url
stringclasses 148
values | commit
stringclasses 148
values | file_path
stringlengths 7
101
| full_name
stringlengths 1
100
| start
stringlengths 6
10
| end
stringlengths 6
11
| tactic
stringlengths 1
11.2k
| state_before
stringlengths 3
2.09M
| state_after
stringlengths 6
2.09M
|
|---|---|---|---|---|---|---|---|---|
null
|
null
|
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lt_of_testBit
|
[116, 1]
|
[138, 29]
|
rw [le_zero_iff] at hm
|
case z
n m✝ i✝ : ℕ
hn✝ : testBit n i✝ = false
hm✝ : testBit m✝ i✝ = true
hnm✝ : ∀ (j : ℕ), i✝ < j → testBit n j = testBit m✝ j
m i : ℕ
hn : testBit 0 i = false
hnm : ∀ (j : ℕ), i < j → testBit 0 j = testBit m j
hm : m ≤ 0
⊢ testBit m i ≠ true
|
case z
n m✝ i✝ : ℕ
hn✝ : testBit n i✝ = false
hm✝ : testBit m✝ i✝ = true
hnm✝ : ∀ (j : ℕ), i✝ < j → testBit n j = testBit m✝ j
m i : ℕ
hn : testBit 0 i = false
hnm : ∀ (j : ℕ), i < j → testBit 0 j = testBit m j
hm : m = 0
⊢ testBit m i ≠ true
|
null
|
null
|
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lt_of_testBit
|
[116, 1]
|
[138, 29]
|
simp [hm]
|
case z
n m✝ i✝ : ℕ
hn✝ : testBit n i✝ = false
hm✝ : testBit m✝ i✝ = true
hnm✝ : ∀ (j : ℕ), i✝ < j → testBit n j = testBit m✝ j
m i : ℕ
hn : testBit 0 i = false
hnm : ∀ (j : ℕ), i < j → testBit 0 j = testBit m j
hm : m = 0
⊢ testBit m i ≠ true
|
no goals
|
null
|
null
|
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lt_of_testBit
|
[116, 1]
|
[138, 29]
|
exact False.elim (Bool.ff_ne_tt ((zero_testBit i).symm.trans hm))
|
case f.z
n✝ m✝ i✝¹ : ℕ
hn✝¹ : testBit n✝ i✝¹ = false
hm✝¹ : testBit m✝ i✝¹ = true
hnm✝¹ : ∀ (j : ℕ), i✝¹ < j → testBit n✝ j = testBit m✝ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m i✝ : ℕ
hn✝ : testBit (bit b n) i✝ = false
hm✝ : testBit m i✝ = true
hnm✝ : ∀ (j : ℕ), i✝ < j → testBit (bit b n) j = testBit m j
i : ℕ
hn : testBit (bit b n) i = false
hm : testBit 0 i = true
hnm : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit 0 j
⊢ bit b n < 0
|
no goals
|
null
|
null
|
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lt_of_testBit
|
[116, 1]
|
[138, 29]
|
subst hi
|
case pos
n✝ m✝¹ i✝¹ : ℕ
hn✝¹ : testBit n✝ i✝¹ = false
hm✝¹ : testBit m✝¹ i✝¹ = true
hnm✝¹ : ∀ (j : ℕ), i✝¹ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i✝ : ℕ
hn✝ : testBit (bit b n) i✝ = false
hm✝ : testBit m✝ i✝ = true
hnm✝ : ∀ (j : ℕ), i✝ < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
i : ℕ
hn : testBit (bit b n) i = false
hm : testBit (bit b' m) i = true
hnm : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit (bit b' m) j
hi : i = 0
⊢ bit b n < bit b' m
|
case pos
n✝ m✝¹ i✝ : ℕ
hn✝¹ : testBit n✝ i✝ = false
hm✝¹ : testBit m✝¹ i✝ = true
hnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i : ℕ
hn✝ : testBit (bit b n) i = false
hm✝ : testBit m✝ i = true
hnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
hn : testBit (bit b n) 0 = false
hm : testBit (bit b' m) 0 = true
hnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j
⊢ bit b n < bit b' m
|
null
|
null
|
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lt_of_testBit
|
[116, 1]
|
[138, 29]
|
simp only [testBit_zero] at hn hm
|
case pos
n✝ m✝¹ i✝ : ℕ
hn✝¹ : testBit n✝ i✝ = false
hm✝¹ : testBit m✝¹ i✝ = true
hnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i : ℕ
hn✝ : testBit (bit b n) i = false
hm✝ : testBit m✝ i = true
hnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
hn : testBit (bit b n) 0 = false
hm : testBit (bit b' m) 0 = true
hnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j
⊢ bit b n < bit b' m
|
case pos
n✝ m✝¹ i✝ : ℕ
hn✝¹ : testBit n✝ i✝ = false
hm✝¹ : testBit m✝¹ i✝ = true
hnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i : ℕ
hn✝ : testBit (bit b n) i = false
hm✝ : testBit m✝ i = true
hnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
hnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j
hn : b = false
hm : b' = true
⊢ bit b n < bit b' m
|
null
|
null
|
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lt_of_testBit
|
[116, 1]
|
[138, 29]
|
have : n = m :=
eq_of_testBit_eq fun i => by convert hnm (i + 1) (Nat.zero_lt_succ _) using 1
<;> rw [testBit_succ]
|
case pos
n✝ m✝¹ i✝ : ℕ
hn✝¹ : testBit n✝ i✝ = false
hm✝¹ : testBit m✝¹ i✝ = true
hnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i : ℕ
hn✝ : testBit (bit b n) i = false
hm✝ : testBit m✝ i = true
hnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
hnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j
hn : b = false
hm : b' = true
⊢ bit b n < bit b' m
|
case pos
n✝ m✝¹ i✝ : ℕ
hn✝¹ : testBit n✝ i✝ = false
hm✝¹ : testBit m✝¹ i✝ = true
hnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i : ℕ
hn✝ : testBit (bit b n) i = false
hm✝ : testBit m✝ i = true
hnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
hnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j
hn : b = false
hm : b' = true
this : n = m
⊢ bit b n < bit b' m
|
null
|
null
|
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lt_of_testBit
|
[116, 1]
|
[138, 29]
|
rw [hn, hm, this, bit_false, bit_true, bit0_val, bit1_val]
|
case pos
n✝ m✝¹ i✝ : ℕ
hn✝¹ : testBit n✝ i✝ = false
hm✝¹ : testBit m✝¹ i✝ = true
hnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i : ℕ
hn✝ : testBit (bit b n) i = false
hm✝ : testBit m✝ i = true
hnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
hnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j
hn : b = false
hm : b' = true
this : n = m
⊢ bit b n < bit b' m
|
case pos
n✝ m✝¹ i✝ : ℕ
hn✝¹ : testBit n✝ i✝ = false
hm✝¹ : testBit m✝¹ i✝ = true
hnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i : ℕ
hn✝ : testBit (bit b n) i = false
hm✝ : testBit m✝ i = true
hnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
hnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j
hn : b = false
hm : b' = true
this : n = m
⊢ 2 * m < 2 * m + 1
|
null
|
null
|
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lt_of_testBit
|
[116, 1]
|
[138, 29]
|
exact lt_add_one _
|
case pos
n✝ m✝¹ i✝ : ℕ
hn✝¹ : testBit n✝ i✝ = false
hm✝¹ : testBit m✝¹ i✝ = true
hnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i : ℕ
hn✝ : testBit (bit b n) i = false
hm✝ : testBit m✝ i = true
hnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
hnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j
hn : b = false
hm : b' = true
this : n = m
⊢ 2 * m < 2 * m + 1
|
no goals
|
null
|
null
|
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lt_of_testBit
|
[116, 1]
|
[138, 29]
|
convert hnm (i + 1) (Nat.zero_lt_succ _) using 1
<;> rw [testBit_succ]
|
n✝ m✝¹ i✝¹ : ℕ
hn✝¹ : testBit n✝ i✝¹ = false
hm✝¹ : testBit m✝¹ i✝¹ = true
hnm✝¹ : ∀ (j : ℕ), i✝¹ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i✝ : ℕ
hn✝ : testBit (bit b n) i✝ = false
hm✝ : testBit m✝ i✝ = true
hnm✝ : ∀ (j : ℕ), i✝ < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
hnm : ∀ (j : ℕ), 0 < j → testBit (bit b n) j = testBit (bit b' m) j
hn : b = false
hm : b' = true
i : ℕ
⊢ testBit n i = testBit m i
|
no goals
|
null
|
null
|
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lt_of_testBit
|
[116, 1]
|
[138, 29]
|
obtain ⟨i', rfl⟩ := exists_eq_succ_of_ne_zero hi
|
case neg
n✝ m✝¹ i✝¹ : ℕ
hn✝¹ : testBit n✝ i✝¹ = false
hm✝¹ : testBit m✝¹ i✝¹ = true
hnm✝¹ : ∀ (j : ℕ), i✝¹ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i✝ : ℕ
hn✝ : testBit (bit b n) i✝ = false
hm✝ : testBit m✝ i✝ = true
hnm✝ : ∀ (j : ℕ), i✝ < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
i : ℕ
hn : testBit (bit b n) i = false
hm : testBit (bit b' m) i = true
hnm : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit (bit b' m) j
hi : ¬i = 0
⊢ bit b n < bit b' m
|
case neg.intro
n✝ m✝¹ i✝ : ℕ
