DEFINITION pr0_gen_void()
TYPE =
∀u1:T
.∀t1:T
.∀x:T
.pr0 (THead (Bind Void) u1 t1) x
→(or
ex3_2 T T λu2:T.λt2:T.eq T x (THead (Bind Void) u2 t2) λu2:T.λ:T.pr0 u1 u2 λ:T.λt2:T.pr0 t1 t2
pr0 t1 (lift (S O) O x))
BODY =
assume u1: T
assume t1: T
assume x: T
suppose H: pr0 (THead (Bind Void) u1 t1) x
assume y: T
suppose H0: pr0 y x
we proceed by induction on H0 to prove
eq T y (THead (Bind Void) u1 t1)
→(or
ex3_2 T T λu2:T.λt2:T.eq T x (THead (Bind Void) u2 t2) λu2:T.λ:T.pr0 u1 u2 λ:T.λt2:T.pr0 t1 t2
pr0 t1 (lift (S O) O x))
case pr0_refl : t:T ⇒
the thesis becomes
∀H1:eq T t (THead (Bind Void) u1 t1)
.or
ex3_2 T T λu2:T.λt2:T.eq T t (THead (Bind Void) u2 t2) λu2:T.λ:T.pr0 u1 u2 λ:T.λt2:T.pr0 t1 t2
pr0 t1 (lift (S O) O t)
suppose H1: eq T t (THead (Bind Void) u1 t1)
(H2)
by (f_equal . . . . . H1)
we proved eq T t (THead (Bind Void) u1 t1)
eq T (λe:T.e t) (λe:T.e (THead (Bind Void) u1 t1))
end of H2
(h1)
by (refl_equal . .)
eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u1 t1)
end of h1
(h2) by (pr0_refl .) we proved pr0 u1 u1
(h3) by (pr0_refl .) we proved pr0 t1 t1
by (ex3_2_intro . . . . . . . h1 h2 h3)
we proved
ex3_2
T
T
λu2:T.λt2:T.eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2 t2)
λu2:T.λ:T.pr0 u1 u2
λ:T.λt2:T.pr0 t1 t2
by (or_introl . . previous)
we proved
or
ex3_2
T
T
λu2:T.λt2:T.eq T (THead (Bind Void) u1 t1) (THead (Bind Void) u2 t2)
λu2:T.λ:T.pr0 u1 u2
λ:T.λt2:T.pr0 t1 t2
pr0 t1 (lift (S O) O (THead (Bind Void) u1 t1))
by (eq_ind_r . . . previous . H2)
we proved
or
ex3_2 T T λu2:T.λt2:T.eq T t (THead (Bind Void) u2 t2) λu2:T.λ:T.pr0 u1 u2 λ:T.λt2:T.pr0 t1 t2
pr0 t1 (lift (S O) O t)
∀H1:eq T t (THead (Bind Void) u1 t1)
.or
ex3_2 T T λu2:T.λt2:T.eq T t (THead (Bind Void) u2 t2) λu2:T.λ:T.pr0 u1 u2 λ:T.λt2:T.pr0 t1 t2
pr0 t1 (lift (S O) O t)
case pr0_comp : u0:T u2:T H1:pr0 u0 u2 t0:T t2:T H3:pr0 t0 t2 k:K ⇒
the thesis becomes
∀H5:eq T (THead k u0 t0) (THead (Bind Void) u1 t1)
.or
ex3_2 T T λu3:T.λt3:T.eq T (THead k u2 t2) (THead (Bind Void) u3 t3) λu3:T.λ:T.pr0 u1 u3 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O (THead k u2 t2))
(H2) by induction hypothesis we know
eq T u0 (THead (Bind Void) u1 t1)
→(or
ex3_2 T T λu3:T.λt2:T.eq T u2 (THead (Bind Void) u3 t2) λu3:T.λ:T.pr0 u1 u3 λ:T.λt2:T.pr0 t1 t2
pr0 t1 (lift (S O) O u2))
(H4) by induction hypothesis we know
eq T t0 (THead (Bind Void) u1 t1)
→(or
ex3_2 T T λu3:T.λt3:T.eq T t2 (THead (Bind Void) u3 t3) λu3:T.λ:T.pr0 u1 u3 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2))
suppose H5: eq T (THead k u0 t0) (THead (Bind Void) u1 t1)
(H6)
by (f_equal . . . . . H5)
we proved
eq
K
<λ:T.K> CASE THead k u0 t0 OF TSort ⇒k | TLRef ⇒k | THead k0 ⇒k0
<λ:T.K> CASE THead (Bind Void) u1 t1 OF TSort ⇒k | TLRef ⇒k | THead k0 ⇒k0
eq
K
λe:T.<λ:T.K> CASE e OF TSort ⇒k | TLRef ⇒k | THead k0 ⇒k0 (THead k u0 t0)
λe:T.<λ:T.K> CASE e OF TSort ⇒k | TLRef ⇒k | THead k0 ⇒k0
