mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: m n : ℤ
h : m.gcd n = 1
hm : m % 2 = 0
hn : n % 2 = 1
H : ¬(m ^ 2 - n ^ 2).gcd (m ^ 2 + n ^ 2) = 1
p : ℕ
hp : Nat.Prime p
hp1 : ↑p ∣ m ^ 2 - n ^ 2
hp2 : ↑p ∣ m ^ 2 + n ^ 2
h2m : ↑p ∣ 2 * m ^ 2
h2n : ↑p ∣ 2 * n ^ 2
hmc : p = 2 ∨ p ∣ m.natAbs
hnc : p = 2 ∨ p ∣ n.natAbs
h2 : p = 2
⊢ (m ^ 2 + n ^ 2) % 2 = 1

Tactic:
simp only [sq, Int.add_emod, Int.mul_emod, hm, hn, dvd_refl, Int.emod_emod_of_dvd]

Goal state after:
m n : ℤ
h : m.gcd n = 1
hm : m % 2 = 0
hn : n % 2 = 1
H : ¬(m ^ 2 - n ^ 2).gcd (m ^ 2 + n ^ 2) = 1
p : ℕ
hp : Nat.Prime p
hp1 : ↑p ∣ m ^ 2 - n ^ 2
hp2 : ↑p ∣ m ^ 2 + n ^ 2
h2m : ↑p ∣ 2 * m ^ 2
h2n : ↑p ∣ 2 * n ^ 2
hmc : p = 2 ∨ p ∣ m.natAbs
hnc : p = 2 ∨ p ∣ n.natAbs
h2 : p = 2
⊢ (0 % 2 % 2 * (0 % 2 % 2) % 2 % 2 + 1 % 2 % 2 * (1 % 2 % 2) % 2 % 2) % 2 = 1
Module: Mathlib.NumberTheory.FLT.Four

Goal state before:
pp: a b c : ℤ
ha : a ≠ 0
⊢ b ≠ 0 → c ≠ 0 → a ^ 4 + b ^ 4 ≠ c ^ 4

Tactic:
hb

Goal state after:
a b c : ℤ
ha : a ≠ 0
hb : b ≠ 0
⊢ c ≠ 0 → a ^ 4 + b ^ 4 ≠ c ^ 4
Module: Mathlib.NumberTheory.FLT.MasonStothers

Goal state before:
pp: k : Type u_1
inst✝¹ : Field k
inst✝ : DecidableEq k
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
hsum : a + b + c = 0
w : k[X] := a.wronskian b
wab : w = a.wronskian b
hbc : IsCoprime b c
hsum' : -(b + c) = a
⊢ IsCoprime c a

Tactic:
← hsum',

Goal state after:
k : Type u_1
inst✝¹ : Field k
inst✝ : DecidableEq k
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
hsum : a + b + c = 0
w : k[X] := a.wronskian b
wab : w = a.wronskian b
hbc : IsCoprime b c
hsum' : -(b + c) = a
⊢ IsCoprime c (-(b + c))
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: x y z : ℤ
h : PythagoreanTriple x y z
hc : x.gcd y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
hQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0
hp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}
q : ℚ := (circleEquivGen hQ).symm ⟨(v, w), hp⟩
ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2)
m : ℤ := ↑q.den
n : ℤ := q.num
hm0 : m ≠ 0
hq2 : q = ↑n / ↑m
hm2n2 : 0 < m ^ 2 + n ^ 2
hm2n20 : ↑m ^ 2 + ↑n ^ 2 ≠ 0
hw2 : w = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
hv2 : v = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hnmcp : n.gcd m = 1
⊢ m.gcd n = 1

Tactic:
rw [Int.gcd_comm]

