||
(λ heq, eq.rec_on heq (eq.rec_on (eq.symm (max_self a)) !le.refl))
|
||||
(λ h : b < a,
|
||||
have aux : a = max a b, from eq_max_left (lt.asymm h),
|
||||
eq.rec_on aux (le_of_lt h)))
|
||||
|
||||
theorem succ_sub_succ_eq_sub (a b : ℕ) : succ a - succ b = a - b :=
|
||||
by induction b with b IH;reflexivity; apply congr (eq.refl pred) IH
|
||||
|
||||
theorem sub_eq_succ_sub_succ (a b : ℕ) : a - b = succ a - succ b :=
|
||||
eq.rec_on (succ_sub_succ_eq_sub a b) rfl
|
||||
|
||||
theorem zero_sub_eq_zero (a : ℕ) : zero - a = zero :=
|
||||
nat.rec_on a
|
||||
rfl
|
||||
(λ a₁ (ih : zero - a₁ = zero), congr (eq.refl pred) ih)
|
||||
|
||||
theorem zero_eq_zero_sub (a : ℕ) : zero = zero - a :=
|
||||
eq.rec_on (zero_sub_eq_zero a) rfl
|
||||
|
||||
theorem sub_lt {a b : ℕ} : zero < a → zero < b → a - b < a :=
|
||||
have aux : Π {a}, zero < a → Π {b}, zero < b → a - b < a, from
|
||||
λa h₁, le.rec_on h₁
|
||||
(λb h₂, le.cases_on h₂
|
||||
(lt.base zero)
|
||||
(λ b₁ bpos,
|
||||
eq.rec_on (sub_eq_succ_sub_succ zero b₁)
|
||||
(eq.rec_on (zero_eq_zero_sub b₁) (lt.base zero))))
|
||||
(λa₁ apos ih b h₂, le.cases_on h₂
|
||||
(lt.base a₁)
|
||||
(λ b₁ bpos,
|
||||
eq.rec_on (sub_eq_succ_sub_succ a₁ b₁)
|
||||
(lt.trans (@ih b₁ bpos) (lt.base a₁)))),
|
||||
λ h₁ h₂, aux h₁ h₂
|
||||
|
||||
theorem sub_le (a b : ℕ) : a - b ≤ a :=
|
||||
nat.rec_on b
|
||||
(le.refl a)
|
||||
(λ b₁ ih, le.trans !pred_le ih)
|
||||
|
||||
end nat
|
||||
exit
|
||||
|
||||
|
||||
definition measure {A : Type} (f : A → nat) : A → A → Prop :=
|
||||
inv_image lt f
|
||||
|
|
|
|||
|
|
@ -2,10 +2,10 @@ namespace nat
|
|||
check induction_on -- ERROR
|
||||
check rec_on -- ERROR
|
||||
check nat.induction_on
|
||||
check lt.rec_on -- OK
|
||||
check nat.lt.rec_on
|
||||
namespace lt
|
||||
check le.rec_on -- OK
|
||||
check nat.le.rec_on
|
||||
namespace le
|
||||
check rec_on -- ERROR
|
||||
check lt.rec_on
|
||||
end lt
|
||||
check le.rec_on
|
||||
end le
|
||||
end nat
|
||||
|
|
|
|||
|
|
@ -1,7 +1,7 @@
|
|||
protected_test.lean:2:8: error: unknown identifier 'induction_on'
|
||||
protected_test.lean:3:8: error: unknown identifier 'rec_on'
|
||||
nat.induction_on : ∀ (n : ℕ), ?C 0 → (∀ (a : ℕ), ?C a → ?C (succ a)) → ?C n
|
||||
lt.rec_on : ?a < ?a_1 → ?C (succ ?a) → (∀ {b : ℕ}, ?a < b → ?C b → ?C (succ b)) → ?C ?a_1
|
||||
lt.rec_on : ?a < ?a_1 → ?C (succ ?a) → (∀ {b : ℕ}, ?a < b → ?C b → ?C (succ b)) → ?C ?a_1
|
||||
le.rec_on : ?a ≤ ?a_1 → ?C ?a → (∀ {b : ℕ}, ?a ≤ b → ?C b → ?C (succ b)) → ?C ?a_1
|
||||
le.rec_on : ?a ≤ ?a_1 → ?C ?a → (∀ {b : ℕ}, ?a ≤ b → ?C b → ?C (succ b)) → ?C ?a_1
|
||||
protected_test.lean:8:10: error: unknown identifier 'rec_on'
|
||||
lt.rec_on : ?a < ?a_1 → ?C (succ ?a) → (∀ {b : ℕ}, ?a < b → ?C b → ?C (succ b)) → ?C ?a_1
|
||||
le.rec_on : ?a ≤ ?a_1 → ?C ?a → (∀ {b : ℕ}, ?a ≤ b → ?C b → ?C (succ b)) → ?C ?a_1
|
||||
|
|
@ -1,8 +1,6 @@
|
|||
import data.list data.nat
|
||||
open nat list eq.ops
|
||||
|
||||
theorem nat.le_of_eq {x y : ℕ} (H : x = y) : x ≤ y := H ▸ !le.refl
|
||||
|
||||
section
|
||||
|
||||
variable {Q : Type}
|
||||
|
|
|
|||
|
|
@ -4,17 +4,17 @@ open nat
|
|||
|
||||
definition lt.trans {a b c : nat} (H₁ : a < b) (H₂ : b < c) : a < c :=
|
||||
have aux : a < b → a < c, from
|
||||
lt.rec_on H₂
|
||||
le.rec_on H₂
|
||||
(λ h₁, lt.step h₁)
|
||||
(λ b₁ bb₁ ih h₁, by constructor; exact ih h₁),
|
||||
aux H₁
|
||||
|
||||
definition succ_lt_succ {a b : nat} (H : a < b) : succ a < succ b :=
|
||||
lt.rec_on H
|
||||
le.rec_on H
|
||||
(by constructor)
|
||||
(λ b hlt ih, lt.trans ih (by constructor))
|
||||
|
||||
definition lt_of_succ_lt {a b : nat} (H : succ a < b) : a < b :=
|
||||
lt.rec_on H
|
||||
le.rec_on H
|
||||
(by constructor; constructor)
|
||||
(λ b h ih, by constructor; exact ih)
|
||||
|
|
|
|||
Loading…
Reference in a new issue