THead (Bind Void) u1 t1
end of H6
(h1)
(H7)
by (f_equal . . . . . H5)
we proved
eq
T
<λ:T.T> CASE THead k u0 t0 OF TSort ⇒u0 | TLRef ⇒u0 | THead t ⇒t
<λ:T.T> CASE THead (Bind Void) u1 t1 OF TSort ⇒u0 | TLRef ⇒u0 | THead t ⇒t
eq
T
λe:T.<λ:T.T> CASE e OF TSort ⇒u0 | TLRef ⇒u0 | THead t ⇒t (THead k u0 t0)
λe:T.<λ:T.T> CASE e OF TSort ⇒u0 | TLRef ⇒u0 | THead t ⇒t
THead (Bind Void) u1 t1
end of H7
(h1)
(H8)
by (f_equal . . . . . H5)
we proved
eq
T
<λ:T.T> CASE THead k u0 t0 OF TSort ⇒t0 | TLRef ⇒t0 | THead t⇒t
<λ:T.T> CASE THead (Bind Void) u1 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 k u0 t0)
λe:T.<λ:T.T> CASE e OF TSort ⇒t0 | TLRef ⇒t0 | THead t⇒t
THead (Bind Void) u1 t1
end of H8
suppose H9: eq T u0 u1
suppose H10: eq K k (Bind Void)
(H12)
consider H8
we proved
eq
T
<λ:T.T> CASE THead k u0 t0 OF TSort ⇒t0 | TLRef ⇒t0 | THead t⇒t
<λ:T.T> CASE THead (Bind Void) u1 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 pr0 t1 t2
case refl_equal : ⇒
the thesis becomes the hypothesis H3
pr0 t1 t2
end of H12
(H14)
we proceed by induction on H9 to prove pr0 u1 u2
case refl_equal : ⇒
the thesis becomes the hypothesis H1
pr0 u1 u2
end of H14
by (refl_equal . .)
we proved eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u2 t2)
by (ex3_2_intro . . . . . . . previous H14 H12)
we proved
ex3_2
T
T
λu3:T.λt3:T.eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)
λu3:T.λ:T.pr0 u1 u3
λ:T.λt3:T.pr0 t1 t3
by (or_introl . . previous)
we proved
or
ex3_2
T
T
λu3:T.λt3:T.eq T (THead (Bind Void) u2 t2) (THead (Bind Void) u3 t3)
λu3:T.λ:T.pr0 u1 u3
λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O (THead (Bind Void) u2 t2))
by (eq_ind_r . . . previous . H10)
we proved
or
ex3_2 T T λu3:T.λt3:T.eq T (THead k u2 t2) (THead (Bind Void) u3 t3) λu3:T.λ:T.pr0 u1 u3 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O (THead k u2 t2))
eq T u0 u1
→(eq K k (Bind Void)
→(or
ex3_2 T T λu3:T.λt3:T.eq T (THead k u2 t2) (THead (Bind Void) u3 t3) λu3:T.λ:T.pr0 u1 u3 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O (THead k u2 t2))))
end of h1
(h2)
consider H7
we proved
eq
T
<λ:T.T> CASE THead k u0 t0 OF TSort ⇒u0 | TLRef ⇒u0 | THead t ⇒t
<λ:T.T> CASE THead (Bind Void) u1 t1 OF TSort ⇒u0 | TLRef ⇒u0 | THead t ⇒t
eq T u0 u1
end of h2
by (h1 h2)
eq K k (Bind Void)
→(or
ex3_2 T T λu3:T.λt3:T.eq T (THead k u2 t2) (THead (Bind Void) u3 t3) λu3:T.λ:T.pr0 u1 u3 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O (THead k u2 t2)))
end of h1
(h2)
consider H6
we proved
eq
K
<λ:T.K> CASE THead k u0 t0 OF TSort ⇒k | TLRef ⇒k | THead k0 ⇒k0
<λ:T.K> CASE THead (Bind Void) u1 t1 OF TSort ⇒k | TLRef ⇒k | THead k0 ⇒k0
eq K k (Bind Void)
end of h2
by (h1 h2)
we proved
or
ex3_2 T T λu3:T.λt3:T.eq T (THead k u2 t2) (THead (Bind Void) u3 t3) λu3:T.λ:T.pr0 u1 u3 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O (THead k u2 t2))
∀H5:eq T (THead k u0 t0) (THead (Bind Void) u1 t1)