Goal state after:
x y z : ℤ
h : PythagoreanTriple x y z
hc : x.gcd y = 1
hyo : y % 2 = 1
hzpos : 0 < z
h0 : ¬x = 0
v : ℚ := ↑x / ↑z
w : ℚ := ↑y / ↑z
hq : v ^ 2 + w ^ 2 = 1
hvz : v ≠ 0
hw1 : w ≠ -1
hQ : ∀ (x : ℚ), 1 + x ^ 2 ≠ 0
hp : (v, w) ∈ {p | p.1 ^ 2 + p.2 ^ 2 = 1 ∧ p.2 ≠ -1}
q : ℚ := (circleEquivGen hQ).symm ⟨(v, w), hp⟩
ht4 : v = 2 * q / (1 + q ^ 2) ∧ w = (1 - q ^ 2) / (1 + q ^ 2)
m : ℤ := ↑q.den
n : ℤ := q.num
hm0 : m ≠ 0
hq2 : q = ↑n / ↑m
hm2n2 : 0 < m ^ 2 + n ^ 2
hm2n20 : ↑m ^ 2 + ↑n ^ 2 ≠ 0
hw2 : w = (↑m ^ 2 - ↑n ^ 2) / (↑m ^ 2 + ↑n ^ 2)
hv2 : v = 2 * ↑m * ↑n / (↑m ^ 2 + ↑n ^ 2)
hnmcp : n.gcd m = 1
⊢ n.gcd m = 1
Module: Mathlib.RingTheory.FractionalIdeal.Norm

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : Free ℤ R
inst✝³ : Module.Finite ℤ R
K : Type u_2
inst✝² : CommRing K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
I : FractionalIdeal R⁰ K
a : ↥R⁰
I₀ : Ideal R
h : a • I.den • ↑I = I.den • Submodule.map (Algebra.linearMap R K) I₀
h' : Ideal.span {↑I.den} * I₀ = Ideal.span {↑a} * I.num
⊢ ↑(Ideal.absNorm I.num * Ideal.absNorm (Ideal.span {↑a})) = ↑(Ideal.absNorm I₀ * Ideal.absNorm (Ideal.span {↑I.den}))

Tactic:
← map_mul,

Goal state after:
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : Free ℤ R
inst✝³ : Module.Finite ℤ R
K : Type u_2
inst✝² : CommRing K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
I : FractionalIdeal R⁰ K
a : ↥R⁰
I₀ : Ideal R
h : a • I.den • ↑I = I.den • Submodule.map (Algebra.linearMap R K) I₀
h' : Ideal.span {↑I.den} * I₀ = Ideal.span {↑a} * I.num
⊢ ↑(Ideal.absNorm (I.num * Ideal.span {↑a})) = ↑(Ideal.absNorm I₀ * Ideal.absNorm (Ideal.span {↑I.den}))
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: case inl
k : Type u_1
inst✝ : Field k
p q r : ℕ
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
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
hbc : IsCoprime b c
hca : IsCoprime c a
hCu : C u ≠ 0
hCv : C v ≠ 0
hCw : C w ≠ 0
hap : a ^ p ≠ 0
hbq : b ^ q ≠ 0
hcr : c ^ r ≠ 0
habp : IsCoprime (C u * a ^ p) (C v * b ^ q)
hbcp : IsCoprime (C v * b ^ q) (C w * c ^ r)
hcap : IsCoprime (C w * c ^ r) (C u * a ^ p)
habcp : IsCoprime (C u * a ^ p * (C v * b ^ q)) (C w * c ^ r)
nd_lt :
  (C u * a ^ p).natDegree + 1 ≤ (radical a * radical (b ^ q) * radical (c ^ r)).natDegree ∧
    (C v * b ^ q).natDegree + 1 ≤ (radical a * radical (b ^ q) * radical (c ^ r)).natDegree ∧
      (C w * c ^ r).natDegree + 1 ≤ (radical a * radical (b ^ q) * radical (c ^ r)).natDegree
⊢ derivative a = 0 ∧ derivative b = 0 ∧ derivative c = 0

Tactic:
radical_pow b hq,

Goal state after:
case inl
k : Type u_1
inst✝ : Field k
p q r : ℕ
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
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
hbc : IsCoprime b c
hca : IsCoprime c a
hCu : C u ≠ 0
hCv : C v ≠ 0
hCw : C w ≠ 0
hap : a ^ p ≠ 0
hbq : b ^ q ≠ 0
hcr : c ^ r ≠ 0
habp : IsCoprime (C u * a ^ p) (C v * b ^ q)
hbcp : IsCoprime (C v * b ^ q) (C w * c ^ r)
hcap : IsCoprime (C w * c ^ r) (C u * a ^ p)
habcp : IsCoprime (C u * a ^ p * (C v * b ^ q)) (C w * c ^ r)
nd_lt :
  (C u * a ^ p).natDegree + 1 ≤ (radical a * radical b * radical (c ^ r)).natDegree ∧
    (C v * b ^ q).natDegree + 1 ≤ (radical a * radical b * radical (c ^ r)).natDegree ∧
      (C w * c ^ r).natDegree + 1 ≤ (radical a * radical b * radical (c ^ r)).natDegree
⊢ derivative a = 0 ∧ derivative b = 0 ∧ derivative c = 0
Module: Mathlib.RingTheory.FractionalIdeal.Norm

