DEFINITION sty0_gen_sort()
TYPE =
∀g:G.∀c:C.∀x:T.∀n:nat.(sty0 g c (TSort n) x)→(eq T x (TSort (next g n)))
BODY =
assume g: G
assume c: C
assume x: T
assume n: nat
suppose H: sty0 g c (TSort n) x
assume y: T
suppose H0: sty0 g c y x
we proceed by induction on H0 to prove (eq T y (TSort n))→(eq T x (TSort (next g n)))
case sty0_sort : :C n0:nat ⇒
the thesis becomes
∀H1:eq T (TSort n0) (TSort n)
.eq T (TSort (next g n0)) (TSort (next g n))
suppose H1: eq T (TSort n0) (TSort n)
(H2)
by (f_equal . . . . . H1)
we proved
eq
nat
<λ:T.nat> CASE TSort n0 OF TSort n1⇒n1 | TLRef ⇒n0 | THead ⇒n0
<λ:T.nat> CASE TSort n OF TSort n1⇒n1 | TLRef ⇒n0 | THead ⇒n0
eq
nat
λe:T.<λ:T.nat> CASE e OF TSort n1⇒n1 | TLRef ⇒n0 | THead ⇒n0 (TSort n0)
λe:T.<λ:T.nat> CASE e OF TSort n1⇒n1 | TLRef ⇒n0 | THead ⇒n0 (TSort n)
end of H2
(h1)
by (refl_equal . .)
eq T (TSort (next g n)) (TSort (next g n))
end of h1
(h2)
consider H2
we proved
eq
nat
<λ:T.nat> CASE TSort n0 OF TSort n1⇒n1 | TLRef ⇒n0 | THead ⇒n0
<λ:T.nat> CASE TSort n OF TSort n1⇒n1 | TLRef ⇒n0 | THead ⇒n0
eq nat n0 n
end of h2
by (eq_ind_r . . . h1 . h2)
we proved eq T (TSort (next g n0)) (TSort (next g n))
∀H1:eq T (TSort n0) (TSort n)
.eq T (TSort (next g n0)) (TSort (next g n))
case sty0_abbr : c0:C d:C v:T i:nat :getl i c0 (CHead d (Bind Abbr) v) w:T :sty0 g d v w ⇒
the thesis becomes
∀H4:eq T (TLRef i) (TSort n)
.eq T (lift (S i) O w) (TSort (next g n))
() by induction hypothesis we know (eq T v (TSort n))→(eq T w (TSort (next g n)))
suppose H4: eq T (TLRef i) (TSort n)
(H5)
we proceed by induction on H4 to prove
<λ:T.Prop>
CASE TSort n OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
case refl_equal : ⇒
the thesis becomes
<λ:T.Prop>
CASE TLRef i OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
consider I
we proved True
<λ:T.Prop>
CASE TLRef i OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
<λ:T.Prop>
CASE TSort n OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
end of H5
consider H5
we proved
<λ:T.Prop>
CASE TSort n OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
that is equivalent to False
we proceed by induction on the previous result to prove eq T (lift (S i) O w) (TSort (next g n))
we proved eq T (lift (S i) O w) (TSort (next g n))
∀H4:eq T (TLRef i) (TSort n)
.eq T (lift (S i) O w) (TSort (next g n))
case sty0_abst : c0:C d:C v:T i:nat :getl i c0 (CHead d (Bind Abst) v) w:T :sty0 g d v w ⇒
the thesis becomes
∀H4:eq T (TLRef i) (TSort n)
.eq T (lift (S i) O v) (TSort (next g n))
() by induction hypothesis we know (eq T v (TSort n))→(eq T w (TSort (next g n)))
suppose H4: eq T (TLRef i) (TSort n)
(H5)
we proceed by induction on H4 to prove
<λ:T.Prop>
CASE TSort n OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
case refl_equal : ⇒
the thesis becomes
<λ:T.Prop>
CASE TLRef i OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
consider I
we proved True
<λ:T.Prop>
CASE TLRef i OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
<λ:T.Prop>
CASE TSort n OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
end of H5
consider H5
we proved
<λ:T.Prop>
CASE TSort n OF
TSort ⇒False
| TLRef ⇒True
| THead ⇒False
that is equivalent to False
we proceed by induction on the previous result to prove eq T (lift (S i) O v) (TSort (next g n))
we proved eq T (lift (S i) O v) (TSort (next g n))
∀H4:eq T (TLRef i) (TSort n)
.eq T (lift (S i) O v) (TSort (next g n))
case sty0_bind : b:B c0:C v:T t1:T t2:T :sty0 g (CHead c0 (Bind b) v) t1 t2 ⇒
the thesis becomes
∀H3:eq T (THead (Bind b) v t1) (TSort n)
.eq T (THead (Bind b) v t2) (TSort (next g n))
