mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Instances.Complex

Goal state before:
pp: case refine_1
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : IsTopologicalDivisionRing ℂ :=
  { toIsTopologicalRing := NormedDivisionRing.to_isTopologicalDivisionRing.toIsTopologicalRing,
    toContinuousInv₀ := NormedDivisionRing.to_isTopologicalDivisionRing.toContinuousInv₀ }
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
⊢ DenseRange ι

Tactic:
·

Goal state after:
case refine_1
K : Subfield ℂ
ψ : ↥K →+* ℂ
hc : UniformContinuous ⇑ψ
this✝ : IsTopologicalDivisionRing ℂ :=
  { toIsTopologicalRing := NormedDivisionRing.to_isTopologicalDivisionRing.toIsTopologicalRing,
    toContinuousInv₀ := NormedDivisionRing.to_isTopologicalDivisionRing.toContinuousInv₀ }
this : IsTopologicalRing ↥K.topologicalClosure := Subring.instIsTopologicalRing K.topologicalClosure.toSubring
ι : ↥K → ↥K.topologicalClosure := ⇑(Subfield.inclusion ⋯)
ui : IsUniformInducing ι
⊢ DenseRange ι
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case mpr
x y z : ℤ
⊢ (∃ m n,
      (x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2) ∧
        (z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)) ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0)) →
    PythagoreanTriple x y z ∧ x.gcd y = 1

Tactic:
·

Goal state after:
case mpr
x y z : ℤ
⊢ (∃ m n,
      (x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2) ∧
        (z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)) ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0)) →
    PythagoreanTriple x y z ∧ x.gcd y = 1
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: case refine_1
k : Type u_1
inst✝ : Field k
a : k[X]
ha : a ≠ 0
hda : derivative a = 0
chn0 : ringChar k ≠ 0
heq : a = (expand k (ringChar k)) (contract (ringChar k) a)
⊢ contract (ringChar k) a ≠ 0

pp: case refine_2
k : Type u_1
inst✝ : Field k
a : k[X]
ha : a ≠ 0
hda : derivative a = 0
chn0 : ringChar k ≠ 0
heq : a = (expand k (ringChar k)) (contract (ringChar k) a)
⊢ a.natDegree = (contract (ringChar k) a).natDegree * ringChar k

Tactic:
·

Goal state after:
case refine_1
k : Type u_1
inst✝ : Field k
a : k[X]
ha : a ≠ 0
hda : derivative a = 0
chn0 : ringChar k ≠ 0
heq : a = (expand k (ringChar k)) (contract (ringChar k) a)
⊢ contract (ringChar k) a ≠ 0
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: case refine_2
k : Type u_1
inst✝ : Field k
a : k[X]
ha : a ≠ 0
hda : derivative a = 0
chn0 : ringChar k ≠ 0
heq : a = (expand k (ringChar k)) (contract (ringChar k) a)
⊢ a.natDegree = (contract (ringChar k) a).natDegree * ringChar k

Tactic:
·

Goal state after:
case refine_2
k : Type u_1
inst✝ : Field k
a : k[X]
ha : a ≠ 0
hda : derivative a = 0
chn0 : ringChar k ≠ 0
heq : a = (expand k (ringChar k)) (contract (ringChar k) a)
⊢ a.natDegree = (contract (ringChar k) a).natDegree * ringChar k
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: case inl
k : Type u_1
inst✝ : Field k
p q r : ℕ
a b c : k[X]
u v w : k
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
ch0 : ringChar k = 0
⊢ a.natDegree = 0

pp: case inr
k : Type u_1
inst✝ : Field k
p q r : ℕ
a b c : k[X]
u v w : k
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
chn0 : ringChar k ≠ 0
⊢ a.natDegree = 0

Tactic:
·

Goal state after:
case inl
k : Type u_1
inst✝ : Field k
p q r : ℕ
a b c : k[X]
u v w : k
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
ch0 : ringChar k = 0
⊢ a.natDegree = 0
Module: Mathlib.RingTheory.FractionalIdeal.Norm

