include "basic_1/s/defs.ma".
-theorem s_S:
+lemma s_S:
\forall (k: K).(\forall (i: nat).(eq nat (s k (S i)) (S (s k i))))
\def
- \lambda (k: K).(let TMP_5 \def (\lambda (k0: K).(\forall (i: nat).(let TMP_1
-\def (S i) in (let TMP_2 \def (s k0 TMP_1) in (let TMP_3 \def (s k0 i) in
-(let TMP_4 \def (S TMP_3) in (eq nat TMP_2 TMP_4))))))) in (let TMP_9 \def
-(\lambda (b: B).(\lambda (i: nat).(let TMP_6 \def (Bind b) in (let TMP_7 \def
-(s TMP_6 i) in (let TMP_8 \def (S TMP_7) in (refl_equal nat TMP_8)))))) in
-(let TMP_13 \def (\lambda (f: F).(\lambda (i: nat).(let TMP_10 \def (Flat f)
-in (let TMP_11 \def (s TMP_10 i) in (let TMP_12 \def (S TMP_11) in
-(refl_equal nat TMP_12)))))) in (K_ind TMP_5 TMP_9 TMP_13 k)))).
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (S
+i)) (S (s k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (s
+(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (s (Flat
+f) i))))) k).
-theorem s_plus:
+lemma s_plus:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j))
(plus (s k i) j))))
\def
- \lambda (k: K).(let TMP_5 \def (\lambda (k0: K).(\forall (i: nat).(\forall
-(j: nat).(let TMP_1 \def (plus i j) in (let TMP_2 \def (s k0 TMP_1) in (let
-TMP_3 \def (s k0 i) in (let TMP_4 \def (plus TMP_3 j) in (eq nat TMP_2
-TMP_4)))))))) in (let TMP_9 \def (\lambda (b: B).(\lambda (i: nat).(\lambda
-(j: nat).(let TMP_6 \def (Bind b) in (let TMP_7 \def (s TMP_6 i) in (let
-TMP_8 \def (plus TMP_7 j) in (refl_equal nat TMP_8))))))) in (let TMP_13 \def
-(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(let TMP_10 \def (Flat f)
-in (let TMP_11 \def (s TMP_10 i) in (let TMP_12 \def (plus TMP_11 j) in
-(refl_equal nat TMP_12))))))) in (K_ind TMP_5 TMP_9 TMP_13 k)))).
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (s k0 (plus i j)) (plus (s k0 i) j))))) (\lambda (b: B).(\lambda
+(i: nat).(\lambda (j: nat).(refl_equal nat (plus (s (Bind b) i) j)))))
+(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (s
+(Flat f) i) j))))) k).
-theorem s_plus_sym:
+lemma s_plus_sym:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (s k (plus i j))
(plus i (s k j)))))
\def
- \lambda (k: K).(let TMP_5 \def (\lambda (k0: K).(\forall (i: nat).(\forall
-(j: nat).(let TMP_1 \def (plus i j) in (let TMP_2 \def (s k0 TMP_1) in (let
-TMP_3 \def (s k0 j) in (let TMP_4 \def (plus i TMP_3) in (eq nat TMP_2
-TMP_4)))))))) in (let TMP_17 \def (\lambda (_: B).(\lambda (i: nat).(\lambda
-(j: nat).(let TMP_6 \def (S j) in (let TMP_7 \def (plus i TMP_6) in (let
-TMP_10 \def (\lambda (n: nat).(let TMP_8 \def (S j) in (let TMP_9 \def (plus
-i TMP_8) in (eq nat n TMP_9)))) in (let TMP_11 \def (S j) in (let TMP_12 \def
-(plus i TMP_11) in (let TMP_13 \def (refl_equal nat TMP_12) in (let TMP_14
-\def (plus i j) in (let TMP_15 \def (S TMP_14) in (let TMP_16 \def (plus_n_Sm
-i j) in (eq_ind_r nat TMP_7 TMP_10 TMP_13 TMP_15 TMP_16))))))))))))) in (let
-TMP_21 \def (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(let TMP_18
-\def (Flat f) in (let TMP_19 \def (s TMP_18 j) in (let TMP_20 \def (plus i
-TMP_19) in (refl_equal nat TMP_20))))))) in (K_ind TMP_5 TMP_17 TMP_21 k)))).
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (s k0 (plus i j)) (plus i (s k0 j)))))) (\lambda (_: B).(\lambda
+(i: nat).(\lambda (j: nat).(eq_ind_r nat (plus i (S j)) (\lambda (n: nat).(eq
+nat n (plus i (S j)))) (refl_equal nat (plus i (S j))) (S (plus i j))
+(plus_n_Sm i j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j:
+nat).(refl_equal nat (plus i (s (Flat f) j)))))) k).
