include "basic_1/T/props.ma".
-theorem cle_r:
+lemma cle_r:
\forall (c: C).(cle c c)
\def
- \lambda (c: C).(let TMP_3 \def (\lambda (c0: C).(let TMP_1 \def (cweight c0)
-in (let TMP_2 \def (cweight c0) in (le TMP_1 TMP_2)))) in (let TMP_4 \def
-(\lambda (_: nat).(le_O_n O)) in (let TMP_8 \def (\lambda (c0: C).(\lambda
-(_: (le (cweight c0) (cweight c0))).(\lambda (_: K).(\lambda (t: T).(let
-TMP_5 \def (cweight c0) in (let TMP_6 \def (tweight t) in (let TMP_7 \def
-(plus TMP_5 TMP_6) in (le_n TMP_7)))))))) in (C_ind TMP_3 TMP_4 TMP_8 c)))).
+ \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).
-theorem cle_head:
+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).(let TMP_1 \def (cweight c1) in (let TMP_2
-\def (cweight c2) in (let TMP_3 \def (tweight u1) in (let TMP_4 \def (tweight
-u2) in (le_plus_plus TMP_1 TMP_2 TMP_3 TMP_4 H H0))))))))))).
+(tweight u2))).(\lambda (_: K).(le_plus_plus (cweight c1) (cweight c2)
+(tweight u1) (tweight u2) H H0))))))).
-theorem cle_trans_head:
+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).(let TMP_1 \def (cweight c1) in (let
-TMP_2 \def (cweight c2) in (let TMP_3 \def (tweight u) in (le_plus_trans
-TMP_1 TMP_2 TMP_3 H)))))))).
+c2))).(\lambda (_: K).(\lambda (u: T).(le_plus_trans (cweight c1) (cweight
+c2) (tweight u) H))))).
-theorem clt_cong:
+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).(let TMP_1 \def (cweight c) in (let
-TMP_2 \def (cweight d) in (let TMP_3 \def (tweight t) in (lt_reg_r TMP_1
-TMP_2 TMP_3 H)))))))).
+d))).(\lambda (_: K).(\lambda (t: T).(lt_reg_r (cweight c) (cweight d)
+(tweight t) H))))).
-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).(let TMP_1 \def (cweight c)
-in (let TMP_2 \def (plus TMP_1 O) in (let TMP_6 \def (\lambda (n: nat).(let
-TMP_3 \def (cweight c) in (let TMP_4 \def (tweight u) in (let TMP_5 \def
-(plus TMP_3 TMP_4) in (lt n TMP_5))))) in (let TMP_7 \def (tweight u) in (let
-TMP_8 \def (cweight c) in (let TMP_9 \def (tweight_lt u) in (let TMP_10 \def
-(lt_reg_l O TMP_7 TMP_8 TMP_9) in (let TMP_11 \def (cweight c) in (let TMP_12
-\def (cweight c) in (let TMP_13 \def (plus_n_O TMP_12) in (eq_ind_r nat TMP_2
-TMP_6 TMP_10 TMP_11 TMP_13))))))))))))).
