Theorem: FOL.NV.Sub.Pred.One.Ind.substitution_theorem
Proof state before:
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds)
h1_phi h1_psi h1_phi' h1_psi' : Formula
a✝¹ : IsSub P zs H h1_phi h1_phi'
a✝ : IsSub P zs H h1_psi h1_psi'
h1_ih_1 :
  ∀ (V : VarAssignment D),
    (∀ (Q : PredName) (ds : List D),
        Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) →
      (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi)
h1_ih_2 :
  ∀ (V : VarAssignment D),
    (∀ (Q : PredName) (ds : List D),
        Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) →
      (Holds D I V E h1_psi' ↔ Holds D J V E h1_psi)
V : VarAssignment D
h2 :
  ∀ (Q : PredName) (ds : List D),
    Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)
⊢ (Holds D I V E h1_phi' ↔ Holds D I V E h1_psi') ↔ (Holds D J V E h1_phi ↔ Holds D J V E h1_psi)
Tactic:
congr! 1
Proof state after:
case a.h.e'_1.a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds)
h1_phi h1_psi h1_phi' h1_psi' : Formula
a✝¹ : IsSub P zs H h1_phi h1_phi'
a✝ : IsSub P zs H h1_psi h1_psi'
h1_ih_1 :
  ∀ (V : VarAssignment D),
    (∀ (Q : PredName) (ds : List D),
        Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) →
      (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi)
h1_ih_2 :
  ∀ (V : VarAssignment D),
    (∀ (Q : PredName) (ds : List D),
        Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) →
      (Holds D I V E h1_psi' ↔ Holds D J V E h1_psi)
V : VarAssignment D
h2 :
  ∀ (Q : PredName) (ds : List D),
    Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)
⊢ Holds D I V E h1_phi' ↔ Holds D J V E h1_phi

case a.h.e'_2.a
D : Type
I J : Interpretation D
E : Env
A : Formula
P : PredName
zs : List VarName
H B : Formula
h3_const : I.pred_const_ = J.pred_const_
h3_var : ∀ (Q : PredName) (ds : List D), ¬(Q = P ∧ ds.length = zs.length) → (I.pred_var_ Q ds ↔ J.pred_var_ Q ds)
h1_phi h1_psi h1_phi' h1_psi' : Formula
a✝¹ : IsSub P zs H h1_phi h1_phi'
a✝ : IsSub P zs H h1_psi h1_psi'
h1_ih_1 :
  ∀ (V : VarAssignment D),
    (∀ (Q : PredName) (ds : List D),
        Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) →
      (Holds D I V E h1_phi' ↔ Holds D J V E h1_phi)
h1_ih_2 :
  ∀ (V : VarAssignment D),
    (∀ (Q : PredName) (ds : List D),
        Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)) →
      (Holds D I V E h1_psi' ↔ Holds D J V E h1_psi)
V : VarAssignment D
h2 :
  ∀ (Q : PredName) (ds : List D),
    Q = P ∧ ds.length = zs.length → (Holds D I (Function.updateListITE V zs ds) E H ↔ J.pred_var_ P ds)
⊢ Holds D I V E h1_psi' ↔ Holds D J V E h1_psi