(* Definition of types *)

Parameter N : Set.Inductive K : Set :=
  | inv : K -> K
  | ka : securite.N -> K
  | ks : securite.N -> K.Inductive K : Set :=
  | inv : K -> K
  | ka : securite.N -> K
  | ks : securite.N -> K.Inductive K : Set :=
  | inv : K -> K
  | ka : securite.N -> K
  | ks : securite.N -> K.Inductive K : Set :=
  | inv : K -> K
  | ka : securite.N -> K
  | ks : securite.N -> K.Parameter D : Set.Parameter Aid Bid : D.Axiom
  D_dec :
    forall d1 d2 : D, {d1 = Aid /\ d2 = Bid} + {~ (d1 = Aid /\ d2 = Bid)}.Inductive B : Set :=
  | K2B : K -> B
  | D2B : D -> B.Inductive B : Set :=
  | K2B : K -> B
  | D2B : D -> B.Inductive B : Set :=
  | K2B : K -> B
  | D2B : D -> B.Inductive C : Set :=
  | Encrypt : C -> K -> C
  | Pair : C -> C -> C
  | B2C : B -> C.Inductive C : Set :=
  | Encrypt : C -> K -> C
  | Pair : C -> C -> C
  | B2C : B -> C.Inductive C : Set :=
  | Encrypt : C -> K -> C
  | Pair : C -> C -> C
  | B2C : B -> C.Inductive C : Set :=
  | Encrypt : C -> K -> C
  | Pair : C -> C -> C
  | B2C : B -> C.Notation SS := (list C) (only parsing).(* <Warning> : Syntax is discontinued *)

(* known_in and not_comp_of predicates, equivS equivalence, setofkeys *)

Inductive known_in : C -> list C -> Prop :=
  | E0 :
      forall (c1 c2 : C) (s : list C), known_in c1 s -> known_in c1 (c2 :: s)
  | EP0 :
      forall (c : C) (s1 s2 : list C), known_in c s1 -> known_in c (s1 ++ s2)
  | A1 :
      forall (c1 c2 : C) (s : list C),
      known_in c1 s -> known_in c2 s -> known_in (Pair c1 c2) s
  | A2A :
      forall (c1 c2 : C) (s : list C),
      known_in (Pair c1 c2) s -> known_in c1 s
  | A2B :
      forall (c1 c2 : C) (s : list C),
      known_in (Pair c1 c2) s -> known_in c2 s
  | A3 :
      forall (c1 : C) (k1 : K) (s : list C),
      known_in c1 s ->
      known_in (B2C (K2B k1)) s -> known_in (Encrypt c1 k1) s
  | A4 :
      forall (c1 : C) (k1 : K) (s : list C),
      known_in (Encrypt c1 k1) s ->
      known_in (B2C (K2B (inv k1))) s -> known_in c1 s.(* <Warning> : Syntax is discontinued *)

(* known_in and not_comp_of predicates, equivS equivalence, setofkeys *)

Inductive known_in : C -> list C -> Prop :=
  | E0 :
      forall (c1 c2 : C) (s : list C), known_in c1 s -> known_in c1 (c2 :: s)
  | EP0 :
      forall (c : C) (s1 s2 : list C), known_in c s1 -> known_in c (s1 ++ s2)
  | A1 :
      forall (c1 c2 : C) (s : list C),
      known_in c1 s -> known_in c2 s -> known_in (Pair c1 c2) s
  | A2A :
      forall (c1 c2 : C) (s : list C),
      known_in (Pair c1 c2) s -> known_in c1 s
  | A2B :
      forall (c1 c2 : C) (s : list C),
      known_in (Pair c1 c2) s -> known_in c2 s
  | A3 :
      forall (c1 : C) (k1 : K) (s : list C),
      known_in c1 s ->
      known_in (B2C (K2B k1)) s -> known_in (Encrypt c1 k1) s
  | A4 :
      forall (c1 : C) (k1 : K) (s : list C),
      known_in (Encrypt c1 k1) s ->
      known_in (B2C (K2B (inv k1))) s -> known_in c1 s.(* <Warning> : Syntax is discontinued *)

