include "basic_1/wf3/getl.ma".
-theorem wf3_pr2_conf:
+lemma wf3_pr2_conf:
\forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr2 c1
t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1
u) \to (pr2 c2 t1 t2)))))))))
\def
\lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (pr2 c1 t1 t2)).(let TMP_1 \def (\lambda (c: C).(\lambda (t: T).(\lambda
-(t0: T).(\forall (c2: C).((wf3 g c c2) \to (\forall (u: T).((ty3 g c t u) \to
-(pr2 c2 t t0)))))))) in (let TMP_2 \def (\lambda (c: C).(\lambda (t3:
-T).(\lambda (t4: T).(\lambda (H0: (pr0 t3 t4)).(\lambda (c2: C).(\lambda (_:
-(wf3 g c c2)).(\lambda (u: T).(\lambda (_: (ty3 g c t3 u)).(pr2_free c2 t3 t4
-H0))))))))) in (let TMP_12 \def (\lambda (c: C).(\lambda (d: C).(\lambda (u:
-T).(\lambda (i: nat).(\lambda (H0: (getl i c (CHead d (Bind Abbr)
-u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 t4)).(\lambda
-(t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2: C).(\lambda (H3: (wf3 g
-c c2)).(\lambda (u0: T).(\lambda (H4: (ty3 g c t3 u0)).(let H_y \def
-(ty3_sred_pr0 t3 t4 H1 g c u0 H4) in (let H_x \def (ty3_getl_subst0 g c t4 u0
-H_y u t i H2 Abbr d u H0) in (let H5 \def H_x in (let TMP_3 \def (\lambda (w:
-T).(ty3 g d u w)) in (let TMP_4 \def (pr2 c2 t3 t) in (let TMP_11 \def
-(\lambda (x: T).(\lambda (H6: (ty3 g d u x)).(let H_x0 \def (wf3_getl_conf
-Abbr i c d u H0 g c2 H3 x H6) in (let H7 \def H_x0 in (let TMP_7 \def
-(\lambda (d2: C).(let TMP_5 \def (Bind Abbr) in (let TMP_6 \def (CHead d2
-TMP_5 u) in (getl i c2 TMP_6)))) in (let TMP_8 \def (\lambda (d2: C).(wf3 g d
-d2)) in (let TMP_9 \def (pr2 c2 t3 t) in (let TMP_10 \def (\lambda (x0:
-C).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(\lambda (_: (wf3 g d
-x0)).(pr2_delta c2 x0 u i H8 t3 t4 H1 t H2)))) in (ex2_ind C TMP_7 TMP_8
-TMP_9 TMP_10 H7))))))))) in (ex_ind T TMP_3 TMP_4 TMP_11
-H5))))))))))))))))))))) in (pr2_ind TMP_1 TMP_2 TMP_12 c1 t1 t2 H)))))))).
+(H: (pr2 c1 t1 t2)).(pr2_ind (\lambda (c: C).(\lambda (t: T).(\lambda (t0:
+T).(\forall (c2: C).((wf3 g c c2) \to (\forall (u: T).((ty3 g c t u) \to (pr2
+c2 t t0)))))))) (\lambda (c: C).(\lambda (t3: T).(\lambda (t4: T).(\lambda
+(H0: (pr0 t3 t4)).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(\lambda (u:
+T).(\lambda (_: (ty3 g c t3 u)).(pr2_free c2 t3 t4 H0))))))))) (\lambda (c:
+C).(\lambda (d: C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c
+(CHead d (Bind Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1:
+(pr0 t3 t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (c2:
+C).(\lambda (H3: (wf3 g c c2)).(\lambda (u0: T).(\lambda (H4: (ty3 g c t3
+u0)).(let H_y \def (ty3_sred_pr0 t3 t4 H1 g c u0 H4) in (let H_x \def
+(ty3_getl_subst0 g c t4 u0 H_y u t i H2 Abbr d u H0) in (let H5 \def H_x in
+(ex_ind T (\lambda (w: T).(ty3 g d u w)) (pr2 c2 t3 t) (\lambda (x:
+T).(\lambda (H6: (ty3 g d u x)).(let H_x0 \def (wf3_getl_conf Abbr i c d u H0
+g c2 H3 x H6) in (let H7 \def H_x0 in (ex2_ind C (\lambda (d2: C).(getl i c2
+(CHead d2 (Bind Abbr) u))) (\lambda (d2: C).(wf3 g d d2)) (pr2 c2 t3 t)
+(\lambda (x0: C).(\lambda (H8: (getl i c2 (CHead x0 (Bind Abbr) u))).(\lambda
+(_: (wf3 g d x0)).(pr2_delta c2 x0 u i H8 t3 t4 H1 t H2)))) H7)))))
+H5)))))))))))))))))) c1 t1 t2 H))))).