hn✝¹ : testBit n✝ i✝ = false
hm✝¹ : testBit m✝¹ i✝ = true
hnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i : ℕ
hn✝ : testBit (bit b n) i = false
hm✝ : testBit m✝ i = true
hnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
i' : ℕ
hn : testBit (bit b n) (succ i') = false
hm : testBit (bit b' m) (succ i') = true
hnm : ∀ (j : ℕ), succ i' < j → testBit (bit b n) j = testBit (bit b' m) j
hi : ¬succ i' = 0
⊢ bit b n < bit b' m
|
null
|
null
|
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lt_of_testBit
|
[116, 1]
|
[138, 29]
|
simp only [testBit_succ] at hn hm
|
case neg.intro
n✝ m✝¹ i✝ : ℕ
hn✝¹ : testBit n✝ i✝ = false
hm✝¹ : testBit m✝¹ i✝ = true
hnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i : ℕ
hn✝ : testBit (bit b n) i = false
hm✝ : testBit m✝ i = true
hnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
i' : ℕ
hn : testBit (bit b n) (succ i') = false
hm : testBit (bit b' m) (succ i') = true
hnm : ∀ (j : ℕ), succ i' < j → testBit (bit b n) j = testBit (bit b' m) j
hi : ¬succ i' = 0
⊢ bit b n < bit b' m
|
case neg.intro
n✝ m✝¹ i✝ : ℕ
hn✝¹ : testBit n✝ i✝ = false
hm✝¹ : testBit m✝¹ i✝ = true
hnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i : ℕ
hn✝ : testBit (bit b n) i = false
hm✝ : testBit m✝ i = true
hnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
i' : ℕ
hnm : ∀ (j : ℕ), succ i' < j → testBit (bit b n) j = testBit (bit b' m) j
hi : ¬succ i' = 0
hn : testBit n i' = false
hm : testBit m i' = true
⊢ bit b n < bit b' m
|
null
|
null
|
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lt_of_testBit
|
[116, 1]
|
[138, 29]
|
have :=
hn' _ hn hm fun j hj => by convert hnm j.succ (succ_lt_succ hj) using 1 <;> rw [testBit_succ]
|
case neg.intro
n✝ m✝¹ i✝ : ℕ
hn✝¹ : testBit n✝ i✝ = false
hm✝¹ : testBit m✝¹ i✝ = true
hnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i : ℕ
hn✝ : testBit (bit b n) i = false
hm✝ : testBit m✝ i = true
hnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
i' : ℕ
hnm : ∀ (j : ℕ), succ i' < j → testBit (bit b n) j = testBit (bit b' m) j
hi : ¬succ i' = 0
hn : testBit n i' = false
hm : testBit m i' = true
⊢ bit b n < bit b' m
|
case neg.intro
n✝ m✝¹ i✝ : ℕ
hn✝¹ : testBit n✝ i✝ = false
hm✝¹ : testBit m✝¹ i✝ = true
hnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i : ℕ
hn✝ : testBit (bit b n) i = false
hm✝ : testBit m✝ i = true
hnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
i' : ℕ
hnm : ∀ (j : ℕ), succ i' < j → testBit (bit b n) j = testBit (bit b' m) j
hi : ¬succ i' = 0
hn : testBit n i' = false
hm : testBit m i' = true
this : n < m
⊢ bit b n < bit b' m
|
null
|
null
|
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lt_of_testBit
|
[116, 1]
|
[138, 29]
|
cases b <;> cases b'
<;> simp only [bit_false, bit_true, bit0_val n, bit1_val n, bit0_val m, bit1_val m]
<;> linarith only [this]
|
case neg.intro
n✝ m✝¹ i✝ : ℕ
hn✝¹ : testBit n✝ i✝ = false
hm✝¹ : testBit m✝¹ i✝ = true
hnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i : ℕ
hn✝ : testBit (bit b n) i = false
hm✝ : testBit m✝ i = true
hnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
i' : ℕ
hnm : ∀ (j : ℕ), succ i' < j → testBit (bit b n) j = testBit (bit b' m) j
hi : ¬succ i' = 0
hn : testBit n i' = false
hm : testBit m i' = true
this : n < m
⊢ bit b n < bit b' m
|
no goals
|
null
|
null
|
Mathlib/Data/Nat/Bitwise.lean
|
Nat.lt_of_testBit
|
[116, 1]
|
[138, 29]
|
convert hnm j.succ (succ_lt_succ hj) using 1 <;> rw [testBit_succ]
|
n✝ m✝¹ i✝ : ℕ
hn✝¹ : testBit n✝ i✝ = false
hm✝¹ : testBit m✝¹ i✝ = true
hnm✝¹ : ∀ (j : ℕ), i✝ < j → testBit n✝ j = testBit m✝¹ j
b : Bool
n : ℕ
hn' :
∀ {m : ℕ} (i : ℕ), testBit n i = false → testBit m i = true → (∀ (j : ℕ), i < j → testBit n j = testBit m j) → n < m
m✝ i : ℕ
hn✝ : testBit (bit b n) i = false
hm✝ : testBit m✝ i = true
hnm✝ : ∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m✝ j
b' : Bool
m : ℕ
hm' :
∀ (i : ℕ),
testBit (bit b n) i = false →
testBit m i = true → (∀ (j : ℕ), i < j → testBit (bit b n) j = testBit m j) → bit b n < m
i' : ℕ
hnm : ∀ (j : ℕ), succ i' < j → testBit (bit b n) j = testBit (bit b' m) j
hi : ¬succ i' = 0
hn : testBit n i' = false
hm : testBit m i' = true
j : ℕ
hj : i' < j
⊢ testBit n j = testBit m j
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/Subsemigroup/Basic.lean
|
Subsemigroup.mem_iSup
|
[451, 1]
|
[454, 43]
|
rw [← closure_singleton_le_iff_mem, le_iSup_iff]
|
M : Type u_2
N : Type ?u.21456
A : Type ?u.21459
inst✝¹ : Mul M
s : Set M
inst✝ : Add A
t : Set A
S : Subsemigroup M
ι : Sort u_1
p : ι → Subsemigroup M
m : M
⊢ (m ∈ ⨆ (i : ι), p i) ↔ ∀ (N : Subsemigroup M), (∀ (i : ι), p i ≤ N) → m ∈ N
|
M : Type u_2
N : Type ?u.21456
A : Type ?u.21459
inst✝¹ : Mul M
s : Set M
inst✝ : Add A
t : Set A
S : Subsemigroup M
ι : Sort u_1
p : ι → Subsemigroup M
m : M
⊢ (∀ (b : Subsemigroup M), (∀ (i : ι), p i ≤ b) → closure {m} ≤ b) ↔
∀ (N : Subsemigroup M), (∀ (i : ι), p i ≤ N) → m ∈ N
|
null
|
null
|
Mathlib/GroupTheory/Subsemigroup/Basic.lean
|
Subsemigroup.mem_iSup
|
[451, 1]
|
[454, 43]
|
simp only [closure_singleton_le_iff_mem]
|
M : Type u_2
N : Type ?u.21456
A : Type ?u.21459
inst✝¹ : Mul M
s : Set M
inst✝ : Add A
t : Set A
S : Subsemigroup M
ι : Sort u_1
p : ι → Subsemigroup M
m : M
⊢ (∀ (b : Subsemigroup M), (∀ (i : ι), p i ≤ b) → closure {m} ≤ b) ↔
∀ (N : Subsemigroup M), (∀ (i : ι), p i ≤ N) → m ∈ N
|
no goals
|
null
|
null
|
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biprod.isoProd_inv
|
[1446, 1]
|
[1448, 50]
|
apply biprod.hom_ext <;> simp [Iso.inv_comp_eq]
|
J : Type w
C : Type u
inst✝² : Category C
inst✝¹ : HasZeroMorphisms C
P Q X Y : C
inst✝ : HasBinaryBiproduct X Y
⊢ (isoProd X Y).inv = lift prod.fst prod.snd
|
no goals
|
null
|
null
|
Mathlib/Data/Set/Prod.lean
|
Set.image_prod_mk_subset_prod
|
[332, 1]
|
[335, 68]
|
rintro _ ⟨x, hx, rfl⟩
|
α : Type u_1
β : Type u_3
γ : Type u_2
δ : Type ?u.83491
s✝ s₁ s₂ : Set α
t t₁ t₂ : Set β
a : α
b : β
f : α → β
g : α → γ
s : Set α
⊢ (fun x => (f x, g x)) '' s ⊆ (f '' s) ×ˢ (g '' s)
|
case intro.intro
α : Type u_1
β : Type u_3
γ : Type u_2
δ : Type ?u.83491
s✝ s₁ s₂ : Set α
t t₁ t₂ : Set β
a : α
b : β
f : α → β
g : α → γ
s : Set α
x : α
hx : x ∈ s
⊢ (fun x => (f x, g x)) x ∈ (f '' s) ×ˢ (g '' s)
|
null
|
null
|
Mathlib/Data/Set/Prod.lean
|
Set.image_prod_mk_subset_prod
|
[332, 1]
|
[335, 68]
|
exact mk_mem_prod (mem_image_of_mem f hx) (mem_image_of_mem g hx)
|
case intro.intro
α : Type u_1
β : Type u_3
γ : Type u_2
δ : Type ?u.83491
s✝ s₁ s₂ : Set α
t t₁ t₂ : Set β
a : α
b : β
f : α → β
g : α → γ
s : Set α
x : α
hx : x ∈ s
⊢ (fun x => (f x, g x)) x ∈ (f '' s) ×ˢ (g '' s)
|
no goals
|
null
|
null
|
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.prod_nonneg
|
[432, 1]
|
[441, 91]
|
revert h
|
ι : Type ?u.134844
α : Type u_1
β : Type ?u.134850
γ : Type ?u.134853
inst✝ : OrderedCommSemiring α
m : Multiset α
h : ∀ (a : α), a ∈ m → 0 ≤ a
⊢ 0 ≤ prod m
|
ι : Type ?u.134844
α : Type u_1
β : Type ?u.134850
γ : Type ?u.134853
inst✝ : OrderedCommSemiring α
m : Multiset α
⊢ (∀ (a : α), a ∈ m → 0 ≤ a) → 0 ≤ prod m
|
null
|
null
|
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.prod_nonneg
|
[432, 1]
|
[441, 91]
|
refine' m.induction_on _ _
|
ι : Type ?u.134844
α : Type u_1
β : Type ?u.134850
γ : Type ?u.134853
inst✝ : OrderedCommSemiring α
m : Multiset α
⊢ (∀ (a : α), a ∈ m → 0 ≤ a) → 0 ≤ prod m
|
case refine'_1
ι : Type ?u.134844
α : Type u_1
β : Type ?u.134850
γ : Type ?u.134853
inst✝ : OrderedCommSemiring α
m : Multiset α
⊢ (∀ (a : α), a ∈ 0 → 0 ≤ a) → 0 ≤ prod 0
case refine'_2
ι : Type ?u.134844
α : Type u_1
β : Type ?u.134850
γ : Type ?u.134853
