mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Perfection

Goal state before:
pp: case refine_1
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
hf : f ≠ 0
hg : g ≠ 0
m n : ℕ
hm : (Perfection.coeff (ModP O p) p (max m n)) f ≠ 0
hn : (Perfection.coeff (ModP O p) p (max m n)) g ≠ 0
⊢ (coeff (max m n + 1)) f ^ p ≠ 0

pp: case refine_2
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
hf : f ≠ 0
hg : g ≠ 0
m n : ℕ
hm : (Perfection.coeff (ModP O p) p (max m n)) f ≠ 0
hn : (Perfection.coeff (ModP O p) p (max m n)) g ≠ 0
⊢ (coeff (max m n + 1)) g ^ p ≠ 0

Tactic:
·

Goal state after:
case refine_1
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
hf : f ≠ 0
hg : g ≠ 0
m n : ℕ
hm : (Perfection.coeff (ModP O p) p (max m n)) f ≠ 0
hn : (Perfection.coeff (ModP O p) p (max m n)) g ≠ 0
⊢ (coeff (max m n + 1)) f ^ p ≠ 0
Module: Mathlib.RingTheory.WittVector.Teichmuller

Goal state before:
pp: p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
r : R
n : ℕ
⊢ (teichmullerFun p r).coeff 0 ^ p ^ n = r ^ p ^ n

pp: case h₀
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
r : R
n : ℕ
⊢ ∀ b ∈ Finset.range (n + 1), b ≠ 0 → ↑p ^ b * (teichmullerFun p r).coeff b ^ p ^ (n - b) = 0

pp: case h₁
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
r : R
n : ℕ
⊢ 0 ∉ Finset.range (n + 1) → ↑p ^ 0 * (teichmullerFun p r).coeff 0 ^ p ^ (n - 0) = 0

Tactic:
·

Goal state after:
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
r : R
n : ℕ
⊢ (teichmullerFun p r).coeff 0 ^ p ^ n = r ^ p ^ n
Module: Mathlib.RingTheory.Perfection

Goal state before:
pp: case refine_2
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
hf : f ≠ 0
hg : g ≠ 0
m n : ℕ
hm : (Perfection.coeff (ModP O p) p (max m n)) f ≠ 0
hn : (Perfection.coeff (ModP O p) p (max m n)) g ≠ 0
⊢ (coeff (max m n + 1)) g ^ p ≠ 0

Tactic:
·

Goal state after:
case refine_2
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
hf : f ≠ 0
hg : g ≠ 0
m n : ℕ
hm : (Perfection.coeff (ModP O p) p (max m n)) f ≠ 0
hn : (Perfection.coeff (ModP O p) p (max m n)) g ≠ 0
⊢ (coeff (max m n + 1)) g ^ p ≠ 0
Module: Mathlib.RingTheory.WittVector.Teichmuller

Goal state before:
pp: case h₀
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
r : R
n : ℕ
⊢ ∀ b ∈ Finset.range (n + 1), b ≠ 0 → ↑p ^ b * (teichmullerFun p r).coeff b ^ p ^ (n - b) = 0

pp: case h₁
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
r : R
n : ℕ
⊢ 0 ∉ Finset.range (n + 1) → ↑p ^ 0 * (teichmullerFun p r).coeff 0 ^ p ^ (n - 0) = 0

Tactic:
·

Goal state after:
case h₀
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
r : R
n : ℕ
⊢ ∀ b ∈ Finset.range (n + 1), b ≠ 0 → ↑p ^ b * (teichmullerFun p r).coeff b ^ p ^ (n - b) = 0
Module: Mathlib.RingTheory.WittVector.Truncated

Goal state before:
pp: case neg
p n : ℕ
R : Type u_1
inst✝ : CommRing R
x : 𝕎 R
i : ℕ
hi : ¬i < n
⊢ 0 = 0

pp: case pos
p n : ℕ
R : Type u_1
inst✝ : CommRing R
x : 𝕎 R
i : ℕ
hi : i < n
⊢ TruncatedWittVector.coeff ⟨i, hi⟩ (truncateFun n x) = x.coeff i

