(* This file was automatically generated: do not edit *********************)
-include "Basic-1/C/defs.ma".
+include "basic_1/C/fwd.ma".
-include "Basic-1/T/props.ma".
+include "basic_1/T/props.ma".
-theorem clt_cong:
+lemma cle_r:
+ \forall (c: C).(cle c c)
+\def
+ \lambda (c: C).(C_ind (\lambda (c0: C).(le (cweight c0) (cweight c0)))
+(\lambda (_: nat).(le_O_n O)) (\lambda (c0: C).(\lambda (_: (le (cweight c0)
+(cweight c0))).(\lambda (_: K).(\lambda (t: T).(le_n (plus (cweight c0)
+(tweight t))))))) c).
+
+lemma cle_head:
+ \forall (c1: C).(\forall (c2: C).((cle c1 c2) \to (\forall (u1: T).(\forall
+(u2: T).((tle u1 u2) \to (\forall (k: K).(cle (CHead c1 k u1) (CHead c2 k
+u2))))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (le (cweight c1) (cweight
+c2))).(\lambda (u1: T).(\lambda (u2: T).(\lambda (H0: (le (tweight u1)
+(tweight u2))).(\lambda (_: K).(le_plus_plus (cweight c1) (cweight c2)
+(tweight u1) (tweight u2) H H0))))))).
+
+lemma cle_trans_head:
+ \forall (c1: C).(\forall (c2: C).((cle c1 c2) \to (\forall (k: K).(\forall
+(u: T).(cle c1 (CHead c2 k u))))))
+\def
+ \lambda (c1: C).(\lambda (c2: C).(\lambda (H: (le (cweight c1) (cweight
+c2))).(\lambda (_: K).(\lambda (u: T).(le_plus_trans (cweight c1) (cweight
+c2) (tweight u) H))))).
+
+lemma clt_cong:
\forall (c: C).(\forall (d: C).((clt c d) \to (\forall (k: K).(\forall (t:
T).(clt (CHead c k t) (CHead d k t))))))
\def
\lambda (c: C).(\lambda (d: C).(\lambda (H: (lt (cweight c) (cweight
d))).(\lambda (_: K).(\lambda (t: T).(lt_reg_r (cweight c) (cweight d)
(tweight t) H))))).
-(* COMMENTS
-Initial nodes: 33
-END *)
-theorem clt_head:
+lemma clt_head:
\forall (k: K).(\forall (c: C).(\forall (u: T).(clt c (CHead c k u))))
\def
\lambda (_: K).(\lambda (c: C).(\lambda (u: T).(eq_ind_r nat (plus (cweight
-c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u))))
-(le_lt_plus_plus (cweight c) (cweight c) O (tweight u) (le_n (cweight c))
-(tweight_lt u)) (cweight c) (plus_n_O (cweight c))))).
-(* COMMENTS
-Initial nodes: 69
-END *)
-
-theorem clt_wf__q_ind:
- \forall (P: ((C \to Prop))).(((\forall (n: nat).((\lambda (P0: ((C \to
-Prop))).(\lambda (n0: nat).(\forall (c: C).((eq nat (cweight c) n0) \to (P0
-c))))) P n))) \to (\forall (c: C).(P c)))
-\def
- let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c:
-C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to
-Prop))).(\lambda (H: ((\forall (n: nat).(\forall (c: C).((eq nat (cweight c)
-n) \to (P c)))))).(\lambda (c: C).(H (cweight c) c (refl_equal nat (cweight
-c)))))).
-(* COMMENTS
-Initial nodes: 61
-END *)
-
-theorem clt_wf_ind:
- \forall (P: ((C \to Prop))).(((\forall (c: C).(((\forall (d: C).((clt d c)
-\to (P d)))) \to (P c)))) \to (\forall (c: C).(P c)))
-\def
- let Q \def (\lambda (P: ((C \to Prop))).(\lambda (n: nat).(\forall (c:
-C).((eq nat (cweight c) n) \to (P c))))) in (\lambda (P: ((C \to
-Prop))).(\lambda (H: ((\forall (c: C).(((\forall (d: C).((lt (cweight d)
-(cweight c)) \to (P d)))) \to (P c))))).(\lambda (c: C).(clt_wf__q_ind
-(\lambda (c0: C).(P c0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (c0:
-C).(P c0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0)
-\to (Q (\lambda (c0: C).(P c0)) m))))).(\lambda (c0: C).(\lambda (H1: (eq nat
-(cweight c0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall
-(m: nat).((lt m n1) \to (\forall (c1: C).((eq nat (cweight c1) m) \to (P
-c1)))))) H0 (cweight c0) H1) in (H c0 (\lambda (d: C).(\lambda (H3: (lt
-(cweight d) (cweight c0))).(H2 (cweight d) H3 d (refl_equal nat (cweight
-d))))))))))))) c)))).
-(* COMMENTS
-Initial nodes: 179
-END *)
+c) O) (\lambda (n: nat).(lt n (plus (cweight c) (tweight u)))) (lt_reg_l O
+(tweight u) (cweight c) (tweight_lt u)) (cweight c) (plus_n_O (cweight c))))).
-theorem chead_ctail:
+lemma chead_ctail:
\forall (c: C).(\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h:
K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c k t) (CTail h u d))))))))
\def
C).(\lambda (u: T).(eq C (CHead (CTail x0 x2 x1) k0 t0) (CTail h u d))))) x0
(CHead x1 k0 t0) x2 (refl_equal C (CHead (CTail x0 x2 x1) k0 t0))) (CHead c0
k t) H1))))) H0))))))))) c).
-(* COMMENTS
-Initial nodes: 395
-END *)
-theorem clt_thead:
+lemma clt_thead:
\forall (k: K).(\forall (u: T).(\forall (c: C).(clt c (CTail k u c))))
\def
\lambda (k: K).(\lambda (u: T).(\lambda (c: C).(C_ind (\lambda (c0: C).(clt
c0 (CTail k u c0))) (\lambda (n: nat).(clt_head k (CSort n) u)) (\lambda (c0:
C).(\lambda (H: (clt c0 (CTail k u c0))).(\lambda (k0: K).(\lambda (t:
T).(clt_cong c0 (CTail k u c0) H k0 t))))) c))).
-(* COMMENTS
-Initial nodes: 71
-END *)
-theorem c_tail_ind:
+lemma c_tail_ind:
\forall (P: ((C \to Prop))).(((\forall (n: nat).(P (CSort n)))) \to
(((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t: T).(P (CTail k t
c))))))) \to (\forall (c: C).(P c))))
(eq_ind C (CHead c1 k t) (\lambda (c2: C).(\forall (d: C).((clt d c2) \to (P
d)))) H2 (CTail x0 x2 x1) H4) in (H0 x1 (H5 x1 (clt_thead x0 x2 x1)) x0 x2))
(CHead c1 k t) H4))))) H3)))))))) c0)) c)))).
-(* COMMENTS
-Initial nodes: 295
-END *)
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