() by induction hypothesis we know (eq T t1 (TSort n))→(eq T t2 (TSort (next g n)))
suppose H3: eq T (THead (Bind b) v t1) (TSort n)
(H4)
we proceed by induction on H3 to prove
<λ:T.Prop>
CASE TSort n OF
TSort ⇒False
| TLRef ⇒False
| THead ⇒True
case refl_equal : ⇒
the thesis becomes
<λ:T.Prop>
CASE THead (Bind b) v t1 OF
TSort ⇒False
| TLRef ⇒False
| THead ⇒True
consider I
we proved True
<λ:T.Prop>
CASE THead (Bind b) v t1 OF
TSort ⇒False
| TLRef ⇒False
| THead ⇒True
<λ:T.Prop>
CASE TSort n OF
TSort ⇒False
| TLRef ⇒False
| THead ⇒True
end of H4
consider H4
we proved
<λ:T.Prop>
CASE TSort n OF
TSort ⇒False
| TLRef ⇒False
| THead ⇒True
that is equivalent to False
we proceed by induction on the previous result to prove eq T (THead (Bind b) v t2) (TSort (next g n))
we proved eq T (THead (Bind b) v t2) (TSort (next g n))
∀H3:eq T (THead (Bind b) v t1) (TSort n)
.eq T (THead (Bind b) v t2) (TSort (next g n))
case sty0_appl : c0:C v:T t1:T t2:T :sty0 g c0 t1 t2 ⇒
the thesis becomes
∀H3:eq T (THead (Flat Appl) v t1) (TSort n)
.eq T (THead (Flat Appl) v t2) (TSort (next g n))
() by induction hypothesis we know (eq T t1 (TSort n))→(eq T t2 (TSort (next g n)))
suppose H3: eq T (THead (Flat Appl) v t1) (TSort n)
(H4)
we proceed by induction on H3 to prove
<λ:T.Prop>
CASE TSort n OF
TSort ⇒False
| TLRef ⇒False
| THead ⇒True
case refl_equal : ⇒
the thesis becomes
<λ:T.Prop>
CASE THead (Flat Appl) v t1 OF
TSort ⇒False
| TLRef ⇒False
| THead ⇒True
consider I
we proved True
<λ:T.Prop>
CASE THead (Flat Appl) v t1 OF
TSort ⇒False
| TLRef ⇒False
| THead ⇒True
<λ:T.Prop>
CASE TSort n OF
TSort ⇒False
| TLRef ⇒False
| THead ⇒True
end of H4
consider H4
we proved
<λ:T.Prop>
CASE TSort n OF
TSort ⇒False
| TLRef ⇒False
| THead ⇒True
that is equivalent to False
we proceed by induction on the previous result to prove eq T (THead (Flat Appl) v t2) (TSort (next g n))
we proved eq T (THead (Flat Appl) v t2) (TSort (next g n))
∀H3:eq T (THead (Flat Appl) v t1) (TSort n)
.eq T (THead (Flat Appl) v t2) (TSort (next g n))
case sty0_cast : c0:C v1:T v2:T :sty0 g c0 v1 v2 t1:T t2:T :sty0 g c0 t1 t2 ⇒
the thesis becomes
∀H5:eq T (THead (Flat Cast) v1 t1) (TSort n)
.eq T (THead (Flat Cast) v2 t2) (TSort (next g n))
() by induction hypothesis we know (eq T v1 (TSort n))→(eq T v2 (TSort (next g n)))
() by induction hypothesis we know (eq T t1 (TSort n))→(eq T t2 (TSort (next g n)))
suppose H5: eq T (THead (Flat Cast) v1 t1) (TSort n)
(H6)
we proceed by induction on H5 to prove
<λ:T.Prop>
CASE TSort n OF
TSort ⇒False
| TLRef ⇒False
| THead ⇒True
case refl_equal : ⇒
the thesis becomes
<λ:T.Prop>
CASE THead (Flat Cast) v1 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead ⇒True
consider I
we proved True
<λ:T.Prop>
CASE THead (Flat Cast) v1 t1 OF
TSort ⇒False
| TLRef ⇒False
| THead ⇒True
<λ:T.Prop>
CASE TSort n OF
TSort ⇒False
| TLRef ⇒False
| THead ⇒True
end of H6
consider H6
we proved
<λ:T.Prop>
CASE TSort n OF
TSort ⇒False
| TLRef ⇒False
| THead ⇒True
that is equivalent to False
we proceed by induction on the previous result to prove eq T (THead (Flat Cast) v2 t2) (TSort (next g n))
we proved eq T (THead (Flat Cast) v2 t2) (TSort (next g n))
∀H5:eq T (THead (Flat Cast) v1 t1) (TSort n)
.eq T (THead (Flat Cast) v2 t2) (TSort (next g n))
we proved (eq T y (TSort n))→(eq T x (TSort (next g n)))
we proved
∀y:T
.sty0 g c y x
→(eq T y (TSort n))→(eq T x (TSort (next g n)))
by (insert_eq . . . . previous H)
we proved eq T x (TSort (next g n))
we proved ∀g:G.∀c:C.∀x:T.∀n:nat.(sty0 g c (TSort n) x)→(eq T x (TSort (next g n)))