(* known_in and not_comp_of predicates, equivS equivalence, setofkeys *)

Inductive known_in : C -> list C -> Prop :=
  | E0 :
      forall (c1 c2 : C) (s : list C), known_in c1 s -> known_in c1 (c2 :: s)
  | EP0 :
      forall (c : C) (s1 s2 : list C), known_in c s1 -> known_in c (s1 ++ s2)
  | A1 :
      forall (c1 c2 : C) (s : list C),
      known_in c1 s -> known_in c2 s -> known_in (Pair c1 c2) s
  | A2A :
      forall (c1 c2 : C) (s : list C),
      known_in (Pair c1 c2) s -> known_in c1 s
  | A2B :
      forall (c1 c2 : C) (s : list C),
      known_in (Pair c1 c2) s -> known_in c2 s
  | A3 :
      forall (c1 : C) (k1 : K) (s : list C),
      known_in c1 s ->
      known_in (B2C (K2B k1)) s -> known_in (Encrypt c1 k1) s
  | A4 :
      forall (c1 : C) (k1 : K) (s : list C),
      known_in (Encrypt c1 k1) s ->
      known_in (B2C (K2B (inv k1))) s -> known_in c1 s.(* <Warning> : Syntax is discontinued *)

(* known_in and not_comp_of predicates, equivS equivalence, setofkeys *)

Inductive known_in : C -> list C -> Prop :=
  | E0 :
      forall (c1 c2 : C) (s : list C), known_in c1 s -> known_in c1 (c2 :: s)
  | EP0 :
      forall (c : C) (s1 s2 : list C), known_in c s1 -> known_in c (s1 ++ s2)
  | A1 :
      forall (c1 c2 : C) (s : list C),
      known_in c1 s -> known_in c2 s -> known_in (Pair c1 c2) s
  | A2A :
      forall (c1 c2 : C) (s : list C),
      known_in (Pair c1 c2) s -> known_in c1 s
  | A2B :
      forall (c1 c2 : C) (s : list C),
      known_in (Pair c1 c2) s -> known_in c2 s
  | A3 :
      forall (c1 : C) (k1 : K) (s : list C),
      known_in c1 s ->
      known_in (B2C (K2B k1)) s -> known_in (Encrypt c1 k1) s
  | A4 :
      forall (c1 : C) (k1 : K) (s : list C),
      known_in (Encrypt c1 k1) s ->
      known_in (B2C (K2B (inv k1))) s -> known_in c1 s.(* <Warning> : Syntax is discontinued *)

(* known_in and not_comp_of predicates, equivS equivalence, setofkeys *)

Inductive known_in : C -> list C -> Prop :=
  | E0 :
      forall (c1 c2 : C) (s : list C), known_in c1 s -> known_in c1 (c2 :: s)
  | EP0 :
      forall (c : C) (s1 s2 : list C), known_in c s1 -> known_in c (s1 ++ s2)
  | A1 :
      forall (c1 c2 : C) (s : list C),
      known_in c1 s -> known_in c2 s -> known_in (Pair c1 c2) s
  | A2A :
      forall (c1 c2 : C) (s : list C),
      known_in (Pair c1 c2) s -> known_in c1 s
  | A2B :
      forall (c1 c2 : C) (s : list C),
      known_in (Pair c1 c2) s -> known_in c2 s
  | A3 :
      forall (c1 : C) (k1 : K) (s : list C),
      known_in c1 s ->
      known_in (B2C (K2B k1)) s -> known_in (Encrypt c1 k1) s
  | A4 :
      forall (c1 : C) (k1 : K) (s : list C),
      known_in (Encrypt c1 k1) s ->
      known_in (B2C (K2B (inv k1))) s -> known_in c1 s.(* <Warning> : Syntax is discontinued *)

