Theorem: Monoid.PushoutI.Reduced.exists_normalWord_prod_eq
File: Mathlib/GroupTheory/PushoutI.lean

tactic: exact ⟨empty, by simp, rfl⟩
state_before: case h_empty
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
hw : Reduced φ Word.empty
⊢ ∃ w', w'.prod = ofCoprodI Word.empty.prod ∧ List.map Sigma.fst w'.toList = List.map Sigma.fst Word.empty.toList
state_after: no goals

tactic: simp
state_before: ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
hw : Reduced φ Word.empty
⊢ empty.prod = ofCoprodI Word.empty.prod
state_after: no goals

tactic: rcases ih (fun _ hg => hw _ (List.mem_cons_of_mem _ hg)) with
  ⟨w', hw'prod, hw'map⟩
state_before: case h_cons
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
i : ι
g : G i
w : Word G
hIdx : w.fstIdx ≠ some i
hg1 : g ≠ 1
ih : Reduced φ w → ∃ w', w'.prod = ofCoprodI w.prod ∧ List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
hw : Reduced φ (Word.cons g w hIdx hg1)
⊢ ∃ w',
    w'.prod = ofCoprodI (Word.cons g w hIdx hg1).prod ∧
      List.map Sigma.fst w'.toList = List.map Sigma.fst (Word.cons g w hIdx hg1).toList
state_after: case h_cons.intro.intro
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
i : ι
g : G i
w : Word G
hIdx : w.fstIdx ≠ some i
hg1 : g ≠ 1
ih : Reduced φ w → ∃ w', w'.prod = ofCoprodI w.prod ∧ List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
hw : Reduced φ (Word.cons g w hIdx hg1)
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
⊢ ∃ w',
    w'.prod = ofCoprodI (Word.cons g w hIdx hg1).prod ∧
      List.map Sigma.fst w'.toList = List.map Sigma.fst (Word.cons g w hIdx hg1).toList

tactic: refine ⟨cons g w' ?_ ?_, ?_⟩
state_before: case h_cons.intro.intro
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
i : ι
g : G i
w : Word G
hIdx : w.fstIdx ≠ some i
hg1 : g ≠ 1
ih : Reduced φ w → ∃ w', w'.prod = ofCoprodI w.prod ∧ List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
hw : Reduced φ (Word.cons g w hIdx hg1)
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
⊢ ∃ w',
    w'.prod = ofCoprodI (Word.cons g w hIdx hg1).prod ∧
      List.map Sigma.fst w'.toList = List.map Sigma.fst (Word.cons g w hIdx hg1).toList
state_after: case h_cons.intro.intro.refine_1
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
i : ι
g : G i
w : Word G
hIdx : w.fstIdx ≠ some i
hg1 : g ≠ 1
ih : Reduced φ w → ∃ w', w'.prod = ofCoprodI w.prod ∧ List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
hw : Reduced φ (Word.cons g w hIdx hg1)
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
⊢ w'.fstIdx ≠ some i

case h_cons.intro.intro.refine_2
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
i : ι
g : G i
w : Word G
hIdx : w.fstIdx ≠ some i
hg1 : g ≠ 1
ih : Reduced φ w → ∃ w', w'.prod = ofCoprodI w.prod ∧ List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
hw : Reduced φ (Word.cons g w hIdx hg1)
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
⊢ g ∉ (φ i).range

case h_cons.intro.intro.refine_3
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
i : ι
g : G i
w : Word G
hIdx : w.fstIdx ≠ some i
hg1 : g ≠ 1
ih : Reduced φ w → ∃ w', w'.prod = ofCoprodI w.prod ∧ List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
hw : Reduced φ (Word.cons g w hIdx hg1)
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
⊢ (NormalWord.cons g w' ?h_cons.intro.intro.refine_1 ?h_cons.intro.intro.refine_2).prod =
      ofCoprodI (Word.cons g w hIdx hg1).prod ∧
    List.map Sigma.fst (NormalWord.cons g w' ?h_cons.intro.intro.refine_1 ?h_cons.intro.intro.refine_2).toList =
      List.map Sigma.fst (Word.cons g w hIdx hg1).toList

tactic: rwa [Word.fstIdx, ← List.head?_map, hw'map, List.head?_map]
state_before: case h_cons.intro.intro.refine_1
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
i : ι
g : G i
w : Word G
hIdx : w.fstIdx ≠ some i
hg1 : g ≠ 1
ih : Reduced φ w → ∃ w', w'.prod = ofCoprodI w.prod ∧ List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
hw : Reduced φ (Word.cons g w hIdx hg1)
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
⊢ w'.fstIdx ≠ some i
state_after: no goals

tactic: exact hw _ (List.mem_cons_self _ _)
state_before: case h_cons.intro.intro.refine_2
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
i : ι
g : G i
w : Word G
hIdx : w.fstIdx ≠ some i
hg1 : g ≠ 1
ih : Reduced φ w → ∃ w', w'.prod = ofCoprodI w.prod ∧ List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
hw : Reduced φ (Word.cons g w hIdx hg1)
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
⊢ g ∉ (φ i).range
state_after: no goals

tactic: simp [hw'prod, hw'map]
state_before: case h_cons.intro.intro.refine_3
ι : Type u_1
G : ι → Type u_2
H : Type u_3
K : Type u_4
inst✝² : Monoid K
inst✝¹ : (i : ι) → Group (G i)
inst✝ : Group H
φ : (i : ι) → H →* G i
d : Transversal φ
i : ι
g : G i
w : Word G
hIdx : w.fstIdx ≠ some i
hg1 : g ≠ 1
ih : Reduced φ w → ∃ w', w'.prod = ofCoprodI w.prod ∧ List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
hw : Reduced φ (Word.cons g w hIdx hg1)
w' : NormalWord d
hw'prod : w'.prod = ofCoprodI w.prod
hw'map : List.map Sigma.fst w'.toList = List.map Sigma.fst w.toList
⊢ (NormalWord.cons g w' ⋯ ⋯).prod = ofCoprodI (Word.cons g w hIdx hg1).prod ∧
    List.map Sigma.fst (NormalWord.cons g w' ⋯ ⋯).toList = List.map Sigma.fst (Word.cons g w hIdx hg1).toList
state_after: no goals