(* This file was automatically generated: do not edit *********************)
-include "Basic-1/subst1/defs.ma".
+include "basic_1/subst1/fwd.ma".
-include "Basic-1/subst0/props.ma".
+include "basic_1/subst0/props.ma".
theorem subst1_head:
\forall (v: T).(\forall (u1: T).(\forall (u2: T).(\forall (i: nat).((subst1
(subst0_fst v t2 u1 i H0 t1 k)) (\lambda (t3: T).(\lambda (H2: (subst0 (s k
i) v t1 t3)).(subst1_single i v (THead k u1 t1) (THead k t2 t3) (subst0_both
v u1 t2 i H0 k t1 t3 H2)))) t0 H1))))))) u2 H))))).
-(* COMMENTS
-Initial nodes: 369
-END *)
-theorem subst1_lift_lt:
+lemma subst1_lift_lt:
\forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).((subst1
i u t1 t2) \to (\forall (d: nat).((lt i d) \to (\forall (h: nat).(subst1 i
(lift h (minus d (S i)) u) (lift h d t1) (lift h d t2)))))))))
nat).(\lambda (H1: (lt i d)).(\lambda (h: nat).(subst1_single i (lift h
(minus d (S i)) u) (lift h d t1) (lift h d t3) (subst0_lift_lt t1 t3 u i H0 d
H1 h))))))) t2 H))))).
-(* COMMENTS
-Initial nodes: 185
-END *)
-theorem subst1_lift_ge:
+lemma subst1_lift_ge:
\forall (t1: T).(\forall (t2: T).(\forall (u: T).(\forall (i: nat).(\forall
(h: nat).((subst1 i u t1 t2) \to (\forall (d: nat).((le d i) \to (subst1
(plus i h) u (lift h d t1) (lift h d t2)))))))))
(lift h d t1)))) (\lambda (t3: T).(\lambda (H0: (subst0 i u t1 t3)).(\lambda
(d: nat).(\lambda (H1: (le d i)).(subst1_single (plus i h) u (lift h d t1)
(lift h d t3) (subst0_lift_ge t1 t3 u i h H0 d H1)))))) t2 H)))))).
-(* COMMENTS
-Initial nodes: 157
-END *)
-theorem subst1_ex:
+lemma subst1_ex:
\forall (u: T).(\forall (t1: T).(\forall (d: nat).(ex T (\lambda (t2:
T).(subst1 d u t1 (lift (S O) d t2))))))
\def
(\lambda (t2: T).(subst1 n u (TLRef n) (lift (S O) n t2))) (lift n O u)
(eq_ind_r T (lift (plus (S O) n) O u) (\lambda (t: T).(subst1 n u (TLRef n)
t)) (subst1_single n u (TLRef n) (lift (S n) O u) (subst0_lref u n)) (lift (S
-O) n (lift n O u)) (lift_free u n (S O) O n (le_n (plus O n)) (le_O_n n)))) d
+O) n (lift n O u)) (lift_free u n (S O) O n (le_plus_r O n) (le_O_n n)))) d
H)) (\lambda (H: (lt d n)).(ex_intro T (\lambda (t2: T).(subst1 d u (TLRef n)
(lift (S O) d t2))) (TLRef (pred n)) (eq_ind_r T (TLRef n) (\lambda (t:
T).(subst1 d u (TLRef n) t)) (subst1_refl d u (TLRef n)) (lift (S O) d (TLRef
d) x0)) (\lambda (t2: T).(subst1 d u (THead k t t0) t2)) (subst1_head u t
(lift (S O) d x) d H2 k t0 (lift (S O) (s k d) x0) H4) (lift (S O) d (THead k
x x0)) (lift_head k x x0 (S O) d))))) H3))))) H1))))))))) t1)).
-(* COMMENTS
-Initial nodes: 925
-END *)
-theorem subst1_lift_S:
+lemma subst1_lift_S:
\forall (u: T).(\forall (i: nat).(\forall (h: nat).((le h i) \to (subst1 i
(TLRef h) (lift (S h) (S i) u) (lift (S h) i u)))))
\def
(\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i (TLRef n)))) (eq_ind_r T
(TLRef n) (\lambda (t: T).(subst1 i (TLRef h) (TLRef n) t)) (subst1_refl i
(TLRef h) (TLRef n)) (lift (S h) i (TLRef n)) (lift_lref_lt n (S h) i H0))
-(lift (S h) (S i) (TLRef n)) (lift_lref_lt n (S h) (S i) (le_S (S n) i H0))))
-(\lambda (H0: (eq nat n i)).(let H1 \def (eq_ind_r nat i (\lambda (n0:
-nat).(le h n0)) H n H0) in (eq_ind nat n (\lambda (n0: nat).(subst1 n0 (TLRef
-h) (lift (S h) (S n0) (TLRef n)) (lift (S h) n0 (TLRef n)))) (eq_ind_r T
-(TLRef n) (\lambda (t: T).(subst1 n (TLRef h) t (lift (S h) n (TLRef n))))
-(eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 n (TLRef h) (TLRef
-n) t)) (eq_ind nat (S (plus n h)) (\lambda (n0: nat).(subst1 n (TLRef h)
