Theorem: FOL.NV.Sub.Pred.One.Rec.substitution_is_valid (FOL/NV/Sub/Pred/One/Rec/Sub.lean)

State before:
F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : F.IsValid
⊢ (replace P zs H F).IsValid
Tactic:
simp only [IsValid] at h2
State after:
F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ (replace P zs H F).IsValid

State before:
F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : ∀ (D : Type) (I : Interpretation D) (V : VarAssignment D) (E : Env), Holds D I V E F
⊢ (replace P zs H F).IsValid
Tactic:
simp only [IsValid]
State after:
F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : ∀ (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 (replace P zs H F)

State before:
F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : ∀ (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 (replace P zs H F)
Tactic:
intro D I V E
State after:
F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : ∀ (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 (replace P zs H F)

State before:
F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : ∀ (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 (replace P zs H F)
Tactic:
simp only [← substitution_theorem D I V E F P zs H h1]
State after:
F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : ∀ (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' D I V E P zs H) V E F

State before:
F : Formula
P : PredName
zs : List VarName
H : Formula
h1 : admits P zs H F
h2 : ∀ (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' D I V E P zs H) V E F
Tactic:
apply h2
State after:
no goals