+ \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)))) (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
- \lambda (c: C).(let TMP_4 \def (\lambda (c0: C).(\forall (t: T).(\forall (k:
-K).(let TMP_3 \def (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(let TMP_1
-\def (CHead c0 k t) in (let TMP_2 \def (CTail h u d) in (eq C TMP_1
-TMP_2)))))) in (ex_3 K C T TMP_3))))) in (let TMP_13 \def (\lambda (n:
-nat).(\lambda (t: T).(\lambda (k: K).(let TMP_8 \def (\lambda (h: K).(\lambda
-(d: C).(\lambda (u: T).(let TMP_5 \def (CSort n) in (let TMP_6 \def (CHead
-TMP_5 k t) in (let TMP_7 \def (CTail h u d) in (eq C TMP_6 TMP_7))))))) in
-(let TMP_9 \def (CSort n) in (let TMP_10 \def (CSort n) in (let TMP_11 \def
-(CHead TMP_10 k t) in (let TMP_12 \def (refl_equal C TMP_11) in (ex_3_intro K
-C T TMP_8 k TMP_9 t TMP_12))))))))) in (let TMP_38 \def (\lambda (c0:
-C).(\lambda (H: ((\forall (t: T).(\forall (k: K).(ex_3 K C T (\lambda (h:
-K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c0 k t) (CTail h u
-d)))))))))).(\lambda (k: K).(\lambda (t: T).(\lambda (t0: T).(\lambda (k0:
-K).(let H_x \def (H t k) in (let H0 \def H_x in (let TMP_16 \def (\lambda (h:
-K).(\lambda (d: C).(\lambda (u: T).(let TMP_14 \def (CHead c0 k t) in (let
-TMP_15 \def (CTail h u d) in (eq C TMP_14 TMP_15)))))) in (let TMP_20 \def
-(\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(let TMP_17 \def (CHead c0 k
-t) in (let TMP_18 \def (CHead TMP_17 k0 t0) in (let TMP_19 \def (CTail h u d)
-in (eq C TMP_18 TMP_19))))))) in (let TMP_21 \def (ex_3 K C T TMP_20) in (let
-TMP_37 \def (\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H1:
-(eq C (CHead c0 k t) (CTail x0 x2 x1))).(let TMP_22 \def (CTail x0 x2 x1) in
-(let TMP_26 \def (\lambda (c1: C).(let TMP_25 \def (\lambda (h: K).(\lambda
-(d: C).(\lambda (u: T).(let TMP_23 \def (CHead c1 k0 t0) in (let TMP_24 \def
-(CTail h u d) in (eq C TMP_23 TMP_24)))))) in (ex_3 K C T TMP_25))) in (let
-TMP_30 \def (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(let TMP_27 \def
-(CTail x0 x2 x1) in (let TMP_28 \def (CHead TMP_27 k0 t0) in (let TMP_29 \def
-(CTail h u d) in (eq C TMP_28 TMP_29))))))) in (let TMP_31 \def (CHead x1 k0
-t0) in (let TMP_32 \def (CTail x0 x2 x1) in (let TMP_33 \def (CHead TMP_32 k0
-t0) in (let TMP_34 \def (refl_equal C TMP_33) in (let TMP_35 \def (ex_3_intro
-K C T TMP_30 x0 TMP_31 x2 TMP_34) in (let TMP_36 \def (CHead c0 k t) in
-(eq_ind_r C TMP_22 TMP_26 TMP_35 TMP_36 H1)))))))))))))) in (ex_3_ind K C T
-TMP_16 TMP_21 TMP_37 H0))))))))))))) in (C_ind TMP_4 TMP_13 TMP_38 c)))).
+ \lambda (c: C).(C_ind (\lambda (c0: C).(\forall (t: T).(\forall (k: K).(ex_3
+K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c0 k t)
+(CTail h u d))))))))) (\lambda (n: nat).(\lambda (t: T).(\lambda (k:
+K).(ex_3_intro K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
+(CHead (CSort n) k t) (CTail h u d))))) k (CSort n) t (refl_equal C (CHead
+(CSort n) k t)))))) (\lambda (c0: C).(\lambda (H: ((\forall (t: T).(\forall
+(k: K).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
+(CHead c0 k t) (CTail h u d)))))))))).(\lambda (k: K).(\lambda (t:
+T).(\lambda (t0: T).(\lambda (k0: K).(let H_x \def (H t k) in (let H0 \def
+H_x in (ex_3_ind K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C
+(CHead c0 k t) (CTail h u d))))) (ex_3 K C T (\lambda (h: K).(\lambda (d:
+C).(\lambda (u: T).(eq C (CHead (CHead c0 k t) k0 t0) (CTail h u d))))))
+(\lambda (x0: K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H1: (eq C (CHead
+c0 k t) (CTail x0 x2 x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c1:
+C).(ex_3 K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead
+c1 k0 t0) (CTail h u d))))))) (ex_3_intro K C T (\lambda (h: K).(\lambda (d:
+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).
-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).(let TMP_2 \def (\lambda (c0:
-C).(let TMP_1 \def (CTail k u c0) in (clt c0 TMP_1))) in (let TMP_4 \def
-(\lambda (n: nat).(let TMP_3 \def (CSort n) in (clt_head k TMP_3 u))) in (let
-TMP_6 \def (\lambda (c0: C).(\lambda (H: (clt c0 (CTail k u c0))).(\lambda
-(k0: K).(\lambda (t: T).(let TMP_5 \def (CTail k u c0) in (clt_cong c0 TMP_5
-H k0 t)))))) in (C_ind TMP_2 TMP_4 TMP_6 c)))))).