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : Free ℤ R
inst✝³ : Module.Finite ℤ R
K : Type u_2
inst✝² : CommRing K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
I : FractionalIdeal R⁰ K
a : ↥R⁰
I₀ : Ideal R
h : a • I.den • ↑I = I.den • Submodule.map (Algebra.linearMap R K) I₀
h' : Ideal.span {↑I.den} * I₀ = Ideal.span {↑a} * I.num
⊢ ↑(Ideal.absNorm (I.num * Ideal.span {↑a})) = ↑(Ideal.absNorm I₀ * Ideal.absNorm (Ideal.span {↑I.den}))

Tactic:
← map_mul,

Goal state after:
R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDedekindDomain R
inst✝⁴ : Free ℤ R
inst✝³ : Module.Finite ℤ R
K : Type u_2
inst✝² : CommRing K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
I : FractionalIdeal R⁰ K
a : ↥R⁰
I₀ : Ideal R
h : a • I.den • ↑I = I.den • Submodule.map (Algebra.linearMap R K) I₀
h' : Ideal.span {↑I.den} * I₀ = Ideal.span {↑a} * I.num
⊢ ↑(Ideal.absNorm (I.num * Ideal.span {↑a})) = ↑(Ideal.absNorm (I₀ * Ideal.span {↑I.den}))
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: case inl
k : Type u_1
inst✝ : Field k
p q r : ℕ
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
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
hbc : IsCoprime b c
hca : IsCoprime c a
hCu : C u ≠ 0
hCv : C v ≠ 0
hCw : C w ≠ 0
hap : a ^ p ≠ 0
hbq : b ^ q ≠ 0
hcr : c ^ r ≠ 0
habp : IsCoprime (C u * a ^ p) (C v * b ^ q)
hbcp : IsCoprime (C v * b ^ q) (C w * c ^ r)
hcap : IsCoprime (C w * c ^ r) (C u * a ^ p)
habcp : IsCoprime (C u * a ^ p * (C v * b ^ q)) (C w * c ^ r)
nd_lt :
  (C u * a ^ p).natDegree + 1 ≤ (radical a * radical b * radical c).natDegree ∧
    (C v * b ^ q).natDegree + 1 ≤ (radical a * radical b * radical c).natDegree ∧
      (C w * c ^ r).natDegree + 1 ≤ (radical a * radical b * radical c).natDegree
⊢ derivative a = 0 ∧ derivative b = 0 ∧ derivative c = 0

Tactic:
natDegree_mul hCu hap,

Goal state after:
case inl
k : Type u_1
inst✝ : Field k
p q r : ℕ
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
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
u v w : k
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
heq : C u * a ^ p + C v * b ^ q + C w * c ^ r = 0
hbc : IsCoprime b c
hca : IsCoprime c a
hCu : C u ≠ 0
hCv : C v ≠ 0
hCw : C w ≠ 0
hap : a ^ p ≠ 0
hbq : b ^ q ≠ 0
hcr : c ^ r ≠ 0
habp : IsCoprime (C u * a ^ p) (C v * b ^ q)
hbcp : IsCoprime (C v * b ^ q) (C w * c ^ r)
hcap : IsCoprime (C w * c ^ r) (C u * a ^ p)
habcp : IsCoprime (C u * a ^ p * (C v * b ^ q)) (C w * c ^ r)
nd_lt :
  (C u).natDegree + (a ^ p).natDegree + 1 ≤ (radical a * radical b * radical c).natDegree ∧
    (C v * b ^ q).natDegree + 1 ≤ (radical a * radical b * radical c).natDegree ∧
      (C w * c ^ r).natDegree + 1 ≤ (radical a * radical b * radical c).natDegree
⊢ derivative a = 0 ∧ derivative b = 0 ∧ derivative c = 0
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case mpr.inr.inl.right
m n : ℤ
co : m.gcd n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
⊢ (2 * m * n).gcd (m ^ 2 - n ^ 2) = 1

