DEFINITION pr3_lift()
TYPE =
∀c:C
.∀e:C
.∀h:nat
.∀d:nat.(drop h d c e)→∀t1:T.∀t2:T.(pr3 e t1 t2)→(pr3 c (lift h d t1) (lift h d t2))
BODY =
assume c: C
assume e: C
assume h: nat
assume d: nat
suppose H: drop h d c e
assume t1: T
assume t2: T
suppose H0: pr3 e t1 t2
we proceed by induction on H0 to prove pr3 c (lift h d t1) (lift h d t2)
case pr3_refl : t:T ⇒
the thesis becomes pr3 c (lift h d t) (lift h d t)
by (pr3_refl . .)
pr3 c (lift h d t) (lift h d t)
case pr3_sing : t0:T t3:T H1:pr2 e t3 t0 t4:T :pr3 e t0 t4 ⇒
the thesis becomes pr3 c (lift h d t3) (lift h d t4)
(H3) by induction hypothesis we know pr3 c (lift h d t0) (lift h d t4)
by (pr2_lift . . . . H . . H1)
we proved pr2 c (lift h d t3) (lift h d t0)
by (pr3_sing . . . previous . H3)
pr3 c (lift h d t3) (lift h d t4)
we proved pr3 c (lift h d t1) (lift h d t2)
we proved
∀c:C
.∀e:C
.∀h:nat
.∀d:nat.(drop h d c e)→∀t1:T.∀t2:T.(pr3 e t1 t2)→(pr3 c (lift h d t1) (lift h d t2))