Theorem: FOL.NV.Sub.Prop.All.Rec.substitution_is_valid (FOL/NV/Sub/Prop/All/Rec/Sub.lean)

State before:
F : Formula
τ : PredName → PredName
h1 : F.IsValid
⊢ (sub τ F).IsValid
Tactic:
simp only [IsValid] at h1
State after:
F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ (sub τ F).IsValid

State before:
F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ (sub τ F).IsValid
Tactic:
simp only [IsValid]
State after:
F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub τ F)

State before:
F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E (sub τ F)
Tactic:
intro D I V E
State after:
F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I V E (sub τ F)

State before:
F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D I V E (sub τ F)
Tactic:
simp only [substitution_theorem D I V E τ F]
State after:
F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D
    { nonempty := ⋯, pred_const_ := I.pred_const_,
      pred_var_ := fun P ds => if ds = [] then Holds D I V E (pred_var_ (τ P) []) else I.pred_var_ P ds }
    V E F

State before:
F : Formula
τ : PredName → PredName
h1 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
D : Type
I : Interpretation D
V : VarAssignment D
E : Env
⊢ Holds D
    { nonempty := ⋯, pred_const_ := I.pred_const_,
      pred_var_ := fun P ds => if ds = [] then Holds D I V E (pred_var_ (τ P) []) else I.pred_var_ P ds }
    V E F
Tactic:
apply h1
State after:
no goals