Tactic:
rw [Int.gcd_comm]

Goal state after:
case mpr.inr.inl.right
m n : ℤ
co : m.gcd n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
⊢ (m ^ 2 - n ^ 2).gcd (2 * m * n) = 1
Module: Mathlib.NumberTheory.PythagoreanTriples

Goal state before:
pp: case mpr.inr.inr.right
m n : ℤ
co : m.gcd n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
⊢ (2 * m * n).gcd (m ^ 2 - n ^ 2) = 1

Tactic:
rw [Int.gcd_comm]

Goal state after:
case mpr.inr.inr.right
m n : ℤ
co : m.gcd n = 1
pp : m % 2 = 0 ∧ n % 2 = 1 ∨ m % 2 = 1 ∧ n % 2 = 0
⊢ (m ^ 2 - n ^ 2).gcd (2 * m * n) = 1
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Basic

Goal state before:
pp: case e'_2.refine_3
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
w : InfinitePlace K
hw : w.IsComplex
x✝ : { x // mk x = w }
φ : K →+* ℂ
hφ : mk φ = w
⊢ mkComplex ⟨φ, ⋯⟩ = ⟨w, hw⟩

Tactic:
rwa [Subtype.ext_iff, mkComplex_coe]

Goal state after:
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: k : Type u_1
inst✝ : Field k
p q r : ℕ
u v w : k
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
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
d : ℕ
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
eq_d : a.natDegree = d + 1
ad : derivative a = 0
bd : derivative b = 0
cd : derivative c = 0
ca : k[X]
ca_nz : ca ≠ 0
cb : k[X]
cb_nz : cb ≠ 0
cc : k[X]
cc_nz : cc ≠ 0
ch : ℕ := ringChar k
chn0 : ch ≠ 0
eq_a : a = (expand k ch) ca
eq_deg_a : a.natDegree = ca.natDegree * ch
eq_b : b = (expand k ch) cb
eq_deg_b : b.natDegree = cb.natDegree * ch
eq_c : c = (expand k ch) cc
eq_deg_c : c.natDegree = cc.natDegree * ch
hnca : ¬ca.natDegree = 0
heq :
  (expand k ch) (C u) * (expand k ch) (ca ^ p) + (expand k ch) (C v) * (expand k ch) (cb ^ q) +
      (expand k ch) (C w) * (expand k ch) (cc ^ r) =
    0
⊢ C u * ca ^ p + C v * cb ^ q + C w * cc ^ r = 0

Tactic:
← map_mul,

Goal state after:
k : Type u_1
inst✝ : Field k
p q r : ℕ
u v w : k
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
hu : u ≠ 0
hv : v ≠ 0
hw : w ≠ 0
d : ℕ
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
hc : c ≠ 0
hab : IsCoprime a b
eq_d : a.natDegree = d + 1
ad : derivative a = 0
bd : derivative b = 0
cd : derivative c = 0
ca : k[X]
ca_nz : ca ≠ 0
cb : k[X]
cb_nz : cb ≠ 0
cc : k[X]
cc_nz : cc ≠ 0
ch : ℕ := ringChar k
chn0 : ch ≠ 0
eq_a : a = (expand k ch) ca
eq_deg_a : a.natDegree = ca.natDegree * ch
eq_b : b = (expand k ch) cb
eq_deg_b : b.natDegree = cb.natDegree * ch
eq_c : c = (expand k ch) cc
eq_deg_c : c.natDegree = cc.natDegree * ch
hnca : ¬ca.natDegree = 0
heq : (expand k ch) (C u * ca ^ p) + (expand k ch) (C v * cb ^ q) + (expand k ch) (C w * cc ^ r) = 0
⊢ C u * ca ^ p + C v * cb ^ q + C w * cc ^ r = 0
Module: Mathlib.NumberTheory.FLT.Polynomial

Goal state before:
pp: k : Type u_1
inst✝ : Field k
n : ℕ
hn : 3 ≤ n
chn : ↑n ≠ 0
a b c : k[X]
ha : a ≠ 0
⊢ b ≠ 0 →
    c ≠ 0 →
      a ^ n + b ^ n = c ^ n → ∃ d a' b' c', (a = a' * d ∧ b = b' * d ∧ c = c' * d) ∧ IsUnit a' ∧ IsUnit b' ∧ IsUnit c'

