DEFINITION ty3_gen_cast()
TYPE =
∀g:G
.∀c:C
.∀t1:T
.∀t2:T
.∀x:T
.ty3 g c (THead (Flat Cast) t2 t1) x
→ex3 T λt0:T.pc3 c (THead (Flat Cast) t0 t2) x λ:T.ty3 g c t1 t2 λt0:T.ty3 g c t2 t0
BODY =
assume g: G
assume c: C
assume t1: T
assume t2: T
assume x: T
suppose H: ty3 g c (THead (Flat Cast) t2 t1) x
assume y: T
suppose H0: ty3 g c y x
we proceed by induction on H0 to prove
eq T y (THead (Flat Cast) t2 t1)
→ex3 T λt3:T.pc3 c (THead (Flat Cast) t3 t2) x λ:T.ty3 g c t1 t2 λt3:T.ty3 g c t2 t3
case ty3_conv : c0:C t0:T t:T :ty3 g c0 t0 t u:T t3:T H3:ty3 g c0 u t3 H5:pc3 c0 t3 t0 ⇒
the thesis becomes
∀H6:eq T u (THead (Flat Cast) t2 t1)
.ex3 T λt4:T.pc3 c0 (THead (Flat Cast) t4 t2) t0 λ:T.ty3 g c0 t1 t2 λt4:T.ty3 g c0 t2 t4
() by induction hypothesis we know
eq T t0 (THead (Flat Cast) t2 t1)
→ex3 T λt3:T.pc3 c0 (THead (Flat Cast) t3 t2) t λ:T.ty3 g c0 t1 t2 λt3:T.ty3 g c0 t2 t3
(H4) by induction hypothesis we know
eq T u (THead (Flat Cast) t2 t1)
→ex3 T λt4:T.pc3 c0 (THead (Flat Cast) t4 t2) t3 λ:T.ty3 g c0 t1 t2 λt4:T.ty3 g c0 t2 t4
suppose H6: eq T u (THead (Flat Cast) t2 t1)
(H7)
by (f_equal . . . . . H6)
we proved eq T u (THead (Flat Cast) t2 t1)
eq T (λe:T.e u) (λe:T.e (THead (Flat Cast) t2 t1))
end of H7
(H8)
we proceed by induction on H7 to prove
eq T (THead (Flat Cast) t2 t1) (THead (Flat Cast) t2 t1)
→ex3 T λt5:T.pc3 c0 (THead (Flat Cast) t5 t2) t3 λ:T.ty3 g c0 t1 t2 λt5:T.ty3 g c0 t2 t5
case refl_equal : ⇒
the thesis becomes the hypothesis H4
eq T (THead (Flat Cast) t2 t1) (THead (Flat Cast) t2 t1)
→ex3 T λt5:T.pc3 c0 (THead (Flat Cast) t5 t2) t3 λ:T.ty3 g c0 t1 t2 λt5:T.ty3 g c0 t2 t5
end of H8
(H10)
by (refl_equal . .)