inst✝ : OrderedCommSemiring α
m : Multiset α
⊢ ∀ ⦃a : α⦄ {s : Multiset α},
((∀ (a : α), a ∈ s → 0 ≤ a) → 0 ≤ prod s) → (∀ (a_2 : α), a_2 ∈ a ::ₘ s → 0 ≤ a_2) → 0 ≤ prod (a ::ₘ s)
|
null
|
null
|
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.prod_nonneg
|
[432, 1]
|
[441, 91]
|
intro a s hs ih
|
case refine'_2
ι : Type ?u.134844
α : Type u_1
β : Type ?u.134850
γ : Type ?u.134853
inst✝ : OrderedCommSemiring α
m : Multiset α
⊢ ∀ ⦃a : α⦄ {s : Multiset α},
((∀ (a : α), a ∈ s → 0 ≤ a) → 0 ≤ prod s) → (∀ (a_2 : α), a_2 ∈ a ::ₘ s → 0 ≤ a_2) → 0 ≤ prod (a ::ₘ s)
|
case refine'_2
ι : Type ?u.134844
α : Type u_1
β : Type ?u.134850
γ : Type ?u.134853
inst✝ : OrderedCommSemiring α
m : Multiset α
a : α
s : Multiset α
hs : (∀ (a : α), a ∈ s → 0 ≤ a) → 0 ≤ prod s
ih : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → 0 ≤ a_1
⊢ 0 ≤ prod (a ::ₘ s)
|
null
|
null
|
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.prod_nonneg
|
[432, 1]
|
[441, 91]
|
rw [prod_cons]
|
case refine'_2
ι : Type ?u.134844
α : Type u_1
β : Type ?u.134850
γ : Type ?u.134853
inst✝ : OrderedCommSemiring α
m : Multiset α
a : α
s : Multiset α
hs : (∀ (a : α), a ∈ s → 0 ≤ a) → 0 ≤ prod s
ih : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → 0 ≤ a_1
⊢ 0 ≤ prod (a ::ₘ s)
|
case refine'_2
ι : Type ?u.134844
α : Type u_1
β : Type ?u.134850
γ : Type ?u.134853
inst✝ : OrderedCommSemiring α
m : Multiset α
a : α
s : Multiset α
hs : (∀ (a : α), a ∈ s → 0 ≤ a) → 0 ≤ prod s
ih : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → 0 ≤ a_1
⊢ 0 ≤ a * prod s
|
null
|
null
|
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.prod_nonneg
|
[432, 1]
|
[441, 91]
|
exact mul_nonneg (ih _ <| mem_cons_self _ _) (hs fun a ha => ih _ <| mem_cons_of_mem ha)
|
case refine'_2
ι : Type ?u.134844
α : Type u_1
β : Type ?u.134850
γ : Type ?u.134853
inst✝ : OrderedCommSemiring α
m : Multiset α
a : α
s : Multiset α
hs : (∀ (a : α), a ∈ s → 0 ≤ a) → 0 ≤ prod s
ih : ∀ (a_1 : α), a_1 ∈ a ::ₘ s → 0 ≤ a_1
⊢ 0 ≤ a * prod s
|
no goals
|
null
|
null
|
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.prod_nonneg
|
[432, 1]
|
[441, 91]
|
rintro -
|
case refine'_1
ι : Type ?u.134844
α : Type u_1
β : Type ?u.134850
γ : Type ?u.134853
inst✝ : OrderedCommSemiring α
m : Multiset α
⊢ (∀ (a : α), a ∈ 0 → 0 ≤ a) → 0 ≤ prod 0
|
case refine'_1
ι : Type ?u.134844
α : Type u_1
β : Type ?u.134850
γ : Type ?u.134853
inst✝ : OrderedCommSemiring α
m : Multiset α
⊢ 0 ≤ prod 0
|
null
|
null
|
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.prod_nonneg
|
[432, 1]
|
[441, 91]
|
rw [prod_zero]
|
case refine'_1
ι : Type ?u.134844
α : Type u_1
β : Type ?u.134850
γ : Type ?u.134853
inst✝ : OrderedCommSemiring α
m : Multiset α
⊢ 0 ≤ prod 0
|
case refine'_1
ι : Type ?u.134844
α : Type u_1
β : Type ?u.134850
γ : Type ?u.134853
inst✝ : OrderedCommSemiring α
m : Multiset α
⊢ 0 ≤ 1
|
null
|
null
|
Mathlib/Algebra/BigOperators/Multiset/Basic.lean
|
Multiset.prod_nonneg
|
[432, 1]
|
[441, 91]
|
exact zero_le_one
|
case refine'_1
ι : Type ?u.134844
α : Type u_1
β : Type ?u.134850
γ : Type ?u.134853
inst✝ : OrderedCommSemiring α
m : Multiset α
⊢ 0 ≤ 1
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/PGroup.lean
|
IsPGroup.to_le
|
[269, 1]
|
[273, 34]
|
change ((Subgroup.inclusion hHK) a : G) = (Subgroup.inclusion hHK) b
|
p : ℕ
G : Type u_1
inst✝ : Group G
H K : Subgroup G
hK : IsPGroup p { x // x ∈ K }
hHK : H ≤ K
a b : { x // x ∈ H }
h : ↑(Subgroup.inclusion hHK) a = ↑(Subgroup.inclusion hHK) b
⊢ ↑a = ↑b
|
p : ℕ
G : Type u_1
inst✝ : Group G
H K : Subgroup G
hK : IsPGroup p { x // x ∈ K }
hHK : H ≤ K
a b : { x // x ∈ H }
h : ↑(Subgroup.inclusion hHK) a = ↑(Subgroup.inclusion hHK) b
⊢ ↑(↑(Subgroup.inclusion hHK) a) = ↑(↑(Subgroup.inclusion hHK) b)
|
null
|
null
|
Mathlib/GroupTheory/PGroup.lean
|
IsPGroup.to_le
|
[269, 1]
|
[273, 34]
|
apply Subtype.ext_iff.mp h
|
p : ℕ
G : Type u_1
inst✝ : Group G
H K : Subgroup G
hK : IsPGroup p { x // x ∈ K }
hHK : H ≤ K
a b : { x // x ∈ H }
h : ↑(Subgroup.inclusion hHK) a = ↑(Subgroup.inclusion hHK) b
⊢ ↑(↑(Subgroup.inclusion hHK) a) = ↑(↑(Subgroup.inclusion hHK) b)
|
no goals
|
null
|
null
|
Mathlib/Data/Real/Irrational.lean
|
irrational_nat_mul_iff
|
[623, 1]
|
[624, 64]
|
rw [← cast_coe_nat, irrational_rat_mul_iff, Nat.cast_ne_zero]
|
q : ℚ
m : ℤ
n : ℕ
x : ℝ
⊢ Irrational (↑n * x) ↔ n ≠ 0 ∧ Irrational x
|
no goals
|
null
|
null
|
Mathlib/Topology/Paracompact.lean
|
normal_of_paracompact_t2
|
[299, 1]
|
[323, 66]
|
have : ∀ s t : Set X, IsClosed s → IsClosed t →
(∀ x ∈ s, ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →
∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v := by
intro s t hs _ H
choose u v hu hv hxu htv huv using SetCoe.forall'.1 H
rcases precise_refinement_set hs u hu fun x hx ↦ mem_iUnion.2 ⟨⟨x, hx⟩, hxu _⟩ with
⟨u', hu'o, hcov', hu'fin, hsub⟩
refine' ⟨⋃ i, u' i, closure (⋃ i, u' i)ᶜ, isOpen_iUnion hu'o, isClosed_closure.isOpen_compl,
hcov', _, disjoint_compl_right.mono le_rfl (compl_le_compl subset_closure)⟩
rw [hu'fin.closure_iUnion, compl_iUnion, subset_iInter_iff]
refine' fun i x hxt hxu ↦
absurd (htv i hxt) (closure_minimal _ (isClosed_compl_iff.2 <| hv _) hxu)
exact fun y hyu hyv ↦ (huv i).le_bot ⟨hsub _ hyu, hyv⟩
|
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
⊢ NormalSpace X
|
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
this :
∀ (s t : Set X),
IsClosed s →
IsClosed t →
(∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →
∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
⊢ NormalSpace X
|
null
|
null
|
Mathlib/Topology/Paracompact.lean
|
normal_of_paracompact_t2
|
[299, 1]
|
[323, 66]
|
refine' ⟨fun s t hs ht hst ↦ this s t hs ht fun x hx ↦ _⟩
|
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
this :
∀ (s t : Set X),
IsClosed s →
IsClosed t →
(∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →
∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
⊢ NormalSpace X
|
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
this :
∀ (s t : Set X),
IsClosed s →
IsClosed t →
(∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →
∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
s t : Set X
hs : IsClosed s
ht : IsClosed t
hst : Disjoint s t
x : X
hx : x ∈ s
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v
|
null
|
null
|
Mathlib/Topology/Paracompact.lean
|
normal_of_paracompact_t2
|
[299, 1]
|
[323, 66]
|
rcases this t {x} ht isClosed_singleton fun y hy ↦ (by
simp_rw [singleton_subset_iff]
exact t2_separation (hst.symm.ne_of_mem hy hx))
with ⟨v, u, hv, hu, htv, hxu, huv⟩
|
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
this :
∀ (s t : Set X),
IsClosed s →
IsClosed t →
(∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →
∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
s t : Set X
hs : IsClosed s
ht : IsClosed t
hst : Disjoint s t
x : X
hx : x ∈ s
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v
|
case intro.intro.intro.intro.intro.intro
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
this :
∀ (s t : Set X),
IsClosed s →
IsClosed t →
(∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →
∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
s t : Set X
hs : IsClosed s
ht : IsClosed t
hst : Disjoint s t
x : X
hx : x ∈ s
v u : Set X
hv : IsOpen v
hu : IsOpen u
htv : t ⊆ v
hxu : {x} ⊆ u
huv : Disjoint v u
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v
|
null
|
null
|
Mathlib/Topology/Paracompact.lean
|
normal_of_paracompact_t2
|
[299, 1]
|
[323, 66]
|
exact ⟨u, v, hu, hv, singleton_subset_iff.1 hxu, htv, huv.symm⟩
|
case intro.intro.intro.intro.intro.intro
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
