Theorem: FOL.NV.Sub.Prop.All.Rec.substitution_theorem
Proof state before:
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula),
    Holds D I V tail✝ (sub τ F) ↔
      Holds D
        { nonempty := ⋯, pred_const_ := I.pred_const_,
          pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
        V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
  ∀ (V : VarAssignment D),
    Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
      Holds D
        { nonempty := ⋯, pred_const_ := I.pred_const_,
          pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
        V (head✝ :: tail✝) phi
psi_ih :
  ∀ (V : VarAssignment D),
    Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
      Holds D
        { nonempty := ⋯, pred_const_ := I.pred_const_,
          pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
        V (head✝ :: tail✝) psi
⊢ (Holds D I V (head✝ :: tail✝) (sub τ phi) ↔ Holds D I V (head✝ :: tail✝) (sub τ psi)) ↔
    (Holds D
        { nonempty := ⋯, pred_const_ := I.pred_const_,
          pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
        V (head✝ :: tail✝) phi ↔
      Holds D
        { nonempty := ⋯, pred_const_ := I.pred_const_,
          pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
        V (head✝ :: tail✝) psi)
Tactic:
congr! 1
Proof state after:
case a.h.e'_1.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula),
    Holds D I V tail✝ (sub τ F) ↔
      Holds D
        { nonempty := ⋯, pred_const_ := I.pred_const_,
          pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
        V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
  ∀ (V : VarAssignment D),
    Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
      Holds D
        { nonempty := ⋯, pred_const_ := I.pred_const_,
          pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
        V (head✝ :: tail✝) phi
psi_ih :
  ∀ (V : VarAssignment D),
    Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
      Holds D
        { nonempty := ⋯, pred_const_ := I.pred_const_,
          pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
        V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
    Holds D
      { nonempty := ⋯, pred_const_ := I.pred_const_,
        pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
      V (head✝ :: tail✝) phi

case a.h.e'_2.a
D : Type
I : Interpretation D
τ : PredName → PredName
head✝ : Definition
tail✝ : List Definition
tail_ih✝ :
  ∀ (V : VarAssignment D) (F : Formula),
    Holds D I V tail✝ (sub τ F) ↔
      Holds D
        { nonempty := ⋯, pred_const_ := I.pred_const_,
          pred_var_ := fun P ds => if ds = [] then Holds D I V tail✝ (pred_var_ (τ P) []) else I.pred_var_ P ds }
        V tail✝ F
phi psi : Formula
V : VarAssignment D
phi_ih :
  ∀ (V : VarAssignment D),
    Holds D I V (head✝ :: tail✝) (sub τ phi) ↔
      Holds D
        { nonempty := ⋯, pred_const_ := I.pred_const_,
          pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
        V (head✝ :: tail✝) phi
psi_ih :
  ∀ (V : VarAssignment D),
    Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
      Holds D
        { nonempty := ⋯, pred_const_ := I.pred_const_,
          pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
        V (head✝ :: tail✝) psi
⊢ Holds D I V (head✝ :: tail✝) (sub τ psi) ↔
    Holds D
      { nonempty := ⋯, pred_const_ := I.pred_const_,
        pred_var_ := fun P ds => if ds = [] then I.pred_var_ (τ P) (List.map V []) else I.pred_var_ P ds }
      V (head✝ :: tail✝) psi