include "basic_1/s/defs.ma".
-theorem r_S:
+lemma r_S:
\forall (k: K).(\forall (i: nat).(eq nat (r k (S i)) (S (r 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 (r k0 TMP_1) in (let TMP_3 \def (r 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
-(r 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 (r 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)))).
-
-theorem r_plus:
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (r k0 (S
+i)) (S (r k0 i))))) (\lambda (b: B).(\lambda (i: nat).(refl_equal nat (S (r
+(Bind b) i))))) (\lambda (f: F).(\lambda (i: nat).(refl_equal nat (S (r (Flat
+f) i))))) k).
+
+lemma r_plus:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
(plus (r 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 (r k0 TMP_1) in (let
-TMP_3 \def (r 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 (r 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 (r 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)))).
-
-theorem r_plus_sym:
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (r k0 (plus i j)) (plus (r k0 i) j))))) (\lambda (b: B).(\lambda
+(i: nat).(\lambda (j: nat).(refl_equal nat (plus (r (Bind b) i) j)))))
+(\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (plus (r
+(Flat f) i) j))))) k).
+
+lemma r_plus_sym:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (r k (plus i j))
(plus i (r 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 (r k0 TMP_1) in (let
-TMP_3 \def (r k0 j) in (let TMP_4 \def (plus i TMP_3) in (eq nat TMP_2
-TMP_4)))))))) in (let TMP_7 \def (\lambda (_: B).(\lambda (i: nat).(\lambda
-(j: nat).(let TMP_6 \def (plus i j) in (refl_equal nat TMP_6))))) in (let
-TMP_8 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i
-j)))) in (K_ind TMP_5 TMP_7 TMP_8 k)))).
-
-theorem r_minus:
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (r k0 (plus i j)) (plus i (r k0 j)))))) (\lambda (_: B).(\lambda
+(i: nat).(\lambda (j: nat).(refl_equal nat (plus i j))))) (\lambda (_:
+F).(\lambda (i: nat).(\lambda (j: nat).(plus_n_Sm i j)))) k).
+
+lemma r_minus:
\forall (i: nat).(\forall (n: nat).((lt n i) \to (\forall (k: K).(eq nat
(minus (r k i) (S n)) (r k (minus i (S n)))))))
\def
\lambda (i: nat).(\lambda (n: nat).(\lambda (H: (lt n i)).(\lambda (k:
-K).(let TMP_7 \def (\lambda (k0: K).(let TMP_1 \def (r k0 i) in (let TMP_2
-\def (S n) in (let TMP_3 \def (minus TMP_1 TMP_2) in (let TMP_4 \def (S n) in
-(let TMP_5 \def (minus i TMP_4) in (let TMP_6 \def (r k0 TMP_5) in (eq nat
-TMP_3 TMP_6)))))))) in (let TMP_10 \def (\lambda (_: B).(let TMP_8 \def (S n)
-in (let TMP_9 \def (minus i TMP_8) in (refl_equal nat TMP_9)))) in (let
-TMP_11 \def (\lambda (_: F).(minus_x_Sy i n H)) in (K_ind TMP_7 TMP_10 TMP_11
-k))))))).
-
-theorem r_dis:
+K).(K_ind (\lambda (k0: K).(eq nat (minus (r k0 i) (S n)) (r k0 (minus i (S
+n))))) (\lambda (_: B).(refl_equal nat (minus i (S n)))) (\lambda (_:
+F).(minus_x_Sy i n H)) k)))).
+
+lemma r_dis:
\forall (k: K).(\forall (P: Prop).(((((\forall (i: nat).(eq nat (r k i) i)))
\to P)) \to (((((\forall (i: nat).(eq nat (r k i) (S i)))) \to P)) \to P)))
\def
- \lambda (k: K).(let TMP_1 \def (\lambda (k0: K).(\forall (P:
-Prop).(((((\forall (i: nat).(eq nat (r k0 i) i))) \to P)) \to (((((\forall
-(i: nat).(eq nat (r k0 i) (S i)))) \to P)) \to P)))) in (let TMP_3 \def
-(\lambda (b: B).(\lambda (P: Prop).(\lambda (H: ((((\forall (i: nat).(eq nat
-(r (Bind b) i) i))) \to P))).(\lambda (_: ((((\forall (i: nat).(eq nat (r
-(Bind b) i) (S i)))) \to P))).(let TMP_2 \def (\lambda (i: nat).(refl_equal
-nat i)) in (H TMP_2)))))) in (let TMP_6 \def (\lambda (f: F).(\lambda (P:
-Prop).(\lambda (_: ((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to
-P))).(\lambda (H0: ((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to
-P))).(let TMP_5 \def (\lambda (i: nat).(let TMP_4 \def (S i) in (refl_equal
-nat TMP_4))) in (H0 TMP_5)))))) in (K_ind TMP_1 TMP_3 TMP_6 k)))).