.or
ex3_2 T T λu3:T.λt3:T.eq T (THead k u2 t2) (THead (Bind Void) u3 t3) λu3:T.λ:T.pr0 u1 u3 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O (THead k u2 t2))
case pr0_beta : u:T v1:T v2:T :pr0 v1 v2 t0:T t2:T :pr0 t0 t2 ⇒
the thesis becomes
∀H5:eq
T
THead (Flat Appl) v1 (THead (Bind Abst) u t0)
THead (Bind Void) u1 t1
.or
ex3_2
T
T
λu2:T.λt3:T.eq T (THead (Bind Abbr) v2 t2) (THead (Bind Void) u2 t3)
λu2:T.λ:T.pr0 u1 u2
λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O (THead (Bind Abbr) v2 t2))
() by induction hypothesis we know
eq T v1 (THead (Bind Void) u1 t1)
→(or
ex3_2 T T λu2:T.λt2:T.eq T v2 (THead (Bind Void) u2 t2) λu2:T.λ:T.pr0 u1 u2 λ:T.λt2:T.pr0 t1 t2
pr0 t1 (lift (S O) O v2))
() by induction hypothesis we know
eq T t0 (THead (Bind Void) u1 t1)
→(or
ex3_2 T T λu2:T.λt3:T.eq T t2 (THead (Bind Void) u2 t3) λu2:T.λ:T.pr0 u1 u2 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2))
suppose H5:
eq
T
THead (Flat Appl) v1 (THead (Bind Abst) u t0)
THead (Bind Void) u1 t1
(H6)
we proceed by induction on H5 to prove
<λ:T.Prop>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒False | Flat ⇒True
case refl_equal : ⇒
the thesis becomes
<λ:T.Prop>
CASE THead (Flat Appl) v1 (THead (Bind Abst) u t0) OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒False | Flat ⇒True
consider I
we proved True
<λ:T.Prop>
CASE THead (Flat Appl) v1 (THead (Bind Abst) u t0) OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒False | Flat ⇒True
<λ:T.Prop>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒False | Flat ⇒True
end of H6
consider H6
we proved
<λ:T.Prop>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒False | Flat ⇒True
that is equivalent to False
we proceed by induction on the previous result to prove
or
ex3_2
T
T
λu2:T.λt3:T.eq T (THead (Bind Abbr) v2 t2) (THead (Bind Void) u2 t3)
λu2:T.λ:T.pr0 u1 u2
λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O (THead (Bind Abbr) v2 t2))
we proved
or
ex3_2
T
T
λu2:T.λt3:T.eq T (THead (Bind Abbr) v2 t2) (THead (Bind Void) u2 t3)
λu2:T.λ:T.pr0 u1 u2
λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O (THead (Bind Abbr) v2 t2))
∀H5:eq
T
THead (Flat Appl) v1 (THead (Bind Abst) u t0)
THead (Bind Void) u1 t1
.or
ex3_2
T
T
λu2:T.λt3:T.eq T (THead (Bind Abbr) v2 t2) (THead (Bind Void) u2 t3)
λu2:T.λ:T.pr0 u1 u2
λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O (THead (Bind Abbr) v2 t2))
case pr0_upsilon : b:B :not (eq B b Abst) v1:T v2:T :pr0 v1 v2 u0:T u2:T :pr0 u0 u2 t0:T t2:T :pr0 t0 t2 ⇒
the thesis becomes
∀H8:eq
T
THead (Flat Appl) v1 (THead (Bind b) u0 t0)
THead (Bind Void) u1 t1
.or
ex3_2
T
T
λu3:T
.λt3:T
.eq
T
THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)
THead (Bind Void) u3 t3
λu3:T.λ:T.pr0 u1 u3
λ:T.λt3:T.pr0 t1 t3
pr0
t1
lift
S O
O
THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)
() by induction hypothesis we know
eq T v1 (THead (Bind Void) u1 t1)
