+
+naxiom T1 : Type[0].
+naxiom T2 : T1 → Type[0].
+naxiom t1 : T1.
+naxiom t2 : ∀x:T1. T2 x.
+
+ninductive I2 : ∀r1:T1.T2 r1 → Type[0] ≝
+| i2c1 : ∀x1:T1.∀x2:T2 x1. I2 x1 x2
+| i2c2 : I2 t1 (t2 t1).
+
+(* nlemma i2d : ∀a,b.∀x,y:I2 a b.
+ ∀e1:a = a.∀e2:R1 T1 a (λz,p.T2 z) b a e1 = b.
+ ∀e: R2 T1 a (λz,p.T2 z) b (λz1,p1,z2,p2.I2 z1 z2) x a e1 b e2 = y.
+ Type[2].
+#a;#b;#x;#y;
+napply (
+match x return (λr1,r2,r.
+ ∀e1:r1 = a. ∀e2:R1 T1 r1 (λz,p. T2 z) r2 a e1 = b.
+ ∀e :R2 T1 r1 (λz,p. T2 z) r2 (λz1,p1,z2,p2. I2 z1 z2) r a e1 b e2 = y. Type[2]) with
+ [ i2c1 x1 x2 ⇒ ?
+ | i2c2 ⇒ ?]
+)
+[napply (match y return (λr1,r2,r.
+ ∀e1: x1 = r1. ∀e2: R1 T1 x1 (λz,p. T2 z) x2 r1 e1 = r2.
+ ∀e : R2 T1 x1 (λz,p.T2 z) x2 (λz1,p1,z2,p2. I2 z1 z2) (i2c1 x1 x2) r1 e1 r2 e2 = r. Type[2]) with
+ [ i2c1 y1 y2 ⇒ ?
+ | i2c2 ⇒ ? ])
+ [#e1; #e2; #e;
+ napply (∀P:Type[1].
+ (∀f1:x1 = y1. ∀f2: R1 T1 x1 (λz,p.T2 z) x2 y1 f1 = y2.
+ ∀f: R2 T1 x1 (λz,p.T2 z) x2
+ (λz1,p1,z2,p2.eq ?
+ (i2c1 (R1 ??? z1 ? (R1 ?? (λm,n.m = y1) f1 ? p1)) ?)
+ (* (R2 ???? (λm1,n1,m2,n2.R1 ?? (λm,n.T2 m) ? ? f1 = y2) f2 ?
+ p1 ? p2)))*)
+(* (R2 ???? (λw1,q1,w2,q2.I2 w1 w2) (i2c1 z1 z2)
+ ? (R1 ?? (λw,q.w = y1) e1 z1 p1)
+ ? (R2 ????
+ (λw1,q1,w2,q2.R1 ?? (λm,n.T2 m) w2 ? q1 = y2)
+ e2 z1 p1 (R1 T1 x1 (λw,q.w = y1) e1 z1 p1) p2))
+ *) (i2c1 y1 y2))
+ ? y1 f1 y2 f2 = refl (I2 y1 y2) (i2c1 y1 y2).P)
+ → P);
+ napply (∀P:Type[1].
+ (∀f1:x1 = y1. ∀f2: R1 T1 x1 (λz,p.T2 z) x2 y1 f1 = y2.
+ ∀f: R2 T1 x1 (λz,p.T2 z) x2
+ (λz1,p1,z2,p2.eq (I2 y1 y2)
+ (R2 T1 z1 (λw,q.T2 w) z2 (λw1,q1,w2,q2.I2 w1 w2) (i2c1 z1 z2)
+ y1 (R1 T1 x1 (λw,q.w = y1) e1 z1 p1)
+ y2 (R2 T1 x1 (λw,q.w = y1) e1
+ (λw1,q1,w2,q2.R1 ??? w2 w1 q1 = y2) e2 z1 p1 (R1 T1 x1 (λw,q.w = y1) e1 z1 p1) p2))
+ (i2c1 y1 y2))
+ e y1 f1 y2 f2 = refl (I2 y1 y2) (i2c1 y1 y2).P)
+ → P);
+
+
+
+ndefinition i2d : ∀a,b.∀x,y:I2 a b.
+ ∀e1:a = a.∀e2:R1 T1 a (λz,p.T2 z) b a e1 = b.
+ ∀e: R2 T1 a (λz,p.T2 z) b (λz1,p1,z2,p2.I2 z1 z2) x a e1 b e2 = y.Type[2] ≝
+λa,b,x,y.
+match x return (λr1,r2,r.
+ ∀e1:r1 = a. ∀e2:R1 T1 r1 (λz,p. T2 z) r2 a e1 = b.
+ ∀e :R2 T1 r1 (λz,p. T2 z) r2 (λz1,p1,z2,p2. I2 z1 z2) r a e1 b e2 = y. Type[2]) with
+ [ i2c1 x1 x2 ⇒
+ match y return (λr1,r2,r.
+ ∀e1: x1 = r1. ∀e2: R1 T1 x1 (λz,p. T2 z) x2 r1 e1 = r2.
+ ∀e : R2 T1 x1 (λz,p.T2 z) x2 (λz1,p1,z2,p2. I2 z1 z2) (i2c1 x1 x2) r1 e1 r2 e2 = r. Type[2]) with
+ [ i2c1 y1 y2 ⇒ λe1,e2,e.∀P:Type[1].
+ (∀f1:x1 = y1. ∀f2: R1 T1 x1 (λz,p.T2 z) x2 y1 f1 = y2.
+ ∀f: R2 T1 x1 (λz,p.T2 z) x2
+ (λz1,p1,z2,p2.eq (I2 y1 y2)
+ (R2 T1 z1 (λw,q.T2 w) z2 (λw1,q1,w2,q2.I2 w1 w2) (i2c1 z1 z2)
+ y1 (R1 T1 x1 (λw,q.w = y1) e1 z1 p1)
+ y2 (R2 T1 x1 (λw,q.w = y1) e1
+ (λw1,q1,w2,q2.R1 ??? w2 w1 q1 = y2) e2 z1 p1 (R1 T1 x1 (λw,q.w = y1) e1 z1 p1) p2))
+ (i2c1 y1 y2))
+ e y1 f1 y2 f2 = refl (I2 y1 y2) (i2c1 y1 y2).P)
+ → P
+ | i2c2 ⇒ λe1,e2,e.∀P:Type[1].P ]
+ | i2c2 ⇒
+ match y return (λr1,r2,r.
+ ∀e1: x1 = r1. ∀e2: R1 ?? (λz,p. T2 z) x2 ? e1 = r2.
+ ∀e : R2 ???? (λz1,p1,z2,p2. I2 z1 z2) i2c2 ? e1 ? e2 = r. Type[2]) with
+ [ i2c1 _ _ ⇒ λe1,e2,e.∀P:Type[1].P
+ | i2c2 ⇒ λe1,e2,e.∀P:Type[1].
+ (∀f: R2 ????
+ (λz1,p1,z2,p2.eq ? i2c2 i2c2)
+ e ? e1 ? e2 = refl ? i2c2.P) → P ] ].
+
+*)
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