(* known_in and not_comp_of predicates, equivS equivalence, setofkeys *)

Inductive known_in : C -> list C -> Prop :=
  | E0 :
      forall (c1 c2 : C) (s : list C), known_in c1 s -> known_in c1 (c2 :: s)
  | EP0 :
      forall (c : C) (s1 s2 : list C), known_in c s1 -> known_in c (s1 ++ s2)
  | A1 :
      forall (c1 c2 : C) (s : list C),
      known_in c1 s -> known_in c2 s -> known_in (Pair c1 c2) s
  | A2A :
      forall (c1 c2 : C) (s : list C),
      known_in (Pair c1 c2) s -> known_in c1 s
  | A2B :
      forall (c1 c2 : C) (s : list C),
      known_in (Pair c1 c2) s -> known_in c2 s
  | A3 :
      forall (c1 : C) (k1 : K) (s : list C),
      known_in c1 s ->
      known_in (B2C (K2B k1)) s -> known_in (Encrypt c1 k1) s
  | A4 :
      forall (c1 : C) (k1 : K) (s : list C),
      known_in (Encrypt c1 k1) s ->
      known_in (B2C (K2B (inv k1))) s -> known_in c1 s.(* <Warning> : Syntax is discontinued *)

(* known_in and not_comp_of predicates, equivS equivalence, setofkeys *)

Inductive known_in : C -> list C -> Prop :=
  | E0 :
      forall (c1 c2 : C) (s : list C), known_in c1 s -> known_in c1 (c2 :: s)
  | EP0 :
      forall (c : C) (s1 s2 : list C), known_in c s1 -> known_in c (s1 ++ s2)
  | A1 :
      forall (c1 c2 : C) (s : list C),
      known_in c1 s -> known_in c2 s -> known_in (Pair c1 c2) s
  | A2A :
      forall (c1 c2 : C) (s : list C),
      known_in (Pair c1 c2) s -> known_in c1 s
  | A2B :
      forall (c1 c2 : C) (s : list C),
      known_in (Pair c1 c2) s -> known_in c2 s
  | A3 :
      forall (c1 : C) (k1 : K) (s : list C),
      known_in c1 s ->
      known_in (B2C (K2B k1)) s -> known_in (Encrypt c1 k1) s
  | A4 :
      forall (c1 : C) (k1 : K) (s : list C),
      known_in (Encrypt c1 k1) s ->
      known_in (B2C (K2B (inv k1))) s -> known_in c1 s.(* <Warning> : Syntax is discontinued *)

(* known_in and not_comp_of predicates, equivS equivalence, setofkeys *)