we proved eq T (THead (Flat Cast) t2 t1) (THead (Flat Cast) t2 t1)
by (H8 previous)
ex3 T λt5:T.pc3 c0 (THead (Flat Cast) t5 t2) t3 λ:T.ty3 g c0 t1 t2 λt5:T.ty3 g c0 t2 t5
end of H10
we proceed by induction on H10 to prove ex3 T λt4:T.pc3 c0 (THead (Flat Cast) t4 t2) t0 λ:T.ty3 g c0 t1 t2 λt4:T.ty3 g c0 t2 t4
case ex3_intro : x0:T H11:pc3 c0 (THead (Flat Cast) x0 t2) t3 H12:ty3 g c0 t1 t2 H13:ty3 g c0 t2 x0 ⇒
the thesis becomes ex3 T λt4:T.pc3 c0 (THead (Flat Cast) t4 t2) t0 λ:T.ty3 g c0 t1 t2 λt4:T.ty3 g c0 t2 t4
by (pc3_t . . . H11 . H5)
we proved pc3 c0 (THead (Flat Cast) x0 t2) t0
by (ex3_intro . . . . . previous H12 H13)
ex3 T λt4:T.pc3 c0 (THead (Flat Cast) t4 t2) t0 λ:T.ty3 g c0 t1 t2 λt4:T.ty3 g c0 t2 t4
we proved ex3 T λt4:T.pc3 c0 (THead (Flat Cast) t4 t2) t0 λ:T.ty3 g c0 t1 t2 λt4:T.ty3 g c0 t2 t4
∀H6:eq T u (THead (Flat Cast) t2 t1)
.ex3 T λt4:T.pc3 c0 (THead (Flat Cast) t4 t2) t0 λ:T.ty3 g c0 t1 t2 λt4:T.ty3 g c0 t2 t4
case ty3_sort : c0:C m:nat ⇒
the thesis becomes
∀H1:eq T (TSort m) (THead (Flat Cast) t2 t1)
.ex3 T λt0:T.pc3 c0 (THead (Flat Cast) t0 t2) (TSort (next g m)) λ:T.ty3 g c0 t1 t2 λt0:T.ty3 g c0 t2 t0
suppose H1: eq T (TSort m) (THead (Flat Cast) t2 t1)
(H2)
we proceed by induction on H1 to prove
<λ:T.Prop>
CASE THead (Flat Cast) t2 t1 OF
TSort ⇒True
| TLRef ⇒False
| THead ⇒False
case refl_equal : ⇒
the thesis becomes
<λ:T.Prop>
CASE TSort m OF
TSort ⇒True
| TLRef ⇒False
| THead ⇒False
consider I
we proved True
<λ:T.Prop>
CASE TSort m OF
TSort ⇒True
| TLRef ⇒False
| THead ⇒False
<λ:T.Prop>
CASE THead (Flat Cast) t2 t1 OF
TSort ⇒True
| TLRef ⇒False
| THead ⇒False
end of H2
consider H2
we proved
<λ:T.Prop>
CASE THead (Flat Cast) t2 t1 OF
TSort ⇒True
| TLRef ⇒False
| THead ⇒False
that is equivalent to False
we proceed by induction on the previous result to prove ex3 T λt0:T.pc3 c0 (THead (Flat Cast) t0 t2) (TSort (next g m)) λ:T.ty3 g c0 t1 t2 λt0:T.ty3 g c0 t2 t0
we proved ex3 T λt0:T.pc3 c0 (THead (Flat Cast) t0 t2) (TSort (next g m)) λ:T.ty3 g c0 t1 t2 λt0:T.ty3 g c0 t2 t0
∀H1:eq T (TSort m) (THead (Flat Cast) t2 t1)
.ex3 T λt0:T.pc3 c0 (THead (Flat Cast) t0 t2) (TSort (next g m)) λ:T.ty3 g c0 t1 t2 λt0:T.ty3 g c0 t2 t0
case ty3_abbr : n:nat c0:C d:C u:T :getl n c0 (CHead d (Bind Abbr) u) t:T :ty3 g d u t ⇒
the thesis becomes
∀H4:eq T (TLRef n) (THead (Flat Cast) t2 t1)
.ex3 T λt0:T.pc3 c0 (THead (Flat Cast) t0 t2) (lift (S n) O t) λ:T.ty3 g c0 t1 t2 λt0:T.ty3 g c0 t2 t0
() by induction hypothesis we know
eq T u (THead (Flat Cast) t2 t1)
→ex3 T λt0:T.pc3 d (THead (Flat Cast) t0 t2) t λ:T.ty3 g d t1 t2 λt0:T.ty3 g d t2 t0
suppose H4: eq T (TLRef n) (THead (Flat Cast) t2 t1)
(H5)
we proceed by induction on H4 to prove
<λ:T.Prop>
CASE THead (Flat Cast) t2 t1 OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
case refl_equal : ⇒
the thesis becomes
<λ:T.Prop>
CASE TLRef n OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
consider I
we proved True
<λ:T.Prop>
