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feat(library/data/nat/sqrt): generalize sqrt_eq theorem

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Leonardo de Moura 2015-04-14 20:10:18 -07:00
parent faca0acd26
commit f7a43c7997
1 changed files with 36 additions and 23 deletions
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59
library/data/nat/sqrt.lean
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@ -56,28 +56,6 @@ theorem sqrt_upper (n : nat) : n ≤ sqrt n * sqrt n + sqrt n + sqrt n :=
have aux : n ≤ n*n + n + n, from le_add_of_le_right (le_add_of_le_left (le.refl n)),
sqrt_aux_upper aux
theorem sqrt_aux_eq : ∀ {s n}, s ≥ n → sqrt_aux s (n*n) = n
| 0 n h :=
assert neqz : n = 0, from eq_zero_of_le_zero h,
by rewrite neqz
| (succ s) n h := by_cases
(λ h₁ : (succ s)*(succ s) ≤ n*n,
assert h₂ : (succ s)*(succ s) ≥ n*n, from mul_le_mul h h,
assert h₃ : (succ s)*(succ s) = n*n, from le.antisymm h₁ h₂,
assert h₄ : ¬ succ s > n, from
assume ssgtn : succ s > n,
assert h₅ : (succ s)*(succ s) > n*n, from mul_lt_mul_of_le_of_le ssgtn ssgtn,
have h₆ : n*n > n*n, by rewrite [h₃ at h₅]; exact h₅,
absurd h₆ !lt.irrefl,
have sslen : succ s ≤ n, from le_of_not_lt h₄,
assert sseqn : succ s = n, from le.antisymm sslen h,
by rewrite [sqrt_aux_succ_of_pos h₁]; exact sseqn)
(λ h₂ : ¬ (succ s)*(succ s) ≤ n*n,
or.elim (eq_or_lt_of_le h)
(λ sseqn, by rewrite [sseqn at h₂]; exact (absurd !le.refl h₂))
(λ sgen : s ≥ n,
by rewrite [sqrt_aux_succ_of_neg h₂]; exact (sqrt_aux_eq sgen)))
private theorem le_squared : ∀ (n : nat), n ≤ n*n
| 0 := !le.refl
| (succ n) :=
@ -85,8 +63,43 @@ private theorem le_squared : ∀ (n : nat), n ≤ n*n
assert aux₂ : 1 * succ n ≤ succ n * succ n, from mul_le_mul aux₁ !le.refl,
by rewrite [one_mul at aux₂]; exact aux₂
theorem sqrt_aux_offset_eq {n k : nat} (h₁ : k ≤ n + n) : ∀ {s}, s ≥ n → sqrt_aux s (n*n + k) = n
| 0 h₂ :=
assert neqz : n = 0, from eq_zero_of_le_zero h₂,
by rewrite neqz
| (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,
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
assume g : succ s > succ n,
have g₁ : (succ s)*(succ s) > (succ n)*(succ n), from mul_lt_mul_of_le_of_le g g,
absurd (lt.trans g₁ l₄) !lt.irrefl,
have sslesn : succ s ≤ succ n, from le_of_not_lt ng,
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,
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,
or.elim (eq_or_lt_of_le h₂)
(λ neqss : n = succ s,
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)))
theorem sqrt_offset_eq {n k : nat} : k ≤ n + n → sqrt (n*n + k) = n :=
assume h,
have h₁ : n ≤ n*n + k, from le.trans !le_squared !le_add_right,
sqrt_aux_offset_eq h h₁
theorem sqrt_eq (n : nat) : sqrt (n*n) = n :=
sqrt_aux_eq !le_squared
sqrt_offset_eq !zero_le
theorem mul_square_cancel {a b : nat} : a*a = b*b → a = b :=
assume h,