DEFINITION csuba_gen_bind_rev()
TYPE =
∀g:G
.∀b1:B
.∀e1:C
.∀c2:C
.∀v1:T
.csuba g c2 (CHead e1 (Bind b1) v1)
→ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C c2 (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
BODY =
assume g: G
assume b1: B
assume e1: C
assume c2: C
assume v1: T
suppose H: csuba g c2 (CHead e1 (Bind b1) v1)
assume y: C
suppose H0: csuba g c2 y
we proceed by induction on H0 to prove
eq C y (CHead e1 (Bind b1) v1)
→ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C c2 (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
case csuba_sort : n:nat ⇒
the thesis becomes
∀H1:eq C (CSort n) (CHead e1 (Bind b1) v1)
.ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C (CSort n) (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
suppose H1: eq C (CSort n) (CHead e1 (Bind b1) v1)
(H2)
we proceed by induction on H1 to prove <λ:C.Prop> CASE CHead e1 (Bind b1) v1 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 (Bind b1) v1 OF CSort ⇒True | CHead ⇒False
end of H2
consider H2
we proved <λ:C.Prop> CASE CHead e1 (Bind b1) v1 OF CSort ⇒True | CHead ⇒False
that is equivalent to False
we proceed by induction on the previous result to prove ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C (CSort n) (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
we proved ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C (CSort n) (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
∀H1:eq C (CSort n) (CHead e1 (Bind b1) v1)
.ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C (CSort n) (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
case csuba_head : c1:C c3:C H1:csuba g c1 c3 k:K u:T ⇒
the thesis becomes
∀H3:eq C (CHead c3 k u) (CHead e1 (Bind b1) v1)
.ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C (CHead c1 k u) (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
(H2) by induction hypothesis we know
eq C c3 (CHead e1 (Bind b1) v1)
→ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C c1 (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
suppose H3: eq C (CHead c3 k u) (CHead e1 (Bind b1) v1)
(H4)
by (f_equal . . . . . H3)
we proved
eq
C
<λ:C.C> CASE CHead c3 k u OF CSort ⇒c3 | CHead c ⇒c
<λ:C.C> CASE CHead e1 (Bind b1) v1 OF CSort ⇒c3 | CHead c ⇒c
eq
C
λe:C.<λ:C.C> CASE e OF CSort ⇒c3 | CHead c ⇒c (CHead c3 k u)
λe:C.<λ:C.C> CASE e OF CSort ⇒c3 | CHead c ⇒c (CHead e1 (Bind b1) v1)
end of H4
(h1)
(H5)
by (f_equal . . . . . H3)
we proved
eq
K
<λ:C.K> CASE CHead c3 k u OF CSort ⇒k | CHead k0 ⇒k0
<λ:C.K> CASE CHead e1 (Bind b1) v1 OF CSort ⇒k | CHead k0 ⇒k0
eq
K
λe:C.<λ:C.K> CASE e OF CSort ⇒k | CHead k0 ⇒k0 (CHead c3 k u)
λe:C.<λ:C.K> CASE e OF CSort ⇒k | CHead k0 ⇒k0 (CHead e1 (Bind b1) v1)
end of H5
(h1)
(H6)
by (f_equal . . . . . H3)
we proved
eq
T
<λ:C.T> CASE CHead c3 k u OF CSort ⇒u | CHead t⇒t
<λ:C.T> CASE CHead e1 (Bind b1) v1 OF CSort ⇒u | CHead t⇒t
eq
T
λe:C.<λ:C.T> CASE e OF CSort ⇒u | CHead t⇒t (CHead c3 k u)
λe:C.<λ:C.T> CASE e OF CSort ⇒u | CHead t⇒t (CHead e1 (Bind b1) v1)
end of H6
suppose H7: eq K k (Bind b1)
suppose H8: eq C c3 e1
(h1)
(H10)
we proceed by induction on H8 to prove csuba g c1 e1
case refl_equal : ⇒
the thesis becomes the hypothesis H1
csuba g c1 e1
end of H10
by (refl_equal . .)