→(or
ex3_2 T T λu2:T.λt2:T.eq T v2 (THead (Bind Void) u2 t2) λu2:T.λ:T.pr0 u1 u2 λ:T.λt2:T.pr0 t1 t2
pr0 t1 (lift (S O) O v2))
() by induction hypothesis we know
eq T u0 (THead (Bind Void) u1 t1)
→(or
ex3_2 T T λu3:T.λt2:T.eq T u2 (THead (Bind Void) u3 t2) λu3:T.λ:T.pr0 u1 u3 λ:T.λt2:T.pr0 t1 t2
pr0 t1 (lift (S O) O u2))
() by induction hypothesis we know
eq T t0 (THead (Bind Void) u1 t1)
→(or
ex3_2 T T λu3:T.λt3:T.eq T t2 (THead (Bind Void) u3 t3) λu3:T.λ:T.pr0 u1 u3 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2))
suppose H8:
eq
T
THead (Flat Appl) v1 (THead (Bind b) u0 t0)
THead (Bind Void) u1 t1
(H9)
we proceed by induction on H8 to prove
<λ:T.Prop>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒False | Flat ⇒True
case refl_equal : ⇒
the thesis becomes
<λ:T.Prop>
CASE THead (Flat Appl) v1 (THead (Bind b) u0 t0) OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒False | Flat ⇒True
consider I
we proved True
<λ:T.Prop>
CASE THead (Flat Appl) v1 (THead (Bind b) u0 t0) OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒False | Flat ⇒True
<λ:T.Prop>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒False | Flat ⇒True
end of H9
consider H9
we proved
<λ:T.Prop>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒False | Flat ⇒True
that is equivalent to False
we proceed by induction on the previous result to prove
or
ex3_2
T
T
λu3:T
.λt3:T
.eq
T
THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)
THead (Bind Void) u3 t3
λu3:T.λ:T.pr0 u1 u3
λ:T.λt3:T.pr0 t1 t3
pr0
t1
lift
S O
O
THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)
we proved
or
ex3_2
T
T
λu3:T
.λt3:T
.eq
T
THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)
THead (Bind Void) u3 t3
λu3:T.λ:T.pr0 u1 u3
λ:T.λt3:T.pr0 t1 t3
pr0
t1
lift
S O
O
THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)
∀H8:eq
T
THead (Flat Appl) v1 (THead (Bind b) u0 t0)
THead (Bind Void) u1 t1
.or
ex3_2
T
T
λu3:T
.λt3:T
.eq
T
THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)
THead (Bind Void) u3 t3
λu3:T.λ:T.pr0 u1 u3
λ:T.λt3:T.pr0 t1 t3
pr0
t1
lift
S O
O
THead (Bind b) u2 (THead (Flat Appl) (lift (S O) O v2) t2)
case pr0_delta : u0:T u2:T :pr0 u0 u2 t0:T t2:T :pr0 t0 t2 w:T :subst0 O u2 t2 w ⇒
the thesis becomes
∀H6:eq T (THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1)
.or
ex3_2
T
T
λu3:T.λt3:T.eq T (THead (Bind Abbr) u2 w) (THead (Bind Void) u3 t3)
λu3:T.λ:T.pr0 u1 u3
λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))
() by induction hypothesis we know
eq T u0 (THead (Bind Void) u1 t1)
→(or
ex3_2 T T λu3:T.λt2:T.eq T u2 (THead (Bind Void) u3 t2) λu3:T.λ:T.pr0 u1 u3 λ:T.λt2:T.pr0 t1 t2
pr0 t1 (lift (S O) O u2))
() by induction hypothesis we know
eq T t0 (THead (Bind Void) u1 t1)
→(or
ex3_2 T T λu3:T.λt3:T.eq T t2 (THead (Bind Void) u3 t3) λu3:T.λ:T.pr0 u1 u3 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2))