-theorem wf3_pr3_conf:
+lemma wf3_pr3_conf:
\forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pr3 c1
t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t1
u) \to (pr3 c2 t1 t2)))))))))
\def
\lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (pr3 c1 t1 t2)).(let TMP_1 \def (\lambda (t: T).(\lambda (t0: T).(\forall
+(H: (pr3 c1 t1 t2)).(pr3_ind c1 (\lambda (t: T).(\lambda (t0: T).(\forall
(c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t u) \to (pr3 c2 t
-t0))))))) in (let TMP_2 \def (\lambda (t: T).(\lambda (c2: C).(\lambda (_:
-(wf3 g c1 c2)).(\lambda (u: T).(\lambda (_: (ty3 g c1 t u)).(pr3_refl c2
-t)))))) in (let TMP_6 \def (\lambda (t3: T).(\lambda (t4: T).(\lambda (H0:
-(pr2 c1 t4 t3)).(\lambda (t5: T).(\lambda (_: (pr3 c1 t3 t5)).(\lambda (H2:
-((\forall (c2: C).((wf3 g c1 c2) \to (\forall (u: T).((ty3 g c1 t3 u) \to
-(pr3 c2 t3 t5))))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c1 c2)).(\lambda
-(u: T).(\lambda (H4: (ty3 g c1 t4 u)).(let TMP_3 \def (wf3_pr2_conf g c1 t4
-t3 H0 c2 H3 u H4) in (let TMP_4 \def (ty3_sred_pr2 c1 t4 t3 H0 g u H4) in
-(let TMP_5 \def (H2 c2 H3 u TMP_4) in (pr3_sing c2 t3 t4 TMP_3 t5
-TMP_5)))))))))))))) in (pr3_ind c1 TMP_1 TMP_2 TMP_6 t1 t2 H)))))))).
+t0))))))) (\lambda (t: T).(\lambda (c2: C).(\lambda (_: (wf3 g c1
+c2)).(\lambda (u: T).(\lambda (_: (ty3 g c1 t u)).(pr3_refl c2 t))))))
+(\lambda (t3: T).(\lambda (t4: T).(\lambda (H0: (pr2 c1 t4 t3)).(\lambda (t5:
+T).(\lambda (_: (pr3 c1 t3 t5)).(\lambda (H2: ((\forall (c2: C).((wf3 g c1
+c2) \to (\forall (u: T).((ty3 g c1 t3 u) \to (pr3 c2 t3 t5))))))).(\lambda
+(c2: C).(\lambda (H3: (wf3 g c1 c2)).(\lambda (u: T).(\lambda (H4: (ty3 g c1
+t4 u)).(pr3_sing c2 t3 t4 (wf3_pr2_conf g c1 t4 t3 H0 c2 H3 u H4) t5 (H2 c2
+H3 u (ty3_sred_pr2 c1 t4 t3 H0 g u H4))))))))))))) t1 t2 H))))).