Tactic:
hb

Goal state after:
k : Type u_1
inst✝ : Field k
n : ℕ
hn : 3 ≤ n
chn : ↑n ≠ 0
a b c : k[X]
ha : a ≠ 0
hb : b ≠ 0
⊢ c ≠ 0 →
    a ^ n + b ^ n = c ^ n → ∃ d a' b' c', (a = a' * d ∧ b = b' * d ∧ c = c' * d) ∧ IsUnit a' ∧ IsUnit b' ∧ IsUnit c'
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
w₀ : { w // w.IsComplex }
inst✝ : NumberField K
B : ℝ≥0
⊢ ENNReal.ofReal 2 * (ENNReal.ofReal 2 * ENNReal.ofReal (↑B ^ 2)) = 4 * ↑B ^ 2

Tactic:
ofReal_pow (coe_nonneg B),

Goal state after:
K : Type u_1
inst✝¹ : Field K
f : InfinitePlace K → ℝ≥0
w₀ : { w // w.IsComplex }
inst✝ : NumberField K
B : ℝ≥0
⊢ ENNReal.ofReal 2 * (ENNReal.ofReal 2 * ENNReal.ofReal ↑B ^ 2) = 4 * ↑B ^ 2
Module: Mathlib.NumberTheory.NumberField.InfinitePlace.Ramification

Goal state before:
pp: case pos
k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra k K
w₁ w₂ : InfinitePlace K
h : w₂.IsComplex ∧ (w₂.comap (algebraMap k K)).IsReal
h_eq : w₁ = w₂
⊢ w₁.embedding ≠ conjugate w₂.embedding

Tactic:
isComplex_iff

Goal state after:
case pos
k : Type u_1
inst✝² : Field k
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra k K
w₁ w₂ : InfinitePlace K
h : ¬ComplexEmbedding.IsReal w₂.embedding ∧ (w₂.comap (algebraMap k K)).IsReal
h_eq : w₁ = w₂
⊢ w₁.embedding ≠ conjugate w₂.embedding
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: case refine_1
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
w : InfinitePlace K
x✝ : w ∈ Finset.univ
⊢ (normAtPlace w) x ≤ ↑w.mult * (normAtPlace w) x

pp: case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
w : InfinitePlace K
x✝ : w ∈ Finset.univ
⊢ 0 ≤ ∑ x_1 ∈ Finset.univ.erase w, ↑x_1.mult * (normAtPlace x_1) x

Tactic:
· exact le_mul_of_one_le_left (normAtPlace_nonneg w x) one_le_mult

Goal state after:
case refine_2
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
x : mixedSpace K
w : InfinitePlace K
x✝ : w ∈ Finset.univ
⊢ 0 ≤ ∑ x_1 ∈ Finset.univ.erase w, ↑x_1.mult * (normAtPlace x_1) x
Module: Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
B : ℝ
hB : 0 < B
this : volume (convexBodySum K 1) = ↑(convexBodySumFactor K)
⊢ volume (convexBodySum K B) = ↑(convexBodySumFactor K) * ENNReal.ofReal B ^ finrank ℚ K

Tactic:
by
      rw [mul_comm]
      convert! addHaar_smul volume B (convexBodySum K 1)
      · simp_rw [← Set.preimage_smul_inv₀ (ne_of_gt hB), Set.preimage_ofPred_eq, convexBodySumFun,
        normAtPlace_smul, abs_inv, abs_eq_self.mpr (le_of_lt hB), ← mul_assoc, mul_comm, mul_assoc,
        ← Finset.mul_sum, inv_mul_le_iff₀ hB, mul_one]
      · rw [abs_pow, ofReal_pow (abs_nonneg _), abs_eq_self.mpr (le_of_lt hB),
          mixedEmbedding.finrank]
      · exact this.symm