suppose H6: eq T (THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1)
(H7)
we proceed by induction on H6 to prove
<λ:T.Prop>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒
<λ:K.Prop>
CASE k OF
Bind b⇒<λ:B.Prop> CASE b OF Abbr⇒True | Abst⇒False | Void⇒False
| Flat ⇒False
case refl_equal : ⇒
the thesis becomes
<λ:T.Prop>
CASE THead (Bind Abbr) u0 t0 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒
<λ:K.Prop>
CASE k OF
Bind b⇒<λ:B.Prop> CASE b OF Abbr⇒True | Abst⇒False | Void⇒False
| Flat ⇒False
consider I
we proved True
<λ:T.Prop>
CASE THead (Bind Abbr) u0 t0 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒
<λ:K.Prop>
CASE k OF
Bind b⇒<λ:B.Prop> CASE b OF Abbr⇒True | Abst⇒False | Void⇒False
| Flat ⇒False
<λ:T.Prop>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒
<λ:K.Prop>
CASE k OF
Bind b⇒<λ:B.Prop> CASE b OF Abbr⇒True | Abst⇒False | Void⇒False
| Flat ⇒False
end of H7
consider H7
we proved
<λ:T.Prop>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒
<λ:K.Prop>
CASE k OF
Bind b⇒<λ:B.Prop> CASE b OF Abbr⇒True | Abst⇒False | Void⇒False
| Flat ⇒False
that is equivalent to False
we proceed by induction on the previous result to prove
or
ex3_2
T
T
λu3:T.λt3:T.eq T (THead (Bind Abbr) u2 w) (THead (Bind Void) u3 t3)
λu3:T.λ:T.pr0 u1 u3
λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))
we proved
or
ex3_2
T
T
λu3:T.λt3:T.eq T (THead (Bind Abbr) u2 w) (THead (Bind Void) u3 t3)
λu3:T.λ:T.pr0 u1 u3
λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))
∀H6:eq T (THead (Bind Abbr) u0 t0) (THead (Bind Void) u1 t1)
.or
ex3_2
T
T
λu3:T.λt3:T.eq T (THead (Bind Abbr) u2 w) (THead (Bind Void) u3 t3)
λu3:T.λ:T.pr0 u1 u3
λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O (THead (Bind Abbr) u2 w))
case pr0_zeta : b:B H1:not (eq B b Abst) t0:T t2:T H2:pr0 t0 t2 u:T ⇒
the thesis becomes
∀H4:eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1)
.or
ex3_2 T T λu2:T.λt3:T.eq T t2 (THead (Bind Void) u2 t3) λu2:T.λ:T.pr0 u1 u2 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2)
(H3) by induction hypothesis we know
eq T t0 (THead (Bind Void) u1 t1)
→(or
ex3_2 T T λu2:T.λt3:T.eq T t2 (THead (Bind Void) u2 t3) λu2:T.λ:T.pr0 u1 u2 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2))
suppose H4:
eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1)
(H5)
by (f_equal . . . . . H4)
we proved
eq
B
<λ:T.B>
CASE THead (Bind b) u (lift (S O) O t0) OF
TSort ⇒b
| TLRef ⇒b
| THead k ⇒<λ:K.B> CASE k OF Bind b0⇒b0 | Flat ⇒b
<λ:T.B>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒b
| TLRef ⇒b
| THead k ⇒<λ:K.B> CASE k OF Bind b0⇒b0 | Flat ⇒b
eq
B
λe:T
.<λ:T.B>
CASE e OF
TSort ⇒b
| TLRef ⇒b
| THead k ⇒<λ:K.B> CASE k OF Bind b0⇒b0 | Flat ⇒b
THead (Bind b) u (lift (S O) O t0)
λe:T
.<λ:T.B>
CASE e OF
TSort ⇒b
| TLRef ⇒b
| THead k ⇒<λ:K.B> CASE k OF Bind b0⇒b0 | Flat ⇒b
THead (Bind Void) u1 t1
end of H5
(h1)
(H6)
by (f_equal . . . . . H4)
we proved
eq
T
<λ:T.T>