Tactic:
·

Goal state after:
case neg
p n : ℕ
R : Type u_1
inst✝ : CommRing R
x : 𝕎 R
i : ℕ
hi : ¬i < n
⊢ 0 = 0
Module: Mathlib.RingTheory.WittVector.Teichmuller

Goal state before:
pp: case h₁
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
r : R
n : ℕ
⊢ 0 ∉ Finset.range (n + 1) → ↑p ^ 0 * (teichmullerFun p r).coeff 0 ^ p ^ (n - 0) = 0

Tactic:
·

Goal state after:
case h₁
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
r : R
n : ℕ
⊢ 0 ∉ Finset.range (n + 1) → ↑p ^ 0 * (teichmullerFun p r).coeff 0 ^ p ^ (n - 0) = 0
Module: Mathlib.RingTheory.WittVector.Truncated

Goal state before:
pp: p n : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Fact (Nat.Prime p)
i : Fin n
⊢ WittVector.coeff 0 ↑i = 0

Tactic:
WittVector.zero_coeff

Goal state after:
p n : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Fact (Nat.Prime p)
i : Fin n
⊢ 0 = 0
Module: Mathlib.RingTheory.WittVector.InitTail

Goal state before:
pp: case pos
p : ℕ
P : ℕ → Prop
hp : Fact (Nat.Prime p)
this : IsPoly p fun {R} [CommRing R] x ↦ select P x + select (fun i ↦ ¬P i) x
R : Type u_1
R._inst : CommRing R
x : 𝕎 R
n m : ℕ
x✝ : m ∈ Finset.range (n + 1)
Pm : P m
⊢ (if P m then X m ^ p ^ (n - m) else 0) + (if ¬P m then X m else 0) ^ p ^ (n - m) = X m ^ p ^ (n - m)

pp: case neg
p : ℕ
P : ℕ → Prop
hp : Fact (Nat.Prime p)
this : IsPoly p fun {R} [CommRing R] x ↦ select P x + select (fun i ↦ ¬P i) x
R : Type u_1
R._inst : CommRing R
x : 𝕎 R
n m : ℕ
x✝ : m ∈ Finset.range (n + 1)
Pm : ¬P m
⊢ (if P m then X m ^ p ^ (n - m) else 0) + (if ¬P m then X m else 0) ^ p ^ (n - m) = X m ^ p ^ (n - m)

Tactic:
·

Goal state after:
case pos
p : ℕ
P : ℕ → Prop
hp : Fact (Nat.Prime p)
this : IsPoly p fun {R} [CommRing R] x ↦ select P x + select (fun i ↦ ¬P i) x
R : Type u_1
R._inst : CommRing R
x : 𝕎 R
n m : ℕ
x✝ : m ∈ Finset.range (n + 1)
Pm : P m
⊢ (if P m then X m ^ p ^ (n - m) else 0) + (if ¬P m then X m else 0) ^ p ^ (n - m) = X m ^ p ^ (n - m)
Module: Mathlib.RingTheory.WittVector.Teichmuller

Goal state before:
pp: case zero
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
r : R
⊢ ((map f) (teichmullerFun p r)).coeff 0 = (teichmullerFun p (f r)).coeff 0

pp: case succ
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
r : R
n✝ : ℕ
⊢ ((map f) (teichmullerFun p r)).coeff (n✝ + 1) = (teichmullerFun p (f r)).coeff (n✝ + 1)

Tactic:
·

Goal state after:
case zero
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
r : R
⊢ ((map f) (teichmullerFun p r)).coeff 0 = (teichmullerFun p (f r)).coeff 0
Module: Mathlib.RingTheory.WittVector.Teichmuller

Goal state before:
pp: case succ
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
r : R
n✝ : ℕ
⊢ ((map f) (teichmullerFun p r)).coeff (n✝ + 1) = (teichmullerFun p (f r)).coeff (n✝ + 1)

Tactic:
·

Goal state after:
case succ
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
r : R
n✝ : ℕ
⊢ ((map f) (teichmullerFun p r)).coeff (n✝ + 1) = (teichmullerFun p (f r)).coeff (n✝ + 1)
Module: Mathlib.RingTheory.WittVector.InitTail