this :
∀ (s t : Set X),
IsClosed s →
IsClosed t →
(∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →
∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
s t : Set X
hs : IsClosed s
ht : IsClosed t
hst : Disjoint s t
x : X
hx : x ∈ s
v u : Set X
hv : IsOpen v
hu : IsOpen u
htv : t ⊆ v
hxu : {x} ⊆ u
huv : Disjoint v u
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v
|
no goals
|
null
|
null
|
Mathlib/Topology/Paracompact.lean
|
normal_of_paracompact_t2
|
[299, 1]
|
[323, 66]
|
intro s t hs _ H
|
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
⊢ ∀ (s t : Set X),
IsClosed s →
IsClosed t →
(∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →
∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
|
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
s t : Set X
hs : IsClosed s
a✝ : IsClosed t
H : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
|
null
|
null
|
Mathlib/Topology/Paracompact.lean
|
normal_of_paracompact_t2
|
[299, 1]
|
[323, 66]
|
choose u v hu hv hxu htv huv using SetCoe.forall'.1 H
|
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
s t : Set X
hs : IsClosed s
a✝ : IsClosed t
H : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
|
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
s t : Set X
hs : IsClosed s
a✝ : IsClosed t
H : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v
u v : ↑s → Set X
hu : ∀ (x : ↑s), IsOpen (u x)
hv : ∀ (x : ↑s), IsOpen (v x)
hxu : ∀ (x : ↑s), ↑x ∈ u x
htv : ∀ (x : ↑s), t ⊆ v x
huv : ∀ (x : ↑s), Disjoint (u x) (v x)
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
|
null
|
null
|
Mathlib/Topology/Paracompact.lean
|
normal_of_paracompact_t2
|
[299, 1]
|
[323, 66]
|
rcases precise_refinement_set hs u hu fun x hx ↦ mem_iUnion.2 ⟨⟨x, hx⟩, hxu _⟩ with
⟨u', hu'o, hcov', hu'fin, hsub⟩
|
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
s t : Set X
hs : IsClosed s
a✝ : IsClosed t
H : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v
u v : ↑s → Set X
hu : ∀ (x : ↑s), IsOpen (u x)
hv : ∀ (x : ↑s), IsOpen (v x)
hxu : ∀ (x : ↑s), ↑x ∈ u x
htv : ∀ (x : ↑s), t ⊆ v x
huv : ∀ (x : ↑s), Disjoint (u x) (v x)
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
|
case intro.intro.intro.intro
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
s t : Set X
hs : IsClosed s
a✝ : IsClosed t
H : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v
u v : ↑s → Set X
hu : ∀ (x : ↑s), IsOpen (u x)
hv : ∀ (x : ↑s), IsOpen (v x)
hxu : ∀ (x : ↑s), ↑x ∈ u x
htv : ∀ (x : ↑s), t ⊆ v x
huv : ∀ (x : ↑s), Disjoint (u x) (v x)
u' : ↑s → Set X
hu'o : ∀ (i : ↑s), IsOpen (u' i)
hcov' : s ⊆ ⋃ (i : ↑s), u' i
hu'fin : LocallyFinite u'
hsub : ∀ (i : ↑s), u' i ⊆ u i
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
|
null
|
null
|
Mathlib/Topology/Paracompact.lean
|
normal_of_paracompact_t2
|
[299, 1]
|
[323, 66]
|
refine' ⟨⋃ i, u' i, closure (⋃ i, u' i)ᶜ, isOpen_iUnion hu'o, isClosed_closure.isOpen_compl,
hcov', _, disjoint_compl_right.mono le_rfl (compl_le_compl subset_closure)⟩
|
case intro.intro.intro.intro
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
s t : Set X
hs : IsClosed s
a✝ : IsClosed t
H : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v
u v : ↑s → Set X
hu : ∀ (x : ↑s), IsOpen (u x)
hv : ∀ (x : ↑s), IsOpen (v x)
hxu : ∀ (x : ↑s), ↑x ∈ u x
htv : ∀ (x : ↑s), t ⊆ v x
huv : ∀ (x : ↑s), Disjoint (u x) (v x)
u' : ↑s → Set X
hu'o : ∀ (i : ↑s), IsOpen (u' i)
hcov' : s ⊆ ⋃ (i : ↑s), u' i
hu'fin : LocallyFinite u'
hsub : ∀ (i : ↑s), u' i ⊆ u i
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
|
case intro.intro.intro.intro
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
s t : Set X
hs : IsClosed s
a✝ : IsClosed t
H : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v
u v : ↑s → Set X
hu : ∀ (x : ↑s), IsOpen (u x)
hv : ∀ (x : ↑s), IsOpen (v x)
hxu : ∀ (x : ↑s), ↑x ∈ u x
htv : ∀ (x : ↑s), t ⊆ v x
huv : ∀ (x : ↑s), Disjoint (u x) (v x)
u' : ↑s → Set X
hu'o : ∀ (i : ↑s), IsOpen (u' i)
hcov' : s ⊆ ⋃ (i : ↑s), u' i
hu'fin : LocallyFinite u'
hsub : ∀ (i : ↑s), u' i ⊆ u i
⊢ t ⊆ closure (⋃ (i : ↑s), u' i)ᶜ
|
null
|
null
|
Mathlib/Topology/Paracompact.lean
|
normal_of_paracompact_t2
|
[299, 1]
|
[323, 66]
|
rw [hu'fin.closure_iUnion, compl_iUnion, subset_iInter_iff]
|
case intro.intro.intro.intro
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
s t : Set X
hs : IsClosed s
a✝ : IsClosed t
H : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v
u v : ↑s → Set X
hu : ∀ (x : ↑s), IsOpen (u x)
hv : ∀ (x : ↑s), IsOpen (v x)
hxu : ∀ (x : ↑s), ↑x ∈ u x
htv : ∀ (x : ↑s), t ⊆ v x
huv : ∀ (x : ↑s), Disjoint (u x) (v x)
u' : ↑s → Set X
hu'o : ∀ (i : ↑s), IsOpen (u' i)
hcov' : s ⊆ ⋃ (i : ↑s), u' i
hu'fin : LocallyFinite u'
hsub : ∀ (i : ↑s), u' i ⊆ u i
⊢ t ⊆ closure (⋃ (i : ↑s), u' i)ᶜ
|
case intro.intro.intro.intro
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
s t : Set X
hs : IsClosed s
a✝ : IsClosed t
H : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v
u v : ↑s → Set X
hu : ∀ (x : ↑s), IsOpen (u x)
hv : ∀ (x : ↑s), IsOpen (v x)
hxu : ∀ (x : ↑s), ↑x ∈ u x
htv : ∀ (x : ↑s), t ⊆ v x
huv : ∀ (x : ↑s), Disjoint (u x) (v x)
u' : ↑s → Set X
hu'o : ∀ (i : ↑s), IsOpen (u' i)
hcov' : s ⊆ ⋃ (i : ↑s), u' i
hu'fin : LocallyFinite u'
hsub : ∀ (i : ↑s), u' i ⊆ u i
⊢ ∀ (i : ↑s), t ⊆ closure (u' i)ᶜ
|
null
|
null
|
Mathlib/Topology/Paracompact.lean
|
normal_of_paracompact_t2
|
[299, 1]
|
[323, 66]
|
refine' fun i x hxt hxu ↦
absurd (htv i hxt) (closure_minimal _ (isClosed_compl_iff.2 <| hv _) hxu)
|
case intro.intro.intro.intro
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
s t : Set X
hs : IsClosed s
a✝ : IsClosed t
H : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v
u v : ↑s → Set X
hu : ∀ (x : ↑s), IsOpen (u x)
hv : ∀ (x : ↑s), IsOpen (v x)
hxu : ∀ (x : ↑s), ↑x ∈ u x
htv : ∀ (x : ↑s), t ⊆ v x
huv : ∀ (x : ↑s), Disjoint (u x) (v x)
u' : ↑s → Set X
hu'o : ∀ (i : ↑s), IsOpen (u' i)
hcov' : s ⊆ ⋃ (i : ↑s), u' i
hu'fin : LocallyFinite u'
hsub : ∀ (i : ↑s), u' i ⊆ u i
⊢ ∀ (i : ↑s), t ⊆ closure (u' i)ᶜ
|
case intro.intro.intro.intro
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
s t : Set X
hs : IsClosed s
a✝ : IsClosed t
H : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v
u v : ↑s → Set X
hu : ∀ (x : ↑s), IsOpen (u x)
hv : ∀ (x : ↑s), IsOpen (v x)
hxu✝ : ∀ (x : ↑s), ↑x ∈ u x
htv : ∀ (x : ↑s), t ⊆ v x
huv : ∀ (x : ↑s), Disjoint (u x) (v x)
u' : ↑s → Set X
hu'o : ∀ (i : ↑s), IsOpen (u' i)
hcov' : s ⊆ ⋃ (i : ↑s), u' i
hu'fin : LocallyFinite u'
hsub : ∀ (i : ↑s), u' i ⊆ u i
i : ↑s
x : X
hxt : x ∈ t
hxu : x ∈ closure (u' i)
⊢ u' i ⊆ v iᶜ
|
null
|
null
|
Mathlib/Topology/Paracompact.lean
|
normal_of_paracompact_t2
|
[299, 1]
|
[323, 66]
|
exact fun y hyu hyv ↦ (huv i).le_bot ⟨hsub _ hyu, hyv⟩
|
case intro.intro.intro.intro
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
s t : Set X
hs : IsClosed s
a✝ : IsClosed t
H : ∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v
u v : ↑s → Set X
hu : ∀ (x : ↑s), IsOpen (u x)
hv : ∀ (x : ↑s), IsOpen (v x)
hxu✝ : ∀ (x : ↑s), ↑x ∈ u x
htv : ∀ (x : ↑s), t ⊆ v x
huv : ∀ (x : ↑s), Disjoint (u x) (v x)
u' : ↑s → Set X
hu'o : ∀ (i : ↑s), IsOpen (u' i)
hcov' : s ⊆ ⋃ (i : ↑s), u' i
hu'fin : LocallyFinite u'
hsub : ∀ (i : ↑s), u' i ⊆ u i
i : ↑s
x : X
hxt : x ∈ t
hxu : x ∈ closure (u' i)
⊢ u' i ⊆ v iᶜ
|
no goals
|
null
|
null
|
Mathlib/Topology/Paracompact.lean
|
normal_of_paracompact_t2
|
[299, 1]
|
[323, 66]
|
simp_rw [singleton_subset_iff]
|
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
this :
∀ (s t : Set X),
IsClosed s →
IsClosed t →
(∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →
∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
s t : Set X
hs : IsClosed s
ht : IsClosed t
hst : Disjoint s t
x : X
hx : x ∈ s
y : X
hy : y ∈ t
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ y ∈ u ∧ {x} ⊆ v ∧ Disjoint u v
|
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
this :
∀ (s t : Set X),
IsClosed s →
IsClosed t →
(∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →
∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
s t : Set X
hs : IsClosed s
ht : IsClosed t
hst : Disjoint s t
x : X
hx : x ∈ s
y : X
hy : y ∈ t
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ y ∈ u ∧ x ∈ v ∧ Disjoint u v
|
null
|
null
|
Mathlib/Topology/Paracompact.lean
|
normal_of_paracompact_t2
|
[299, 1]
|
[323, 66]
|
exact t2_separation (hst.symm.ne_of_mem hy hx)
|
ι : Type u
X : Type v
Y : Type w
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : T2Space X
inst✝ : ParacompactSpace X
this :
∀ (s t : Set X),
IsClosed s →
IsClosed t →
(∀ (x : X), x ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ t ⊆ v ∧ Disjoint u v) →
∃ u v, IsOpen u ∧ IsOpen v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v
s t : Set X
hs : IsClosed s
ht : IsClosed t
hst : Disjoint s t
x : X
hx : x ∈ s
y : X
hy : y ∈ t
⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ y ∈ u ∧ x ∈ v ∧ Disjoint u v
|
no goals
|
null
|
null
|
Mathlib/Data/Nat/Factorization/PrimePow.lean
|
isPrimePow_of_minFac_pow_factorization_eq
|
[30, 1]
|
[37, 23]
|
rcases eq_or_ne n 0 with (rfl | hn')
|
R : Type ?u.706
inst✝ : CommMonoidWithZero R
n✝ p : R
k n : ℕ
h : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n
hn : n ≠ 1
⊢ IsPrimePow n
|
case inl
R : Type ?u.706
inst✝ : CommMonoidWithZero R
n p : R
k : ℕ
h : Nat.minFac 0 ^ ↑(Nat.factorization 0) (Nat.minFac 0) = 0
hn : 0 ≠ 1
⊢ IsPrimePow 0
case inr
R : Type ?u.706
inst✝ : CommMonoidWithZero R
n✝ p : R
k n : ℕ
h : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n
hn : n ≠ 1
hn' : n ≠ 0
⊢ IsPrimePow n
|
null
|
null
|
Mathlib/Data/Nat/Factorization/PrimePow.lean
|
isPrimePow_of_minFac_pow_factorization_eq
|
[30, 1]
|
[37, 23]
|
refine' ⟨_, _, (Nat.minFac_prime hn).prime, _, h⟩
|
case inr
R : Type ?u.706
inst✝ : CommMonoidWithZero R
n✝ p : R
k n : ℕ
h : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n
hn : n ≠ 1
hn' : n ≠ 0
⊢ IsPrimePow n
|
case inr
R : Type ?u.706
inst✝ : CommMonoidWithZero R
n✝ p : R
k n : ℕ
h : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n
hn : n ≠ 1
hn' : n ≠ 0
⊢ 0 < ↑(Nat.factorization n) (Nat.minFac n)
|
null
|
null
|
Mathlib/Data/Nat/Factorization/PrimePow.lean
|
isPrimePow_of_minFac_pow_factorization_eq
|
[30, 1]
|
[37, 23]
|
rw [pos_iff_ne_zero, ← Finsupp.mem_support_iff, Nat.factor_iff_mem_factorization,
Nat.mem_factors_iff_dvd hn' (Nat.minFac_prime hn)]
|
case inr
R : Type ?u.706
inst✝ : CommMonoidWithZero R
n✝ p : R
k n : ℕ
h : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n
hn : n ≠ 1
hn' : n ≠ 0
⊢ 0 < ↑(Nat.factorization n) (Nat.minFac n)
|
case inr
R : Type ?u.706
inst✝ : CommMonoidWithZero R
n✝ p : R
k n : ℕ
h : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n
hn : n ≠ 1
hn' : n ≠ 0
⊢ Nat.minFac n ∣ n
|
null
|
null
|
Mathlib/Data/Nat/Factorization/PrimePow.lean
|
isPrimePow_of_minFac_pow_factorization_eq
|
[30, 1]
|
[37, 23]
|
apply Nat.minFac_dvd
|
case inr
R : Type ?u.706
inst✝ : CommMonoidWithZero R
n✝ p : R
k n : ℕ
h : Nat.minFac n ^ ↑(Nat.factorization n) (Nat.minFac n) = n
hn : n ≠ 1
hn' : n ≠ 0
⊢ Nat.minFac n ∣ n
|
no goals
|
null
|
null
|
Mathlib/Data/Nat/Factorization/PrimePow.lean
|
isPrimePow_of_minFac_pow_factorization_eq
|
[30, 1]
|
[37, 23]
|
simp_all
|
case inl
R : Type ?u.706
inst✝ : CommMonoidWithZero R
n p : R
k : ℕ
h : Nat.minFac 0 ^ ↑(Nat.factorization 0) (Nat.minFac 0) = 0
hn : 0 ≠ 1
⊢ IsPrimePow 0
|
no goals
|
null
|
null
|
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.smulRight_one_pow
|
[1432, 1]
|
[1437, 72]
|
induction' n with n ihn
|
R : Type u_1
inst✝¹⁵ : Ring R
R₂ : Type ?u.888280
inst✝¹⁴ : Ring R₂
R₃ : Type ?u.888286
inst✝¹³ : Ring R₃
M : Type ?u.888292
inst✝¹² : TopologicalSpace M
inst✝¹¹ : AddCommGroup M
M₂ : Type ?u.888301
inst✝¹⁰ : TopologicalSpace M₂
inst✝⁹ : AddCommGroup M₂
M₃ : Type ?u.888310
inst✝⁸ : TopologicalSpace M₃
inst✝⁷ : AddCommGroup M₃
M₄ : Type ?u.888319
inst✝⁶ : TopologicalSpace M₄
inst✝⁵ : AddCommGroup M₄
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
inst✝² : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalRing R
c : R
n : ℕ
⊢ smulRight 1 c ^ n = smulRight 1 (c ^ n)
|
case zero
R : Type u_1
inst✝¹⁵ : Ring R
R₂ : Type ?u.888280
inst✝¹⁴ : Ring R₂
R₃ : Type ?u.888286
inst✝¹³ : Ring R₃
M : Type ?u.888292
inst✝¹² : TopologicalSpace M
inst✝¹¹ : AddCommGroup M
M₂ : Type ?u.888301
inst✝¹⁰ : TopologicalSpace M₂
inst✝⁹ : AddCommGroup M₂
M₃ : Type ?u.888310
inst✝⁸ : TopologicalSpace M₃
inst✝⁷ : AddCommGroup M₃
M₄ : Type ?u.888319
inst✝⁶ : TopologicalSpace M₄
inst✝⁵ : AddCommGroup M₄
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
inst✝² : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalRing R
c : R
⊢ smulRight 1 c ^ Nat.zero = smulRight 1 (c ^ Nat.zero)
case succ
R : Type u_1
inst✝¹⁵ : Ring R
R₂ : Type ?u.888280
inst✝¹⁴ : Ring R₂
R₃ : Type ?u.888286
inst✝¹³ : Ring R₃
M : Type ?u.888292
inst✝¹² : TopologicalSpace M
inst✝¹¹ : AddCommGroup M
M₂ : Type ?u.888301
inst✝¹⁰ : TopologicalSpace M₂
inst✝⁹ : AddCommGroup M₂
M₃ : Type ?u.888310
inst✝⁸ : TopologicalSpace M₃
inst✝⁷ : AddCommGroup M₃
M₄ : Type ?u.888319
inst✝⁶ : TopologicalSpace M₄
inst✝⁵ : AddCommGroup M₄
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
inst✝² : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalRing R
c : R
n : ℕ
ihn : smulRight 1 c ^ n = smulRight 1 (c ^ n)
⊢ smulRight 1 c ^ Nat.succ n = smulRight 1 (c ^ Nat.succ n)
|
null
|
null
|
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.smulRight_one_pow
|
[1432, 1]
|
[1437, 72]
|
ext
|
case zero
R : Type u_1
inst✝¹⁵ : Ring R
R₂ : Type ?u.888280
inst✝¹⁴ : Ring R₂
R₃ : Type ?u.888286
inst✝¹³ : Ring R₃
M : Type ?u.888292
inst✝¹² : TopologicalSpace M
inst✝¹¹ : AddCommGroup M
M₂ : Type ?u.888301
inst✝¹⁰ : TopologicalSpace M₂
inst✝⁹ : AddCommGroup M₂
M₃ : Type ?u.888310
inst✝⁸ : TopologicalSpace M₃
inst✝⁷ : AddCommGroup M₃
M₄ : Type ?u.888319
inst✝⁶ : TopologicalSpace M₄
inst✝⁵ : AddCommGroup M₄
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
inst✝² : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalRing R
c : R
⊢ smulRight 1 c ^ Nat.zero = smulRight 1 (c ^ Nat.zero)
|
case zero.h
R : Type u_1
inst✝¹⁵ : Ring R
R₂ : Type ?u.888280
inst✝¹⁴ : Ring R₂
R₃ : Type ?u.888286
inst✝¹³ : Ring R₃
M : Type ?u.888292
inst✝¹² : TopologicalSpace M
inst✝¹¹ : AddCommGroup M
M₂ : Type ?u.888301
inst✝¹⁰ : TopologicalSpace M₂
inst✝⁹ : AddCommGroup M₂
M₃ : Type ?u.888310
inst✝⁸ : TopologicalSpace M₃
inst✝⁷ : AddCommGroup M₃
M₄ : Type ?u.888319
inst✝⁶ : TopologicalSpace M₄
inst✝⁵ : AddCommGroup M₄
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
inst✝² : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalRing R
c : R
⊢ ↑(smulRight 1 c ^ Nat.zero) 1 = ↑(smulRight 1 (c ^ Nat.zero)) 1
|
null
|
null
|
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.smulRight_one_pow
|
[1432, 1]
|
[1437, 72]
|
simp
|
case zero.h
R : Type u_1
inst✝¹⁵ : Ring R
R₂ : Type ?u.888280
inst✝¹⁴ : Ring R₂
R₃ : Type ?u.888286
inst✝¹³ : Ring R₃
M : Type ?u.888292
inst✝¹² : TopologicalSpace M
inst✝¹¹ : AddCommGroup M
M₂ : Type ?u.888301
inst✝¹⁰ : TopologicalSpace M₂
inst✝⁹ : AddCommGroup M₂
M₃ : Type ?u.888310
inst✝⁸ : TopologicalSpace M₃
inst✝⁷ : AddCommGroup M₃
M₄ : Type ?u.888319
inst✝⁶ : TopologicalSpace M₄