Inductive known_in : C -> list C -> Prop :=
  | E0 :
      forall (c1 c2 : C) (s : list C), known_in c1 s -> known_in c1 (c2 :: s)
  | EP0 :
      forall (c : C) (s1 s2 : list C), known_in c s1 -> known_in c (s1 ++ s2)
  | A1 :
      forall (c1 c2 : C) (s : list C),
      known_in c1 s -> known_in c2 s -> known_in (Pair c1 c2) s
  | A2A :
      forall (c1 c2 : C) (s : list C),
      known_in (Pair c1 c2) s -> known_in c1 s
  | A2B :
      forall (c1 c2 : C) (s : list C),
      known_in (Pair c1 c2) s -> known_in c2 s
  | A3 :
      forall (c1 : C) (k1 : K) (s : list C),
      known_in c1 s ->
      known_in (B2C (K2B k1)) s -> known_in (Encrypt c1 k1) s
  | A4 :
      forall (c1 : C) (k1 : K) (s : list C),
      known_in (Encrypt c1 k1) s ->
      known_in (B2C (K2B (inv k1))) s -> known_in c1 s.Inductive equivS : list C -> list C -> Prop :=
  | equivS1a : equivS nil nil
  | equivS1b :
      forall (s1 s2 : list C) (c : C),
      equivS s1 s2 -> equivS (c :: s1) (c :: s2)
  | equivS2 :
      forall (s : list C) (c1 c2 : C), equivS (c1 :: c2 :: s) (c2 :: c1 :: s)
  | equivS4 :
      forall s1 s2 s3 : list C, equivS s1 s2 -> equivS s2 s3 -> equivS s1 s3
  | AlreadyIn1 : forall (c1 : C) (s : list C), In c1 s -> equivS s (c1 :: s)
  | AlreadyIn1b : forall (c1 : C) (s : list C), In c1 s -> equivS (c1 :: s) s
  | AlreadyIn :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS s (c1 :: s)
  | AlreadyInb :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS (c1 :: s) s.Inductive equivS : list C -> list C -> Prop :=
  | equivS1a : equivS nil nil
  | equivS1b :
      forall (s1 s2 : list C) (c : C),
      equivS s1 s2 -> equivS (c :: s1) (c :: s2)
  | equivS2 :
      forall (s : list C) (c1 c2 : C), equivS (c1 :: c2 :: s) (c2 :: c1 :: s)
  | equivS4 :
      forall s1 s2 s3 : list C, equivS s1 s2 -> equivS s2 s3 -> equivS s1 s3
  | AlreadyIn1 : forall (c1 : C) (s : list C), In c1 s -> equivS s (c1 :: s)
  | AlreadyIn1b : forall (c1 : C) (s : list C), In c1 s -> equivS (c1 :: s) s
  | AlreadyIn :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS s (c1 :: s)
  | AlreadyInb :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS (c1 :: s) s.Inductive equivS : list C -> list C -> Prop :=
  | equivS1a : equivS nil nil
  | equivS1b :
      forall (s1 s2 : list C) (c : C),
      equivS s1 s2 -> equivS (c :: s1) (c :: s2)
  | equivS2 :
      forall (s : list C) (c1 c2 : C), equivS (c1 :: c2 :: s) (c2 :: c1 :: s)
  | equivS4 :
      forall s1 s2 s3 : list C, equivS s1 s2 -> equivS s2 s3 -> equivS s1 s3
  | AlreadyIn1 : forall (c1 : C) (s : list C), In c1 s -> equivS s (c1 :: s)
  | AlreadyIn1b : forall (c1 : C) (s : list C), In c1 s -> equivS (c1 :: s) s
  | AlreadyIn :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS s (c1 :: s)
  | AlreadyInb :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS (c1 :: s) s.Inductive equivS : list C -> list C -> Prop :=
  | equivS1a : equivS nil nil
  | equivS1b :
      forall (s1 s2 : list C) (c : C),
      equivS s1 s2 -> equivS (c :: s1) (c :: s2)
  | equivS2 :
      forall (s : list C) (c1 c2 : C), equivS (c1 :: c2 :: s) (c2 :: c1 :: s)
  | equivS4 :
      forall s1 s2 s3 : list C, equivS s1 s2 -> equivS s2 s3 -> equivS s1 s3
  | AlreadyIn1 : forall (c1 : C) (s : list C), In c1 s -> equivS s (c1 :: s)
  | AlreadyIn1b : forall (c1 : C) (s : list C), In c1 s -> equivS (c1 :: s) s
  | AlreadyIn :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS s (c1 :: s)