+ \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))).
-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))))
\def
\lambda (P: ((C \to Prop))).(\lambda (H: ((\forall (n: nat).(P (CSort
n))))).(\lambda (H0: ((\forall (c: C).((P c) \to (\forall (k: K).(\forall (t:
-T).(P (CTail k t c)))))))).(\lambda (c: C).(let TMP_1 \def (\lambda (c0:
-C).(P c0)) in (let TMP_20 \def (\lambda (c0: C).(let TMP_2 \def (\lambda (c1:
-C).(((\forall (d: C).((clt d c1) \to (P d)))) \to (P c1))) in (let TMP_3 \def
-(\lambda (n: nat).(\lambda (_: ((\forall (d: C).((clt d (CSort n)) \to (P
-d))))).(H n))) in (let TMP_19 \def (\lambda (c1: C).(\lambda (_: ((((\forall
-(d: C).((clt d c1) \to (P d)))) \to (P c1)))).(\lambda (k: K).(\lambda (t:
-T).(\lambda (H2: ((\forall (d: C).((clt d (CHead c1 k t)) \to (P d))))).(let
-H_x \def (chead_ctail c1 t k) in (let H3 \def H_x in (let TMP_6 \def (\lambda
-(h: K).(\lambda (d: C).(\lambda (u: T).(let TMP_4 \def (CHead c1 k t) in (let
-TMP_5 \def (CTail h u d) in (eq C TMP_4 TMP_5)))))) in (let TMP_7 \def (CHead
-c1 k t) in (let TMP_8 \def (P TMP_7) in (let TMP_18 \def (\lambda (x0:
-K).(\lambda (x1: C).(\lambda (x2: T).(\lambda (H4: (eq C (CHead c1 k t)
-(CTail x0 x2 x1))).(let TMP_9 \def (CTail x0 x2 x1) in (let TMP_10 \def
-(\lambda (c2: C).(P c2)) in (let TMP_11 \def (CHead c1 k t) in (let TMP_12
-\def (\lambda (c2: C).(\forall (d: C).((clt d c2) \to (P d)))) in (let TMP_13
-\def (CTail x0 x2 x1) in (let H5 \def (eq_ind C TMP_11 TMP_12 H2 TMP_13 H4)
-in (let TMP_14 \def (clt_thead x0 x2 x1) in (let TMP_15 \def (H5 x1 TMP_14)
-in (let TMP_16 \def (H0 x1 TMP_15 x0 x2) in (let TMP_17 \def (CHead c1 k t)
-in (eq_ind_r C TMP_9 TMP_10 TMP_16 TMP_17 H4))))))))))))))) in (ex_3_ind K C
-T TMP_6 TMP_8 TMP_18 H3)))))))))))) in (C_ind TMP_2 TMP_3 TMP_19 c0))))) in
-(clt_wf_ind TMP_1 TMP_20 c)))))).
+T).(P (CTail k t c)))))))).(\lambda (c: C).(clt_wf_ind (\lambda (c0: C).(P
+c0)) (\lambda (c0: C).(C_ind (\lambda (c1: C).(((\forall (d: C).((clt d c1)
+\to (P d)))) \to (P c1))) (\lambda (n: nat).(\lambda (_: ((\forall (d:
+C).((clt d (CSort n)) \to (P d))))).(H n))) (\lambda (c1: C).(\lambda (_:
+((((\forall (d: C).((clt d c1) \to (P d)))) \to (P c1)))).(\lambda (k:
+K).(\lambda (t: T).(\lambda (H2: ((\forall (d: C).((clt d (CHead c1 k t)) \to
+(P d))))).(let H_x \def (chead_ctail c1 t k) in (let H3 \def H_x in (ex_3_ind
+K C T (\lambda (h: K).(\lambda (d: C).(\lambda (u: T).(eq C (CHead c1 k t)
+(CTail h u d))))) (P (CHead c1 k t)) (\lambda (x0: K).(\lambda (x1:
+C).(\lambda (x2: T).(\lambda (H4: (eq C (CHead c1 k t) (CTail x0 x2
+x1))).(eq_ind_r C (CTail x0 x2 x1) (\lambda (c2: C).(P c2)) (let H5 \def
+(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)))).
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