CASE THead (Bind b) u (lift (S O) O t0) OF
TSort ⇒u
| TLRef ⇒u
| THead t ⇒t
<λ:T.T>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒u
| TLRef ⇒u
| THead t ⇒t
eq
T
λe:T.<λ:T.T> CASE e OF TSort ⇒u | TLRef ⇒u | THead t ⇒t
THead (Bind b) u (lift (S O) O t0)
λe:T.<λ:T.T> CASE e OF TSort ⇒u | TLRef ⇒u | THead t ⇒t
THead (Bind Void) u1 t1
end of H6
(h1)
(H7)
by (f_equal . . . . . H4)
we proved
eq
T
<λ:T.T>
CASE THead (Bind b) u (lift (S O) O t0) OF
TSort ⇒
FIXlref_map{
lref_map:(nat→nat)→nat→T→T
:=λf:nat→nat
.λd:nat
.λt:T
.<λt3:T.T>
CASE t OF
TSort n⇒TSort n
| TLRef i⇒TLRef <λb1:bool.nat> CASE blt i d OF true⇒i | false⇒f i
| THead k u0 t3⇒THead k (lref_map f d u0) (lref_map f (s k d) t3)
}
λx0:nat.plus x0 (S O)
O
t0
| TLRef ⇒
FIXlref_map{
lref_map:(nat→nat)→nat→T→T
:=λf:nat→nat
.λd:nat
.λt:T
.<λt3:T.T>
CASE t OF
TSort n⇒TSort n
| TLRef i⇒TLRef <λb1:bool.nat> CASE blt i d OF true⇒i | false⇒f i
| THead k u0 t3⇒THead k (lref_map f d u0) (lref_map f (s k d) t3)
}
λx0:nat.plus x0 (S O)
O
t0
| THead t⇒t
<λ:T.T>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒
FIXlref_map{
lref_map:(nat→nat)→nat→T→T
:=λf:nat→nat
.λd:nat
.λt:T
.<λt3:T.T>
CASE t OF
TSort n⇒TSort n
| TLRef i⇒TLRef <λb1:bool.nat> CASE blt i d OF true⇒i | false⇒f i
| THead k u0 t3⇒THead k (lref_map f d u0) (lref_map f (s k d) t3)
}
λx0:nat.plus x0 (S O)
O
t0
| TLRef ⇒
FIXlref_map{
lref_map:(nat→nat)→nat→T→T
:=λf:nat→nat
.λd:nat
.λt:T
.<λt3:T.T>
CASE t OF
TSort n⇒TSort n
| TLRef i⇒TLRef <λb1:bool.nat> CASE blt i d OF true⇒i | false⇒f i
| THead k u0 t3⇒THead k (lref_map f d u0) (lref_map f (s k d) t3)
}
λx0:nat.plus x0 (S O)
O
t0
| THead t⇒t
eq
T
λe:T
.<λ:T.T>
CASE e OF
TSort ⇒
FIXlref_map{
lref_map:(nat→nat)→nat→T→T
:=λf:nat→nat
.λd:nat
.λt:T
.<λt3:T.T>
CASE t OF
TSort n⇒TSort n
| TLRef i⇒TLRef <λb1:bool.nat> CASE blt i d OF true⇒i | false⇒f i
| THead k u0 t3⇒THead k (lref_map f d u0) (lref_map f (s k d) t3)
}
λx0:nat.plus x0 (S O)
O
t0
| TLRef ⇒
FIXlref_map{
lref_map:(nat→nat)→nat→T→T
:=λf:nat→nat
.λd:nat
.λt:T
.<λt3:T.T>
CASE t OF
TSort n⇒TSort n
| TLRef i⇒TLRef <λb1:bool.nat> CASE blt i d OF true⇒i | false⇒f i
| THead k u0 t3⇒THead k (lref_map f d u0) (lref_map f (s k d) t3)
}
λx0:nat.plus x0 (S O)
O
t0
| THead t⇒t
THead (Bind b) u (lift (S O) O t0)
λe:T
.<λ:T.T>
CASE e OF
TSort ⇒
FIXlref_map{
lref_map:(nat→nat)→nat→T→T
:=λf:nat→nat
.λd:nat
.λt:T
.<λt3:T.T>
CASE t OF
TSort n⇒TSort n
| TLRef i⇒TLRef <λb1:bool.nat> CASE blt i d OF true⇒i | false⇒f i
| THead k u0 t3⇒THead k (lref_map f d u0) (lref_map f (s k d) t3)
}
λx0:nat.plus x0 (S O)
O
t0
| TLRef ⇒
FIXlref_map{
lref_map:(nat→nat)→nat→T→T
:=λf:nat→nat
.λd:nat
.λt:T
.<λt3:T.T>
CASE t OF
TSort n⇒TSort n
| TLRef i⇒TLRef <λb1:bool.nat> CASE blt i d OF true⇒i | false⇒f i
| THead k u0 t3⇒THead k (lref_map f d u0) (lref_map f (s k d) t3)
}
λx0:nat.plus x0 (S O)
O
t0
| THead t⇒t
THead (Bind Void) u1 t1
end of H7
suppose : eq T u u1