-theorem s_minus:
+lemma s_minus:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).((le j i) \to (eq nat (s
k (minus i j)) (minus (s k i) j)))))
\def
- \lambda (k: K).(let TMP_5 \def (\lambda (k0: K).(\forall (i: nat).(\forall
-(j: nat).((le j i) \to (let TMP_1 \def (minus i j) in (let TMP_2 \def (s k0
-TMP_1) in (let TMP_3 \def (s k0 i) in (let TMP_4 \def (minus TMP_3 j) in (eq
-nat TMP_2 TMP_4))))))))) in (let TMP_17 \def (\lambda (_: B).(\lambda (i:
-nat).(\lambda (j: nat).(\lambda (H: (le j i)).(let TMP_6 \def (S i) in (let
-TMP_7 \def (minus TMP_6 j) in (let TMP_10 \def (\lambda (n: nat).(let TMP_8
-\def (S i) in (let TMP_9 \def (minus TMP_8 j) in (eq nat n TMP_9)))) in (let
-TMP_11 \def (S i) in (let TMP_12 \def (minus TMP_11 j) in (let TMP_13 \def
-(refl_equal nat TMP_12) in (let TMP_14 \def (minus i j) in (let TMP_15 \def
-(S TMP_14) in (let TMP_16 \def (minus_Sn_m i j H) in (eq_ind_r nat TMP_7
-TMP_10 TMP_13 TMP_15 TMP_16)))))))))))))) in (let TMP_21 \def (\lambda (f:
-F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (_: (le j i)).(let TMP_18
-\def (Flat f) in (let TMP_19 \def (s TMP_18 i) in (let TMP_20 \def (minus
-TMP_19 j) in (refl_equal nat TMP_20)))))))) in (K_ind TMP_5 TMP_17 TMP_21
-k)))).
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((le j i) \to (eq nat (s k0 (minus i j)) (minus (s k0 i) j))))))
+(\lambda (_: B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le j
+i)).(eq_ind_r nat (minus (S i) j) (\lambda (n: nat).(eq nat n (minus (S i)
+j))) (refl_equal nat (minus (S i) j)) (S (minus i j)) (minus_Sn_m i j H))))))
+(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (_: (le j
+i)).(refl_equal nat (minus (s (Flat f) i) j)))))) k).
-theorem minus_s_s:
+lemma minus_s_s:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (s k i) (s
k j)) (minus i j))))
\def
- \lambda (k: K).(let TMP_5 \def (\lambda (k0: K).(\forall (i: nat).(\forall
-(j: nat).(let TMP_1 \def (s k0 i) in (let TMP_2 \def (s k0 j) in (let TMP_3
-\def (minus TMP_1 TMP_2) in (let TMP_4 \def (minus i j) in (eq nat TMP_3
-TMP_4)))))))) in (let TMP_7 \def (\lambda (_: B).(\lambda (i: nat).(\lambda
-(j: nat).(let TMP_6 \def (minus i j) in (refl_equal nat TMP_6))))) in (let
-TMP_9 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(let TMP_8
-\def (minus i j) in (refl_equal nat TMP_8))))) in (K_ind TMP_5 TMP_7 TMP_9
-k)))).
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (minus (s k0 i) (s k0 j)) (minus i j))))) (\lambda (_:
+B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i j)))))
+(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i
+j))))) k).
-theorem s_le:
+lemma s_le:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).((le i j) \to (le (s k i)
(s k j)))))
\def
- \lambda (k: K).(let TMP_3 \def (\lambda (k0: K).(\forall (i: nat).(\forall
-(j: nat).((le i j) \to (let TMP_1 \def (s k0 i) in (let TMP_2 \def (s k0 j)
-in (le TMP_1 TMP_2))))))) in (let TMP_4 \def (\lambda (_: B).(\lambda (i:
-nat).(\lambda (j: nat).(\lambda (H: (le i j)).(le_n_S i j H))))) in (let
-TMP_5 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H:
-(le i j)).H)))) in (K_ind TMP_3 TMP_4 TMP_5 k)))).
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((le i j) \to (le (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i:
+nat).(\lambda (j: nat).(\lambda (H: (le i j)).(le_n_S i j H))))) (\lambda (_:
+F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (le i j)).H)))) k).
-theorem s_lt:
+lemma s_lt:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt i j) \to (lt (s k i)
(s k j)))))
\def
- \lambda (k: K).(let TMP_3 \def (\lambda (k0: K).(\forall (i: nat).(\forall
-(j: nat).((lt i j) \to (let TMP_1 \def (s k0 i) in (let TMP_2 \def (s k0 j)
-in (lt TMP_1 TMP_2))))))) in (let TMP_4 \def (\lambda (_: B).(\lambda (i:
-nat).(\lambda (j: nat).(\lambda (H: (lt i j)).(lt_n_S i j H))))) in (let
-TMP_5 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H:
-(lt i j)).H)))) in (K_ind TMP_3 TMP_4 TMP_5 k)))).
