Module: Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity

Goal state before:
pp: case succ.refine_1
R : Type u_2
inst✝ : CommRing R
n✝ n : ℕ
IH :
  ∀ {M : Submodule ℤ R},
    1 ∈ M →
      ∀ (k : ℕ) (d : Multiset (Fin n)) (S : ConstructibleSetData (MvPolynomial (Fin n) R)),
        (∀ C ∈ S, C.n ≤ k) →
          (∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) d)) →
            ∃ T,
              comap MvPolynomial.C '' S.toSet = T.toSet ∧
                ∀ C ∈ T,
                  C.n ≤ numBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n ∧
                    ∀ (i : Fin C.n), C.g i ∈ M ^ degBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n
M : Submodule ℤ R
hM : 1 ∈ M
k : ℕ
d : Multiset (Fin (n + 1))
S : ConstructibleSetData (MvPolynomial (Fin (n + 1)) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS :
  ∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin (n + 1)) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin (n + 1)) d)
e : MvPolynomial (Fin (n + 1)) R ≃ₐ[R] (MvPolynomial (Fin n) R)[X] := MvPolynomial.finSuccEquiv R n
S' : ConstructibleSetData (MvPolynomial (Fin n) R)[X] := ConstructibleSetData.map e.toRingEquiv.toRingHom S
hS' : S'.degBound ≤ k * (1 + Multiset.count 0 d)
B : Multiset (Fin n) := Finsupp.toMultiset (Finsupp.comapDomain Fin.succ (Multiset.toFinsupp d) ⋯)
T : ConstructibleSetData (MvPolynomial (Fin n) R)
hT₁ : comap Polynomial.C '' S'.toSet = T.toSet
hT₂ :
  ∀ C ∈ T,
    C.n ≤ S'.degBound ∧
      ∀ (i : Fin C.n),
        C.g i ∈ (coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) B)) ^ S'.degBound ^ S'.degBound
N : ℕ := (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d))
this✝ :
  ∀ C ∈ T, ∀ (a : Fin C.n), C.g a ∈ coeffsIn (Fin n) (M ^ N) ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) (N • B))
h1M : 1 ≤ M
U : ConstructibleSetData R
hU₁ : comap MvPolynomial.C '' T.toSet = U.toSet
hU₂ :
  ∀ C ∈ U,
    C.n ≤
        numBound S'.degBound
          (fun i ↦
            1 +
              (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
                Multiset.count i (Multiset.map Fin.val B))
          n ∧
      ∀ (i : Fin C.n),
        C.g i ∈
          M ^
            ((k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
              degBound S'.degBound
                (fun i ↦
                  1 +
                    (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
                      Multiset.count i (Multiset.map Fin.val B))
                n)
this :
  ∀ i < n + 1,
    (Nat.casesOn i (1 + Multiset.count 0 d) fun x ↦ 1 + Multiset.count x (Multiset.map Fin.val B)) ≤
      1 + Multiset.count i (Multiset.map Fin.val d)
⊢ comap MvPolynomial.C '' S.toSet = U.toSet

pp: case succ.refine_2
R : Type u_2
inst✝ : CommRing R
n✝ n : ℕ
IH :
  ∀ {M : Submodule ℤ R},
    1 ∈ M →
      ∀ (k : ℕ) (d : Multiset (Fin n)) (S : ConstructibleSetData (MvPolynomial (Fin n) R)),
        (∀ C ∈ S, C.n ≤ k) →
          (∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) d)) →
            ∃ T,
              comap MvPolynomial.C '' S.toSet = T.toSet ∧
                ∀ C ∈ T,
                  C.n ≤ numBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n ∧
                    ∀ (i : Fin C.n), C.g i ∈ M ^ degBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n
M : Submodule ℤ R
hM : 1 ∈ M
k : ℕ
d : Multiset (Fin (n + 1))
S : ConstructibleSetData (MvPolynomial (Fin (n + 1)) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS :
  ∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin (n + 1)) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin (n + 1)) d)
e : MvPolynomial (Fin (n + 1)) R ≃ₐ[R] (MvPolynomial (Fin n) R)[X] := MvPolynomial.finSuccEquiv R n