suppose H9: eq B b Void
consider H7
we proved
eq
T
<λ:T.T>
CASE THead (Bind b) u (lift (S O) O t0) OF
TSort ⇒
FIXlref_map{
lref_map:(nat→nat)→nat→T→T
:=λf:nat→nat
.λd:nat
.λt:T
.<λt3:T.T>
CASE t OF
TSort n⇒TSort n
| TLRef i⇒TLRef <λb1:bool.nat> CASE blt i d OF true⇒i | false⇒f i
| THead k u0 t3⇒THead k (lref_map f d u0) (lref_map f (s k d) t3)
}
λx0:nat.plus x0 (S O)
O
t0
| TLRef ⇒
FIXlref_map{
lref_map:(nat→nat)→nat→T→T
:=λf:nat→nat
.λd:nat
.λt:T
.<λt3:T.T>
CASE t OF
TSort n⇒TSort n
| TLRef i⇒TLRef <λb1:bool.nat> CASE blt i d OF true⇒i | false⇒f i
| THead k u0 t3⇒THead k (lref_map f d u0) (lref_map f (s k d) t3)
}
λx0:nat.plus x0 (S O)
O
t0
| THead t⇒t
<λ:T.T>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒
FIXlref_map{
lref_map:(nat→nat)→nat→T→T
:=λf:nat→nat
.λd:nat
.λt:T
.<λt3:T.T>
CASE t OF
TSort n⇒TSort n
| TLRef i⇒TLRef <λb1:bool.nat> CASE blt i d OF true⇒i | false⇒f i
| THead k u0 t3⇒THead k (lref_map f d u0) (lref_map f (s k d) t3)
}
λx0:nat.plus x0 (S O)
O
t0
| TLRef ⇒
FIXlref_map{
lref_map:(nat→nat)→nat→T→T
:=λf:nat→nat
.λd:nat
.λt:T
.<λt3:T.T>
CASE t OF
TSort n⇒TSort n
| TLRef i⇒TLRef <λb1:bool.nat> CASE blt i d OF true⇒i | false⇒f i
| THead k u0 t3⇒THead k (lref_map f d u0) (lref_map f (s k d) t3)
}
λx0:nat.plus x0 (S O)
O
t0
| THead t⇒t
that is equivalent to eq T (lift (S O) O t0) t1
we proceed by induction on the previous result to prove
or
ex3_2 T T λu2:T.λt3:T.eq T t2 (THead (Bind Void) u2 t3) λu2:T.λ:T.pr0 u1 u2 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2)
case refl_equal : ⇒
the thesis becomes
or
ex3_2 T T λu2:T.λt3:T.eq T t2 (THead (Bind Void) u2 t3) λu2:T.λ:T.pr0 u1 u2 λ:T.λt3:T.pr0 (lift (S O) O t0) t3
pr0 (lift (S O) O t0) (lift (S O) O t2)
by (pr0_lift . . H2 . .)
we proved pr0 (lift (S O) O t0) (lift (S O) O t2)
by (or_intror . . previous)
or
ex3_2 T T λu2:T.λt3:T.eq T t2 (THead (Bind Void) u2 t3) λu2:T.λ:T.pr0 u1 u2 λ:T.λt3:T.pr0 (lift (S O) O t0) t3
pr0 (lift (S O) O t0) (lift (S O) O t2)
we proved
or
ex3_2 T T λu2:T.λt3:T.eq T t2 (THead (Bind Void) u2 t3) λu2:T.λ:T.pr0 u1 u2 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2)
eq T u u1
→(eq B b Void
→(or
ex3_2 T T λu2:T.λt3:T.eq T t2 (THead (Bind Void) u2 t3) λu2:T.λ:T.pr0 u1 u2 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2)))
end of h1
(h2)
consider H6
we proved
eq
T
<λ:T.T>
CASE THead (Bind b) u (lift (S O) O t0) OF
TSort ⇒u
| TLRef ⇒u
| THead t ⇒t
<λ:T.T>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒u
| TLRef ⇒u
| THead t ⇒t
eq T u u1
end of h2
by (h1 h2)
eq B b Void
→(or
ex3_2 T T λu2:T.λt3:T.eq T t2 (THead (Bind Void) u2 t3) λu2:T.λ:T.pr0 u1 u2 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2))
end of h1
(h2)
consider H5
we proved
eq
B
<λ:T.B>
CASE THead (Bind b) u (lift (S O) O t0) OF
TSort ⇒b
| TLRef ⇒b
| THead k ⇒<λ:K.B> CASE k OF Bind b0⇒b0 | Flat ⇒b
<λ:T.B>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒b
| TLRef ⇒b
