Module: Mathlib.ModelTheory.Semantics

Goal state before:
pp: case imp
L : Language
M : Type w
inst✝¹ : L.Structure M
α : Type u'
β : Type v'
inst✝ : DecidableEq α
n : ℕ
v : β → M
v' : α → M
n✝ : ℕ
f₁✝ f₂✝ : L.BoundedFormula α n✝
ih1 :
  ∀ {f : ↥f₁✝.freeVarFinset → β} {xs : Fin n✝ → M},
    (∀ (a : ↥f₁✝.freeVarFinset), v (f a) = v' ↑a) → ((f₁✝.restrictFreeVar f).Realize v xs ↔ f₁✝.Realize v' xs)
ih2 :
  ∀ {f : ↥f₂✝.freeVarFinset → β} {xs : Fin n✝ → M},
    (∀ (a : ↥f₂✝.freeVarFinset), v (f a) = v' ↑a) → ((f₂✝.restrictFreeVar f).Realize v xs ↔ f₂✝.Realize v' xs)
f : ↥(f₁✝ ⟹ f₂✝).freeVarFinset → β
xs : Fin n✝ → M
hv' : ∀ (a : ↥(f₁✝ ⟹ f₂✝).freeVarFinset), v (f a) = v' ↑a
⊢ f₁✝.Realize v' xs → (f₂✝.restrictFreeVar (f ∘ Set.inclusion ⋯)).Realize v xs ↔ f₁✝.Realize v' xs → f₂✝.Realize v' xs

pp: case imp
L : Language
M : Type w
inst✝¹ : L.Structure M
α : Type u'
β : Type v'
inst✝ : DecidableEq α
n : ℕ
v : β → M
v' : α → M
n✝ : ℕ
f₁✝ f₂✝ : L.BoundedFormula α n✝
ih1 :
  ∀ {f : ↥f₁✝.freeVarFinset → β} {xs : Fin n✝ → M},
    (∀ (a : ↥f₁✝.freeVarFinset), v (f a) = v' ↑a) → ((f₁✝.restrictFreeVar f).Realize v xs ↔ f₁✝.Realize v' xs)
ih2 :
  ∀ {f : ↥f₂✝.freeVarFinset → β} {xs : Fin n✝ → M},
    (∀ (a : ↥f₂✝.freeVarFinset), v (f a) = v' ↑a) → ((f₂✝.restrictFreeVar f).Realize v xs ↔ f₂✝.Realize v' xs)
f : ↥(f₁✝ ⟹ f₂✝).freeVarFinset → β
xs : Fin n✝ → M
hv' : ∀ (a : ↥(f₁✝ ⟹ f₂✝).freeVarFinset), v (f a) = v' ↑a
⊢ ∀ (a : ↥f₁✝.freeVarFinset), v ((f ∘ Set.inclusion ⋯) a) = v' ↑a

Tactic:
ih2

Goal state after:
case imp
L : Language
M : Type w
inst✝¹ : L.Structure M
α : Type u'
β : Type v'
inst✝ : DecidableEq α
n : ℕ
v : β → M
v' : α → M
n✝ : ℕ
f₁✝ f₂✝ : L.BoundedFormula α n✝
ih1 :
  ∀ {f : ↥f₁✝.freeVarFinset → β} {xs : Fin n✝ → M},
    (∀ (a : ↥f₁✝.freeVarFinset), v (f a) = v' ↑a) → ((f₁✝.restrictFreeVar f).Realize v xs ↔ f₁✝.Realize v' xs)
ih2 :
  ∀ {f : ↥f₂✝.freeVarFinset → β} {xs : Fin n✝ → M},
    (∀ (a : ↥f₂✝.freeVarFinset), v (f a) = v' ↑a) → ((f₂✝.restrictFreeVar f).Realize v xs ↔ f₂✝.Realize v' xs)
f : ↥(f₁✝ ⟹ f₂✝).freeVarFinset → β
xs : Fin n✝ → M
hv' : ∀ (a : ↥(f₁✝ ⟹ f₂✝).freeVarFinset), v (f a) = v' ↑a
⊢ f₁✝.Realize v' xs → f₂✝.Realize v' xs ↔ f₁✝.Realize v' xs → f₂✝.Realize v' xs

case imp
L : Language
M : Type w
inst✝¹ : L.Structure M
α : Type u'
β : Type v'
inst✝ : DecidableEq α
n : ℕ
v : β → M
v' : α → M
n✝ : ℕ
f₁✝ f₂✝ : L.BoundedFormula α n✝
ih1 :
  ∀ {f : ↥f₁✝.freeVarFinset → β} {xs : Fin n✝ → M},
    (∀ (a : ↥f₁✝.freeVarFinset), v (f a) = v' ↑a) → ((f₁✝.restrictFreeVar f).Realize v xs ↔ f₁✝.Realize v' xs)
ih2 :
  ∀ {f : ↥f₂✝.freeVarFinset → β} {xs : Fin n✝ → M},
    (∀ (a : ↥f₂✝.freeVarFinset), v (f a) = v' ↑a) → ((f₂✝.restrictFreeVar f).Realize v xs ↔ f₂✝.Realize v' xs)
f : ↥(f₁✝ ⟹ f₂✝).freeVarFinset → β
xs : Fin n✝ → M
hv' : ∀ (a : ↥(f₁✝ ⟹ f₂✝).freeVarFinset), v (f a) = v' ↑a
⊢ ∀ (a : ↥f₂✝.freeVarFinset), v ((f ∘ Set.inclusion ⋯) a) = v' ↑a

case imp
L : Language
M : Type w
inst✝¹ : L.Structure M
α : Type u'
β : Type v'
inst✝ : DecidableEq α
n : ℕ
v : β → M
v' : α → M
n✝ : ℕ
f₁✝ f₂✝ : L.BoundedFormula α n✝
ih1 :
  ∀ {f : ↥f₁✝.freeVarFinset → β} {xs : Fin n✝ → M},
    (∀ (a : ↥f₁✝.freeVarFinset), v (f a) = v' ↑a) → ((f₁✝.restrictFreeVar f).Realize v xs ↔ f₁✝.Realize v' xs)
ih2 :
  ∀ {f : ↥f₂✝.freeVarFinset → β} {xs : Fin n✝ → M},
    (∀ (a : ↥f₂✝.freeVarFinset), v (f a) = v' ↑a) → ((f₂✝.restrictFreeVar f).Realize v xs ↔ f₂✝.Realize v' xs)
f : ↥(f₁✝ ⟹ f₂✝).freeVarFinset → β
xs : Fin n✝ → M
hv' : ∀ (a : ↥(f₁✝ ⟹ f₂✝).freeVarFinset), v (f a) = v' ↑a
⊢ ∀ (a : ↥f₁✝.freeVarFinset), v ((f ∘ Set.inclusion ⋯) a) = v' ↑a