Goal state before:
pp: case neg
p : ℕ
P : ℕ → Prop
hp : Fact (Nat.Prime p)
this : IsPoly p fun {R} [CommRing R] x ↦ select P x + select (fun i ↦ ¬P i) x
R : Type u_1
R._inst : CommRing R
x : 𝕎 R
n m : ℕ
x✝ : m ∈ Finset.range (n + 1)
Pm : ¬P m
⊢ (if P m then X m ^ p ^ (n - m) else 0) + (if ¬P m then X m else 0) ^ p ^ (n - m) = X m ^ p ^ (n - m)

Tactic:
·

Goal state after:
case neg
p : ℕ
P : ℕ → Prop
hp : Fact (Nat.Prime p)
this : IsPoly p fun {R} [CommRing R] x ↦ select P x + select (fun i ↦ ¬P i) x
R : Type u_1
R._inst : CommRing R
x : 𝕎 R
n m : ℕ
x✝ : m ∈ Finset.range (n + 1)
Pm : ¬P m
⊢ (if P m then X m ^ p ^ (n - m) else 0) + (if ¬P m then X m else 0) ^ p ^ (n - m) = X m ^ p ^ (n - m)
Module: Mathlib.RingTheory.WittVector.Teichmuller

Goal state before:
pp: case zero
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ (teichmullerFun p 1).coeff 0 = coeff 1 0

pp: case succ
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
n✝ : ℕ
⊢ (teichmullerFun p 1).coeff (n✝ + 1) = coeff 1 (n✝ + 1)

Tactic:
·

Goal state after:
case zero
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
⊢ (teichmullerFun p 1).coeff 0 = coeff 1 0
Module: Mathlib.RingTheory.WittVector.Teichmuller

Goal state before:
pp: case succ
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
n✝ : ℕ
⊢ (teichmullerFun p 1).coeff (n✝ + 1) = coeff 1 (n✝ + 1)

Tactic:
·

Goal state after:
case succ
p : ℕ
R : Type u_1
S : Type u_2
hp : Fact (Nat.Prime p)
inst✝¹ : CommRing R
inst✝ : CommRing S
n✝ : ℕ
⊢ (teichmullerFun p 1).coeff (n✝ + 1) = coeff 1 (n✝ + 1)
Module: Mathlib.RingTheory.WittVector.Teichmuller

Goal state before:
pp: case zero
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
⊢ ((teichmuller p) 0).coeff 0 = coeff 0 0

Tactic:
·

Goal state after:
case zero
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
⊢ ((teichmuller p) 0).coeff 0 = coeff 0 0
Module: Mathlib.RingTheory.WittVector.Teichmuller

Goal state before:
pp: case succ
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
n✝ : ℕ
⊢ ((teichmuller p) 0).coeff (n✝ + 1) = coeff 0 (n✝ + 1)

Tactic:
·

Goal state after:
case succ
p : ℕ
R : Type u_1
hp : Fact (Nat.Prime p)
inst✝ : CommRing R
n✝ : ℕ
⊢ ((teichmuller p) 0).coeff (n✝ + 1) = coeff 0 (n✝ + 1)
Module: Mathlib.RingTheory.WittVector.Truncated

Goal state before:
pp: p n : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Fact (Nat.Prime p)
m : ℕ
hm : n ≤ m
i : Fin n
x : TruncatedWittVector p m R
⊢ coeff i ((truncate hm) x) = coeff (Fin.castLE hm i) x

Tactic:
by
  obtain ⟨y, rfl⟩ := @WittVector.truncate_surjective p _ _ _ _ x
  simp only [truncate_wittVector_truncate, WittVector.coeff_truncate, Fin.val_castLE]

Goal state after:
Module: Mathlib.RingTheory.Perfection

Goal state before:
pp: case inl
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
g : PreTilt O p
⊢ valAux K v O p (0 + g) ≤ max (valAux K v O p 0) (valAux K v O p g)

pp: case inr
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
hf : f ≠ 0
⊢ valAux K v O p (f + g) ≤ max (valAux K v O p f) (valAux K v O p g)