Goal state before:
pp: R : Type u_1
inst✝⁹ : CommRing R
inst✝⁸ : IsDedekindDomain R
inst✝⁷ : Free ℤ R
inst✝⁶ : Module.Finite ℤ R
K : Type u_2
inst✝⁵ : CommRing K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid R ℤ⁰) K
inst✝¹ : Algebra ℚ K
inst✝ : Module.Finite ℚ K
x : K
this : IsDomain K
d : ↥R⁰
r : R
hr : (algebraMap R K) r = ↑d • x
⊢ ↑(Ideal.absNorm (Ideal.span {r})) / ↑|(Algebra.norm ℤ) ↑d| = |(Algebra.norm ℚ) x|

pp: R : Type u_1
inst✝⁹ : CommRing R
inst✝⁸ : IsDedekindDomain R
inst✝⁷ : Free ℤ R
inst✝⁶ : Module.Finite ℤ R
K : Type u_2
inst✝⁵ : CommRing K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid R ℤ⁰) K
inst✝¹ : Algebra ℚ K
inst✝ : Module.Finite ℚ K
x : K
this : IsDomain K
d : ↥R⁰
r : R
hr : (algebraMap R K) r = ↑d • x
⊢ d • ↑(spanSingleton R⁰ x) = Submodule.map (Algebra.linearMap R K) (Ideal.span {r})

Tactic:
·

Goal state after:
R : Type u_1
inst✝⁹ : CommRing R
inst✝⁸ : IsDedekindDomain R
inst✝⁷ : Free ℤ R
inst✝⁶ : Module.Finite ℤ R
K : Type u_2
inst✝⁵ : CommRing K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid R ℤ⁰) K
inst✝¹ : Algebra ℚ K
inst✝ : Module.Finite ℚ K
x : K
this : IsDomain K
d : ↥R⁰
r : R
hr : (algebraMap R K) r = ↑d • x
⊢ ↑(Ideal.absNorm (Ideal.span {r})) / ↑|(Algebra.norm ℤ) ↑d| = |(Algebra.norm ℚ) x|
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: case inr
k : Type u_1
inst✝ : Field k
p q r : ℕ
a b c : k[X]
u v w : k
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
chn0 : ringChar k ≠ 0
⊢ a.natDegree = 0

Tactic:
·

Goal state after:
case inr
k : Type u_1
inst✝ : Field k
p q r : ℕ
a b c : k[X]
u v w : k
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
hp : p ≠ 0
hq : q ≠ 0
hr : r ≠ 0
hineq : q * r + r * p + p * q ≤ p * q * r
chp : ↑p ≠ 0
chq : ↑q ≠ 0
chr : ↑r ≠ 0
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
chn0 : ringChar k ≠ 0
⊢ a.natDegree = 0
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: case mp
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
⊢ mk φ = mk ψ → φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ

pp: case mpr
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
⊢ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ → mk φ = mk ψ

Tactic:
·

Goal state after:
case mp
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
⊢ mk φ = mk ψ → φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case inl
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht1 : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
⊢ ∃ m n,
    x = m ^ 2 - n ^ 2 ∧
      y = 2 * m * n ∧ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m

pp: case inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht1 : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
⊢ ∃ m n,
    x = m ^ 2 - n ^ 2 ∧
      y = 2 * m * n ∧ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m

Tactic:
·

Goal state after:
case inl
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht1 : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
⊢ ∃ m n,
    x = m ^ 2 - n ^ 2 ∧
      y = 2 * m * n ∧ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case h.inl
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ x = m ^ 2 - n ^ 2 ∧
    y = 2 * m * n ∧ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m

pp: case h.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_even : x = 2 * m * n ∧ y = m ^ 2 - n ^ 2
⊢ x = m ^ 2 - n ^ 2 ∧
    y = 2 * m * n ∧ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m

Tactic:
·

Goal state after:
case h.inl
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ x = m ^ 2 - n ^ 2 ∧
    y = 2 * m * n ∧ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case h.inl
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ y = 2 * m * n ∧ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m

Tactic:
apply And.intro h_odd.2

Goal state after:
case h.inl
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case h.inl.inl
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos✝ : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_pos : z = m ^ 2 + n ^ 2
⊢ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m

pp: case h.inl.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_neg : z = -(m ^ 2 + n ^ 2)
⊢ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m