-
-theorem s_r:
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (P: Prop).(((((\forall (i:
+nat).(eq nat (r k0 i) i))) \to P)) \to (((((\forall (i: nat).(eq nat (r k0 i)
+(S i)))) \to P)) \to P)))) (\lambda (b: B).(\lambda (P: Prop).(\lambda (H:
+((((\forall (i: nat).(eq nat (r (Bind b) i) i))) \to P))).(\lambda (_:
+((((\forall (i: nat).(eq nat (r (Bind b) i) (S i)))) \to P))).(H (\lambda (i:
+nat).(refl_equal nat i))))))) (\lambda (f: F).(\lambda (P: Prop).(\lambda (_:
+((((\forall (i: nat).(eq nat (r (Flat f) i) i))) \to P))).(\lambda (H0:
+((((\forall (i: nat).(eq nat (r (Flat f) i) (S i)))) \to P))).(H0 (\lambda
+(i: nat).(refl_equal nat (S i)))))))) k).
+
+lemma s_r:
\forall (k: K).(\forall (i: nat).(eq nat (s k (r k i)) (S i)))
\def
- \lambda (k: K).(let TMP_4 \def (\lambda (k0: K).(\forall (i: nat).(let TMP_1
-\def (r k0 i) in (let TMP_2 \def (s k0 TMP_1) in (let TMP_3 \def (S i) in (eq
-nat TMP_2 TMP_3)))))) in (let TMP_6 \def (\lambda (_: B).(\lambda (i:
-nat).(let TMP_5 \def (S i) in (refl_equal nat TMP_5)))) in (let TMP_8 \def
-(\lambda (_: F).(\lambda (i: nat).(let TMP_7 \def (S i) in (refl_equal nat
-TMP_7)))) in (K_ind TMP_4 TMP_6 TMP_8 k)))).
-
-theorem r_arith0:
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(eq nat (s k0 (r k0
+i)) (S i)))) (\lambda (_: B).(\lambda (i: nat).(refl_equal nat (S i))))
+(\lambda (_: F).(\lambda (i: nat).(refl_equal nat (S i)))) k).
+
+lemma r_arith0:
\forall (k: K).(\forall (i: nat).(eq nat (minus (r k (S i)) (S O)) (r k i)))
\def
- \lambda (k: K).(\lambda (i: nat).(let TMP_1 \def (r k i) in (let TMP_2 \def
-(S TMP_1) in (let TMP_6 \def (\lambda (n: nat).(let TMP_3 \def (S O) in (let
-TMP_4 \def (minus n TMP_3) in (let TMP_5 \def (r k i) in (eq nat TMP_4
-TMP_5))))) in (let TMP_7 \def (r k i) in (let TMP_9 \def (\lambda (n:
-nat).(let TMP_8 \def (r k i) in (eq nat n TMP_8))) in (let TMP_10 \def (r k
-i) in (let TMP_11 \def (refl_equal nat TMP_10) in (let TMP_12 \def (r k i) in
-(let TMP_13 \def (S TMP_12) in (let TMP_14 \def (S O) in (let TMP_15 \def
-(minus TMP_13 TMP_14) in (let TMP_16 \def (r k i) in (let TMP_17 \def
-(minus_Sx_SO TMP_16) in (let TMP_18 \def (eq_ind_r nat TMP_7 TMP_9 TMP_11
-TMP_15 TMP_17) in (let TMP_19 \def (S i) in (let TMP_20 \def (r k TMP_19) in
-(let TMP_21 \def (r_S k i) in (eq_ind_r nat TMP_2 TMP_6 TMP_18 TMP_20
-TMP_21))))))))))))))))))).
-
-theorem r_arith1:
+ \lambda (k: K).(\lambda (i: nat).(eq_ind_r nat (S (r k i)) (\lambda (n:
+nat).(eq nat (minus n (S O)) (r k i))) (eq_ind_r nat (r k i) (\lambda (n:
+nat).(eq nat n (r k i))) (refl_equal nat (r k i)) (minus (S (r k i)) (S O))
+(minus_Sx_SO (r k i))) (r k (S i)) (r_S k i))).