inst✝⁵ : AddCommGroup M₄
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
inst✝² : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalRing R
c : R
⊢ ↑(smulRight 1 c ^ Nat.zero) 1 = ↑(smulRight 1 (c ^ Nat.zero)) 1
|
no goals
|
null
|
null
|
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.smulRight_one_pow
|
[1432, 1]
|
[1437, 72]
|
rw [pow_succ, ihn, mul_def, smulRight_comp, smul_eq_mul, pow_succ']
|
case succ
R : Type u_1
inst✝¹⁵ : Ring R
R₂ : Type ?u.888280
inst✝¹⁴ : Ring R₂
R₃ : Type ?u.888286
inst✝¹³ : Ring R₃
M : Type ?u.888292
inst✝¹² : TopologicalSpace M
inst✝¹¹ : AddCommGroup M
M₂ : Type ?u.888301
inst✝¹⁰ : TopologicalSpace M₂
inst✝⁹ : AddCommGroup M₂
M₃ : Type ?u.888310
inst✝⁸ : TopologicalSpace M₃
inst✝⁷ : AddCommGroup M₃
M₄ : Type ?u.888319
inst✝⁶ : TopologicalSpace M₄
inst✝⁵ : AddCommGroup M₄
inst✝⁴ : Module R M
inst✝³ : Module R₂ M₂
inst✝² : Module R₃ M₃
σ₁₂ : R →+* R₂
σ₂₃ : R₂ →+* R₃
σ₁₃ : R →+* R₃
inst✝¹ : TopologicalSpace R
inst✝ : TopologicalRing R
c : R
n : ℕ
ihn : smulRight 1 c ^ n = smulRight 1 (c ^ n)
⊢ smulRight 1 c ^ Nat.succ n = smulRight 1 (c ^ Nat.succ n)
|
no goals
|
null
|
null
|
Mathlib/Probability/ConditionalProbability.lean
|
ProbabilityTheory.cond_univ
|
[100, 1]
|
[101, 51]
|
simp [cond, measure_univ, Measure.restrict_univ]
|
Ω : Type u_1
m : MeasurableSpace Ω
μ : MeasureTheory.Measure Ω
s t : Set Ω
inst✝ : IsProbabilityMeasure μ
⊢ μ[|Set.univ] = μ
|
no goals
|
null
|
null
|
Mathlib/NumberTheory/Padics/PadicVal.lean
|
padicValNat.div'
|
[439, 11]
|
[442, 18]
|
rw [padicValNat.div_of_dvd dvd, eq_zero_of_not_dvd (hp.out.coprime_iff_not_dvd.mp cpm),
Nat.sub_zero]
|
p a b✝ : ℕ
hp : Fact (Nat.Prime p)
m : ℕ
cpm : coprime p m
b : ℕ
dvd : m ∣ b
⊢ padicValNat p (b / m) = padicValNat p b
|
no goals
|
null
|
null
|
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ico_diff_Iio
|
[1815, 1]
|
[1816, 53]
|
rw [diff_eq, compl_Iio, Ico_inter_Ici, sup_eq_max]
|
α : Type u_1
β : Type ?u.197269
inst✝¹ : LinearOrder α
inst✝ : LinearOrder β
f : α → β
a a₁ a₂ b b₁ b₂ c d : α
⊢ Ico a b \ Iio c = Ico (max a c) b
|
no goals
|
null
|
null
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_zero
|
[171, 1]
|
[171, 57]
|
simp
|
α : Type u_1
β : Type ?u.74361
γ : Type ?u.74364
δ : Type ?u.74367
m : MeasurableSpace α
μ ν : Measure α
⊢ (∫⁻ (x : α), 0 ∂μ) = 0
|
no goals
|
null
|
null
|
Mathlib/LinearAlgebra/Matrix/Adjugate.lean
|
Matrix.adjugate_pow
|
[511, 1]
|
[514, 81]
|
induction' k with k IH
|
m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
k : ℕ
⊢ adjugate (A ^ k) = adjugate A ^ k
|
case zero
m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
⊢ adjugate (A ^ Nat.zero) = adjugate A ^ Nat.zero
case succ
m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
k : ℕ
IH : adjugate (A ^ k) = adjugate A ^ k
⊢ adjugate (A ^ Nat.succ k) = adjugate A ^ Nat.succ k
|
null
|
null
|
Mathlib/LinearAlgebra/Matrix/Adjugate.lean
|
Matrix.adjugate_pow
|
[511, 1]
|
[514, 81]
|
simp
|
case zero
m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
⊢ adjugate (A ^ Nat.zero) = adjugate A ^ Nat.zero
|
no goals
|
null
|
null
|
Mathlib/LinearAlgebra/Matrix/Adjugate.lean
|
Matrix.adjugate_pow
|
[511, 1]
|
[514, 81]
|
rw [pow_succ', mul_eq_mul, adjugate_mul_distrib, IH, ← mul_eq_mul, pow_succ]
|
case succ
m : Type u
n : Type v
α : Type w
inst✝⁴ : DecidableEq n
inst✝³ : Fintype n
inst✝² : DecidableEq m
inst✝¹ : Fintype m
inst✝ : CommRing α
A : Matrix n n α
k : ℕ
IH : adjugate (A ^ k) = adjugate A ^ k
⊢ adjugate (A ^ Nat.succ k) = adjugate A ^ Nat.succ k
|
no goals
|
null
|
null
|
Mathlib/LinearAlgebra/SesquilinearForm.lean
|
LinearMap.flip_separatingLeft
|
[691, 1]
|
[692, 91]
|
rw [← flip_separatingRight, flip_flip]
|
R : Type u_2
R₁ : Type u_1
R₂ : Type u_5
R₃ : Type ?u.540765
M : Type ?u.540768
M₁ : Type u_3
M₂ : Type u_4
Mₗ₁ : Type ?u.540777
Mₗ₁' : Type ?u.540780
Mₗ₂ : Type ?u.540783
Mₗ₂' : Type ?u.540786
K : Type ?u.540789
K₁ : Type ?u.540792
K₂ : Type ?u.540795
V : Type ?u.540798
V₁ : Type ?u.540801
V₂ : Type ?u.540804
n : Type ?u.540807
inst✝⁶ : CommSemiring R
inst✝⁵ : CommSemiring R₁
inst✝⁴ : AddCommMonoid M₁
inst✝³ : Module R₁ M₁
inst✝² : CommSemiring R₂
inst✝¹ : AddCommMonoid M₂
inst✝ : Module R₂ M₂
I₁ : R₁ →+* R
I₂ : R₂ →+* R
I₁' : R₁ →+* R
B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] R
⊢ SeparatingLeft (flip B) ↔ SeparatingRight B
|
no goals
|
null
|
null
|
Mathlib/MeasureTheory/Measure/Haar/Basic.lean
|
MeasureTheory.Measure.haar.haarContent_outerMeasure_self_pos
|
[581, 1]
|
[586, 28]
|
refine' zero_lt_one.trans_le _
|
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
⊢ 0 < ↑(Content.outerMeasure (haarContent K₀)) ↑K₀
|
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
⊢ 1 ≤ ↑(Content.outerMeasure (haarContent K₀)) ↑K₀
|
null
|
null
|
Mathlib/MeasureTheory/Measure/Haar/Basic.lean
|
MeasureTheory.Measure.haar.haarContent_outerMeasure_self_pos
|
[581, 1]
|
[586, 28]
|
rw [Content.outerMeasure_eq_iInf]
|
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
⊢ 1 ≤ ↑(Content.outerMeasure (haarContent K₀)) ↑K₀
|
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
⊢ 1 ≤
⨅ (U : Set G) (hU : IsOpen U) (_ : ↑K₀ ⊆ U), Content.innerContent (haarContent K₀) { carrier := U, is_open' := hU }
|
null
|
null
|
Mathlib/MeasureTheory/Measure/Haar/Basic.lean
|
MeasureTheory.Measure.haar.haarContent_outerMeasure_self_pos
|
[581, 1]
|
[586, 28]
|
refine' le_iInf₂ fun U hU => le_iInf fun hK₀ => le_trans _ <| le_iSup₂ K₀.toCompacts hK₀
|
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
⊢ 1 ≤
⨅ (U : Set G) (hU : IsOpen U) (_ : ↑K₀ ⊆ U), Content.innerContent (haarContent K₀) { carrier := U, is_open' := hU }
|
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
U : Set G
hU : IsOpen U
hK₀ : ↑K₀ ⊆ U
⊢ 1 ≤ (fun s => ↑(Content.toFun (haarContent K₀) s)) K₀.toCompacts
|
null
|
null
|
Mathlib/MeasureTheory/Measure/Haar/Basic.lean
|
MeasureTheory.Measure.haar.haarContent_outerMeasure_self_pos
|
[581, 1]
|
[586, 28]
|
exact haarContent_self.ge
|
G : Type u_1
inst✝³ : Group G
inst✝² : TopologicalSpace G
inst✝¹ : TopologicalGroup G
inst✝ : T2Space G
K₀ : PositiveCompacts G
U : Set G
hU : IsOpen U
hK₀ : ↑K₀ ⊆ U
⊢ 1 ≤ (fun s => ↑(Content.toFun (haarContent K₀) s)) K₀.toCompacts
|
no goals
|
null
|
null
|
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.aleph'_omega
|
[231, 1]
|
[234, 90]
|
simp only [aleph'_le_of_limit omega_isLimit, lt_omega, exists_imp, aleph0_le]
|
c : Cardinal
⊢ aleph' ω ≤ c ↔ ℵ₀ ≤ c
|
c : Cardinal
⊢ (∀ (o' : Ordinal) (x : ℕ), o' = ↑x → aleph' o' ≤ c) ↔ ∀ (n : ℕ), ↑n ≤ c
|
null
|
null
|
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.aleph'_omega
|
[231, 1]
|
[234, 90]
|
exact forall_swap.trans (forall_congr' fun n => by simp only [forall_eq, aleph'_nat])
|
c : Cardinal
⊢ (∀ (o' : Ordinal) (x : ℕ), o' = ↑x → aleph' o' ≤ c) ↔ ∀ (n : ℕ), ↑n ≤ c
|
no goals
|
null
|
null
|
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.aleph'_omega
|
[231, 1]
|
[234, 90]
|
simp only [forall_eq, aleph'_nat]
|
c : Cardinal
n : ℕ
⊢ (∀ (x : Ordinal), x = ↑n → aleph' x ≤ c) ↔ ↑n ≤ c
|
no goals
|
null
|
null
|
Mathlib/Algebra/CharP/Basic.lean
|
CharP.addOrderOf_one
|
[129, 1]
|
[130, 77]
|
rw [← Nat.smul_one_eq_coe, addOrderOf_dvd_iff_nsmul_eq_zero]
|
R✝ : Type ?u.43695
R : Type u_1
inst✝ : Semiring R
n : ℕ
⊢ ↑n = 0 ↔ addOrderOf 1 ∣ n
|
no goals
|
null
|
null
|
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.blsub_le_iff
|
[1805, 1]
|
[1808, 24]
|
convert bsup_le_iff.{_, v} (f := fun a ha => succ (f a ha)) (a := a) using 2
|
α : Type ?u.349869
β : Type ?u.349872
γ : Type ?u.349875