S' : ConstructibleSetData (MvPolynomial (Fin n) R)[X] := ConstructibleSetData.map e.toRingEquiv.toRingHom S
hS' : S'.degBound ≤ k * (1 + Multiset.count 0 d)
B : Multiset (Fin n) := Finsupp.toMultiset (Finsupp.comapDomain Fin.succ (Multiset.toFinsupp d) ⋯)
T : ConstructibleSetData (MvPolynomial (Fin n) R)
hT₁ : comap Polynomial.C '' S'.toSet = T.toSet
hT₂ :
  ∀ C ∈ T,
    C.n ≤ S'.degBound ∧
      ∀ (i : Fin C.n),
        C.g i ∈ (coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) B)) ^ S'.degBound ^ S'.degBound
N : ℕ := (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d))
this✝ :
  ∀ C ∈ T, ∀ (a : Fin C.n), C.g a ∈ coeffsIn (Fin n) (M ^ N) ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) (N • B))
h1M : 1 ≤ M
U : ConstructibleSetData R
hU₁ : comap MvPolynomial.C '' T.toSet = U.toSet
hU₂ :
  ∀ C ∈ U,
    C.n ≤
        numBound S'.degBound
          (fun i ↦
            1 +
              (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
                Multiset.count i (Multiset.map Fin.val B))
          n ∧
      ∀ (i : Fin C.n),
        C.g i ∈
          M ^
            ((k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
              degBound S'.degBound
                (fun i ↦
                  1 +
                    (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
                      Multiset.count i (Multiset.map Fin.val B))
                n)
this :
  ∀ i < n + 1,
    (Nat.casesOn i (1 + Multiset.count 0 d) fun x ↦ 1 + Multiset.count x (Multiset.map Fin.val B)) ≤
      1 + Multiset.count i (Multiset.map Fin.val d)
C : BasicConstructibleSetData R
hCU : C ∈ U
⊢ numBound S'.degBound
      (fun i ↦
        1 + (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) * Multiset.count i (Multiset.map Fin.val B))
      n ≤
    numBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) (n + 1)

pp: case succ.refine_3
R : Type u_2
inst✝ : CommRing R
n✝ n : ℕ
IH :
  ∀ {M : Submodule ℤ R},
    1 ∈ M →
      ∀ (k : ℕ) (d : Multiset (Fin n)) (S : ConstructibleSetData (MvPolynomial (Fin n) R)),
        (∀ C ∈ S, C.n ≤ k) →
          (∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) d)) →
            ∃ T,
              comap MvPolynomial.C '' S.toSet = T.toSet ∧
                ∀ C ∈ T,
                  C.n ≤ numBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n ∧
                    ∀ (i : Fin C.n), C.g i ∈ M ^ degBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n
M : Submodule ℤ R
hM : 1 ∈ M
k : ℕ
d : Multiset (Fin (n + 1))
S : ConstructibleSetData (MvPolynomial (Fin (n + 1)) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS :
  ∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin (n + 1)) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin (n + 1)) d)
e : MvPolynomial (Fin (n + 1)) R ≃ₐ[R] (MvPolynomial (Fin n) R)[X] := MvPolynomial.finSuccEquiv R n
S' : ConstructibleSetData (MvPolynomial (Fin n) R)[X] := ConstructibleSetData.map e.toRingEquiv.toRingHom S
hS' : S'.degBound ≤ k * (1 + Multiset.count 0 d)
B : Multiset (Fin n) := Finsupp.toMultiset (Finsupp.comapDomain Fin.succ (Multiset.toFinsupp d) ⋯)
T : ConstructibleSetData (MvPolynomial (Fin n) R)
hT₁ : comap Polynomial.C '' S'.toSet = T.toSet
hT₂ :
  ∀ C ∈ T,
    C.n ≤ S'.degBound ∧
      ∀ (i : Fin C.n),
        C.g i ∈ (coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) B)) ^ S'.degBound ^ S'.degBound
N : ℕ := (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d))
this✝ :
  ∀ C ∈ T, ∀ (a : Fin C.n), C.g a ∈ coeffsIn (Fin n) (M ^ N) ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) (N • B))
h1M : 1 ≤ M
U : ConstructibleSetData R
