From d51f0e2aaa8f9ee6a5b1bd759bb351a86edca162 Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Wed, 7 Feb 2007 18:29:01 +0000 Subject: [PATCH] refactoring --- .../LAMBDA-TYPES/Base-1/Base/blt/defs.ma | 27 + .../LAMBDA-TYPES/Base-1/Base/blt/props.ma | 102 +++ .../LAMBDA-TYPES/Base-1/Base/ext/arith.ma | 588 ++++++++++++++++++ .../LAMBDA-TYPES/Base-1/Base/ext/tactics.ma | 42 ++ .../LAMBDA-TYPES/Base-1/Base/makefile | 39 ++ .../LAMBDA-TYPES/Base-1/Base/plist/defs.ma | 45 ++ .../LAMBDA-TYPES/Base-1/Base/plist/props.ma | 33 + .../LAMBDA-TYPES/Base-1/Base/preamble.ma | 160 +++++ .../LAMBDA-TYPES/Base-1/Base/spare.ma | 20 + .../LAMBDA-TYPES/Base-1/Base/theory.ma | 28 + .../LAMBDA-TYPES/Base-1/Base/types/defs.ma | 150 +++++ .../LAMBDA-TYPES/Base-1/Base/types/props.ma | 32 + 12 files changed, 1266 insertions(+) create mode 100644 matita/contribs/LAMBDA-TYPES/Base-1/Base/blt/defs.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Base-1/Base/blt/props.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Base-1/Base/ext/arith.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Base-1/Base/ext/tactics.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Base-1/Base/makefile create mode 100644 matita/contribs/LAMBDA-TYPES/Base-1/Base/plist/defs.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Base-1/Base/plist/props.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Base-1/Base/preamble.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Base-1/Base/spare.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Base-1/Base/theory.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Base-1/Base/types/defs.ma create mode 100644 matita/contribs/LAMBDA-TYPES/Base-1/Base/types/props.ma diff --git a/matita/contribs/LAMBDA-TYPES/Base-1/Base/blt/defs.ma b/matita/contribs/LAMBDA-TYPES/Base-1/Base/blt/defs.ma new file mode 100644 index 000000000..4864a2c86 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Base-1/Base/blt/defs.ma @@ -0,0 +1,27 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/blt/defs". + +include "preamble.ma". + +definition blt: + nat \to (nat \to bool) +\def + let rec blt (m: nat) (n: nat) on n: bool \def (match n with [O \Rightarrow +false | (S n0) \Rightarrow (match m with [O \Rightarrow true | (S m0) +\Rightarrow (blt m0 n0)])]) in blt. + diff --git a/matita/contribs/LAMBDA-TYPES/Base-1/Base/blt/props.ma b/matita/contribs/LAMBDA-TYPES/Base-1/Base/blt/props.ma new file mode 100644 index 000000000..c7952ebd2 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Base-1/Base/blt/props.ma @@ -0,0 +1,102 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/blt/props". + +include "blt/defs.ma". + +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 return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat +n O) \to (eq bool (blt y O) true)))) 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 return (\lambda (_: nat).Prop) 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 return (\lambda (_: nat).Prop) +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: + \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 return (\lambda (n0: +nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to (eq bool (blt O (S n)) +false)))) 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 return (\lambda (_: +nat).Prop) 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 return (\lambda (_: nat).Prop) 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: + \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 return (\lambda (b: bool).(\lambda (_: (eq +? ? b)).((eq bool b true) \to (lt y O)))) 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 return (\lambda (_: bool).Prop) 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_le_S (S n0) +(S n) (lt_n_S n0 n (H n0 H1)))))) y)))) 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 return (\lambda +(b: bool).