∀b1: T1 b0 e0.
∀e1:R1 ??? a1 ? e0 = b1.
T2 b0 e0 b1 e1.
-intros 9;intro e1;
+intros (T0 a0 T1 a1 T2 a2);
apply (eq_rect' ????? e1);
apply (R1 ?? ? ?? e0);
simplify;assumption;
∀a2:T2 a0 (refl_eq ? a0) a1 (refl_eq ? a1).
∀T3:∀x0:T0. ∀p0:a0=x0. ∀x1:T1 x0 p0.∀p1:R1 ??? a1 ? p0 = x1.
∀x2:T2 x0 p0 x1 p1.R2 T0 a0 T1 a1 T2 a2 ? p0 ? p1 = x2→ Type.
+ ∀a3:T3 a0 (refl_eq ? a0) a1 (refl_eq ? a1) a2 (refl_eq ? a2).
∀b0:T0.
∀e0:a0 = b0.
∀b1: T1 b0 e0.
∀e1:R1 ??? a1 ? e0 = b1.
∀b2: T2 b0 e0 b1 e1.
∀e2:R2 T0 a0 T1 a1 T2 a2 ? e0 ? e1 = b2.
- ∀a3:T3 a0 (refl_eq ? a0) a1 (refl_eq ? a1) a2 (refl_eq ? a2).T3 b0 e0 b1 e1 b2 e2.
-intros 12;intros 2 (e2 H);
+ T3 b0 e0 b1 e1 b2 e2.
+intros (T0 a0 T1 a1 T2 a2 T3 a3);
apply (eq_rect' ????? e2);
apply (R2 ?? ? ???? e0 ? e1);
simplify;assumption;
inductive I3 : Type ≝
| kI3 : ∀x1:nat.∀x2:I1 x1.∀x3:I2 x1 x2.I3.
-definition I3d: I3 → I3 → Type ≝
-λx,y.match x with
+(* lemma idfof : (∀t1,t2,t3,u1,u2,u3.((λy1:ℕ.λp1:t1=y1.λy2:I1 y1.λp2:R1 ℕ t1 (λy1:ℕ.λp1:t1=y1.I1 y1) t2 y1 p1 =y2.
+ λy3:I2 y1 y2.λp3:R2 ℕ t1 (λy1:ℕ.λp1:t1=y1.I1 y1) t2 (λy1:ℕ.λp1:t1=y1.λy2:I1 y1.λp2:R1 ℕ t1 (λy1:ℕ.λp1:t1 =y1.I1 y1) t2 y1 p1 =y2.I2 y1 y2) t3 y1 p1 y2 p2 =y3.
+ kI3 y1 y2 y3 =kI3 u1 u2 u3)
+t1 (refl_eq ℕ t1) t2 (refl_eq ((λy1:ℕ.λp1:t1=y1.I1 y1) t1 (refl_eq ℕ t1)) t2)
+ t3 (refl_eq ((λy1:ℕ.λp1:t1=y1.λy2:I1 y1.λp2:R1 ℕ t1 (λy1:ℕ.λp1:t1=y1.I1 y1) t2 y1 p1 =y2.I2 y1 y2) t1 (refl_eq ℕ t1) t2 (refl_eq ((λy1:ℕ.λp1:t1=y1.I1 y1) t1 (refl_eq ℕ t1)) t2)) t3)
+ )
+ → True).
+simplify; *)
+
+definition I3d : ∀x,y:I3.x = y → Type ≝
+λx,y.match x return (λx:I3.x = y → Type) with
+[ kI3 x1 x2 x3 ⇒ match y return (λy:I3.kI3 x1 x2 x3 = y → Type) with
+ [ kI3 y1 y2 y3 ⇒
+ λe:kI3 x1 x2 x3 = kI3 y1 y2 y3.
+ ∀P:Prop.(∀e1: x1 = y1.
+ ∀e2: R1 ?? (λz1,p1.I1 z1) ?? e1 = y2.
+ ∀e3: R2 ???? (λz1,p1,z2,p2.I2 z1 z2) x3 ? e1 ? e2 = y3.
+ R3 ??????
+ (λz1,p1,z2,p2,z3,p3.
+ eq ? (kI3 z1 z2 z3) (kI3 y1 y2 y3)) e y1 e1 y2 e2 y3 e3
+ = refl_eq ? (kI3 y1 y2 y3)
+ → P) → P]].
+
+definition I3d : ∀x,y:I3.x=y → Type.
+intros 2;cases x;cases y;intro;
+apply (∀P:Prop.(∀e1: x1 = x3.
+ ∀e2: R1 ?? (λy1,p1.I1 y1) ?? e1 = x4.
+ ∀e3: R2 ???? (λy1,p1,y2,p2.I2 y1 y2) i ? e1 ? e2 = i1.
+ R3 ??????
+ (λy1,p1,y2,p2,y3,p3.
+ eq ? (kI3 y1 y2 y3) (kI3 x3 x4 i1)) H x3 e1 x4 e2 i1 e3
+ = refl_eq ? (kI3 x3 x4 i1)
+ → P) → P);
+qed.
+
+(* definition I3d : ∀x,y:nat. x = y → Type ≝
+λx,y.