Goal state after:
Module: Mathlib.RingTheory.IntegralClosure.IntegralRestrict

Goal state before:
pp: A : Type u_1
B : Type u_6
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
inst✝⁶ : Algebra A B
inst✝⁵ : IsDomain A
inst✝⁴ : IsIntegrallyClosed A
inst✝³ : IsDomain B
inst✝² : IsIntegrallyClosed B
inst✝¹ : Module.Finite A B
inst✝ : IsTorsionFree A B
x : B
this✝² : IsIntegralClosure B A (FractionRing B)
this✝¹ : Algebra.IsAlgebraic (FractionRing A) (FractionRing B)
this✝ : IsLocalization (algebraMapSubmonoid B A⁰) (FractionRing B)
this : FiniteDimensional (FractionRing A) (FractionRing B)
⊢ (algebraMap A (FractionRing A)) ((intTrace A B) x) =
    (trace (FractionRing A) (FractionRing B)) ((algebraMap B (FractionRing B)) x)

Tactic:
exact Algebra.map_intTraceAux x

Goal state after:
Module: Mathlib.RingTheory.IntegralClosure.IntegralRestrict

Goal state before:
pp: case neg
A : Type u_1
B : Type u_6
inst✝²⁴ : CommRing A
inst✝²³ : CommRing B
inst✝²² : Algebra A B
Aₘ : Type u_9
Bₘ : Type u_10
inst✝²¹ : CommRing Aₘ
inst✝²⁰ : CommRing Bₘ
inst✝¹⁹ : Algebra Aₘ Bₘ
inst✝¹⁸ : Algebra A Aₘ
inst✝¹⁷ : Algebra B Bₘ
inst✝¹⁶ : Algebra A Bₘ
inst✝¹⁵ : IsScalarTower A Aₘ Bₘ
inst✝¹⁴ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹³ : IsLocalization M Aₘ
inst✝¹² : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsIntegrallyClosed A
inst✝⁹ : IsDomain B
inst✝⁸ : IsIntegrallyClosed B
inst✝⁷ : Module.Finite A B
inst✝⁶ : IsTorsionFree A B
inst✝⁵ : IsDomain Aₘ
inst✝⁴ : IsIntegrallyClosed Aₘ
inst✝³ : IsDomain Bₘ
inst✝² : IsIntegrallyClosed Bₘ
inst✝¹ : IsTorsionFree Aₘ Bₘ
inst✝ : Module.Finite Aₘ Bₘ
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_6 := FractionRing B
this✝¹ : IsIntegralClosure B A L
this✝ : Algebra.IsAlgebraic (FractionRing A) (FractionRing B)
this : IsLocalization (algebraMapSubmonoid B A⁰) L
⊢ (algebraMap A Aₘ) ((intTrace A B) x) = (intTrace Aₘ Bₘ) ((algebraMap B Bₘ) x)

Tactic:
let f : Aₘ →+* K := IsLocalization.map _ (T := A⁰) (RingHom.id A) hM

Goal state after:
case neg
A : Type u_1
B : Type u_6
inst✝²⁴ : CommRing A
inst✝²³ : CommRing B
inst✝²² : Algebra A B
Aₘ : Type u_9
Bₘ : Type u_10
inst✝²¹ : CommRing Aₘ
inst✝²⁰ : CommRing Bₘ
inst✝¹⁹ : Algebra Aₘ Bₘ
inst✝¹⁸ : Algebra A Aₘ
inst✝¹⁷ : Algebra B Bₘ
inst✝¹⁶ : Algebra A Bₘ
inst✝¹⁵ : IsScalarTower A Aₘ Bₘ
inst✝¹⁴ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹³ : IsLocalization M Aₘ
inst✝¹² : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsIntegrallyClosed A
inst✝⁹ : IsDomain B
inst✝⁸ : IsIntegrallyClosed B
inst✝⁷ : Module.Finite A B
inst✝⁶ : IsTorsionFree A B
inst✝⁵ : IsDomain Aₘ
inst✝⁴ : IsIntegrallyClosed Aₘ
inst✝³ : IsDomain Bₘ
inst✝² : IsIntegrallyClosed Bₘ
inst✝¹ : IsTorsionFree Aₘ Bₘ
inst✝ : Module.Finite Aₘ Bₘ
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_6 := FractionRing B
this✝¹ : IsIntegralClosure B A L
this✝ : Algebra.IsAlgebraic (FractionRing A) (FractionRing B)
this : IsLocalization (algebraMapSubmonoid B A⁰) L
f : Aₘ →+* K := IsLocalization.map K (RingHom.id A) hM
⊢ (algebraMap A Aₘ) ((intTrace A B) x) = (intTrace Aₘ Bₘ) ((algebraMap B Bₘ) x)
Module: Mathlib.NumberTheory.NumberField.Discriminant.Basic