CASE TLRef n OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
<λ:T.Prop>
CASE THead (Flat Cast) t2 t1 OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
end of H5
consider H5
we proved
<λ:T.Prop>
CASE THead (Flat Cast) t2 t1 OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
that is equivalent to False
we proceed by induction on the previous result to prove ex3 T λt0:T.pc3 c0 (THead (Flat Cast) t0 t2) (lift (S n) O t) λ:T.ty3 g c0 t1 t2 λt0:T.ty3 g c0 t2 t0
we proved ex3 T λt0:T.pc3 c0 (THead (Flat Cast) t0 t2) (lift (S n) O t) λ:T.ty3 g c0 t1 t2 λt0:T.ty3 g c0 t2 t0
∀H4:eq T (TLRef n) (THead (Flat Cast) t2 t1)
.ex3 T λt0:T.pc3 c0 (THead (Flat Cast) t0 t2) (lift (S n) O t) λ:T.ty3 g c0 t1 t2 λt0:T.ty3 g c0 t2 t0
case ty3_abst : n:nat c0:C d:C u:T :getl n c0 (CHead d (Bind Abst) u) t:T :ty3 g d u t ⇒
the thesis becomes
∀H4:eq T (TLRef n) (THead (Flat Cast) t2 t1)
.ex3 T λt0:T.pc3 c0 (THead (Flat Cast) t0 t2) (lift (S n) O u) λ:T.ty3 g c0 t1 t2 λt0:T.ty3 g c0 t2 t0
() by induction hypothesis we know
eq T u (THead (Flat Cast) t2 t1)
→ex3 T λt0:T.pc3 d (THead (Flat Cast) t0 t2) t λ:T.ty3 g d t1 t2 λt0:T.ty3 g d t2 t0
suppose H4: eq T (TLRef n) (THead (Flat Cast) t2 t1)
(H5)
we proceed by induction on H4 to prove
<λ:T.Prop>
CASE THead (Flat Cast) t2 t1 OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
case refl_equal : ⇒
the thesis becomes
<λ:T.Prop>
CASE TLRef n OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
consider I
we proved True
<λ:T.Prop>
CASE TLRef n OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
<λ:T.Prop>
CASE THead (Flat Cast) t2 t1 OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
end of H5
consider H5
we proved
<λ:T.Prop>
CASE THead (Flat Cast) t2 t1 OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
that is equivalent to False
we proceed by induction on the previous result to prove ex3 T λt0:T.pc3 c0 (THead (Flat Cast) t0 t2) (lift (S n) O u) λ:T.ty3 g c0 t1 t2 λt0:T.ty3 g c0 t2 t0
we proved ex3 T λt0:T.pc3 c0 (THead (Flat Cast) t0 t2) (lift (S n) O u) λ:T.ty3 g c0 t1 t2 λt0:T.ty3 g c0 t2 t0
∀H4:eq T (TLRef n) (THead (Flat Cast) t2 t1)
.ex3 T λt0:T.pc3 c0 (THead (Flat Cast) t0 t2) (lift (S n) O u) λ:T.ty3 g c0 t1 t2 λt0:T.ty3 g c0 t2 t0
case ty3_bind : c0:C u:T t:T :ty3 g c0 u t b:B t0:T t3:T :ty3 g (CHead c0 (Bind b) u) t0 t3 ⇒
the thesis becomes
∀H5:eq T (THead (Bind b) u t0) (THead (Flat Cast) t2 t1)
.ex3 T λt4:T.pc3 c0 (THead (Flat Cast) t4 t2) (THead (Bind b) u t3) λ:T.ty3 g c0 t1 t2 λt4:T.ty3 g c0 t2 t4
() by induction hypothesis we know
eq T u (THead (Flat Cast) t2 t1)
→ex3 T λt0:T.pc3 c0 (THead (Flat Cast) t0 t2) t λ:T.ty3 g c0 t1 t2 λt0:T.ty3 g c0 t2 t0
() by induction hypothesis we know
eq T t0 (THead (Flat Cast) t2 t1)
→(ex3
T
λt4:T.pc3 (CHead c0 (Bind b) u) (THead (Flat Cast) t4 t2) t3
λ:T.ty3 g (CHead c0 (Bind b) u) t1 t2
λt4:T.ty3 g (CHead c0 (Bind b) u) t2 t4)
suppose H5: eq T (THead (Bind b) u t0) (THead (Flat Cast) t2 t1)
(H6)