+match x
+ return (λx.x = y → Type)
+ with
+[ O ⇒ match y return (λy.O = y → Type) with
+ [ O ⇒ λe:O = O.∀P.P → P
+ | S q ⇒ λe: O = S q. ∀P.P]
+| S p ⇒ match y return (λy.S p = y → Type) with
+ [ O ⇒ λe:S p = O.∀P.P
+ | S q ⇒ λe: S p = S q. ∀P.(p = q → P) → P]].
+
+definition I3d:
+ ∀x,y:I3. x = y → Type
+ ≝
+λx,y.
+match x with
+[ kI3 t1 t2 (t3:I2 t1 t2) ⇒ match y with
+ [ kI3 u1 u2 u3 ⇒ λe:kI3 t1 t2 t3 = kI3 u1 u2 u3.∀P:Type.
+ (∀e1: t1 = u1.
+ ∀e2: R1 nat t1 (λy1:nat.λp1:y1 = u1.I1 y1) t2 ? e1 = u2.
+ ∀e3: R2 nat t1 (λy1,p1.I1 y1) t2 (λy1,p1,y2,p2.I2 y1 y2) t3 ? e1 ? e2 = u3.
+ (* ∀e: kI3 t1 t2 t3 = kI3 u1 u2 u3.*)
+ R3 nat t1 (λy1,p1.I1 y1) t2 (λy1,p1,y2,p2.I2 y1 y2) t3
+ (λy1,p1,y2,p2,y3,p3.eq I3 (kI3 y1 y2 y3) (kI3 u1 u2 u3)) e u1 e1 u2 e2 u3 e3 = refl_eq I3 (kI3 u1 u2 u3)
+ → P)
+ → P]].
+
+definition I3d:
+ ∀x,y:I3.
+ (∀x,y.match x with [ kI3 t1 t2 t3 ⇒
+ match y with [ kI3 u1 u2 u3 ⇒ kI3 t1 t2 t3 = kI3 u1 u2 u3]]) → Type
+ ≝
+λx,y.λe:
+ (∀x,y.match x with [ kI3 t1 t2 t3 ⇒
+ match y with [ kI3 u1 u2 u3 ⇒ kI3 t1 t2 t3 = kI3 u1 u2 u3]]).
+match x with
[ kI3 t1 t2 (t3:I2 t1 t2) ⇒ match y with
[ kI3 u1 u2 u3 ⇒ ∀P:Type.
(∀e1: t1 = u1.
∀e2: R1 ?? (λy1,p1.I1 y1) ?? e1 = u2.
- ∀e3: R2 ???? (λy1,p1,y2,p2.I2 y1 y2) ? e1 ? e2 t3 = u3.P) → P]].
-
-lemma I3nc : ∀a,b.a = b → I3d a b.
-intros;rewrite > H;elim b;simplify;intros;apply f;reflexivity;
+ ∀e3: R2 ???? (λy1,p1,y2,p2.I2 y1 y2) t3 ? e1 ? e2 = u3.
+ (* ∀e: kI3 t1 t2 t3 = kI3 u1 u2 u3.*)
+ R3 nat t1 (λy1,p1.I1 y1) t2 (λy1,p1,y2,p2.I2 y1 y2) t3
+ (λy1,p1,y2,p2,y3,p3.eq I3 (kI3 y1 y2 y3) (kI3 u1 u2 u3)) (e (kI3 t1 t2 t3) (kI3 u1 u2 u3)) u1 e1 u2 e2 u3 e3 = refl_eq ? (kI3 u1 u2 u3)
+ → P)
+ → P]].*)
+
+lemma I3nc : ∀a,b.∀e:a = b. I3d a b e.
+intros;apply (R1 ????? e);elim a;whd;intros;apply H;reflexivity;
qed.
(*lemma R1_r : ΠA:Type.Πx:A.ΠP:Πy:A.y=x→Type.P x (refl_eq A x)→Πy:A.Πp:y=x.P y p.
simplify;assumption;
qed.*)
-definition I3prova : ∀a,b,c,d,e,f.kI3 a b c = kI3 d e f → ∃P.P d e f.
-intros;apply (I3nc ?? H);clear H;
+definition I3prova : ∀a,b,c,d,e,f.∀Heq:kI3 a b c = kI3 d e f.
+ ∀P:? → ? → ? → ? → Prop.
+ P d e f Heq →
+ P a b c (refl_eq ??).
+intros;apply (I3nc ?? Heq);
simplify;intro;
+generalize in match H as H;generalize in match Heq as Heq;
generalize in match f as f;generalize in match e as e;
-generalize in match c as c;generalize in match b as b;
-clear f e c b;
+clear H Heq f e;
apply (R1 ????? e1);intros 5;simplify in e2;
-generalize in match f as f;generalize in match c as c;
-clear f c;
-apply (R1 ????? e2);intros;simplify in H;
-elim daemon;
+generalize in match H as H;generalize in match Heq as Heq;
+generalize in match f as f;
+clear H Heq f;
+apply (R1 ????? e2);intros 4;simplify in e3;
+generalize in match H as H;generalize in match Heq as Heq;
+clear H Heq;
+apply (R1 ????? e3);intros;simplify in H1;
+apply (R1 ????? H1);
+assumption;
qed.