-theorem wf3_pc3_conf:
+lemma wf3_pc3_conf:
\forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((pc3 c1
t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (\forall (u1: T).((ty3 g c1 t1
u1) \to (\forall (u2: T).((ty3 g c1 t2 u2) \to (pc3 c2 t1 t2)))))))))))
\lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
(H: (pc3 c1 t1 t2)).(\lambda (c2: C).(\lambda (H0: (wf3 g c1 c2)).(\lambda
(u1: T).(\lambda (H1: (ty3 g c1 t1 u1)).(\lambda (u2: T).(\lambda (H2: (ty3 g
-c1 t2 u2)).(let H3 \def H in (let TMP_1 \def (\lambda (t: T).(pr3 c1 t1 t))
-in (let TMP_2 \def (\lambda (t: T).(pr3 c1 t2 t)) in (let TMP_3 \def (pc3 c2
-t1 t2) in (let TMP_6 \def (\lambda (x: T).(\lambda (H4: (pr3 c1 t1
-x)).(\lambda (H5: (pr3 c1 t2 x)).(let TMP_4 \def (wf3_pr3_conf g c1 t1 x H4
-c2 H0 u1 H1) in (let TMP_5 \def (wf3_pr3_conf g c1 t2 x H5 c2 H0 u2 H2) in
-(pc3_pr3_t c2 t1 x TMP_4 t2 TMP_5)))))) in (ex2_ind T TMP_1 TMP_2 TMP_3 TMP_6
-H3)))))))))))))))).
+c1 t2 u2)).(let H3 \def H in (ex2_ind T (\lambda (t: T).(pr3 c1 t1 t))
+(\lambda (t: T).(pr3 c1 t2 t)) (pc3 c2 t1 t2) (\lambda (x: T).(\lambda (H4:
+(pr3 c1 t1 x)).(\lambda (H5: (pr3 c1 t2 x)).(pc3_pr3_t c2 t1 x (wf3_pr3_conf
+g c1 t1 x H4 c2 H0 u1 H1) t2 (wf3_pr3_conf g c1 t2 x H5 c2 H0 u2 H2)))))
+H3)))))))))))).
-theorem wf3_ty3_conf:
+lemma wf3_ty3_conf:
\forall (g: G).(\forall (c1: C).(\forall (t1: T).(\forall (t2: T).((ty3 g c1
t1 t2) \to (\forall (c2: C).((wf3 g c1 c2) \to (ty3 g c2 t1 t2)))))))
\def
\lambda (g: G).(\lambda (c1: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda
-(H: (ty3 g c1 t1 t2)).(let TMP_1 \def (\lambda (c: C).(\lambda (t:
-T).(\lambda (t0: T).(\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t t0))))))
-in (let TMP_9 \def (\lambda (c: C).(\lambda (t3: T).(\lambda (t: T).(\lambda
-(H0: (ty3 g c t3 t)).(\lambda (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g
-c2 t3 t))))).(\lambda (u: T).(\lambda (t4: T).(\lambda (H2: (ty3 g c u
-t4)).(\lambda (H3: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 u
-t4))))).(\lambda (H4: (pc3 c t4 t3)).(\lambda (c2: C).(\lambda (H5: (wf3 g c
-c2)).(let TMP_2 \def (\lambda (t0: T).(ty3 g c t4 t0)) in (let TMP_3 \def
-(ty3 g c2 u t3) in (let TMP_7 \def (\lambda (x: T).(\lambda (H6: (ty3 g c t4
-x)).(let TMP_4 \def (H1 c2 H5) in (let TMP_5 \def (H3 c2 H5) in (let TMP_6
-\def (wf3_pc3_conf g c t4 t3 H4 c2 H5 x H6 t H0) in (ty3_conv g c2 t3 t TMP_4
-u t4 TMP_5 TMP_6)))))) in (let TMP_8 \def (ty3_correct g c u t4 H2) in
-(ex_ind T TMP_2 TMP_3 TMP_7 TMP_8))))))))))))))))) in (let TMP_10 \def
-(\lambda (c: C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (wf3 g c
-c2)).(ty3_sort g c2 m))))) in (let TMP_21 \def (\lambda (n: nat).(\lambda (c:
-C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c (CHead d (Bind
-Abbr) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u t)).(\lambda (H2:
-((\forall (c2: C).((wf3 g d c2) \to (ty3 g c2 u t))))).(\lambda (c2:
-C).(\lambda (H3: (wf3 g c c2)).(let H_x \def (wf3_getl_conf Abbr n c d u H0 g
-c2 H3 t H1) in (let H4 \def H_x in (let TMP_13 \def (\lambda (d2: C).(let
-TMP_11 \def (Bind Abbr) in (let TMP_12 \def (CHead d2 TMP_11 u) in (getl n c2