we proved eq C (CHead c1 (Bind b1) v1) (CHead c1 (Bind b1) v1)
by (ex2_3_intro . . . . . . . . previous H10)
we proved
ex2_3
B
C
T
λb2:B.λe2:C.λv2:T.eq C (CHead c1 (Bind b1) v1) (CHead e2 (Bind b2) v2)
λ:B.λe2:C.λ:T.csuba g e2 e1
by (eq_ind_r . . . previous . H7)
ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C (CHead c1 k v1) (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
end of h1
(h2)
consider H6
we proved
eq
T
<λ:C.T> CASE CHead c3 k u OF CSort ⇒u | CHead t⇒t
<λ:C.T> CASE CHead e1 (Bind b1) v1 OF CSort ⇒u | CHead t⇒t
eq T u v1
end of h2
by (eq_ind_r . . . h1 . h2)
we proved ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C (CHead c1 k u) (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
eq K k (Bind b1)
→(eq C c3 e1
→ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C (CHead c1 k u) (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1)
end of h1
(h2)
consider H5
we proved
eq
K
<λ:C.K> CASE CHead c3 k u OF CSort ⇒k | CHead k0 ⇒k0
<λ:C.K> CASE CHead e1 (Bind b1) v1 OF CSort ⇒k | CHead k0 ⇒k0
eq K k (Bind b1)
end of h2
by (h1 h2)
eq C c3 e1
→ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C (CHead c1 k u) (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
end of h1
(h2)
consider H4
we proved
eq
C
<λ:C.C> CASE CHead c3 k u OF CSort ⇒c3 | CHead c ⇒c
<λ:C.C> CASE CHead e1 (Bind b1) v1 OF CSort ⇒c3 | CHead c ⇒c
eq C c3 e1
end of h2
by (h1 h2)
we proved ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C (CHead c1 k u) (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
∀H3:eq C (CHead c3 k u) (CHead e1 (Bind b1) v1)
.ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C (CHead c1 k u) (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
case csuba_void : c1:C c3:C H1:csuba g c1 c3 b:B H3:not (eq B b Void) u1:T u2:T ⇒
the thesis becomes
∀H4:eq C (CHead c3 (Bind b) u2) (CHead e1 (Bind b1) v1)
.ex2_3
B
C
T
λb2:B.λe2:C.λv2:T.eq C (CHead c1 (Bind Void) u1) (CHead e2 (Bind b2) v2)
λ:B.λe2:C.λ:T.csuba g e2 e1
(H2) by induction hypothesis we know
eq C c3 (CHead e1 (Bind b1) v1)
→ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C c1 (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
suppose H4: eq C (CHead c3 (Bind b) u2) (CHead e1 (Bind b1) v1)
(H5)
by (f_equal . . . . . H4)
we proved
eq
C
<λ:C.C> CASE CHead c3 (Bind b) u2 OF CSort ⇒c3 | CHead c ⇒c
<λ:C.C> CASE CHead e1 (Bind b1) v1 OF CSort ⇒c3 | CHead c ⇒c
eq
C
λe:C.<λ:C.C> CASE e OF CSort ⇒c3 | CHead c ⇒c (CHead c3 (Bind b) u2)
λe:C.<λ:C.C> CASE e OF CSort ⇒c3 | CHead c ⇒c (CHead e1 (Bind b1) v1)
end of H5
(h1)
(H6)
by (f_equal . . . . . H4)
we proved
eq
B
<λ:C.B>
CASE CHead c3 (Bind b) u2 OF
CSort ⇒b
| CHead k ⇒<λ:K.B> CASE k OF Bind b0⇒b0 | Flat ⇒b
<λ:C.B>
CASE CHead e1 (Bind b1) v1 OF
CSort ⇒b
| CHead k ⇒<λ:K.B> CASE k OF Bind b0⇒b0 | Flat ⇒b
eq
B
λe:C.<λ:C.B> CASE e OF CSort ⇒b | CHead k ⇒<λ:K.B> CASE k OF Bind b0⇒b0 | Flat ⇒b
CHead c3 (Bind b) u2
λe:C.<λ:C.B> CASE e OF CSort ⇒b | CHead k ⇒<λ:K.B> CASE k OF Bind b0⇒b0 | Flat ⇒b
CHead e1 (Bind b1) v1
end of H6
(H8)
consider H6
we proved
eq
B
<λ:C.B>
CASE CHead c3 (Bind b) u2 OF
CSort ⇒b
| CHead k ⇒<λ:K.B> CASE k OF Bind b0⇒b0 | Flat ⇒b
<λ:C.B>
CASE CHead e1 (Bind b1) v1 OF
CSort ⇒b
| CHead k ⇒<λ:K.B> CASE k OF Bind b0⇒b0 | Flat ⇒b
eq B b b1
end of H8
suppose H9: eq C c3 e1
(H12)
we proceed by induction on H9 to prove csuba g c1 e1
case refl_equal : ⇒
the thesis becomes the hypothesis H1
csuba g c1 e1
end of H12
by (refl_equal . .)
