include "ground_1/blt/defs.ma".
-theorem lt_blt:
+lemma lt_blt:
\forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true)))
\def
- \lambda (x: nat).(let TMP_2 \def (\lambda (n: nat).(\forall (y: nat).((lt y
-n) \to (let TMP_1 \def (blt y n) in (eq bool TMP_1 true))))) in (let TMP_13
-\def (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0 \def (match H with
-[le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let TMP_8 \def (S y) in
-(let TMP_9 \def (\lambda (e: nat).(match e with [O \Rightarrow False | (S _)
-\Rightarrow True])) in (let H1 \def (eq_ind nat TMP_8 TMP_9 I O H0) in (let
-TMP_10 \def (blt y O) in (let TMP_11 \def (eq bool TMP_10 true) in (False_ind
-TMP_11 H1))))))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m)
-O)).(let TMP_3 \def (S m) in (let TMP_4 \def (\lambda (e: nat).(match e with
-[O \Rightarrow False | (S _) \Rightarrow True])) in (let H2 \def (eq_ind nat
-TMP_3 TMP_4 I O H1) in (let TMP_6 \def ((le (S y) m) \to (let TMP_5 \def (blt
-y O) in (eq bool TMP_5 true))) in (let TMP_7 \def (False_ind TMP_6 H2) in
-(TMP_7 H0)))))))]) in (let TMP_12 \def (refl_equal nat O) in (H0 TMP_12)))))
-in (let TMP_21 \def (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y
-n) \to (eq bool (blt y n) true))))).(\lambda (y: nat).(let TMP_16 \def
-(\lambda (n0: nat).((lt n0 (S n)) \to (let TMP_14 \def (S n) in (let TMP_15
-\def (blt n0 TMP_14) in (eq bool TMP_15 true))))) in (let TMP_17 \def
-(\lambda (_: (lt O (S n))).(refl_equal bool true)) in (let TMP_20 \def
-(\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) \to (eq bool (match n0 with
-[O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true)))).(\lambda (H1:
-(lt (S n0) (S n))).(let TMP_18 \def (S n0) in (let TMP_19 \def (le_S_n TMP_18
-n H1) in (H n0 TMP_19)))))) in (nat_ind TMP_16 TMP_17 TMP_20 y))))))) in
-(nat_ind TMP_2 TMP_13 TMP_21 x)))).
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to
+(eq bool (blt y n) true)))) (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0
+\def (match H with [le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let H1
+\def (eq_ind nat (S y) (\lambda (e: nat).(match e with [O \Rightarrow False |
+(S _) \Rightarrow True])) I O H0) in (False_ind (eq bool (blt y O) true)
+H1))) | (le_S m H0) \Rightarrow (\lambda (H1: (eq nat (S m) O)).((let H2 \def
+(eq_ind nat (S m) (\lambda (e: nat).(match e with [O \Rightarrow False | (S
+_) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq bool (blt
+y O) true)) H2)) H0))]) in (H0 (refl_equal nat O))))) (\lambda (n:
+nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq bool (blt y n)
+true))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((lt n0 (S n)) \to
+(eq bool (blt n0 (S n)) true))) (\lambda (_: (lt O (S n))).(refl_equal bool
+true)) (\lambda (n0: nat).(\lambda (_: (((lt n0 (S n)) \to (eq bool (match n0
+with [O \Rightarrow true | (S m) \Rightarrow (blt m n)]) true)))).(\lambda
+(H1: (lt (S n0) (S n))).(H n0 (le_S_n (S n0) n H1))))) y)))) x).
-theorem le_bge:
+lemma le_bge:
\forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false)))
\def
- \lambda (x: nat).(let TMP_2 \def (\lambda (n: nat).(\forall (y: nat).((le n
-y) \to (let TMP_1 \def (blt y n) in (eq bool TMP_1 false))))) in (let TMP_3
-\def (\lambda (y: nat).(\lambda (_: (le O y)).(refl_equal bool false))) in
-(let TMP_22 \def (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((le n y)
-\to (eq bool (blt y n) false))))).(\lambda (y: nat).(let TMP_6 \def (\lambda
-(n0: nat).((le (S n) n0) \to (let TMP_4 \def (S n) in (let TMP_5 \def (blt n0
-TMP_4) in (eq bool TMP_5 false))))) in (let TMP_19 \def (\lambda (H0: (le (S
-n) O)).(let H1 \def (match H0 with [le_n \Rightarrow (\lambda (H1: (eq nat (S
-n) O)).(let TMP_13 \def (S n) in (let TMP_14 \def (\lambda (e: nat).(match e
-with [O \Rightarrow False | (S _) \Rightarrow True])) in (let H2 \def (eq_ind
-nat TMP_13 TMP_14 I O H1) in (let TMP_15 \def (S n) in (let TMP_16 \def (blt
-O TMP_15) in (let TMP_17 \def (eq bool TMP_16 false) in (False_ind TMP_17
-H2)))))))) | (le_S m H1) \Rightarrow (\lambda (H2: (eq nat (S m) O)).(let
-TMP_7 \def (S m) in (let TMP_8 \def (\lambda (e: nat).(match e with [O
-\Rightarrow False | (S _) \Rightarrow True])) in (let H3 \def (eq_ind nat
-TMP_7 TMP_8 I O H2) in (let TMP_11 \def ((le (S n) m) \to (let TMP_9 \def (S
-n) in (let TMP_10 \def (blt O TMP_9) in (eq bool TMP_10 false)))) in (let
-TMP_12 \def (False_ind TMP_11 H3) in (TMP_12 H1)))))))]) in (let TMP_18 \def
-(refl_equal nat O) in (H1 TMP_18)))) in (let TMP_21 \def (\lambda (n0:
-nat).(\lambda (_: (((le (S n) n0) \to (eq bool (blt n0 (S n))
-false)))).(\lambda (H1: (le (S n) (S n0))).(let TMP_20 \def (le_S_n n n0 H1)
-in (H n0 TMP_20))))) in (nat_ind TMP_6 TMP_19 TMP_21 y))))))) in (nat_ind
-TMP_2 TMP_3 TMP_22 x)))).
