theorem lt_blt:
\forall (x: nat).(\forall (y: nat).((lt y x) \to (eq bool (blt y x) true)))
\def
- \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 in le with [le_n \Rightarrow (\lambda (H0: (eq nat (S y)
-O)).(let H1 \def (eq_ind nat (S y) (\lambda (e: nat).(match e in nat 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 in nat 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).
+ \lambda (x: nat).(let TMP_793 \def (\lambda (n: nat).(\forall (y: nat).((lt
+y n) \to (let TMP_792 \def (blt y n) in (eq bool TMP_792 true))))) in (let
+TMP_791 \def (\lambda (y: nat).(\lambda (H: (lt y O)).(let H0 \def (match H
+in le with [le_n \Rightarrow (\lambda (H0: (eq nat (S y) O)).(let TMP_787
+\def (S y) in (let TMP_786 \def (\lambda (e: nat).(match e in nat with [O
+\Rightarrow False | (S _) \Rightarrow True])) in (let H1 \def (eq_ind nat
+TMP_787 TMP_786 I O H0) in (let TMP_788 \def (blt y O) in (let TMP_789 \def
+(eq bool TMP_788 true) in (False_ind TMP_789 H1))))))) | (le_S m H0)
+\Rightarrow (\lambda (H1: (eq nat (S m) O)).(let TMP_782 \def (S m) in (let
+TMP_781 \def (\lambda (e: nat).(match e in nat with [O \Rightarrow False | (S
+_) \Rightarrow True])) in (let H2 \def (eq_ind nat TMP_782 TMP_781 I O H1) in
+(let TMP_784 \def ((le (S y) m) \to (let TMP_783 \def (blt y O) in (eq bool
+TMP_783 true))) in (let TMP_785 \def (False_ind TMP_784 H2) in (TMP_785
+H0)))))))]) in (let TMP_790 \def (refl_equal nat O) in (H0 TMP_790))))) in
+(let TMP_780 \def (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n)
+\to (eq bool (blt y n) true))))).(\lambda (y: nat).(let TMP_779 \def (\lambda
+(n0: nat).((lt n0 (S n)) \to (let TMP_777 \def (S n) in (let TMP_778 \def
+(blt n0 TMP_777) in (eq bool TMP_778 true))))) in (let TMP_776 \def (\lambda
+(_: (lt O (S n))).(refl_equal bool true)) in (let TMP_775 \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_773 \def (S n0) in (let TMP_774 \def (le_S_n TMP_773 n H1) in
+(H n0 TMP_774)))))) in (nat_ind TMP_779 TMP_776 TMP_775 y))))))) in (nat_ind
+TMP_793 TMP_791 TMP_780 x)))).
theorem le_bge:
\forall (x: nat).(\forall (y: nat).((le x y) \to (eq bool (blt y x) false)))
\def
- \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 in le with [le_n
-\Rightarrow (\lambda (H1: (eq nat (S n) O)).(let H2 \def (eq_ind nat (S n)
-(\lambda (e: nat).(match e in nat 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 in nat 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:
+ \lambda (x: nat).(let TMP_815 \def (\lambda (n: nat).(\forall (y: nat).((le
+n y) \to (let TMP_814 \def (blt y n) in (eq bool TMP_814 false))))) in (let
+TMP_813 \def (\lambda (y: nat).(\lambda (_: (le O y)).(refl_equal bool
+false))) in (let TMP_812 \def (\lambda (n: nat).(\lambda (H: ((\forall (y:
+nat).((le n y) \to (eq bool (blt y n) false))))).(\lambda (y: nat).(let
+TMP_811 \def (\lambda (n0: nat).((le (S n) n0) \to (let TMP_809 \def (S n) in
+(let TMP_810 \def (blt n0 TMP_809) in (eq bool TMP_810 false))))) in (let
+TMP_808 \def (\lambda (H0: (le (S n) O)).(let H1 \def (match H0 in le with
+[le_n \Rightarrow (\lambda (H1: (eq nat (S n) O)).(let TMP_803 \def (S n) in
+(let TMP_802 \def (\lambda (e: nat).(match e in nat with [O \Rightarrow False
+| (S _) \Rightarrow True])) in (let H2 \def (eq_ind nat TMP_803 TMP_802 I O
+H1) in (let TMP_804 \def (S n) in (let TMP_805 \def (blt O TMP_804) in (let
+TMP_806 \def (eq bool TMP_805 false) in (False_ind TMP_806 H2)))))))) | (le_S
+m H1) \Rightarrow (\lambda (H2: (eq nat (S m) O)).(let TMP_797 \def (S m) in
+(let TMP_796 \def (\lambda (e: nat).(match e in nat with [O \Rightarrow False
+| (S _) \Rightarrow True])) in (let H3 \def (eq_ind nat TMP_797 TMP_796 I O
+H2) in (let TMP_800 \def ((le (S n) m) \to (let TMP_798 \def (S n) in (let
+TMP_799 \def (blt O TMP_798) in (eq bool TMP_799 false)))) in (let TMP_801
+\def (False_ind TMP_800 H3) in (TMP_801 H1)))))))]) in (let TMP_807 \def
+(refl_equal nat O) in (H1 TMP_807)))) in (let TMP_795 \def (\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).