| THead k ⇒<λ:K.B> CASE k OF Bind b0⇒b0 | Flat ⇒b
eq B b Void
end of h2
by (h1 h2)
we proved
or
ex3_2 T T λu2:T.λt3:T.eq T t2 (THead (Bind Void) u2 t3) λu2:T.λ:T.pr0 u1 u2 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2)
∀H4:eq T (THead (Bind b) u (lift (S O) O t0)) (THead (Bind Void) u1 t1)
.or
ex3_2 T T λu2:T.λt3:T.eq T t2 (THead (Bind Void) u2 t3) λu2:T.λ:T.pr0 u1 u2 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2)
case pr0_tau : t0:T t2:T :pr0 t0 t2 u:T ⇒
the thesis becomes
∀H3:eq T (THead (Flat Cast) u t0) (THead (Bind Void) u1 t1)
.or
ex3_2 T T λu2:T.λt3:T.eq T t2 (THead (Bind Void) u2 t3) λu2:T.λ:T.pr0 u1 u2 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2)
() by induction hypothesis we know
eq T t0 (THead (Bind Void) u1 t1)
→(or
ex3_2 T T λu2:T.λt3:T.eq T t2 (THead (Bind Void) u2 t3) λu2:T.λ:T.pr0 u1 u2 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2))
suppose H3: eq T (THead (Flat Cast) u t0) (THead (Bind Void) u1 t1)
(H4)
we proceed by induction on H3 to prove
<λ:T.Prop>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒False | Flat ⇒True
case refl_equal : ⇒
the thesis becomes
<λ:T.Prop>
CASE THead (Flat Cast) u t0 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒False | Flat ⇒True
consider I
we proved True
<λ:T.Prop>
CASE THead (Flat Cast) u t0 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒False | Flat ⇒True
<λ:T.Prop>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒False | Flat ⇒True
end of H4
consider H4
we proved
<λ:T.Prop>
CASE THead (Bind Void) u1 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead k ⇒<λ:K.Prop> CASE k OF Bind ⇒False | Flat ⇒True
that is equivalent to False
we proceed by induction on the previous result to prove
or
ex3_2 T T λu2:T.λt3:T.eq T t2 (THead (Bind Void) u2 t3) λu2:T.λ:T.pr0 u1 u2 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2)
we proved
or
ex3_2 T T λu2:T.λt3:T.eq T t2 (THead (Bind Void) u2 t3) λu2:T.λ:T.pr0 u1 u2 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2)
∀H3:eq T (THead (Flat Cast) u t0) (THead (Bind Void) u1 t1)
.or
ex3_2 T T λu2:T.λt3:T.eq T t2 (THead (Bind Void) u2 t3) λu2:T.λ:T.pr0 u1 u2 λ:T.λt3:T.pr0 t1 t3
pr0 t1 (lift (S O) O t2)
we proved
eq T y (THead (Bind Void) u1 t1)
→(or
ex3_2 T T λu2:T.λt2:T.eq T x (THead (Bind Void) u2 t2) λu2:T.λ:T.pr0 u1 u2 λ:T.λt2:T.pr0 t1 t2
pr0 t1 (lift (S O) O x))
we proved
∀y:T
.pr0 y x
→(eq T y (THead (Bind Void) u1 t1)
→(or
ex3_2 T T λu2:T.λt2:T.eq T x (THead (Bind Void) u2 t2) λu2:T.λ:T.pr0 u1 u2 λ:T.λt2:T.pr0 t1 t2
pr0 t1 (lift (S O) O x)))
by (insert_eq . . . . previous H)
we proved
or
ex3_2 T T λu2:T.λt2:T.eq T x (THead (Bind Void) u2 t2) λu2:T.λ:T.pr0 u1 u2 λ:T.λt2:T.pr0 t1 t2
pr0 t1 (lift (S O) O x)
we proved
∀u1:T
.∀t1:T
.∀x:T
.pr0 (THead (Bind Void) u1 t1) x
→(or
ex3_2 T T λu2:T.λt2:T.eq T x (THead (Bind Void) u2 t2) λu2:T.λ:T.pr0 u1 u2 λ:T.λt2:T.pr0 t1 t2
pr0 t1 (lift (S O) O x))