we proved eq C (CHead c1 (Bind Void) u1) (CHead c1 (Bind Void) u1)
by (ex2_3_intro . . . . . . . . previous H12)
we proved
ex2_3
B
C
T
λb2:B.λe2:C.λv2:T.eq C (CHead c1 (Bind Void) u1) (CHead e2 (Bind b2) v2)
λ:B.λe2:C.λ:T.csuba g e2 e1
eq C c3 e1
→(ex2_3
B
C
T
λb2:B.λe2:C.λv2:T.eq C (CHead c1 (Bind Void) u1) (CHead e2 (Bind b2) v2)
λ:B.λe2:C.λ:T.csuba g e2 e1)
end of h1
(h2)
consider H5
we proved
eq
C
<λ:C.C> CASE CHead c3 (Bind b) u2 OF CSort ⇒c3 | CHead c ⇒c
<λ:C.C> CASE CHead e1 (Bind b1) v1 OF CSort ⇒c3 | CHead c ⇒c
eq C c3 e1
end of h2
by (h1 h2)
we proved
ex2_3
B
C
T
λb2:B.λe2:C.λv2:T.eq C (CHead c1 (Bind Void) u1) (CHead e2 (Bind b2) v2)
λ:B.λe2:C.λ:T.csuba g e2 e1
∀H4:eq C (CHead c3 (Bind b) u2) (CHead e1 (Bind b1) v1)
.ex2_3
B
C
T
λb2:B.λe2:C.λv2:T.eq C (CHead c1 (Bind Void) u1) (CHead e2 (Bind b2) v2)
λ:B.λe2:C.λ:T.csuba g e2 e1
case csuba_abst : c1:C c3:C H1:csuba g c1 c3 t:T a:A :arity g c1 t (asucc g a) u:T H4:arity g c3 u a ⇒
the thesis becomes
∀H5:eq C (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1)
.ex2_3
B
C
T
λb2:B.λe2:C.λv2:T.eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) v2)
λ:B.λe2:C.λ:T.csuba g e2 e1
(H2) by induction hypothesis we know
eq C c3 (CHead e1 (Bind b1) v1)
→ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C c1 (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
suppose H5: eq C (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1)
(H6)
by (f_equal . . . . . H5)
we proved
eq
C
<λ:C.C> CASE CHead c3 (Bind Abbr) u OF CSort ⇒c3 | CHead c ⇒c
<λ:C.C> CASE CHead e1 (Bind b1) v1 OF CSort ⇒c3 | CHead c ⇒c
eq
C
λe:C.<λ:C.C> CASE e OF CSort ⇒c3 | CHead c ⇒c (CHead c3 (Bind Abbr) u)
λe:C.<λ:C.C> CASE e OF CSort ⇒c3 | CHead c ⇒c (CHead e1 (Bind b1) v1)
end of H6
(h1)
(H7)
by (f_equal . . . . . H5)
we proved
eq
B
<λ:C.B>
CASE CHead c3 (Bind Abbr) u OF
CSort ⇒Abbr
| CHead k ⇒<λ:K.B> CASE k OF Bind b⇒b | Flat ⇒Abbr
<λ:C.B>
CASE CHead e1 (Bind b1) v1 OF
CSort ⇒Abbr
| CHead k ⇒<λ:K.B> CASE k OF Bind b⇒b | Flat ⇒Abbr
eq
B
λe:C
.<λ:C.B>
CASE e OF
CSort ⇒Abbr
| CHead k ⇒<λ:K.B> CASE k OF Bind b⇒b | Flat ⇒Abbr
CHead c3 (Bind Abbr) u
λe:C
.<λ:C.B>
CASE e OF
CSort ⇒Abbr
| CHead k ⇒<λ:K.B> CASE k OF Bind b⇒b | Flat ⇒Abbr
CHead e1 (Bind b1) v1
end of H7
(h1)
(H8)
by (f_equal . . . . . H5)
we proved
eq
T
<λ:C.T> CASE CHead c3 (Bind Abbr) u OF CSort ⇒u | CHead t0⇒t0
<λ:C.T> CASE CHead e1 (Bind b1) v1 OF CSort ⇒u | CHead t0⇒t0
eq
T
λe:C.<λ:C.T> CASE e OF CSort ⇒u | CHead t0⇒t0 (CHead c3 (Bind Abbr) u)
λe:C.<λ:C.T> CASE e OF CSort ⇒u | CHead t0⇒t0 (CHead e1 (Bind b1) v1)
end of H8
suppose H9: eq B Abbr b1
suppose H10: eq C c3 e1
(H11)
consider H8
we proved
eq
T
<λ:C.T> CASE CHead c3 (Bind Abbr) u OF CSort ⇒u | CHead t0⇒t0
<λ:C.T> CASE CHead e1 (Bind b1) v1 OF CSort ⇒u | CHead t0⇒t0