we proceed by induction on H5 to prove
<λ:T.Prop>
CASE THead (Flat Cast) t2 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒True | Flat ⇒False
case refl_equal : ⇒
the thesis becomes
<λ:T.Prop>
CASE THead (Bind b) u t0 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒True | Flat ⇒False
consider I
we proved True
<λ:T.Prop>
CASE THead (Bind b) u t0 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒True | Flat ⇒False
<λ:T.Prop>
CASE THead (Flat Cast) t2 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒True | Flat ⇒False
end of H6
consider H6
we proved
<λ:T.Prop>
CASE THead (Flat Cast) t2 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒True | Flat ⇒False
that is equivalent to False
we proceed by induction on the previous result to prove ex3 T λt4:T.pc3 c0 (THead (Flat Cast) t4 t2) (THead (Bind b) u t3) λ:T.ty3 g c0 t1 t2 λt4:T.ty3 g c0 t2 t4
we proved ex3 T λt4:T.pc3 c0 (THead (Flat Cast) t4 t2) (THead (Bind b) u t3) λ:T.ty3 g c0 t1 t2 λt4:T.ty3 g c0 t2 t4
∀H5:eq T (THead (Bind b) u t0) (THead (Flat Cast) t2 t1)
.ex3 T λt4:T.pc3 c0 (THead (Flat Cast) t4 t2) (THead (Bind b) u t3) λ:T.ty3 g c0 t1 t2 λt4:T.ty3 g c0 t2 t4
case ty3_appl : c0:C w:T u:T :ty3 g c0 w u v:T t:T :ty3 g c0 v (THead (Bind Abst) u t) ⇒
the thesis becomes
∀H5:eq T (THead (Flat Appl) w v) (THead (Flat Cast) t2 t1)
.ex3
T
λt0:T
.pc3
c0
THead (Flat Cast) t0 t2
THead (Flat Appl) w (THead (Bind Abst) u t)
λ:T.ty3 g c0 t1 t2
λt0:T.ty3 g c0 t2 t0
() by induction hypothesis we know
eq T w (THead (Flat Cast) t2 t1)
→ex3 T λt0:T.pc3 c0 (THead (Flat Cast) t0 t2) u λ:T.ty3 g c0 t1 t2 λt0:T.ty3 g c0 t2 t0
() by induction hypothesis we know
eq T v (THead (Flat Cast) t2 t1)
→(ex3
T
λt0:T.pc3 c0 (THead (Flat Cast) t0 t2) (THead (Bind Abst) u t)
λ:T.ty3 g c0 t1 t2
λt0:T.ty3 g c0 t2 t0)
suppose H5: eq T (THead (Flat Appl) w v) (THead (Flat Cast) t2 t1)
(H6)
we proceed by induction on H5 to prove
<λ:T.Prop>
CASE THead (Flat Cast) t2 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒
<λ:K.Prop>
CASE k OF
Bind ⇒False
| Flat f⇒<λ:F.Prop> CASE f OF Appl⇒True | Cast⇒False
case refl_equal : ⇒
the thesis becomes
<λ:T.Prop>
CASE THead (Flat Appl) w v OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒
<λ:K.Prop>
CASE k OF
Bind ⇒False
| Flat f⇒<λ:F.Prop> CASE f OF Appl⇒True | Cast⇒False
consider I
we proved True
<λ:T.Prop>
CASE THead (Flat Appl) w v OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒
<λ:K.Prop>
CASE k OF
Bind ⇒False
| Flat f⇒<λ:F.Prop> CASE f OF Appl⇒True | Cast⇒False
<λ:T.Prop>
CASE THead (Flat Cast) t2 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒
<λ:K.Prop>
CASE k OF
Bind ⇒False
| Flat f⇒<λ:F.Prop> CASE f OF Appl⇒True | Cast⇒False
end of H6
consider H6
we proved
<λ:T.Prop>
CASE THead (Flat Cast) t2 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒
<λ:K.Prop>
CASE k OF
Bind ⇒False
| Flat f⇒<λ:F.Prop> CASE f OF Appl⇒True | Cast⇒False
that is equivalent to False
we proceed by induction on the previous result to prove