Tactic:
·

Goal state after:
case inl
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
g : PreTilt O p
⊢ valAux K v O p (0 + g) ≤ max (valAux K v O p 0) (valAux K v O p g)
Module: Mathlib.RingTheory.WittVector.InitTail

Goal state before:
pp: case pos
p n✝ : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n ↦ y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n
n : ℕ
hn : P n
⊢ x.coeff n = x.coeff n

pp: case neg
p n✝ : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n ↦ y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n
n : ℕ
hn : ¬P n
⊢ 0 = x.coeff n

Tactic:
·

Goal state after:
case pos
p n✝ : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n ↦ y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n
n : ℕ
hn : P n
⊢ x.coeff n = x.coeff n
Module: Mathlib.RingTheory.Perfection

Goal state before:
pp: case inr.inl
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f : PreTilt O p
hf : f ≠ 0
⊢ valAux K v O p (f + 0) ≤ max (valAux K v O p f) (valAux K v O p 0)

pp: case inr.inr
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
hf : f ≠ 0
hg : g ≠ 0
⊢ valAux K v O p (f + g) ≤ max (valAux K v O p f) (valAux K v O p g)

Tactic:
·

Goal state after:
case inr.inl
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f : PreTilt O p
hf : f ≠ 0
⊢ valAux K v O p (f + 0) ≤ max (valAux K v O p f) (valAux K v O p 0)
Module: Mathlib.RingTheory.WittVector.InitTail

Goal state before:
pp: case neg
p n✝ : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n ↦ y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n
n : ℕ
hn : ¬P n
⊢ 0 = x.coeff n

Tactic:
·

Goal state after:
case neg
p n✝ : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n ↦ y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n
n : ℕ
hn : ¬P n
⊢ 0 = x.coeff n
Module: Mathlib.RingTheory.WittVector.Truncated

Goal state before:
pp: p : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : Fact (Nat.Prime p)
⊢ ⨅ i, RingHom.ker (WittVector.truncate i) = ⊥

Tactic:
by
  rw [Submodule.eq_bot_iff]
  intro x hx
  ext
  simp only [WittVector.mem_ker_truncate, Ideal.mem_iInf, WittVector.zero_coeff] at hx ⊢
  exact hx _ _ (Nat.lt_succ_self _)

Goal state after:
Module: Mathlib.RingTheory.Perfection

Goal state before:
pp: case pos
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
hf : f ≠ 0
hg : g ≠ 0
hfg : f + g = 0
⊢ valAux K v O p (f + g) ≤ max (valAux K v O p f) (valAux K v O p g)

pp: case neg
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
hf : f ≠ 0
hg : g ≠ 0
hfg : ¬f + g = 0
⊢ valAux K v O p (f + g) ≤ max (valAux K v O p f) (valAux K v O p g)

Tactic:
·

Goal state after:
case pos
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
hf : f ≠ 0
hg : g ≠ 0
hfg : f + g = 0
⊢ valAux K v O p (f + g) ≤ max (valAux K v O p f) (valAux K v O p g)
Module: Mathlib.RingTheory.WittVector.InitTail

Goal state before:
pp: case pos
p n✝ : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n ↦ y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n
hx : select P z = x
n : ℕ
hn : P n
⊢ 0 = y.coeff n

pp: case neg
p n✝ : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n ↦ y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n
hx : select P z = x
n : ℕ
hn : ¬P n
⊢ y.coeff n = y.coeff n

Tactic:
·

Goal state after:
case pos
p n✝ : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n ↦ y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n
hx : select P z = x
n : ℕ
hn : P n
⊢ 0 = y.coeff n
Module: Mathlib.RingTheory.WittVector.InitTail

Goal state before:
pp: case neg
p n✝ : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n ↦ y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n
hx : select P z = x
n : ℕ
hn : ¬P n
⊢ y.coeff n = y.coeff n

Tactic:
·

Goal state after:
case neg
p n✝ : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n ↦ y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n
hx : select P z = x
n : ℕ
hn : ¬P n
⊢ y.coeff n = y.coeff n
Module: Mathlib.RingTheory.WittVector.InitTail