-TMP_12)))) in (let TMP_14 \def (\lambda (d2: C).(wf3 g d d2)) in (let TMP_15
-\def (TLRef n) in (let TMP_16 \def (S n) in (let TMP_17 \def (lift TMP_16 O
-t) in (let TMP_18 \def (ty3 g c2 TMP_15 TMP_17) in (let TMP_20 \def (\lambda
-(x: C).(\lambda (H5: (getl n c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (wf3
-g d x)).(let TMP_19 \def (H2 x H6) in (ty3_abbr g n c2 x u H5 t TMP_19)))))
-in (ex2_ind C TMP_13 TMP_14 TMP_18 TMP_20 H4)))))))))))))))))))) in (let
-TMP_32 \def (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u:
-T).(\lambda (H0: (getl n c (CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda
+(H: (ty3 g c1 t1 t2)).(ty3_ind g (\lambda (c: C).(\lambda (t: T).(\lambda
+(t0: T).(\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t t0)))))) (\lambda (c:
+C).(\lambda (t3: T).(\lambda (t: T).(\lambda (H0: (ty3 g c t3 t)).(\lambda
+(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t))))).(\lambda (u:
+T).(\lambda (t4: T).(\lambda (H2: (ty3 g c u t4)).(\lambda (H3: ((\forall
+(c2: C).((wf3 g c c2) \to (ty3 g c2 u t4))))).(\lambda (H4: (pc3 c t4
+t3)).(\lambda (c2: C).(\lambda (H5: (wf3 g c c2)).(ex_ind T (\lambda (t0:
+T).(ty3 g c t4 t0)) (ty3 g c2 u t3) (\lambda (x: T).(\lambda (H6: (ty3 g c t4
+x)).(ty3_conv g c2 t3 t (H1 c2 H5) u t4 (H3 c2 H5) (wf3_pc3_conf g c t4 t3 H4
+c2 H5 x H6 t H0)))) (ty3_correct g c u t4 H2)))))))))))))) (\lambda (c:
+C).(\lambda (m: nat).(\lambda (c2: C).(\lambda (_: (wf3 g c c2)).(ty3_sort g
+c2 m))))) (\lambda (n: nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u:
+T).(\lambda (H0: (getl n c (CHead d (Bind Abbr) u))).(\lambda (t: T).(\lambda
(H1: (ty3 g d u t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g
c2 u t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def
-(wf3_getl_conf Abst n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (let
-TMP_24 \def (\lambda (d2: C).(let TMP_22 \def (Bind Abst) in (let TMP_23 \def
-(CHead d2 TMP_22 u) in (getl n c2 TMP_23)))) in (let TMP_25 \def (\lambda
-(d2: C).(wf3 g d d2)) in (let TMP_26 \def (TLRef n) in (let TMP_27 \def (S n)
-in (let TMP_28 \def (lift TMP_27 O u) in (let TMP_29 \def (ty3 g c2 TMP_26
-TMP_28) in (let TMP_31 \def (\lambda (x: C).(\lambda (H5: (getl n c2 (CHead x
-(Bind Abst) u))).(\lambda (H6: (wf3 g d x)).(let TMP_30 \def (H2 x H6) in
-(ty3_abst g n c2 x u H5 t TMP_30))))) in (ex2_ind C TMP_24 TMP_25 TMP_29
-TMP_31 H4)))))))))))))))))))) in (let TMP_38 \def (\lambda (c: C).(\lambda
-(u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c u t)).(\lambda (H1: ((\forall
-(c2: C).((wf3 g c c2) \to (ty3 g c2 u t))))).(\lambda (b: B).(\lambda (t3:
-T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u) t3 t4)).(\lambda
-(H3: ((\forall (c2: C).((wf3 g (CHead c (Bind b) u) c2) \to (ty3 g c2 t3
-t4))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c c2)).(let TMP_33 \def (H1 c2
-H4) in (let TMP_34 \def (Bind b) in (let TMP_35 \def (CHead c2 TMP_34 u) in
-(let TMP_36 \def (wf3_bind g c c2 H4 u t H0 b) in (let TMP_37 \def (H3 TMP_35
-TMP_36) in (ty3_bind g c2 u t TMP_33 b t3 t4 TMP_37)))))))))))))))))) in (let