ex3
T
λt0:T
.pc3
c0
THead (Flat Cast) t0 t2
THead (Flat Appl) w (THead (Bind Abst) u t)
λ:T.ty3 g c0 t1 t2
λt0:T.ty3 g c0 t2 t0
we proved
ex3
T
λt0:T
.pc3
c0
THead (Flat Cast) t0 t2
THead (Flat Appl) w (THead (Bind Abst) u t)
λ:T.ty3 g c0 t1 t2
λt0:T.ty3 g c0 t2 t0
∀H5:eq T (THead (Flat Appl) w v) (THead (Flat Cast) t2 t1)
.ex3
T
λt0:T
.pc3
c0
THead (Flat Cast) t0 t2
THead (Flat Appl) w (THead (Bind Abst) u t)
λ:T.ty3 g c0 t1 t2
λt0:T.ty3 g c0 t2 t0
case ty3_cast : c0:C t0:T t3:T H1:ty3 g c0 t0 t3 t4:T H3:ty3 g c0 t3 t4 ⇒
the thesis becomes
∀H5:eq T (THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1)
.ex3
T
λt5:T.pc3 c0 (THead (Flat Cast) t5 t2) (THead (Flat Cast) t4 t3)
λ:T.ty3 g c0 t1 t2
λt5:T.ty3 g c0 t2 t5
(H2) by induction hypothesis we know
eq T t0 (THead (Flat Cast) t2 t1)
→ex3 T λt4:T.pc3 c0 (THead (Flat Cast) t4 t2) t3 λ:T.ty3 g c0 t1 t2 λt4:T.ty3 g c0 t2 t4
(H4) by induction hypothesis we know
eq T t3 (THead (Flat Cast) t2 t1)
→ex3 T λt5:T.pc3 c0 (THead (Flat Cast) t5 t2) t4 λ:T.ty3 g c0 t1 t2 λt5:T.ty3 g c0 t2 t5
suppose H5: eq T (THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1)
(H6)
by (f_equal . . . . . H5)
we proved
eq
T
<λ:T.T> CASE THead (Flat Cast) t3 t0 OF TSort ⇒t3 | TLRef ⇒t3 | THead t ⇒t
<λ:T.T> CASE THead (Flat Cast) t2 t1 OF TSort ⇒t3 | TLRef ⇒t3 | THead t ⇒t
eq
T
λe:T.<λ:T.T> CASE e OF TSort ⇒t3 | TLRef ⇒t3 | THead t ⇒t
THead (Flat Cast) t3 t0
λe:T.<λ:T.T> CASE e OF TSort ⇒t3 | TLRef ⇒t3 | THead t ⇒t
THead (Flat Cast) t2 t1
end of H6
(h1)
(H7)
by (f_equal . . . . . H5)
we proved
eq
T
<λ:T.T> CASE THead (Flat Cast) t3 t0 OF TSort ⇒t0 | TLRef ⇒t0 | THead t⇒t
<λ:T.T> CASE THead (Flat Cast) t2 t1 OF TSort ⇒t0 | TLRef ⇒t0 | THead t⇒t
eq
T
λe:T.<λ:T.T> CASE e OF TSort ⇒t0 | TLRef ⇒t0 | THead t⇒t
THead (Flat Cast) t3 t0
λe:T.<λ:T.T> CASE e OF TSort ⇒t0 | TLRef ⇒t0 | THead t⇒t
THead (Flat Cast) t2 t1
end of H7
suppose H8: eq T t3 t2
(H10)
we proceed by induction on H8 to prove ty3 g c0 t2 t4
case refl_equal : ⇒
the thesis becomes the hypothesis H3
ty3 g c0 t2 t4
end of H10
(H11)
we proceed by induction on H8 to prove
eq T t0 (THead (Flat Cast) t2 t1)
→ex3 T λt5:T.pc3 c0 (THead (Flat Cast) t5 t2) t2 λ:T.ty3 g c0 t1 t2 λt5:T.ty3 g c0 t2 t5
case refl_equal : ⇒
the thesis becomes the hypothesis H2
eq T t0 (THead (Flat Cast) t2 t1)
→ex3 T λt5:T.pc3 c0 (THead (Flat Cast) t5 t2) t2 λ:T.ty3 g c0 t1 t2 λt5:T.ty3 g c0 t2 t5
end of H11
(H12)
we proceed by induction on H8 to prove ty3 g c0 t0 t2
case refl_equal : ⇒
the thesis becomes the hypothesis H1
ty3 g c0 t0 t2
end of H12
(H14)
consider H7
we proved
eq
T
<λ:T.T> CASE THead (Flat Cast) t3 t0 OF TSort ⇒t0 | TLRef ⇒t0 | THead t⇒t
<λ:T.T> CASE THead (Flat Cast) t2 t1 OF TSort ⇒t0 | TLRef ⇒t0 | THead t⇒t
that is equivalent to eq T t0 t1
we proceed by induction on the previous result to prove ty3 g c0 t1 t2
case refl_equal : ⇒
the thesis becomes the hypothesis H12
ty3 g c0 t1 t2
end of H14
by (pc3_refl . .)
