ninductive eq (A: Type[0]) (a: A) : A → CProp[0] ≝
refl: eq A a a.
-nlemma eq_rect_Type0_r':
- ∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → P x p.
- #A; #a; #x; #p; ncases p; #P; #H; nassumption.
-nqed.
-
-nlemma eq_rect_Type0_r:
- ∀A.∀a.∀P: ∀x:A. eq ? x a → Type[0]. P a (refl A a) → ∀x.∀p:eq ? x a.P x p.
- #A; #a; #P; #p; #x0; #p0; napply (eq_rect_Type0_r' ??? p0); nassumption.
-nqed.
-
nlemma eq_rect_CProp0_r':
∀A.∀a,x.∀p:eq ? x a.∀P: ∀x:A. eq ? x a → CProp[0]. P a (refl A a) → P x p.
#A; #a; #x; #p; ncases p; #P; #H; nassumption.
napply a4;
nqed.
-naxiom streicherK : ∀T:Type[0].∀t:T.∀P:t = t → Type[2].P (refl ? t) → ∀p.P p.
-
ndefinition EQ: ∀A:Type[0]. equivalence_relation A.
#A; napply mk_equivalence_relation
[ napply eq
| napply refl
| #x; #y; #H; nrewrite < H; napply refl
| #x; #y; #z; #Hyx; #Hxz; nrewrite < Hxz; nassumption]
-nqed.
-
-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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+nqed.
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