Tactic:
·

Goal state after:
case h.inl.inl
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos✝ : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_pos : z = m ^ 2 + n ^ 2
⊢ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case h.inl.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_neg : z = -(m ^ 2 + n ^ 2)
⊢ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m

Tactic:
·

Goal state after:
case h.inl.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : 0 ≤ m
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_neg : z = -(m ^ 2 + n ^ 2)
⊢ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m
Module: Mathlib.RingTheory.FractionalIdeal.Norm

Goal state before:
pp: R : Type u_1
inst✝⁹ : CommRing R
inst✝⁸ : IsDedekindDomain R
inst✝⁷ : Free ℤ R
inst✝⁶ : Module.Finite ℤ R
K : Type u_2
inst✝⁵ : CommRing K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid R ℤ⁰) K
inst✝¹ : Algebra ℚ K
inst✝ : Module.Finite ℚ K
x : K
this : IsDomain K
d : ↥R⁰
r : R
hr : (algebraMap R K) r = ↑d • x
⊢ d • ↑(spanSingleton R⁰ x) = Submodule.map (Algebra.linearMap R K) (Ideal.span {r})

Tactic:
·

Goal state after:
R : Type u_1
inst✝⁹ : CommRing R
inst✝⁸ : IsDedekindDomain R
inst✝⁷ : Free ℤ R
inst✝⁶ : Module.Finite ℤ R
K : Type u_2
inst✝⁵ : CommRing K
inst✝⁴ : Algebra R K
inst✝³ : IsFractionRing R K
inst✝² : IsLocalization (Algebra.algebraMapSubmonoid R ℤ⁰) K
inst✝¹ : Algebra ℚ K
inst✝ : Module.Finite ℚ K
x : K
this : IsDomain K
d : ↥R⁰
r : R
hr : (algebraMap R K) r = ↑d • x
⊢ d • ↑(spanSingleton R⁰ x) = Submodule.map (Algebra.linearMap R K) (Ideal.span {r})
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht1 : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
⊢ ∃ m n,
    x = m ^ 2 - n ^ 2 ∧
      y = 2 * m * n ∧ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m

Tactic:
·

Goal state after:
case inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht1 : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n ∨ x = 2 * m * n ∧ y = m ^ 2 - n ^ 2
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
⊢ ∃ m n,
    x = m ^ 2 - n ^ 2 ∧
      y = 2 * m * n ∧ z = m ^ 2 + n ^ 2 ∧ m.gcd n = 1 ∧ (m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0) ∧ 0 ≤ m
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case h.inl
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ x = (-m) ^ 2 - (-n) ^ 2 ∧
    y = 2 * -m * -n ∧
      z = (-m) ^ 2 + (-n) ^ 2 ∧ (-m).gcd (-n) = 1 ∧ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m

pp: case h.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_even : x = 2 * m * n ∧ y = m ^ 2 - n ^ 2
⊢ x = (-m) ^ 2 - (-n) ^ 2 ∧
    y = 2 * -m * -n ∧
      z = (-m) ^ 2 + (-n) ^ 2 ∧ (-m).gcd (-n) = 1 ∧ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m

Tactic:
·

Goal state after:
case h.inl
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ x = (-m) ^ 2 - (-n) ^ 2 ∧
    y = 2 * -m * -n ∧
      z = (-m) ^ 2 + (-n) ^ 2 ∧ (-m).gcd (-n) = 1 ∧ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: case mpr
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
⊢ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ → mk φ = mk ψ

Tactic:
·

Goal state after:
case mpr
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
⊢ φ = ψ ∨ ComplexEmbedding.conjugate φ = ψ → mk φ = mk ψ
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: case mpr.inl
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h : φ = ψ
⊢ mk φ = mk ψ

pp: case mpr.inr
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h : ComplexEmbedding.conjugate φ = ψ
⊢ mk φ = mk ψ

Tactic:
·

Goal state after:
case mpr.inl
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h : φ = ψ
⊢ mk φ = mk ψ
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: case mpr.inr
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h : ComplexEmbedding.conjugate φ = ψ
⊢ mk φ = mk ψ