hU₁ : comap MvPolynomial.C '' T.toSet = U.toSet
hU₂ :
  ∀ C ∈ U,
    C.n ≤
        numBound S'.degBound
          (fun i ↦
            1 +
              (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
                Multiset.count i (Multiset.map Fin.val B))
          n ∧
      ∀ (i : Fin C.n),
        C.g i ∈
          M ^
            ((k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
              degBound S'.degBound
                (fun i ↦
                  1 +
                    (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
                      Multiset.count i (Multiset.map Fin.val B))
                n)
this :
  ∀ i < n + 1,
    (Nat.casesOn i (1 + Multiset.count 0 d) fun x ↦ 1 + Multiset.count x (Multiset.map Fin.val B)) ≤
      1 + Multiset.count i (Multiset.map Fin.val d)
C : BasicConstructibleSetData R
hCU : C ∈ U
i : Fin C.n
⊢ (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
      degBound S'.degBound
        (fun i ↦
          1 +
            (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) * Multiset.count i (Multiset.map Fin.val B))
        n ≤
    degBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) (n + 1)

Tactic:
· unfold S' at hT₁
    rw [← hU₁, ← hT₁, ← Set.image_comp, ← comap_comp,
      ConstructibleSetData.toSet_map]
    change _ = _ '' ((comapEquiv e.toRingEquiv).symm ⁻¹' _)
    rw [← OrderIso.image_eq_preimage_symm, Set.image_image]
    simp only [comapEquiv_apply, ← comap_comp_apply]
    congr!
    exact e.symm.toAlgHom.comp_algebraMap.symm

Goal state after:
case succ.refine_2
R : Type u_2
inst✝ : CommRing R
n✝ n : ℕ
IH :
  ∀ {M : Submodule ℤ R},
    1 ∈ M →
      ∀ (k : ℕ) (d : Multiset (Fin n)) (S : ConstructibleSetData (MvPolynomial (Fin n) R)),
        (∀ C ∈ S, C.n ≤ k) →
          (∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) d)) →
            ∃ T,
              comap MvPolynomial.C '' S.toSet = T.toSet ∧
                ∀ C ∈ T,
                  C.n ≤ numBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n ∧
                    ∀ (i : Fin C.n), C.g i ∈ M ^ degBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n
M : Submodule ℤ R
hM : 1 ∈ M
k : ℕ
d : Multiset (Fin (n + 1))
S : ConstructibleSetData (MvPolynomial (Fin (n + 1)) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS :
  ∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin (n + 1)) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin (n + 1)) d)
e : MvPolynomial (Fin (n + 1)) R ≃ₐ[R] (MvPolynomial (Fin n) R)[X] := MvPolynomial.finSuccEquiv R n
S' : ConstructibleSetData (MvPolynomial (Fin n) R)[X] := ConstructibleSetData.map e.toRingEquiv.toRingHom S
hS' : S'.degBound ≤ k * (1 + Multiset.count 0 d)
B : Multiset (Fin n) := Finsupp.toMultiset (Finsupp.comapDomain Fin.succ (Multiset.toFinsupp d) ⋯)
T : ConstructibleSetData (MvPolynomial (Fin n) R)
hT₁ : comap Polynomial.C '' S'.toSet = T.toSet
hT₂ :
  ∀ C ∈ T,
    C.n ≤ S'.degBound ∧
      ∀ (i : Fin C.n),
        C.g i ∈ (coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) B)) ^ S'.degBound ^ S'.degBound
N : ℕ := (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d))
this✝ :
  ∀ C ∈ T, ∀ (a : Fin C.n), C.g a ∈ coeffsIn (Fin n) (M ^ N) ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) (N • B))
h1M : 1 ≤ M
U : ConstructibleSetData R
hU₁ : comap MvPolynomial.C '' T.toSet = U.toSet
hU₂ :
  ∀ C ∈ U,
    C.n ≤
        numBound S'.degBound
          (fun i ↦
            1 +
              (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
                Multiset.count i (Multiset.map Fin.val B))
          n ∧
      ∀ (i : Fin C.n),
        C.g i ∈
          M ^