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal
f : (a : Ordinal) → a < o → Ordinal
a : Ordinal
⊢ blsub o f ≤ a ↔ ∀ (i : Ordinal) (h : i < o), f i h < a
|
case h.e'_2.h.a
α : Type ?u.349869
β : Type ?u.349872
γ : Type ?u.349875
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal
f : (a : Ordinal) → a < o → Ordinal
a : Ordinal
a✝ : Ordinal
⊢ (∀ (h : a✝ < o), f a✝ h < a) ↔ ∀ (h : a✝ < o), succ (f a✝ h) ≤ a
|
null
|
null
|
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.blsub_le_iff
|
[1805, 1]
|
[1808, 24]
|
simp_rw [succ_le_iff]
|
case h.e'_2.h.a
α : Type ?u.349869
β : Type ?u.349872
γ : Type ?u.349875
r : α → α → Prop
s : β → β → Prop
t : γ → γ → Prop
o : Ordinal
f : (a : Ordinal) → a < o → Ordinal
a : Ordinal
a✝ : Ordinal
⊢ (∀ (h : a✝ < o), f a✝ h < a) ↔ ∀ (h : a✝ < o), succ (f a✝ h) ≤ a
|
no goals
|
null
|
null
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.map_dirac
|
[2004, 1]
|
[2005, 72]
|
simp [hs, map_apply hf hs, hf hs, indicator_apply]
|
α : Type u_1
β : Type u_2
γ : Type ?u.329647
δ : Type ?u.329650
ι : Type ?u.329653
R : Type ?u.329656
R' : Type ?u.329659
m0 : MeasurableSpace α
inst✝² : MeasurableSpace β
inst✝¹ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
inst✝ : MeasurableSpace α
f : α → β
hf : Measurable f
a : α
s : Set β
hs : MeasurableSet s
⊢ ↑↑(map f (dirac a)) s = ↑↑(dirac (f a)) s
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/Perm/List.lean
|
List.formPerm_apply_mem_eq_self_iff
|
[383, 1]
|
[394, 31]
|
obtain ⟨k, hk, rfl⟩ := nthLe_of_mem hx
|
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x✝ : α
hl : Nodup l
x : α
hx : x ∈ l
⊢ ↑(formPerm l) x = x ↔ length l ≤ 1
|
case intro.intro
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk : k < length l
hx : nthLe l k hk ∈ l
⊢ ↑(formPerm l) (nthLe l k hk) = nthLe l k hk ↔ length l ≤ 1
|
null
|
null
|
Mathlib/GroupTheory/Perm/List.lean
|
List.formPerm_apply_mem_eq_self_iff
|
[383, 1]
|
[394, 31]
|
rw [formPerm_apply_nthLe _ hl, hl.nthLe_inj_iff]
|
case intro.intro
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk : k < length l
hx : nthLe l k hk ∈ l
⊢ ↑(formPerm l) (nthLe l k hk) = nthLe l k hk ↔ length l ≤ 1
|
case intro.intro
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk : k < length l
hx : nthLe l k hk ∈ l
⊢ (k + 1) % length l = k ↔ length l ≤ 1
|
null
|
null
|
Mathlib/GroupTheory/Perm/List.lean
|
List.formPerm_apply_mem_eq_self_iff
|
[383, 1]
|
[394, 31]
|
cases hn : l.length
|
case intro.intro
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk : k < length l
hx : nthLe l k hk ∈ l
⊢ (k + 1) % length l = k ↔ length l ≤ 1
|
case intro.intro.zero
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk : k < length l
hx : nthLe l k hk ∈ l
hn : length l = Nat.zero
⊢ (k + 1) % Nat.zero = k ↔ Nat.zero ≤ 1
case intro.intro.succ
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk : k < length l
hx : nthLe l k hk ∈ l
n✝ : ℕ
hn : length l = Nat.succ n✝
⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1
|
null
|
null
|
Mathlib/GroupTheory/Perm/List.lean
|
List.formPerm_apply_mem_eq_self_iff
|
[383, 1]
|
[394, 31]
|
exact absurd k.zero_le (hk.trans_le hn.le).not_le
|
case intro.intro.zero
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk : k < length l
hx : nthLe l k hk ∈ l
hn : length l = Nat.zero
⊢ (k + 1) % Nat.zero = k ↔ Nat.zero ≤ 1
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/Perm/List.lean
|
List.formPerm_apply_mem_eq_self_iff
|
[383, 1]
|
[394, 31]
|
rw [hn] at hk
|
case intro.intro.succ
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk : k < length l
hx : nthLe l k hk ∈ l
n✝ : ℕ
hn : length l = Nat.succ n✝
⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1
|
case intro.intro.succ
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk✝ : k < length l
hx : nthLe l k hk✝ ∈ l
n✝ : ℕ
hk : k < Nat.succ n✝
hn : length l = Nat.succ n✝
⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1
|
null
|
null
|
Mathlib/GroupTheory/Perm/List.lean
|
List.formPerm_apply_mem_eq_self_iff
|
[383, 1]
|
[394, 31]
|
cases' (Nat.le_of_lt_succ hk).eq_or_lt with hk' hk'
|
case intro.intro.succ
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk✝ : k < length l
hx : nthLe l k hk✝ ∈ l
n✝ : ℕ
hk : k < Nat.succ n✝
hn : length l = Nat.succ n✝
⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1
|
case intro.intro.succ.inl
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk✝ : k < length l
hx : nthLe l k hk✝ ∈ l
n✝ : ℕ
hk : k < Nat.succ n✝
hn : length l = Nat.succ n✝
hk' : k = n✝
⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1
case intro.intro.succ.inr
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk✝ : k < length l
hx : nthLe l k hk✝ ∈ l
n✝ : ℕ
hk : k < Nat.succ n✝
hn : length l = Nat.succ n✝
hk' : k < n✝
⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1
|
null
|
null
|
Mathlib/GroupTheory/Perm/List.lean
|
List.formPerm_apply_mem_eq_self_iff
|
[383, 1]
|
[394, 31]
|
simp [← hk', Nat.succ_le_succ_iff, eq_comm]
|
case intro.intro.succ.inl
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk✝ : k < length l
hx : nthLe l k hk✝ ∈ l
n✝ : ℕ
hk : k < Nat.succ n✝
hn : length l = Nat.succ n✝
hk' : k = n✝
⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1
|
no goals
|
null
|
null
|
Mathlib/GroupTheory/Perm/List.lean
|
List.formPerm_apply_mem_eq_self_iff
|
[383, 1]
|
[394, 31]
|
simpa [Nat.mod_eq_of_lt (Nat.succ_lt_succ hk'), Nat.succ_lt_succ_iff] using
k.zero_le.trans_lt hk'
|
case intro.intro.succ.inr
α : Type u_1
β : Type ?u.813892
inst✝ : DecidableEq α
l : List α
x : α
hl : Nodup l
k : ℕ
hk✝ : k < length l
hx : nthLe l k hk✝ ∈ l
n✝ : ℕ
hk : k < Nat.succ n✝
hn : length l = Nat.succ n✝
hk' : k < n✝
⊢ (k + 1) % Nat.succ n✝ = k ↔ Nat.succ n✝ ≤ 1
|
no goals
|
null
|
null
|
Mathlib/Analysis/SpecialFunctions/Exp.lean
|
Real.isLittleO_exp_comp_exp_comp
|
[383, 1]
|
[386, 55]
|
simp only [isLittleO_iff_tendsto, exp_ne_zero, ← exp_sub, ← tendsto_neg_atTop_iff, false_imp_iff,
imp_true_iff, tendsto_exp_comp_nhds_zero, neg_sub]
|
α : Type u_1
x y z : ℝ
l : Filter α
f g : α → ℝ
⊢ ((fun x => exp (f x)) =o[l] fun x => exp (g x)) ↔ Tendsto (fun x => g x - f x) l atTop
|
no goals
|
null
|
null
|
Mathlib/MeasureTheory/Function/Jacobian.lean
|
MeasureTheory.integral_image_eq_integral_abs_deriv_smul
|
[1251, 1]
|
[1256, 54]
|
simpa only [det_one_smulRight] using
integral_image_eq_integral_abs_det_fderiv_smul volume hs
(fun x hx => (hf' x hx).hasFDerivWithinAt) hf g
|
E : Type ?u.1055778
F : Type u_1
inst✝⁸ : NormedAddCommGroup E
inst✝⁷ : NormedSpace ℝ E
inst✝⁶ : FiniteDimensional ℝ E
inst✝⁵ : NormedAddCommGroup F
inst✝⁴ : NormedSpace ℝ F
s✝ : Set E
f✝ : E → E
f'✝ : E → E →L[ℝ] E
inst✝³ : MeasurableSpace E
inst✝² : BorelSpace E
μ : Measure E
inst✝¹ : IsAddHaarMeasure μ
s : Set ℝ
f f' : ℝ → ℝ
inst✝ : CompleteSpace F
hs : MeasurableSet s
hf' : ∀ (x : ℝ), x ∈ s → HasDerivWithinAt f (f' x) s x
hf : InjOn f s
g : ℝ → F
⊢ (∫ (x : ℝ) in f '' s, g x) = ∫ (x : ℝ) in s, abs (f' x) • g (f x)
|
no goals
|
null
|
null
|
Mathlib/Data/Num/Lemmas.lean
|
ZNum.add_one
|
[1170, 1]
|
[1173, 32]
|
cases p <;> rfl
|
α : Type ?u.723974
p : PosNum
⊢ neg p + 1 = succ (neg p)
|
no goals
|
null
|
null
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.map_toOuterMeasure
|
[1234, 1]
|
[1238, 63]
|
rw [← trimmed, OuterMeasure.trim_eq_trim_iff]
|
α : Type u_1
β : Type u_2
γ : Type ?u.213751
δ : Type ?u.213754
ι : Type ?u.213757
R : Type ?u.213760
R' : Type ?u.213763
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
hf : AEMeasurable f
⊢ ↑(map f μ) = OuterMeasure.trim (↑(OuterMeasure.map f) ↑μ)
|
α : Type u_1
β : Type u_2
γ : Type ?u.213751
δ : Type ?u.213754