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : 1 < finrank ℚ K
a : ℕ → ℝ := fun n ↦ ↑n ^ (n * 2) / ((4 / π) ^ n * ↑n.factorial ^ 2)
⊢ 4 / 9 * (3 * π / 4) ^ finrank ℚ K ≤
    ↑(finrank ℚ K) ^ (finrank ℚ K * 2) / ((4 / π) ^ (2 * nrComplexPlaces K) * ↑(finrank ℚ K).factorial ^ 2)

Tactic:
suffices ∀ n, 2 ≤ n → (4 / 9 : ℝ) * (3 * π / 4) ^ n ≤ a n by
    refine le_trans (this (finrank ℚ K) h) ?_
    simp only [a]
    gcongr
    · exact (one_le_div Real.pi_pos).2 Real.pi_le_four
    · rw [← card_add_two_mul_card_eq_rank, mul_comm]
      exact Nat.le_add_left _ _

Goal state after:
K : Type u_1
inst✝¹ : Field K
inst✝ : NumberField K
h : 1 < finrank ℚ K
a : ℕ → ℝ := fun n ↦ ↑n ^ (n * 2) / ((4 / π) ^ n * ↑n.factorial ^ 2)
⊢ ∀ (n : ℕ), 2 ≤ n → 4 / 9 * (3 * π / 4) ^ n ≤ a n
Module: Mathlib.RingTheory.IntegralClosure.IntegralRestrict

Goal state before:
pp: case neg
A : Type u_1
B : Type u_6
inst✝²⁵ : CommRing A
inst✝²⁴ : CommRing B
inst✝²³ : Algebra A B
Aₘ : Type u_9
Bₘ : Type u_10
inst✝²² : CommRing Aₘ
inst✝²¹ : CommRing Bₘ
inst✝²⁰ : Algebra Aₘ Bₘ
inst✝¹⁹ : Algebra A Aₘ
inst✝¹⁸ : Algebra B Bₘ
inst✝¹⁷ : Algebra A Bₘ
inst✝¹⁶ : IsScalarTower A Aₘ Bₘ
inst✝¹⁵ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁴ : IsLocalization M Aₘ
inst✝¹³ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsDomain B
inst✝⁹ : IsIntegrallyClosed B
inst✝⁸ : Algebra.IsIntegral A B
inst✝⁷ : IsTorsionFree A B
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : IsTorsionFree Aₘ Bₘ
inst✝¹ : Algebra.IsIntegral Aₘ Bₘ
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_6 := FractionRing B
⊢ (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)

Tactic:
let f : Aₘ →+* K := IsLocalization.map _ (T := A⁰) (RingHom.id A) hM

Goal state after:
case neg
A : Type u_1
B : Type u_6
inst✝²⁵ : CommRing A
inst✝²⁴ : CommRing B
inst✝²³ : Algebra A B
Aₘ : Type u_9
Bₘ : Type u_10
inst✝²² : CommRing Aₘ
inst✝²¹ : CommRing Bₘ
inst✝²⁰ : Algebra Aₘ Bₘ
inst✝¹⁹ : Algebra A Aₘ
inst✝¹⁸ : Algebra B Bₘ
inst✝¹⁷ : Algebra A Bₘ
inst✝¹⁶ : IsScalarTower A Aₘ Bₘ
inst✝¹⁵ : IsScalarTower A B Bₘ
M : Submonoid A
inst✝¹⁴ : IsLocalization M Aₘ
inst✝¹³ : IsLocalization (algebraMapSubmonoid B M) Bₘ
inst✝¹² : IsIntegrallyClosed A
inst✝¹¹ : IsDomain A
inst✝¹⁰ : IsDomain B
inst✝⁹ : IsIntegrallyClosed B
inst✝⁸ : Algebra.IsIntegral A B
inst✝⁷ : IsTorsionFree A B
inst✝⁶ : IsDomain Aₘ
inst✝⁵ : IsIntegrallyClosed Aₘ
inst✝⁴ : IsDomain Bₘ
inst✝³ : IsIntegrallyClosed Bₘ
inst✝² : IsTorsionFree Aₘ Bₘ
inst✝¹ : Algebra.IsIntegral Aₘ Bₘ
inst✝ : FiniteDimensional (FractionRing A) (FractionRing B)
x : B
hM : M ≤ A⁰
K : Type u_1 := FractionRing A
L : Type u_6 := FractionRing B
f : Aₘ →+* K := IsLocalization.map K (RingHom.id A) hM
⊢ (algebraMap A Aₘ) ((intNorm A B) x) = (intNorm Aₘ Bₘ) ((algebraMap B Bₘ) x)
Module: Mathlib.RingTheory.Ideal.Quotient.HasFiniteQuotients