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to
+(eq bool (blt y n) false)))) (\lambda (y: nat).(\lambda (_: (le O
+y)).(refl_equal bool false))) (\lambda (n: nat).(\lambda (H: ((\forall (y:
+nat).((le n y) \to (eq bool (blt y n) false))))).(\lambda (y: nat).(nat_ind
+(\lambda (n0: nat).((le (S n) n0) \to (eq bool (blt n0 (S n)) false)))
+(\lambda (H0: (le (S n) O)).(let H1 \def (match H0 with [le_n \Rightarrow
+(\lambda (H1: (eq nat (S n) O)).(let H2 \def (eq_ind nat (S n) (\lambda (e:
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1)
+in (False_ind (eq bool (blt O (S n)) false) H2))) | (le_S m H1) \Rightarrow
+(\lambda (H2: (eq nat (S m) O)).((let H3 \def (eq_ind nat (S m) (\lambda (e:
+nat).(match e with [O \Rightarrow False | (S _) \Rightarrow True])) I O H2)
+in (False_ind ((le (S n) m) \to (eq bool (blt O (S n)) false)) H3)) H1))]) in
+(H1 (refl_equal nat O)))) (\lambda (n0: nat).(\lambda (_: (((le (S n) n0) \to
+(eq bool (blt n0 (S n)) false)))).(\lambda (H1: (le (S n) (S n0))).(H n0
+(le_S_n n n0 H1))))) y)))) x).
-theorem blt_lt:
+lemma blt_lt:
\forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x)))
\def
- \lambda (x: nat).(let TMP_1 \def (\lambda (n: nat).(\forall (y: nat).((eq
-bool (blt y n) true) \to (lt y n)))) in (let TMP_6 \def (\lambda (y:
-nat).(\lambda (H: (eq bool (blt y O) true)).(let H0 \def (match H with
-[refl_equal \Rightarrow (\lambda (H0: (eq bool (blt y O) true)).(let TMP_2
-\def (blt y O) in (let TMP_3 \def (\lambda (e: bool).(match e with [true
-\Rightarrow False | false \Rightarrow True])) in (let H1 \def (eq_ind bool
-TMP_2 TMP_3 I true H0) in (let TMP_4 \def (lt y O) in (False_ind TMP_4
-H1))))))]) in (let TMP_5 \def (refl_equal bool true) in (H0 TMP_5))))) in
-(let TMP_19 \def (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq bool
-(blt y n) true) \to (lt y n))))).(\lambda (y: nat).(let TMP_8 \def (\lambda
-(n0: nat).((eq bool (blt n0 (S n)) true) \to (let TMP_7 \def (S n) in (lt n0
-TMP_7)))) in (let TMP_16 \def (\lambda (_: (eq bool true true)).(let TMP_9
-\def (S O) in (let TMP_10 \def (S n) in (let TMP_11 \def (S O) in (let TMP_12
-\def (S n) in (let TMP_13 \def (le_O_n n) in (let TMP_14 \def (le_n_S O n
-TMP_13) in (let TMP_15 \def (le_n_S TMP_11 TMP_12 TMP_14) in (le_S_n TMP_9
-TMP_10 TMP_15))))))))) in (let TMP_18 \def (\lambda (n0: nat).(\lambda (_:
-(((eq bool (match n0 with [O \Rightarrow true | (S m) \Rightarrow (blt m n)])
-true) \to (lt n0 (S n))))).(\lambda (H1: (eq bool (blt n0 n) true)).(let
-TMP_17 \def (H n0 H1) in (lt_n_S n0 n TMP_17))))) in (nat_ind TMP_8 TMP_16
-TMP_18 y))))))) in (nat_ind TMP_1 TMP_6 TMP_19 x)))).