  | AlreadyInb :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS (c1 :: s) s.Inductive equivS : list C -> list C -> Prop :=
  | equivS1a : equivS nil nil
  | equivS1b :
      forall (s1 s2 : list C) (c : C),
      equivS s1 s2 -> equivS (c :: s1) (c :: s2)
  | equivS2 :
      forall (s : list C) (c1 c2 : C), equivS (c1 :: c2 :: s) (c2 :: c1 :: s)
  | equivS4 :
      forall s1 s2 s3 : list C, equivS s1 s2 -> equivS s2 s3 -> equivS s1 s3
  | AlreadyIn1 : forall (c1 : C) (s : list C), In c1 s -> equivS s (c1 :: s)
  | AlreadyIn1b : forall (c1 : C) (s : list C), In c1 s -> equivS (c1 :: s) s
  | AlreadyIn :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS s (c1 :: s)
  | AlreadyInb :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS (c1 :: s) s.Inductive equivS : list C -> list C -> Prop :=
  | equivS1a : equivS nil nil
  | equivS1b :
      forall (s1 s2 : list C) (c : C),
      equivS s1 s2 -> equivS (c :: s1) (c :: s2)
  | equivS2 :
      forall (s : list C) (c1 c2 : C), equivS (c1 :: c2 :: s) (c2 :: c1 :: s)
  | equivS4 :
      forall s1 s2 s3 : list C, equivS s1 s2 -> equivS s2 s3 -> equivS s1 s3
  | AlreadyIn1 : forall (c1 : C) (s : list C), In c1 s -> equivS s (c1 :: s)
  | AlreadyIn1b : forall (c1 : C) (s : list C), In c1 s -> equivS (c1 :: s) s
  | AlreadyIn :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS s (c1 :: s)
  | AlreadyInb :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS (c1 :: s) s.Inductive equivS : list C -> list C -> Prop :=
  | equivS1a : equivS nil nil
  | equivS1b :
      forall (s1 s2 : list C) (c : C),
      equivS s1 s2 -> equivS (c :: s1) (c :: s2)
  | equivS2 :
      forall (s : list C) (c1 c2 : C), equivS (c1 :: c2 :: s) (c2 :: c1 :: s)
  | equivS4 :
      forall s1 s2 s3 : list C, equivS s1 s2 -> equivS s2 s3 -> equivS s1 s3
  | AlreadyIn1 : forall (c1 : C) (s : list C), In c1 s -> equivS s (c1 :: s)
  | AlreadyIn1b : forall (c1 : C) (s : list C), In c1 s -> equivS (c1 :: s) s
  | AlreadyIn :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS s (c1 :: s)
  | AlreadyInb :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS (c1 :: s) s.Inductive equivS : list C -> list C -> Prop :=
  | equivS1a : equivS nil nil
  | equivS1b :
      forall (s1 s2 : list C) (c : C),
      equivS s1 s2 -> equivS (c :: s1) (c :: s2)
  | equivS2 :
      forall (s : list C) (c1 c2 : C), equivS (c1 :: c2 :: s) (c2 :: c1 :: s)
  | equivS4 :
      forall s1 s2 s3 : list C, equivS s1 s2 -> equivS s2 s3 -> equivS s1 s3
  | AlreadyIn1 : forall (c1 : C) (s : list C), In c1 s -> equivS s (c1 :: s)
  | AlreadyIn1b : forall (c1 : C) (s : list C), In c1 s -> equivS (c1 :: s) s
  | AlreadyIn :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS s (c1 :: s)
  | AlreadyInb :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS (c1 :: s) s.Inductive equivS : list C -> list C -> Prop :=
  | equivS1a : equivS nil nil
  | equivS1b :
      forall (s1 s2 : list C) (c : C),
      equivS s1 s2 -> equivS (c :: s1) (c :: s2)
  | equivS2 :
      forall (s : list C) (c1 c2 : C), equivS (c1 :: c2 :: s) (c2 :: c1 :: s)
  | equivS4 :
      forall s1 s2 s3 : list C, equivS s1 s2 -> equivS s2 s3 -> equivS s1 s3
  | AlreadyIn1 : forall (c1 : C) (s : list C), In c1 s -> equivS s (c1 :: s)
  | AlreadyIn1b : forall (c1 : C) (s : list C), In c1 s -> equivS (c1 :: s) s
  | AlreadyIn :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS s (c1 :: s)
  | AlreadyInb :
      forall (c1 : C) (s : list C), known_in c1 s -> equivS (c1 :: s) s.

Lemma equivS3 : forall s1 s2 : list C, equivS s1 s2 -> equivS s2 s1.