DEFINITION csubt_gen_flat()
TYPE =
∀g:G
.∀e1:C
.∀c2:C
.∀v:T
.∀f:F
.csubt g (CHead e1 (Flat f) v) c2
→ex2 C λe2:C.eq C c2 (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
BODY =
assume g: G
assume e1: C
assume c2: C
assume v: T
assume f: F
suppose H: csubt g (CHead e1 (Flat f) v) c2
assume y: C
suppose H0: csubt g y c2
we proceed by induction on H0 to prove
eq C y (CHead e1 (Flat f) v)
→ex2 C λe2:C.eq C c2 (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
case csubt_sort : n:nat ⇒
the thesis becomes
∀H1:eq C (CSort n) (CHead e1 (Flat f) v)
.ex2 C λe2:C.eq C (CSort n) (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
suppose H1: eq C (CSort n) (CHead e1 (Flat f) v)
(H2)
we proceed by induction on H1 to prove <λ:C.Prop> CASE CHead e1 (Flat f) v OF CSort ⇒True | CHead ⇒False
case refl_equal : ⇒
the thesis becomes <λ:C.Prop> CASE CSort n OF CSort ⇒True | CHead ⇒False
consider I
we proved True
<λ:C.Prop> CASE CSort n OF CSort ⇒True | CHead ⇒False
<λ:C.Prop> CASE CHead e1 (Flat f) v OF CSort ⇒True | CHead ⇒False
end of H2
consider H2
we proved <λ:C.Prop> CASE CHead e1 (Flat f) v OF CSort ⇒True | CHead ⇒False
that is equivalent to False
we proceed by induction on the previous result to prove ex2 C λe2:C.eq C (CSort n) (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
we proved ex2 C λe2:C.eq C (CSort n) (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
∀H1:eq C (CSort n) (CHead e1 (Flat f) v)
.ex2 C λe2:C.eq C (CSort n) (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
case csubt_head : c1:C c3:C H1:csubt g c1 c3 k:K u:T ⇒
the thesis becomes
∀H3:eq C (CHead c1 k u) (CHead e1 (Flat f) v)
.ex2 C λe2:C.eq C (CHead c3 k u) (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
(H2) by induction hypothesis we know
eq C c1 (CHead e1 (Flat f) v)
→ex2 C λe2:C.eq C c3 (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
suppose H3: eq C (CHead c1 k u) (CHead e1 (Flat f) v)
(H4)
by (f_equal . . . . . H3)
we proved
eq
C
<λ:C.C> CASE CHead c1 k u OF CSort ⇒c1 | CHead c ⇒c
<λ:C.C> CASE CHead e1 (Flat f) v OF CSort ⇒c1 | CHead c ⇒c
eq
C
λe:C.<λ:C.C> CASE e OF CSort ⇒c1 | CHead c ⇒c (CHead c1 k u)
λe:C.<λ:C.C> CASE e OF CSort ⇒c1 | CHead c ⇒c (CHead e1 (Flat f) v)
end of H4
(h1)
(H5)
by (f_equal . . . . . H3)
we proved
eq
K
<λ:C.K> CASE CHead c1 k u OF CSort ⇒k | CHead k0 ⇒k0
<λ:C.K> CASE CHead e1 (Flat f) v OF CSort ⇒k | CHead k0 ⇒k0
eq
K
λe:C.<λ:C.K> CASE e OF CSort ⇒k | CHead k0 ⇒k0 (CHead c1 k u)
λe:C.<λ:C.K> CASE e OF CSort ⇒k | CHead k0 ⇒k0 (CHead e1 (Flat f) v)
end of H5
(h1)
(H6)
by (f_equal . . . . . H3)
we proved
eq
T
<λ:C.T> CASE CHead c1 k u OF CSort ⇒u | CHead t⇒t
<λ:C.T> CASE CHead e1 (Flat f) v OF CSort ⇒u | CHead t⇒t
eq
T
λe:C.<λ:C.T> CASE e OF CSort ⇒u | CHead t⇒t (CHead c1 k u)
λe:C.<λ:C.T> CASE e OF CSort ⇒u | CHead t⇒t (CHead e1 (Flat f) v)
end of H6
suppose H7: eq K k (Flat f)
suppose H8: eq C c1 e1
(h1)
(H10)
we proceed by induction on H8 to prove csubt g e1 c3
case refl_equal : ⇒
the thesis becomes the hypothesis H1
csubt g e1 c3
end of H10
by (refl_equal . .)