Goal state before:
pp: case pos
p n : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n ↦ y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n
hx : select P z = x
hy : select (fun i ↦ ¬P i) z = y
y0 : P n
⊢ x.coeff n = x.coeff n + y.coeff n

pp: case neg
p n : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n ↦ y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n
hx : select P z = x
hy : select (fun i ↦ ¬P i) z = y
y0 : ¬P n
⊢ y.coeff n = x.coeff n + y.coeff n

Tactic:
·

Goal state after:
case pos
p n : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n ↦ y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n
hx : select P z = x
hy : select (fun i ↦ ¬P i) z = y
y0 : P n
⊢ x.coeff n = x.coeff n + y.coeff n
Module: Mathlib.RingTheory.WittVector.InitTail

Goal state before:
pp: case neg
p n : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n ↦ y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n
hx : select P z = x
hy : select (fun i ↦ ¬P i) z = y
y0 : ¬P n
⊢ y.coeff n = x.coeff n + y.coeff n

Tactic:
·

Goal state after:
case neg
p n : ℕ
R : Type u_1
inst✝ : CommRing R
hp : Fact (Nat.Prime p)
x y : 𝕎 R
h : ∀ (n : ℕ), x.coeff n = 0 ∨ y.coeff n = 0
P : ℕ → Prop := fun n ↦ y.coeff n = 0
this : DecidablePred P
z : 𝕎 R := mk p fun n ↦ if P n then x.coeff n else y.coeff n
hx : select P z = x
hy : select (fun i ↦ ¬P i) z = y
y0 : ¬P n
⊢ y.coeff n = x.coeff n + y.coeff n
Module: Mathlib.RingTheory.Perfection

Goal state before:
pp: case neg.inl
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
m n k : ℕ
hm : (Perfection.coeff (ModP O p) p (max (max m n) k)) f ≠ 0
hn : (Perfection.coeff (ModP O p) p (max (max m n) k)) g ≠ 0
hk : (Perfection.coeff (ModP O p) p (max (max m n) k)) (f + g) ≠ 0
h :
  ModP.preVal K v O p ((coeff (max (max m n) k)) f + (coeff (max (max m n) k)) g) ≤
    ModP.preVal K v O p ((coeff (max (max m n) k)) f)
⊢ ModP.preVal K v O p ((coeff (max (max m n) k)) f + (coeff (max (max m n) k)) g) ^ p ^ max (max m n) k ≤
    max (ModP.preVal K v O p ((coeff (max (max m n) k)) f) ^ p ^ max (max m n) k)
      (ModP.preVal K v O p ((coeff (max (max m n) k)) g) ^ p ^ max (max m n) k)

pp: case neg.inr
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
m n k : ℕ
hm : (Perfection.coeff (ModP O p) p (max (max m n) k)) f ≠ 0
hn : (Perfection.coeff (ModP O p) p (max (max m n) k)) g ≠ 0
hk : (Perfection.coeff (ModP O p) p (max (max m n) k)) (f + g) ≠ 0
h :
  ModP.preVal K v O p ((coeff (max (max m n) k)) f + (coeff (max (max m n) k)) g) ≤
    ModP.preVal K v O p ((coeff (max (max m n) k)) g)
⊢ ModP.preVal K v O p ((coeff (max (max m n) k)) f + (coeff (max (max m n) k)) g) ^ p ^ max (max m n) k ≤
    max (ModP.preVal K v O p ((coeff (max (max m n) k)) f) ^ p ^ max (max m n) k)
      (ModP.preVal K v O p ((coeff (max (max m n) k)) g) ^ p ^ max (max m n) k)