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt
+y n) true) \to (lt y n)))) (\lambda (y: nat).(\lambda (H: (eq bool (blt y O)
+true)).(let H0 \def (match H with [refl_equal \Rightarrow (\lambda (H0: (eq
+bool (blt y O) true)).(let H1 \def (eq_ind bool (blt y O) (\lambda (e:
+bool).(match e with [true \Rightarrow False | false \Rightarrow True])) I
+true H0) in (False_ind (lt y O) H1)))]) in (H0 (refl_equal bool true)))))
+(\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq bool (blt y n) true)
+\to (lt y n))))).(\lambda (y: nat).(nat_ind (\lambda (n0: nat).((eq bool (blt
+n0 (S n)) true) \to (lt n0 (S n)))) (\lambda (_: (eq bool true true)).(le_S_n
+(S O) (S n) (le_n_S (S O) (S n) (le_n_S O n (le_O_n n))))) (\lambda (n0:
+nat).(\lambda (_: (((eq bool (match n0 with [O \Rightarrow true | (S m)
+\Rightarrow (blt m n)]) true) \to (lt n0 (S n))))).(\lambda (H1: (eq bool
+(blt n0 n) true)).(lt_n_S n0 n (H n0 H1))))) y)))) x).
-theorem bge_le:
+lemma bge_le:
\forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y)))
\def
- \lambda (x: nat).(let TMP_1 \def (\lambda (n: nat).(\forall (y: nat).((eq
-bool (blt y n) false) \to (le n y)))) in (let TMP_2 \def (\lambda (y:
-nat).(\lambda (_: (eq bool (blt y O) false)).(le_O_n y))) in (let TMP_20 \def
-(\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq bool (blt y n) false)
-\to (le n y))))).(\lambda (y: nat).(let TMP_4 \def (\lambda (n0: nat).((eq
-bool (blt n0 (S n)) false) \to (let TMP_3 \def (S n) in (le TMP_3 n0)))) in
-(let TMP_11 \def (\lambda (H0: (eq bool (blt O (S n)) false)).(let H1 \def
-(match H0 with [refl_equal \Rightarrow (\lambda (H1: (eq bool (blt O (S n))
-false)).(let TMP_5 \def (S n) in (let TMP_6 \def (blt O TMP_5) in (let TMP_7
-\def (\lambda (e: bool).(match e with [true \Rightarrow True | false
-\Rightarrow False])) in (let H2 \def (eq_ind bool TMP_6 TMP_7 I false H1) in
-(let TMP_8 \def (S n) in (let TMP_9 \def (le TMP_8 O) in (False_ind TMP_9
-H2))))))))]) in (let TMP_10 \def (refl_equal bool false) in (H1 TMP_10)))) in
-(let TMP_19 \def (\lambda (n0: nat).(\lambda (_: (((eq bool (blt n0 (S n))
-false) \to (le (S n) n0)))).(\lambda (H1: (eq bool (blt (S n0) (S n))
-false)).(let TMP_12 \def (S n) in (let TMP_13 \def (S n0) in (let TMP_14 \def
-(S n) in (let TMP_15 \def (S n0) in (let TMP_16 \def (H n0 H1) in (let TMP_17
-\def (le_n_S n n0 TMP_16) in (let TMP_18 \def (le_n_S TMP_14 TMP_15 TMP_17)
-in (le_S_n TMP_12 TMP_13 TMP_18))))))))))) in (nat_ind TMP_4 TMP_11 TMP_19
-y))))))) in (nat_ind TMP_1 TMP_2 TMP_20 x)))).
+ \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq bool (blt
+y n) false) \to (le n y)))) (\lambda (y: nat).(\lambda (_: (eq bool (blt y O)
+false)).(le_O_n y))) (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq
+bool (blt y n) false) \to (le n y))))).(\lambda (y: nat).(nat_ind (\lambda
+(n0: nat).((eq bool (blt n0 (S n)) false) \to (le (S n) n0))) (\lambda (H0:
+(eq bool (blt O (S n)) false)).(let H1 \def (match H0 with [refl_equal
+\Rightarrow (\lambda (H1: (eq bool (blt O (S n)) false)).(let H2 \def (eq_ind
+bool (blt O (S n)) (\lambda (e: bool).(match e with [true \Rightarrow True |
+false \Rightarrow False])) I false H1) in (False_ind (le (S n) O) H2)))]) in
+(H1 (refl_equal bool false)))) (\lambda (n0: nat).(\lambda (_: (((eq bool
+(blt n0 (S n)) false) \to (le (S n) n0)))).(\lambda (H1: (eq bool (blt (S n0)
+(S n)) false)).(le_S_n (S n) (S n0) (le_n_S (S n) (S n0) (le_n_S n n0 (H n0
+H1))))))) y)))) x).
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