+
+lemma r_arith1:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k (S
i)) (S j)) (minus (r k i) j))))
\def
- \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(let TMP_1 \def (r k i)
-in (let TMP_2 \def (S TMP_1) in (let TMP_7 \def (\lambda (n: nat).(let TMP_3
-\def (S j) in (let TMP_4 \def (minus n TMP_3) in (let TMP_5 \def (r k i) in
-(let TMP_6 \def (minus TMP_5 j) in (eq nat TMP_4 TMP_6)))))) in (let TMP_8
-\def (r k i) in (let TMP_9 \def (minus TMP_8 j) in (let TMP_10 \def
-(refl_equal nat TMP_9) in (let TMP_11 \def (S i) in (let TMP_12 \def (r k
-TMP_11) in (let TMP_13 \def (r_S k i) in (eq_ind_r nat TMP_2 TMP_7 TMP_10
-TMP_12 TMP_13)))))))))))).
-
-theorem r_arith2:
+ \lambda (k: K).(\lambda (i: nat).(\lambda (j: nat).(eq_ind_r nat (S (r k i))
+(\lambda (n: nat).(eq nat (minus n (S j)) (minus (r k i) j))) (refl_equal nat
+(minus (r k i) j)) (r k (S i)) (r_S k i)))).
+
+lemma r_arith2:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (S i) (s k j)) \to
(le (r k i) j))))
\def
- \lambda (k: K).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall
-(j: nat).((le (S i) (s k0 j)) \to (let TMP_1 \def (r k0 i) in (le TMP_1
-j)))))) in (let TMP_3 \def (\lambda (_: B).(\lambda (i: nat).(\lambda (j:
-nat).(\lambda (H: (le (S i) (S j))).(let H_y \def (le_S_n i j H) in H_y)))))
-in (let TMP_4 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
-nat).(\lambda (H: (le (S i) j)).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
-
-theorem r_arith3:
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((le (S i) (s k0 j)) \to (le (r k0 i) j))))) (\lambda (_: B).(\lambda
+(i: nat).(\lambda (j: nat).(\lambda (H: (le (S i) (S j))).(let H_y \def
+(le_S_n i j H) in H_y))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
+nat).(\lambda (H: (le (S i) j)).H)))) k).
+
+lemma r_arith3:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).((le (s k j) (S i)) \to
(le j (r k i)))))
\def
- \lambda (k: K).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall
-(j: nat).((le (s k0 j) (S i)) \to (let TMP_1 \def (r k0 i) in (le j
-TMP_1)))))) in (let TMP_3 \def (\lambda (_: B).(\lambda (i: nat).(\lambda (j:
-nat).(\lambda (H: (le (S j) (S i))).(let H_y \def (le_S_n j i H) in H_y)))))
-in (let TMP_4 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
-nat).(\lambda (H: (le j (S i))).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
-
-theorem r_arith4:
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((le (s k0 j) (S i)) \to (le j (r k0 i)))))) (\lambda (_: B).(\lambda
+(i: nat).(\lambda (j: nat).(\lambda (H: (le (S j) (S i))).(let H_y \def
+(le_S_n j i H) in H_y))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
+nat).(\lambda (H: (le j (S i))).H)))) k).
+
+lemma r_arith4:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (S i) (s k
j)) (minus (r k i) j))))
\def
- \lambda (k: K).(let TMP_6 \def (\lambda (k0: K).(\forall (i: nat).(\forall
-(j: nat).(let TMP_1 \def (S i) in (let TMP_2 \def (s k0 j) in (let TMP_3 \def
-(minus TMP_1 TMP_2) in (let TMP_4 \def (r k0 i) in (let TMP_5 \def (minus
-TMP_4 j) in (eq nat TMP_3 TMP_5))))))))) in (let TMP_10 \def (\lambda (b:
-B).(\lambda (i: nat).(\lambda (j: nat).(let TMP_7 \def (Bind b) in (let TMP_8
-\def (r TMP_7 i) in (let TMP_9 \def (minus TMP_8 j) in (refl_equal nat
-TMP_9))))))) in (let TMP_14 \def (\lambda (f: F).(\lambda (i: nat).(\lambda
-(j: nat).(let TMP_11 \def (Flat f) in (let TMP_12 \def (r TMP_11 i) in (let
-TMP_13 \def (minus TMP_12 j) in (refl_equal nat TMP_13))))))) in (K_ind TMP_6
-TMP_10 TMP_14 k)))).