-(TLRef n) (TLRef n0))) (eq_ind_r nat (plus h n) (\lambda (n0: nat).(subst1 n
-(TLRef h) (TLRef n) (TLRef (S n0)))) (eq_ind nat (plus h (S n)) (\lambda (n0:
-nat).(subst1 n (TLRef h) (TLRef n) (TLRef n0))) (eq_ind T (lift (S n) O
-(TLRef h)) (\lambda (t: T).(subst1 n (TLRef h) (TLRef n) t)) (subst1_single n
-(TLRef h) (TLRef n) (lift (S n) O (TLRef h)) (subst0_lref (TLRef h) n))
-(TLRef (plus h (S n))) (lift_lref_ge h (S n) O (le_O_n h))) (S (plus h n))
-(sym_eq nat (S (plus h n)) (plus h (S n)) (plus_n_Sm h n))) (plus n h)
-(plus_sym n h)) (plus n (S h)) (plus_n_Sm n h)) (lift (S h) n (TLRef n))
-(lift_lref_ge n (S h) n (le_n n))) (lift (S h) (S n) (TLRef n)) (lift_lref_lt
-n (S h) (S n) (le_n (S n)))) i H0))) (\lambda (H0: (lt i n)).(eq_ind_r T
-(TLRef (plus n (S h))) (\lambda (t: T).(subst1 i (TLRef h) t (lift (S h) i
-(TLRef n)))) (eq_ind_r T (TLRef (plus n (S h))) (\lambda (t: T).(subst1 i
-(TLRef h) (TLRef (plus n (S h))) t)) (subst1_refl i (TLRef h) (TLRef (plus n
-(S h)))) (lift (S h) i (TLRef n)) (lift_lref_ge n (S h) i (le_S_n i n (le_S
-(S i) n H0)))) (lift (S h) (S i) (TLRef n)) (lift_lref_ge n (S h) (S i)
-H0)))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (i:
+(lift (S h) (S i) (TLRef n)) (lift_lref_lt n (S h) (S i) (le_S_n (S n) (S i)
+(le_S (S (S n)) (S i) (le_n_S (S n) i H0)))))) (\lambda (H0: (eq nat n
+i)).(let H1 \def (eq_ind_r nat i (\lambda (n0: nat).(le h n0)) H n H0) in
+(eq_ind nat n (\lambda (n0: nat).(subst1 n0 (TLRef h) (lift (S h) (S n0)
+(TLRef n)) (lift (S h) n0 (TLRef n)))) (eq_ind_r T (TLRef n) (\lambda (t:
+T).(subst1 n (TLRef h) t (lift (S h) n (TLRef n)))) (eq_ind_r T (TLRef (plus
+n (S h))) (\lambda (t: T).(subst1 n (TLRef h) (TLRef n) t)) (eq_ind nat (S
+(plus n h)) (\lambda (n0: nat).(subst1 n (TLRef h) (TLRef n) (TLRef n0)))
+(eq_ind_r nat (plus h n) (\lambda (n0: nat).(subst1 n (TLRef h) (TLRef n)
+(TLRef (S n0)))) (eq_ind nat (plus h (S n)) (\lambda (n0: nat).(subst1 n
+(TLRef h) (TLRef n) (TLRef n0))) (eq_ind T (lift (S n) O (TLRef h)) (\lambda
+(t: T).(subst1 n (TLRef h) (TLRef n) t)) (subst1_single n (TLRef h) (TLRef n)
+(lift (S n) O (TLRef h)) (subst0_lref (TLRef h) n)) (TLRef (plus h (S n)))
+(lift_lref_ge h (S n) O (le_O_n h))) (S (plus h n)) (sym_eq nat (S (plus h
+n)) (plus h (S n)) (plus_n_Sm h n))) (plus n h) (plus_sym n h)) (plus n (S
+h)) (plus_n_Sm n h)) (lift (S h) n (TLRef n)) (lift_lref_ge n (S h) n (le_n
+n))) (lift (S h) (S n) (TLRef n)) (lift_lref_lt n (S h) (S n) (le_n (S n))))
+i H0))) (\lambda (H0: (lt i n)).(eq_ind_r T (TLRef (plus n (S h))) (\lambda
+(t: T).(subst1 i (TLRef h) t (lift (S h) i (TLRef n)))) (eq_ind_r T (TLRef
+(plus n (S h))) (\lambda (t: T).(subst1 i (TLRef h) (TLRef (plus n (S h)))
+t)) (subst1_refl i (TLRef h) (TLRef (plus n (S h)))) (lift (S h) i (TLRef n))
+(lift_lref_ge n (S h) i (le_S_n i n (le_S_n (S i) (S n) (le_S (S (S i)) (S n)
+(le_n_S (S i) n H0)))))) (lift (S h) (S i) (TLRef n)) (lift_lref_ge n (S h)
+(S i) H0)))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (H: ((\forall (i:
nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i) t)
(lift (S h) i t))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (i:
nat).(\forall (h: nat).((le h i) \to (subst1 i (TLRef h) (lift (S h) (S i)
(S h) (s k i) t0))) (H0 (s k i) h (le_trans h i (s k i) H1 (s_inc k i))) (s k
(S i)) (s_S k i))) (lift (S h) i (THead k t t0)) (lift_head k t t0 (S h) i))
(lift (S h) (S i) (THead k t t0)) (lift_head k t t0 (S h) (S i))))))))))) u).
-(* COMMENTS
-Initial nodes: 1421
-END *)
projects / helm.git / blobdiff