Tactic:
·

Goal state after:
case mpr.inr
K : Type u_1
inst✝ : Field K
φ ψ : K →+* ℂ
h : ComplexEmbedding.conjugate φ = ψ
⊢ mk φ = mk ψ
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case h.inl.left
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ y = 2 * -m * -n

pp: case h.inl.right
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ z = m ^ 2 + n ^ 2 ∧ (-m).gcd (-n) = 1 ∧ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m

Tactic:
·

Goal state after:
case h.inl.left
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht2 : z = m ^ 2 + n ^ 2 ∨ z = -(m ^ 2 + n ^ 2)
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
⊢ y = 2 * -m * -n
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case h.inl.right.inl
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos✝ : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_pos : z = m ^ 2 + n ^ 2
⊢ z = m ^ 2 + n ^ 2 ∧ (-m).gcd (-n) = 1 ∧ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m

pp: case h.inl.right.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_neg : z = -(m ^ 2 + n ^ 2)
⊢ z = m ^ 2 + n ^ 2 ∧ (-m).gcd (-n) = 1 ∧ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m

Tactic:
·

Goal state after:
case h.inl.right.inl
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos✝ : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_pos : z = m ^ 2 + n ^ 2
⊢ z = m ^ 2 + n ^ 2 ∧ (-m).gcd (-n) = 1 ∧ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case h.inl.right.inl.left
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos✝ : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_pos : z = m ^ 2 + n ^ 2
⊢ (-m).gcd (-n) = 1

pp: case h.inl.right.inl.right
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos✝ : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_pos : z = m ^ 2 + n ^ 2
⊢ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m

Tactic:
·

Goal state after:
case h.inl.right.inl.left
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos✝ : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_pos : z = m ^ 2 + n ^ 2
⊢ (-m).gcd (-n) = 1
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case h.inl.right.inl.right
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos✝ : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_pos : z = m ^ 2 + n ^ 2
⊢ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m

Tactic:
·

Goal state after:
case h.inl.right.inl.right
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos✝ : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_pos : z = m ^ 2 + n ^ 2
⊢ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: case pos
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : w.IsReal
⊢ #({w.embedding} ∪ {ComplexEmbedding.conjugate w.embedding}) = 1

pp: case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : ¬w.IsReal
⊢ #({w.embedding} ∪ {ComplexEmbedding.conjugate w.embedding}) = 2

Tactic:
·

Goal state after:
case pos
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : w.IsReal
⊢ #({w.embedding} ∪ {ComplexEmbedding.conjugate w.embedding}) = 1
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case h.inl.right.inl.right
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos✝ : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_pos : z = m ^ 2 + n ^ 2
⊢ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m

Tactic:
rw [Int.neg_emod_two, Int.neg_emod_two]
          apply And.intro ht4
          lia

Goal state after:
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : ¬w.IsReal
⊢ #({w.embedding} ∪ {ComplexEmbedding.conjugate w.embedding}) = 2

Tactic:
·

Goal state after:
case neg
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : ¬w.IsReal
⊢ #({w.embedding} ∪ {ComplexEmbedding.conjugate w.embedding}) = 2
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case h.inl.right.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_neg : z = -(m ^ 2 + n ^ 2)
⊢ z = m ^ 2 + n ^ 2 ∧ (-m).gcd (-n) = 1 ∧ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m

Tactic:
·

Goal state after:
case h.inl.right.inr
x y z : ℤ
h : PythagoreanTriple x y z
h_coprime : x.gcd y = 1
h_parity : x % 2 = 1
h_pos : 0 < z
m n : ℤ
ht3 : m.gcd n = 1
ht4 : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
hm : m < 0
h_odd : x = m ^ 2 - n ^ 2 ∧ y = 2 * m * n
h_neg : z = -(m ^ 2 + n ^ 2)
⊢ z = m ^ 2 + n ^ 2 ∧ (-m).gcd (-n) = 1 ∧ (-m % 2 = 0 ∧ -n % 2 = 1 ∨ -m % 2 = 1 ∧ -n % 2 = 0) ∧ 0 ≤ -m