-
-theorem r_arith5:
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (minus (S i) (s k0 j)) (minus (r k0 i) j))))) (\lambda (b:
+B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus (r (Bind b) i)
+j))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat
+(minus (r (Flat f) i) j))))) k).
+
+lemma r_arith5:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).((lt (s k j) (S i)) \to
(lt j (r k i)))))
\def
- \lambda (k: K).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall
-(j: nat).((lt (s k0 j) (S i)) \to (let TMP_1 \def (r k0 i) in (lt j
-TMP_1)))))) in (let TMP_3 \def (\lambda (_: B).(\lambda (i: nat).(\lambda (j:
-nat).(\lambda (H: (lt (S j) (S i))).(lt_S_n j i H))))) in (let TMP_4 \def
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((lt (s k0 j) (S i)) \to (lt j (r k0 i)))))) (\lambda (_: B).(\lambda
+(i: nat).(\lambda (j: nat).(\lambda (H: (lt (S j) (S i))).(lt_S_n j i H)))))
(\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (lt j (S
-i))).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
+i))).H)))) k).
-theorem r_arith6:
+lemma r_arith6:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).(eq nat (minus (r k i) (S
j)) (minus i (s k j)))))
\def
- \lambda (k: K).(let TMP_6 \def (\lambda (k0: K).(\forall (i: nat).(\forall
-(j: nat).(let TMP_1 \def (r k0 i) in (let TMP_2 \def (S j) in (let TMP_3 \def
-(minus TMP_1 TMP_2) in (let TMP_4 \def (s k0 j) in (let TMP_5 \def (minus i
-TMP_4) in (eq nat TMP_3 TMP_5))))))))) in (let TMP_10 \def (\lambda (b:
-B).(\lambda (i: nat).(\lambda (j: nat).(let TMP_7 \def (Bind b) in (let TMP_8
-\def (s TMP_7 j) in (let TMP_9 \def (minus i TMP_8) in (refl_equal nat
-TMP_9))))))) in (let TMP_14 \def (\lambda (f: F).(\lambda (i: nat).(\lambda
-(j: nat).(let TMP_11 \def (Flat f) in (let TMP_12 \def (s TMP_11 j) in (let
-TMP_13 \def (minus i TMP_12) in (refl_equal nat TMP_13))))))) in (K_ind TMP_6
-TMP_10 TMP_14 k)))).
-
-theorem r_arith7:
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).(eq nat (minus (r k0 i) (S j)) (minus i (s k0 j)))))) (\lambda (b:
+B).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat (minus i (s (Bind b)
+j)))))) (\lambda (f: F).(\lambda (i: nat).(\lambda (j: nat).(refl_equal nat
+(minus i (s (Flat f) j)))))) k).
+
+lemma r_arith7:
\forall (k: K).(\forall (i: nat).(\forall (j: nat).((eq nat (S i) (s k j))
\to (eq nat (r k i) j))))
\def
- \lambda (k: K).(let TMP_2 \def (\lambda (k0: K).(\forall (i: nat).(\forall
-(j: nat).((eq nat (S i) (s k0 j)) \to (let TMP_1 \def (r k0 i) in (eq nat
-TMP_1 j)))))) in (let TMP_3 \def (\lambda (_: B).(\lambda (i: nat).(\lambda
-(j: nat).(\lambda (H: (eq nat (S i) (S j))).(eq_add_S i j H))))) in (let
-TMP_4 \def (\lambda (_: F).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H:
-(eq nat (S i) j)).H)))) in (K_ind TMP_2 TMP_3 TMP_4 k)))).
+ \lambda (k: K).(K_ind (\lambda (k0: K).(\forall (i: nat).(\forall (j:
+nat).((eq nat (S i) (s k0 j)) \to (eq nat (r k0 i) j))))) (\lambda (_:
+B).(\lambda (i: nat).(\lambda (j: nat).(\lambda (H: (eq nat (S i) (S
+j))).(eq_add_S i j H))))) (\lambda (_: F).(\lambda (i: nat).(\lambda (j:
+nat).(\lambda (H: (eq nat (S i) j)).H)))) k).
projects / helm.git / blobdiff