+false)))).(\lambda (H1: (le (S n) (S n0))).(let TMP_794 \def (le_S_n n n0 H1)
+in (H n0 TMP_794))))) in (nat_ind TMP_811 TMP_808 TMP_795 y))))))) in
+(nat_ind TMP_815 TMP_813 TMP_812 x)))).
theorem blt_lt:
\forall (x: nat).(\forall (y: nat).((eq bool (blt y x) true) \to (lt y x)))
\def
- \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 in eq 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 in bool 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).
+ \lambda (x: nat).(let TMP_834 \def (\lambda (n: nat).(\forall (y: nat).((eq
+bool (blt y n) true) \to (lt y n)))) in (let TMP_833 \def (\lambda (y:
+nat).(\lambda (H: (eq bool (blt y O) true)).(let H0 \def (match H in eq with
+[refl_equal \Rightarrow (\lambda (H0: (eq bool (blt y O) true)).(let TMP_830
+\def (blt y O) in (let TMP_829 \def (\lambda (e: bool).(match e in bool with
+[true \Rightarrow False | false \Rightarrow True])) in (let H1 \def (eq_ind
+bool TMP_830 TMP_829 I true H0) in (let TMP_831 \def (lt y O) in (False_ind
+TMP_831 H1))))))]) in (let TMP_832 \def (refl_equal bool true) in (H0
+TMP_832))))) in (let TMP_828 \def (\lambda (n: nat).(\lambda (H: ((\forall
+(y: nat).((eq bool (blt y n) true) \to (lt y n))))).(\lambda (y: nat).(let
+TMP_827 \def (\lambda (n0: nat).((eq bool (blt n0 (S n)) true) \to (let
+TMP_826 \def (S n) in (lt n0 TMP_826)))) in (let TMP_825 \def (\lambda (_:
+(eq bool true true)).(let TMP_824 \def (S O) in (let TMP_823 \def (S n) in
+(let TMP_821 \def (S O) in (let TMP_820 \def (S n) in (let TMP_818 \def
+(le_O_n n) in (let TMP_819 \def (le_n_S O n TMP_818) in (let TMP_822 \def
+(le_n_S TMP_821 TMP_820 TMP_819) in (le_S_n TMP_824 TMP_823 TMP_822)))))))))
+in (let TMP_817 \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_816 \def (H n0 H1)
+in (lt_n_S n0 n TMP_816))))) in (nat_ind TMP_827 TMP_825 TMP_817 y))))))) in
+(nat_ind TMP_834 TMP_833 TMP_828 x)))).
theorem bge_le:
\forall (x: nat).(\forall (y: nat).((eq bool (blt y x) false) \to (le x y)))
\def
- \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 in eq 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 in bool 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).
+ \lambda (x: nat).(let TMP_854 \def (\lambda (n: nat).(\forall (y: nat).((eq
+bool (blt y n) false) \to (le n y)))) in (let TMP_853 \def (\lambda (y:
+nat).(\lambda (_: (eq bool (blt y O) false)).(le_O_n y))) in (let TMP_852
+\def (\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((eq bool (blt y n)
+false) \to (le n y))))).(\lambda (y: nat).(let TMP_851 \def (\lambda (n0:
+nat).((eq bool (blt n0 (S n)) false) \to (let TMP_850 \def (S n) in (le
+TMP_850 n0)))) in (let TMP_849 \def (\lambda (H0: (eq bool (blt O (S n))
+false)).(let H1 \def (match H0 in eq with [refl_equal \Rightarrow (\lambda
+(H1: (eq bool (blt O (S n)) false)).(let TMP_844 \def (S n) in (let TMP_845
+\def (blt O TMP_844) in (let TMP_843 \def (\lambda (e: bool).(match e in bool
+with [true \Rightarrow True | false \Rightarrow False])) in (let H2 \def
+(eq_ind bool TMP_845 TMP_843 I false H1) in (let TMP_846 \def (S n) in (let
+TMP_847 \def (le TMP_846 O) in (False_ind TMP_847 H2))))))))]) in (let
+TMP_848 \def (refl_equal bool false) in (H1 TMP_848)))) in (let TMP_842 \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_841 \def
+(S n) in (let TMP_840 \def (S n0) in (let TMP_838 \def (S n) in (let TMP_837
+\def (S n0) in (let TMP_835 \def (H n0 H1) in (let TMP_836 \def (le_n_S n n0
+TMP_835) in (let TMP_839 \def (le_n_S TMP_838 TMP_837 TMP_836) in (le_S_n
+TMP_841 TMP_840 TMP_839))))))))))) in (nat_ind TMP_851 TMP_849 TMP_842
+y))))))) in (nat_ind TMP_854 TMP_853 TMP_852 x)))).
projects / helm.git / blobdiff