ι : Type ?u.213757
R : Type ?u.213760
R' : Type ?u.213763
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
hf : AEMeasurable f
⊢ ∀ (s : Set β), MeasurableSet s → ↑↑(map f μ) s = ↑(↑(OuterMeasure.map f) ↑μ) s
|
null
|
null
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.map_toOuterMeasure
|
[1234, 1]
|
[1238, 63]
|
intro s hs
|
α : Type u_1
β : Type u_2
γ : Type ?u.213751
δ : Type ?u.213754
ι : Type ?u.213757
R : Type ?u.213760
R' : Type ?u.213763
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s s' t : Set α
f : α → β
hf : AEMeasurable f
⊢ ∀ (s : Set β), MeasurableSet s → ↑↑(map f μ) s = ↑(↑(OuterMeasure.map f) ↑μ) s
|
α : Type u_1
β : Type u_2
γ : Type ?u.213751
δ : Type ?u.213754
ι : Type ?u.213757
R : Type ?u.213760
R' : Type ?u.213763
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f : α → β
hf : AEMeasurable f
s : Set β
hs : MeasurableSet s
⊢ ↑↑(map f μ) s = ↑(↑(OuterMeasure.map f) ↑μ) s
|
null
|
null
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.map_toOuterMeasure
|
[1234, 1]
|
[1238, 63]
|
rw [map_apply_of_aemeasurable hf hs, OuterMeasure.map_apply]
|
α : Type u_1
β : Type u_2
γ : Type ?u.213751
δ : Type ?u.213754
ι : Type ?u.213757
R : Type ?u.213760
R' : Type ?u.213763
m0 : MeasurableSpace α
inst✝¹ : MeasurableSpace β
inst✝ : MeasurableSpace γ
μ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α
s✝ s' t : Set α
f : α → β
hf : AEMeasurable f
s : Set β
hs : MeasurableSet s
⊢ ↑↑(map f μ) s = ↑(↑(OuterMeasure.map f) ↑μ) s
|
no goals
|
null
|
null
|
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.isEquiv_iff_val_eq_one
|
[450, 1]
|
[483, 27]
|
constructor
|
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
⊢ IsEquiv v v' ↔ ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
|
case mp
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
⊢ IsEquiv v v' → ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
case mpr
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
⊢ (∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1) → IsEquiv v v'
|
null
|
null
|
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.isEquiv_iff_val_eq_one
|
[450, 1]
|
[483, 27]
|
intro h x
|
case mp
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
⊢ IsEquiv v v' → ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
|
case mp
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : IsEquiv v v'
x : K
⊢ ↑v x = 1 ↔ ↑v' x = 1
|
null
|
null
|
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.isEquiv_iff_val_eq_one
|
[450, 1]
|
[483, 27]
|
simpa using @IsEquiv.val_eq _ _ _ _ _ _ v v' h x 1
|
case mp
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : IsEquiv v v'
x : K
⊢ ↑v x = 1 ↔ ↑v' x = 1
|
no goals
|
null
|
null
|
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.isEquiv_iff_val_eq_one
|
[450, 1]
|
[483, 27]
|
intro h
|
case mpr
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
⊢ (∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1) → IsEquiv v v'
|
case mpr
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
⊢ IsEquiv v v'
|
null
|
null
|
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.isEquiv_iff_val_eq_one
|
[450, 1]
|
[483, 27]
|
apply isEquiv_of_val_le_one
|
case mpr
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
⊢ IsEquiv v v'
|
case mpr.h
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
⊢ ∀ {x : K}, ↑v x ≤ 1 ↔ ↑v' x ≤ 1
|
null
|
null
|
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.isEquiv_iff_val_eq_one
|
[450, 1]
|
[483, 27]
|
intro x
|
case mpr.h
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
⊢ ∀ {x : K}, ↑v x ≤ 1 ↔ ↑v' x ≤ 1
|
case mpr.h
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
⊢ ↑v x ≤ 1 ↔ ↑v' x ≤ 1
|
null
|
null
|
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.isEquiv_iff_val_eq_one
|
[450, 1]
|
[483, 27]
|
constructor
|
case mpr.h
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
⊢ ↑v x ≤ 1 ↔ ↑v' x ≤ 1
|
case mpr.h.mp
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
⊢ ↑v x ≤ 1 → ↑v' x ≤ 1
case mpr.h.mpr
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
⊢ ↑v' x ≤ 1 → ↑v x ≤ 1
|
null
|
null
|
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.isEquiv_iff_val_eq_one
|
[450, 1]
|
[483, 27]
|
intro hx
|
case mpr.h.mp
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
⊢ ↑v x ≤ 1 → ↑v' x ≤ 1
|
case mpr.h.mp
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
hx : ↑v x ≤ 1
⊢ ↑v' x ≤ 1
|
null
|
null
|
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.isEquiv_iff_val_eq_one
|
[450, 1]
|
[483, 27]
|
cases' lt_or_eq_of_le hx with hx' hx'
|
case mpr.h.mp
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
hx : ↑v x ≤ 1
⊢ ↑v' x ≤ 1
|
case mpr.h.mp.inl
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
hx : ↑v x ≤ 1
hx' : ↑v x < 1
⊢ ↑v' x ≤ 1
case mpr.h.mp.inr
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
hx : ↑v x ≤ 1
hx' : ↑v x = 1
⊢ ↑v' x ≤ 1
|
null
|
null
|
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.isEquiv_iff_val_eq_one
|
[450, 1]
|
[483, 27]
|
have : v (1 + x) = 1 := by
rw [← v.map_one]
apply map_add_eq_of_lt_left
simpa
|
case mpr.h.mp.inl
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
hx : ↑v x ≤ 1
hx' : ↑v x < 1
⊢ ↑v' x ≤ 1
|
case mpr.h.mp.inl
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
hx : ↑v x ≤ 1
hx' : ↑v x < 1
this : ↑v (1 + x) = 1
⊢ ↑v' x ≤ 1
|
null
|
null
|
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.isEquiv_iff_val_eq_one
|
[450, 1]
|
[483, 27]
|
rw [h] at this
|
case mpr.h.mp.inl
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
hx : ↑v x ≤ 1
hx' : ↑v x < 1
this : ↑v (1 + x) = 1
⊢ ↑v' x ≤ 1
|
case mpr.h.mp.inl
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
hx : ↑v x ≤ 1
hx' : ↑v x < 1
this : ↑v' (1 + x) = 1
⊢ ↑v' x ≤ 1
|
null
|
null
|
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.isEquiv_iff_val_eq_one
|
[450, 1]
|
[483, 27]
|
rw [show x = -1 + (1 + x) by simp]
|
case mpr.h.mp.inl
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
hx : ↑v x ≤ 1
hx' : ↑v x < 1
this : ↑v' (1 + x) = 1
⊢ ↑v' x ≤ 1
|
case mpr.h.mp.inl
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
hx : ↑v x ≤ 1
hx' : ↑v x < 1
this : ↑v' (1 + x) = 1
⊢ ↑v' (-1 + (1 + x)) ≤ 1
|
null
|
null
|
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.isEquiv_iff_val_eq_one
|
[450, 1]
|
[483, 27]
|
refine' le_trans (v'.map_add _ _) _
|
case mpr.h.mp.inl
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
hx : ↑v x ≤ 1
hx' : ↑v x < 1
this : ↑v' (1 + x) = 1
⊢ ↑v' (-1 + (1 + x)) ≤ 1
|
case mpr.h.mp.inl
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
hx : ↑v x ≤ 1
hx' : ↑v x < 1
this : ↑v' (1 + x) = 1
⊢ max (↑v' (-1)) (↑v' (1 + x)) ≤ 1
|
null
|
null
|
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.isEquiv_iff_val_eq_one
|
[450, 1]
|
[483, 27]
|
simp [this]
|
case mpr.h.mp.inl
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
hx : ↑v x ≤ 1
hx' : ↑v x < 1
this : ↑v' (1 + x) = 1
⊢ max (↑v' (-1)) (↑v' (1 + x)) ≤ 1
|
no goals
|
null
|
null
|
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.isEquiv_iff_val_eq_one
|
[450, 1]
|
[483, 27]
|
rw [← v.map_one]
|
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
hx : ↑v x ≤ 1
hx' : ↑v x < 1
⊢ ↑v (1 + x) = 1
|
K : Type u_3
F : Type ?u.3253919
R : Type ?u.3253922
inst✝³ : DivisionRing K
Γ₀ : Type u_1
Γ'₀ : Type u_2
Γ''₀ : Type ?u.3253934
inst✝² : LinearOrderedCommMonoidWithZero Γ''₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ'₀
v : Valuation K Γ₀
v' : Valuation K Γ'₀
h : ∀ {x : K}, ↑v x = 1 ↔ ↑v' x = 1
x : K
hx : ↑v x ≤ 1
hx' : ↑v x < 1
⊢ ↑v (1 + x) = ↑v 1
|
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