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((lt i j) \to (lt (s k0 i) (s k0 j)))))) (\lambda (_: B).(\lambda (i:
+nat).(\lambda (j: nat).(\lambda (H: (lt i j)).(lt_n_S i j H))))) (\lambda (_:
+F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt i j)).H)))) k).
-theorem s_inc:
+lemma s_inc:
\forall (k: K).(\forall (i: nat).(le i (s k i)))
\def
- \lambda (k: K).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(let TMP_1
-\def (s k0 i) in (le i TMP_1)))) in (let TMP_30 \def (\lambda (b: B).(\lambda
-(i: nat).(let TMP_3 \def (Bind b) in (let TMP_4 \def (s TMP_3 i) in (let
-TMP_5 \def (S i) in (let TMP_6 \def (Bind b) in (let TMP_7 \def (s TMP_6 i)
-in (let TMP_8 \def (S TMP_7) in (let TMP_9 \def (S i) in (let TMP_10 \def (S
-TMP_9) in (let TMP_11 \def (Bind b) in (let TMP_12 \def (s TMP_11 i) in (let
-TMP_13 \def (S TMP_12) in (let TMP_14 \def (S TMP_13) in (let TMP_15 \def (S
-i) in (let TMP_16 \def (S TMP_15) in (let TMP_17 \def (S TMP_16) in (let
-TMP_18 \def (Bind b) in (let TMP_19 \def (s TMP_18 i) in (let TMP_20 \def (S
-TMP_19) in (let TMP_21 \def (S TMP_20) in (let TMP_22 \def (Bind b) in (let
-TMP_23 \def (s TMP_22 i) in (let TMP_24 \def (S TMP_23) in (let TMP_25 \def
-(S TMP_24) in (let TMP_26 \def (le_n TMP_25) in (let TMP_27 \def (le_S TMP_17
-TMP_21 TMP_26) in (let TMP_28 \def (le_S_n TMP_10 TMP_14 TMP_27) in (let
-TMP_29 \def (le_S_n TMP_5 TMP_8 TMP_28) in (le_S_n i TMP_4
-TMP_29)))))))))))))))))))))))))))))) in (let TMP_33 \def (\lambda (f:
-F).(\lambda (i: nat).(let TMP_31 \def (Flat f) in (let TMP_32 \def (s TMP_31
-i) in (le_n TMP_32))))) in (K_ind TMP_2 TMP_30 TMP_33 k)))).
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(le i (s k0 i))))
+(\lambda (b: B).(\lambda (i: nat).(le_S_n i (s (Bind b) i) (le_S_n (S i) (S
+(s (Bind b) i)) (le_S_n (S (S i)) (S (S (s (Bind b) i))) (le_S (S (S (S i)))
+(S (S (s (Bind b) i))) (le_n (S (S (s (Bind b) i)))))))))) (\lambda (f:
+F).(\lambda (i: nat).(le_n (s (Flat f) i)))) k).
-theorem s_arith0:
+lemma s_arith0:
\forall (k: K).(\forall (i: nat).(eq nat (minus (s k i) (s k O)) i))
\def
- \lambda (k: K).(\lambda (i: nat).(let TMP_1 \def (minus i O) in (let TMP_2
-\def (\lambda (n: nat).(eq nat n i)) in (let TMP_3 \def (\lambda (n: nat).(eq
-nat n i)) in (let TMP_4 \def (refl_equal nat i) in (let TMP_5 \def (minus i
-O) in (let TMP_6 \def (minus_n_O i) in (let TMP_7 \def (eq_ind nat i TMP_3
-TMP_4 TMP_5 TMP_6) in (let TMP_8 \def (s k i) in (let TMP_9 \def (s k O) in
-(let TMP_10 \def (minus TMP_8 TMP_9) in (let TMP_11 \def (minus_s_s k i O) in
-(eq_ind_r nat TMP_1 TMP_2 TMP_7 TMP_10 TMP_11))))))))))))).
+ \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (minus i O) (\lambda (n:
+nat).(eq nat n i)) (eq_ind nat i (\lambda (n: nat).(eq nat n i)) (refl_equal
+nat i) (minus i O) (minus_n_O i)) (minus (s k i) (s k O)) (minus_s_s k i O))).
-theorem s_arith1:
+lemma s_arith1:
\forall (b: B).(\forall (i: nat).(eq nat (minus (s (Bind b) i) (S O)) i))
\def
- \lambda (_: B).(\lambda (i: nat).(let TMP_1 \def (\lambda (n: nat).(eq nat n
-i)) in (let TMP_2 \def (refl_equal nat i) in (let TMP_3 \def (minus i O) in
-(let TMP_4 \def (minus_n_O i) in (eq_ind nat i TMP_1 TMP_2 TMP_3 TMP_4)))))).
+ \lambda (_: B).(\lambda (i: nat).(eq_ind nat i (\lambda (n: nat).(eq nat n
+i)) (refl_equal nat i) (minus i O) (minus_n_O i))).
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