            ((k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
              degBound S'.degBound
                (fun i ↦
                  1 +
                    (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
                      Multiset.count i (Multiset.map Fin.val B))
                n)
this :
  ∀ i < n + 1,
    (Nat.casesOn i (1 + Multiset.count 0 d) fun x ↦ 1 + Multiset.count x (Multiset.map Fin.val B)) ≤
      1 + Multiset.count i (Multiset.map Fin.val d)
C : BasicConstructibleSetData R
hCU : C ∈ U
⊢ numBound S'.degBound
      (fun i ↦
        1 + (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) * Multiset.count i (Multiset.map Fin.val B))
      n ≤
    numBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) (n + 1)

case succ.refine_3
R : Type u_2
inst✝ : CommRing R
n✝ n : ℕ
IH :
  ∀ {M : Submodule ℤ R},
    1 ∈ M →
      ∀ (k : ℕ) (d : Multiset (Fin n)) (S : ConstructibleSetData (MvPolynomial (Fin n) R)),
        (∀ C ∈ S, C.n ≤ k) →
          (∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) d)) →
            ∃ T,
              comap MvPolynomial.C '' S.toSet = T.toSet ∧
                ∀ C ∈ T,
                  C.n ≤ numBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n ∧
                    ∀ (i : Fin C.n), C.g i ∈ M ^ degBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) n
M : Submodule ℤ R
hM : 1 ∈ M
k : ℕ
d : Multiset (Fin (n + 1))
S : ConstructibleSetData (MvPolynomial (Fin (n + 1)) R)
hSn : ∀ C ∈ S, C.n ≤ k
hS :
  ∀ C ∈ S, ∀ (j : Fin C.n), C.g j ∈ coeffsIn (Fin (n + 1)) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin (n + 1)) d)
e : MvPolynomial (Fin (n + 1)) R ≃ₐ[R] (MvPolynomial (Fin n) R)[X] := MvPolynomial.finSuccEquiv R n
S' : ConstructibleSetData (MvPolynomial (Fin n) R)[X] := ConstructibleSetData.map e.toRingEquiv.toRingHom S
hS' : S'.degBound ≤ k * (1 + Multiset.count 0 d)
B : Multiset (Fin n) := Finsupp.toMultiset (Finsupp.comapDomain Fin.succ (Multiset.toFinsupp d) ⋯)
T : ConstructibleSetData (MvPolynomial (Fin n) R)
hT₁ : comap Polynomial.C '' S'.toSet = T.toSet
hT₂ :
  ∀ C ∈ T,
    C.n ≤ S'.degBound ∧
      ∀ (i : Fin C.n),
        C.g i ∈ (coeffsIn (Fin n) M ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) B)) ^ S'.degBound ^ S'.degBound
N : ℕ := (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d))
this✝ :
  ∀ C ∈ T, ∀ (a : Fin C.n), C.g a ∈ coeffsIn (Fin n) (M ^ N) ⊓ Submodule.restrictScalars ℤ (degreesLE R (Fin n) (N • B))
h1M : 1 ≤ M
U : ConstructibleSetData R
hU₁ : comap MvPolynomial.C '' T.toSet = U.toSet
hU₂ :
  ∀ C ∈ U,
    C.n ≤
        numBound S'.degBound
          (fun i ↦
            1 +
              (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
                Multiset.count i (Multiset.map Fin.val B))
          n ∧
      ∀ (i : Fin C.n),
        C.g i ∈
          M ^
            ((k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
              degBound S'.degBound
                (fun i ↦
                  1 +
                    (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
                      Multiset.count i (Multiset.map Fin.val B))
                n)
this :
  ∀ i < n + 1,
    (Nat.casesOn i (1 + Multiset.count 0 d) fun x ↦ 1 + Multiset.count x (Multiset.map Fin.val B)) ≤
      1 + Multiset.count i (Multiset.map Fin.val d)
C : BasicConstructibleSetData R
hCU : C ∈ U
i : Fin C.n
⊢ (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) *
      degBound S'.degBound
        (fun i ↦
          1 +
            (k * (1 + Multiset.count 0 d)) ^ (k * (1 + Multiset.count 0 d)) * Multiset.count i (Multiset.map Fin.val B))
        n ≤
    degBound k (fun i ↦ 1 + Multiset.count i (Multiset.map Fin.val d)) (n + 1)