that is equivalent to eq T u v1
we proceed by induction on the previous result to prove arity g c3 v1 a
case refl_equal : ⇒
the thesis becomes the hypothesis H4
arity g c3 v1 a
end of H11
(H13)
we proceed by induction on H10 to prove
eq C e1 (CHead e1 (Bind b1) v1)
→ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C c1 (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
case refl_equal : ⇒
the thesis becomes the hypothesis H2
eq C e1 (CHead e1 (Bind b1) v1)
→ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C c1 (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
end of H13
(H14)
we proceed by induction on H10 to prove csuba g c1 e1
case refl_equal : ⇒
the thesis becomes the hypothesis H1
csuba g c1 e1
end of H14
by (refl_equal . .)
we proved eq C (CHead c1 (Bind Abst) t) (CHead c1 (Bind Abst) t)
by (ex2_3_intro . . . . . . . . previous H14)
we proved
ex2_3
B
C
T
λb2:B.λe2:C.λv2:T.eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) v2)
λ:B.λe2:C.λ:T.csuba g e2 e1
eq B Abbr b1
→(eq C c3 e1
→(ex2_3
B
C
T
λb2:B.λe2:C.λv2:T.eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) v2)
λ:B.λe2:C.λ:T.csuba g e2 e1))
end of h1
(h2)
consider H7
we proved
eq
B
<λ:C.B>
CASE CHead c3 (Bind Abbr) u OF
CSort ⇒Abbr
| CHead k ⇒<λ:K.B> CASE k OF Bind b⇒b | Flat ⇒Abbr
<λ:C.B>
CASE CHead e1 (Bind b1) v1 OF
CSort ⇒Abbr
| CHead k ⇒<λ:K.B> CASE k OF Bind b⇒b | Flat ⇒Abbr
eq B Abbr b1
end of h2
by (h1 h2)
eq C c3 e1
→(ex2_3
B
C
T
λb2:B.λe2:C.λv2:T.eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) v2)
λ:B.λe2:C.λ:T.csuba g e2 e1)
end of h1
(h2)
consider H6
we proved
eq
C
<λ:C.C> CASE CHead c3 (Bind Abbr) u OF CSort ⇒c3 | CHead c ⇒c
<λ:C.C> CASE CHead e1 (Bind b1) v1 OF CSort ⇒c3 | CHead c ⇒c
eq C c3 e1
end of h2
by (h1 h2)
we proved
ex2_3
B
C
T
λb2:B.λe2:C.λv2:T.eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) v2)
λ:B.λe2:C.λ:T.csuba g e2 e1
∀H5:eq C (CHead c3 (Bind Abbr) u) (CHead e1 (Bind b1) v1)
.ex2_3
B
C
T
λb2:B.λe2:C.λv2:T.eq C (CHead c1 (Bind Abst) t) (CHead e2 (Bind b2) v2)
λ:B.λe2:C.λ:T.csuba g e2 e1
we proved
eq C y (CHead e1 (Bind b1) v1)
→ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C c2 (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
we proved
∀y:C
.csuba g c2 y
→(eq C y (CHead e1 (Bind b1) v1)
→ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C c2 (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1)
by (insert_eq . . . . previous H)
we proved ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C c2 (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1
we proved
∀g:G
.∀b1:B
.∀e1:C
.∀c2:C
.∀v1:T
.csuba g c2 (CHead e1 (Bind b1) v1)
→ex2_3 B C T λb2:B.λe2:C.λv2:T.eq C c2 (CHead e2 (Bind b2) v2) λ:B.λe2:C.λ:T.csuba g e2 e1