-TMP_41 \def (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda (_: (ty3
-g c w u)).(\lambda (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 w
-u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead (Bind
-Abst) u t))).(\lambda (H3: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 v
-(THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c
-c2)).(let TMP_39 \def (H1 c2 H4) in (let TMP_40 \def (H3 c2 H4) in (ty3_appl
-g c2 w u TMP_39 v t TMP_40)))))))))))))) in (let TMP_44 \def (\lambda (c:
+(wf3_getl_conf Abbr n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind
+C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abbr) u))) (\lambda (d2:
+C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O t)) (\lambda (x:
+C).(\lambda (H5: (getl n c2 (CHead x (Bind Abbr) u))).(\lambda (H6: (wf3 g d
+x)).(ty3_abbr g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (n:
+nat).(\lambda (c: C).(\lambda (d: C).(\lambda (u: T).(\lambda (H0: (getl n c
+(CHead d (Bind Abst) u))).(\lambda (t: T).(\lambda (H1: (ty3 g d u
+t)).(\lambda (H2: ((\forall (c2: C).((wf3 g d c2) \to (ty3 g c2 u
+t))))).(\lambda (c2: C).(\lambda (H3: (wf3 g c c2)).(let H_x \def
+(wf3_getl_conf Abst n c d u H0 g c2 H3 t H1) in (let H4 \def H_x in (ex2_ind
+C (\lambda (d2: C).(getl n c2 (CHead d2 (Bind Abst) u))) (\lambda (d2:
+C).(wf3 g d d2)) (ty3 g c2 (TLRef n) (lift (S n) O u)) (\lambda (x:
+C).(\lambda (H5: (getl n c2 (CHead x (Bind Abst) u))).(\lambda (H6: (wf3 g d
+x)).(ty3_abst g n c2 x u H5 t (H2 x H6))))) H4))))))))))))) (\lambda (c:
+C).(\lambda (u: T).(\lambda (t: T).(\lambda (H0: (ty3 g c u t)).(\lambda (H1:
+((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 u t))))).(\lambda (b:
+B).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g (CHead c (Bind b) u)
+t3 t4)).(\lambda (H3: ((\forall (c2: C).((wf3 g (CHead c (Bind b) u) c2) \to
+(ty3 g c2 t3 t4))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c c2)).(ty3_bind g
+c2 u t (H1 c2 H4) b t3 t4 (H3 (CHead c2 (Bind b) u) (wf3_bind g c c2 H4 u t
+H0 b))))))))))))))) (\lambda (c: C).(\lambda (w: T).(\lambda (u: T).(\lambda
+(_: (ty3 g c w u)).(\lambda (H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g
+c2 w u))))).(\lambda (v: T).(\lambda (t: T).(\lambda (_: (ty3 g c v (THead
+(Bind Abst) u t))).(\lambda (H3: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g
+c2 v (THead (Bind Abst) u t)))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c
+c2)).(ty3_appl g c2 w u (H1 c2 H4) v t (H3 c2 H4))))))))))))) (\lambda (c:
C).(\lambda (t3: T).(\lambda (t4: T).(\lambda (_: (ty3 g c t3 t4)).(\lambda
(H1: ((\forall (c2: C).((wf3 g c c2) \to (ty3 g c2 t3 t4))))).(\lambda (t0:
T).(\lambda (_: (ty3 g c t4 t0)).(\lambda (H3: ((\forall (c2: C).((wf3 g c
c2) \to (ty3 g c2 t4 t0))))).(\lambda (c2: C).(\lambda (H4: (wf3 g c
-c2)).(let TMP_42 \def (H1 c2 H4) in (let TMP_43 \def (H3 c2 H4) in (ty3_cast
-g c2 t3 t4 TMP_42 t0 TMP_43))))))))))))) in (ty3_ind g TMP_1 TMP_9 TMP_10
-TMP_21 TMP_32 TMP_38 TMP_41 TMP_44 c1 t1 t2 H))))))))))))).
+c2)).(ty3_cast g c2 t3 t4 (H1 c2 H4) t0 (H3 c2 H4)))))))))))) c1 t1 t2 H))))).
projects / helm.git / blobdiff