we proved eq C (CHead c3 (Flat f) v) (CHead c3 (Flat f) v)
by (ex_intro2 . . . . previous H10)
we proved
ex2 C λe2:C.eq C (CHead c3 (Flat f) v) (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
by (eq_ind_r . . . previous . H7)
ex2 C λe2:C.eq C (CHead c3 k v) (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
end of h1
(h2)
consider H6
we proved
eq
T
<λ:C.T> CASE CHead c1 k u OF CSort ⇒u | CHead t⇒t
<λ:C.T> CASE CHead e1 (Flat f) v OF CSort ⇒u | CHead t⇒t
eq T u v
end of h2
by (eq_ind_r . . . h1 . h2)
we proved ex2 C λe2:C.eq C (CHead c3 k u) (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
eq K k (Flat f)
→(eq C c1 e1
→ex2 C λe2:C.eq C (CHead c3 k u) (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2)
end of h1
(h2)
consider H5
we proved
eq
K
<λ:C.K> CASE CHead c1 k u OF CSort ⇒k | CHead k0 ⇒k0
<λ:C.K> CASE CHead e1 (Flat f) v OF CSort ⇒k | CHead k0 ⇒k0
eq K k (Flat f)
end of h2
by (h1 h2)
eq C c1 e1
→ex2 C λe2:C.eq C (CHead c3 k u) (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
end of h1
(h2)
consider H4
we proved
eq
C
<λ:C.C> CASE CHead c1 k u OF CSort ⇒c1 | CHead c ⇒c
<λ:C.C> CASE CHead e1 (Flat f) v OF CSort ⇒c1 | CHead c ⇒c
eq C c1 e1
end of h2
by (h1 h2)
we proved ex2 C λe2:C.eq C (CHead c3 k u) (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
∀H3:eq C (CHead c1 k u) (CHead e1 (Flat f) v)
.ex2 C λe2:C.eq C (CHead c3 k u) (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
case csubt_void : c1:C c3:C :csubt g c1 c3 b:B :not (eq B b Void) u1:T u2:T ⇒
the thesis becomes
∀H4:eq C (CHead c1 (Bind Void) u1) (CHead e1 (Flat f) v)
.ex2 C λe2:C.eq C (CHead c3 (Bind b) u2) (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
() by induction hypothesis we know
eq C c1 (CHead e1 (Flat f) v)
→ex2 C λe2:C.eq C c3 (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
suppose H4: eq C (CHead c1 (Bind Void) u1) (CHead e1 (Flat f) v)
(H5)
we proceed by induction on H4 to prove
<λ:C.Prop>
CASE CHead e1 (Flat f) v OF
CSort ⇒False
| CHead k ⇒<λ:K.Prop> CASE k OF Bind ⇒True | Flat ⇒False
case refl_equal : ⇒
the thesis becomes
<λ:C.Prop>
CASE CHead c1 (Bind Void) u1 OF
CSort ⇒False
| CHead k ⇒<λ:K.Prop> CASE k OF Bind ⇒True | Flat ⇒False
consider I
we proved True
<λ:C.Prop>
CASE CHead c1 (Bind Void) u1 OF
CSort ⇒False
| CHead k ⇒<λ:K.Prop> CASE k OF Bind ⇒True | Flat ⇒False
<λ:C.Prop>
CASE CHead e1 (Flat f) v OF
CSort ⇒False
| CHead k ⇒<λ:K.Prop> CASE k OF Bind ⇒True | Flat ⇒False
end of H5
consider H5
we proved
<λ:C.Prop>
CASE CHead e1 (Flat f) v OF
CSort ⇒False
| CHead 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 ex2 C λe2:C.eq C (CHead c3 (Bind b) u2) (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
we proved ex2 C λe2:C.eq C (CHead c3 (Bind b) u2) (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
∀H4:eq C (CHead c1 (Bind Void) u1) (CHead e1 (Flat f) v)
.ex2 C λe2:C.eq C (CHead c3 (Bind b) u2) (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
case csubt_abst : c1:C c3:C :csubt g c1 c3 u:T t:T :ty3 g c1 u t :ty3 g c3 u t ⇒
the thesis becomes
∀H5:eq C (CHead c1 (Bind Abst) t) (CHead e1 (Flat f) v)
.ex2
C
λe2:C.eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Flat f) v)
λe2:C.csubt g e1 e2
() by induction hypothesis we know
eq C c1 (CHead e1 (Flat f) v)
→ex2 C λe2:C.eq C c3 (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
suppose H5: eq C (CHead c1 (Bind Abst) t) (CHead e1 (Flat f) v)
(H6)
we proceed by induction on H5 to prove
<λ:C.Prop>
CASE CHead e1 (Flat f) v OF
CSort ⇒False
| CHead k ⇒<λ:K.Prop> CASE k OF Bind ⇒True | Flat ⇒False
case refl_equal : ⇒
the thesis becomes
<λ:C.Prop>
CASE CHead c1 (Bind Abst) t OF
CSort ⇒False
| CHead k ⇒<λ:K.Prop> CASE k OF Bind ⇒True | Flat ⇒False
consider I
we proved True
<λ:C.Prop>
CASE CHead c1 (Bind Abst) t OF
CSort ⇒False
| CHead k ⇒<λ:K.Prop> CASE k OF Bind ⇒True | Flat ⇒False
<λ:C.Prop>
CASE CHead e1 (Flat f) v OF
CSort ⇒False
| CHead k ⇒<λ:K.Prop> CASE k OF Bind ⇒True | Flat ⇒False
end of H6
consider H6
we proved
<λ:C.Prop>
CASE CHead e1 (Flat f) v OF
CSort ⇒False
| CHead 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
ex2
C
λe2:C.eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Flat f) v)
λe2:C.csubt g e1 e2
we proved
ex2
C
λe2:C.eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Flat f) v)
λe2:C.csubt g e1 e2
∀H5:eq C (CHead c1 (Bind Abst) t) (CHead e1 (Flat f) v)
.ex2
C
λe2:C.eq C (CHead c3 (Bind Abbr) u) (CHead e2 (Flat f) v)
λe2:C.csubt g e1 e2
we proved
eq C y (CHead e1 (Flat f) v)
→ex2 C λe2:C.eq C c2 (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
we proved
∀y:C
.csubt g y c2
→(eq C y (CHead e1 (Flat f) v)
→ex2 C λe2:C.eq C c2 (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2)
by (insert_eq . . . . previous H)
we proved ex2 C λe2:C.eq C c2 (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2
we proved
∀g:G
.∀e1:C
.∀c2:C
.∀v:T
.∀f:F
.csubt g (CHead e1 (Flat f) v) c2
→ex2 C λe2:C.eq C c2 (CHead e2 (Flat f) v) λe2:C.csubt g e1 e2