Tactic:
·

Goal state after:
case neg.inl
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
m n k : ℕ
hm : (Perfection.coeff (ModP O p) p (max (max m n) k)) f ≠ 0
hn : (Perfection.coeff (ModP O p) p (max (max m n) k)) g ≠ 0
hk : (Perfection.coeff (ModP O p) p (max (max m n) k)) (f + g) ≠ 0
h :
  ModP.preVal K v O p ((coeff (max (max m n) k)) f + (coeff (max (max m n) k)) g) ≤
    ModP.preVal K v O p ((coeff (max (max m n) k)) f)
⊢ ModP.preVal K v O p ((coeff (max (max m n) k)) f + (coeff (max (max m n) k)) g) ^ p ^ max (max m n) k ≤
    max (ModP.preVal K v O p ((coeff (max (max m n) k)) f) ^ p ^ max (max m n) k)
      (ModP.preVal K v O p ((coeff (max (max m n) k)) g) ^ p ^ max (max m n) k)
Module: Mathlib.RingTheory.WittVector.Complete

Goal state before:
pp: case refine_1
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
x y : 𝕎 k
n : ℕ
h : ∀ i < n, x.coeff i = y.coeff i
⊢ (truncate n) x = (truncate n) y

pp: case refine_2
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
x y : 𝕎 k
n : ℕ
h : (truncate n) x = (truncate n) y
i : ℕ
hi : i < n
⊢ x.coeff i = y.coeff i

Tactic:
·

Goal state after:
case refine_1
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
x y : 𝕎 k
n : ℕ
h : ∀ i < n, x.coeff i = y.coeff i
⊢ (truncate n) x = (truncate n) y
Module: Mathlib.RingTheory.WittVector.Complete

Goal state before:
pp: case refine_2
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
x y : 𝕎 k
n : ℕ
h : (truncate n) x = (truncate n) y
i : ℕ
hi : i < n
⊢ x.coeff i = y.coeff i

Tactic:
·

Goal state after:
case refine_2
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
x y : 𝕎 k
n : ℕ
h : (truncate n) x = (truncate n) y
i : ℕ
hi : i < n
⊢ x.coeff i = y.coeff i
Module: Mathlib.RingTheory.Perfection

Goal state before:
pp: case neg.inr
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
m n k : ℕ
hm : (Perfection.coeff (ModP O p) p (max (max m n) k)) f ≠ 0
hn : (Perfection.coeff (ModP O p) p (max (max m n) k)) g ≠ 0
hk : (Perfection.coeff (ModP O p) p (max (max m n) k)) (f + g) ≠ 0
h :
  ModP.preVal K v O p ((coeff (max (max m n) k)) f + (coeff (max (max m n) k)) g) ≤
    ModP.preVal K v O p ((coeff (max (max m n) k)) g)
⊢ ModP.preVal K v O p ((coeff (max (max m n) k)) f + (coeff (max (max m n) k)) g) ^ p ^ max (max m n) k ≤
    max (ModP.preVal K v O p ((coeff (max (max m n) k)) f) ^ p ^ max (max m n) k)
      (ModP.preVal K v O p ((coeff (max (max m n) k)) g) ^ p ^ max (max m n) k)

Tactic:
·

Goal state after:
case neg.inr
K : Type u₁
inst✝⁴ : Field K
v : Valuation K ℝ≥0
O : Type u₂
inst✝³ : CommRing O
inst✝² : Algebra O K
hv : v.Integers O
p : ℕ
inst✝¹ : Fact (Nat.Prime p)
inst✝ : Fact ¬IsUnit ↑p
f g : PreTilt O p
m n k : ℕ
hm : (Perfection.coeff (ModP O p) p (max (max m n) k)) f ≠ 0
hn : (Perfection.coeff (ModP O p) p (max (max m n) k)) g ≠ 0
hk : (Perfection.coeff (ModP O p) p (max (max m n) k)) (f + g) ≠ 0
h :
  ModP.preVal K v O p ((coeff (max (max m n) k)) f + (coeff (max (max m n) k)) g) ≤
    ModP.preVal K v O p ((coeff (max (max m n) k)) g)
⊢ ModP.preVal K v O p ((coeff (max (max m n) k)) f + (coeff (max (max m n) k)) g) ^ p ^ max (max m n) k ≤
    max (ModP.preVal K v O p ((coeff (max (max m n) k)) f) ^ p ^ max (max m n) k)
      (ModP.preVal K v O p ((coeff (max (max m n) k)) g) ^ p ^ max (max m n) k)