we proved pc3 c0 (THead (Flat Cast) t4 t2) (THead (Flat Cast) t4 t2)
by (ex3_intro . . . . . previous H14 H10)
we proved
ex3
T
λt5:T.pc3 c0 (THead (Flat Cast) t5 t2) (THead (Flat Cast) t4 t2)
λ:T.ty3 g c0 t1 t2
λt5:T.ty3 g c0 t2 t5
by (eq_ind_r . . . previous . H8)
we proved
ex3
T
λt5:T.pc3 c0 (THead (Flat Cast) t5 t2) (THead (Flat Cast) t4 t3)
λ:T.ty3 g c0 t1 t2
λt5:T.ty3 g c0 t2 t5
eq T t3 t2
→(ex3
T
λt5:T.pc3 c0 (THead (Flat Cast) t5 t2) (THead (Flat Cast) t4 t3)
λ:T.ty3 g c0 t1 t2
λt5:T.ty3 g c0 t2 t5)
end of h1
(h2)
consider H6
we proved
eq
T
<λ:T.T> CASE THead (Flat Cast) t3 t0 OF TSort ⇒t3 | TLRef ⇒t3 | THead t ⇒t
<λ:T.T> CASE THead (Flat Cast) t2 t1 OF TSort ⇒t3 | TLRef ⇒t3 | THead t ⇒t
eq T t3 t2
end of h2
by (h1 h2)
we proved
ex3
T
λt5:T.pc3 c0 (THead (Flat Cast) t5 t2) (THead (Flat Cast) t4 t3)
λ:T.ty3 g c0 t1 t2
λt5:T.ty3 g c0 t2 t5
∀H5:eq T (THead (Flat Cast) t3 t0) (THead (Flat Cast) t2 t1)
.ex3
T
λt5:T.pc3 c0 (THead (Flat Cast) t5 t2) (THead (Flat Cast) t4 t3)
λ:T.ty3 g c0 t1 t2
λt5:T.ty3 g c0 t2 t5
we proved
eq T y (THead (Flat Cast) t2 t1)
→ex3 T λt3:T.pc3 c (THead (Flat Cast) t3 t2) x λ:T.ty3 g c t1 t2 λt3:T.ty3 g c t2 t3
we proved
∀y:T
.ty3 g c y x
→(eq T y (THead (Flat Cast) t2 t1)
→ex3 T λt3:T.pc3 c (THead (Flat Cast) t3 t2) x λ:T.ty3 g c t1 t2 λt3:T.ty3 g c t2 t3)
by (insert_eq . . . . previous H)
we proved ex3 T λt0:T.pc3 c (THead (Flat Cast) t0 t2) x λ:T.ty3 g c t1 t2 λt0:T.ty3 g c t2 t0
we proved
∀g:G
.∀c:C
.∀t1:T
.∀t2:T
.∀x:T
.ty3 g c (THead (Flat Cast) t2 t1) x
→ex3 T λt0:T.pc3 c (THead (Flat Cast) t0 t2) x λ:T.ty3 g c t1 t2 λt0:T.ty3 g c t2 t0