Goal state before:
pp: case inr
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : HasFiniteQuotients R
inst✝² : IsDomain R
inst✝¹ : PerfectField (FractionRing R)
P : Ideal R
inst✝ : P.IsPrime
hP : P ≠ ⊥
⊢ PerfectField P.ResidueField

Tactic:
· have : Finite (R ⧸ P) := Ring.HasFiniteQuotients.finiteQuotient hP
    infer_instance

Goal state after:
Module: Mathlib.RingTheory.FractionalIdeal.Extended

Goal state before:
pp: case neg
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁸ : CommRing A
inst✝¹⁷ : IsDomain A
inst✝¹⁶ : CommRing B
inst✝¹⁵ : IsDomain B
inst✝¹⁴ : Algebra A B
inst✝¹³ : IsTorsionFree A B
inst✝¹² : Field K
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra A K
inst✝⁹ : Algebra B L
inst✝⁸ : IsFractionRing A K
inst✝⁷ : IsFractionRing B L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra A L
inst✝⁴ : IsScalarTower A B L
inst✝³ : IsScalarTower A K L
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsDedekindDomain A
inst✝ : IsDedekindDomain B
I J : FractionalIdeal A⁰ K
h : (extendedHom L B) I * (extendedHom L B) J⁻¹ = 1
hI : ¬I = 0
hJ : ¬J = 0
⊢ I = J

Tactic:
← map_mul,

Goal state after:
case neg
A : Type u_1
K : Type u_2
L : Type u_3
B : Type u_4
inst✝¹⁸ : CommRing A
inst✝¹⁷ : IsDomain A
inst✝¹⁶ : CommRing B
inst✝¹⁵ : IsDomain B
inst✝¹⁴ : Algebra A B
inst✝¹³ : IsTorsionFree A B
inst✝¹² : Field K
inst✝¹¹ : Field L
inst✝¹⁰ : Algebra A K
inst✝⁹ : Algebra B L
inst✝⁸ : IsFractionRing A K
inst✝⁷ : IsFractionRing B L
inst✝⁶ : Algebra K L
inst✝⁵ : Algebra A L
inst✝⁴ : IsScalarTower A B L
inst✝³ : IsScalarTower A K L
inst✝² : Algebra.IsIntegral A B
inst✝¹ : IsDedekindDomain A
inst✝ : IsDedekindDomain B
I J : FractionalIdeal A⁰ K
h : (extendedHom L B) (I * J⁻¹) = 1
hI : ¬I = 0
hJ : ¬J = 0
⊢ I = J
Module: Mathlib.RingTheory.DedekindDomain.PID

Goal state before:
pp: R : Type u_2
A : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
I : (FractionalIdeal S A)ˣ
v : A
hv : v ∈ ↑I⁻¹
h : Submodule.comap (Algebra.linearMap R A) (↑↑I * (R ∙ v)) = ⊤
hinv : ↑I * ↑I⁻¹ = 1
⊢ (↑↑I).IsPrincipal

Tactic:
set J := Submodule.comap (Algebra.linearMap R A) ((I : Submodule R A) * Submodule.span R {v})

Goal state after:
R : Type u_2
A : Type u_3
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
S : Submonoid R
inst✝ : IsLocalization S A
I : (FractionalIdeal S A)ˣ
v : A
hv : v ∈ ↑I⁻¹
hinv : ↑I * ↑I⁻¹ = 1
J : Submodule R R := Submodule.comap (Algebra.linearMap R A) (↑↑I * (R ∙ v))
h : J = ⊤
⊢ (↑↑I).IsPrincipal