(\lambda (_: (eq ? ? b)).((eq bool b false) \to (le (S n) O)))) +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 return (\lambda (_: bool).Prop) 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). + diff --git a/matita/contribs/LAMBDA-TYPES/Base-1/Base/ext/arith.ma b/matita/contribs/LAMBDA-TYPES/Base-1/Base/ext/arith.ma new file mode 100644 index 000000000..1ce93fd7f --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Base-1/Base/ext/arith.ma @@ -0,0 +1,588 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/ext/arith". + +include "preamble.ma". + +theorem nat_dec: + \forall (n1: nat).(\forall (n2: nat).(or (eq nat n1 n2) ((eq nat n1 n2) \to +(\forall (P: Prop).P)))) +\def + \lambda (n1: nat).(nat_ind (\lambda (n: nat).(\forall (n2: nat).(or (eq nat +n n2) ((eq nat n n2) \to (\forall (P: Prop).P))))) (\lambda (n2: +nat).(nat_ind (\lambda (n: nat).(or (eq nat O n) ((eq nat O n) \to (\forall +(P: Prop).P)))) (or_introl (eq nat O O) ((eq nat O O) \to (\forall (P: +Prop).P)) (refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (eq nat O n) +((eq nat O n) \to (\forall (P: Prop).P)))).(or_intror (eq nat O (S n)) ((eq +nat O (S n)) \to (\forall (P: Prop).P)) (\lambda (H0: (eq nat O (S +n))).(\lambda (P: Prop).(let H1 \def (eq_ind nat O (\lambda (ee: nat).(match +ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True | (S _) +\Rightarrow False])) I (S n) H0) in (False_ind P H1))))))) n2)) (\lambda (n: +nat).(\lambda (H: ((\forall (n2: nat).(or (eq nat n n2) ((eq nat n n2) \to +(\forall (P: Prop).P)))))).(\lambda (n2: nat).(nat_ind (\lambda (n0: nat).(or +(eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall (P: Prop).P)))) (or_intror +(eq nat (S n) O) ((eq nat (S n) O) \to (\forall (P: Prop).P)) (\lambda (H0: +(eq nat (S n) O)).(\lambda (P: Prop).(let H1 \def (eq_ind nat (S n) (\lambda +(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow +False | (S _) \Rightarrow True])) I O H0) in (False_ind P H1))))) (\lambda +(n0: nat).(\lambda (H0: (or (eq nat (S n) n0) ((eq nat (S n) n0) \to (\forall +(P: Prop).P)))).(or_ind (eq nat n n0) ((eq nat n n0) \to (\forall (P: +Prop).P)) (or (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to (\forall (P: +Prop).P))) (\lambda (H1: (eq nat n n0)).(let H2 \def (eq_ind_r nat n0 +(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P: +Prop).P)))) H0 n H1) in (eq_ind nat n (\lambda (n3: nat).(or (eq nat (S n) (S +n3)) ((eq nat (S n) (S n3)) \to (\forall (P: Prop).P)))) (or_introl (eq nat +(S n) (S n)) ((eq nat (S n) (S n)) \to (\forall (P: Prop).P)) (refl_equal nat +(S n))) n0 H1))) (\lambda (H1: (((eq nat n n0) \to (\forall (P: +Prop).P)))).(or_intror (eq nat (S n) (S n0)) ((eq nat (S n) (S n0)) \to +(\forall (P: Prop).P)) (\lambda (H2: (eq nat (S n) (S n0))).(\lambda (P: +Prop).(let H3 \def (f_equal nat nat (\lambda (e: nat).(match e in nat return +(\lambda (_: nat).nat) with [O \Rightarrow n | (S n3) \Rightarrow n3])) (S n) +(S n0) H2) in (let H4 \def (eq_ind_r nat n0 (\lambda (n3: nat).((eq nat n n3) +\to (\forall (P0: Prop).P0))) H1 n H3) in (let H5 \def (eq_ind_r nat n0 +(\lambda (n3: nat).(or (eq nat (S n) n3) ((eq nat (S n) n3) \to (\forall (P0: +Prop).P0)))) H0 n H3) in (H4 (refl_equal nat n) P)))))))) (H n0)))) n2)))) +n1). + +theorem simpl_plus_r: + \forall (n: nat).(\forall (m: nat).(\forall (p: nat).((eq nat (plus m n) +(plus p n)) \to (eq nat m p)))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (p: nat).(\lambda (H: (eq nat +(plus m n) (plus p n))).(plus_reg_l n m p (eq_ind_r nat (plus m n) (\lambda +(n0: nat).(eq nat n0 (plus n p))) (eq_ind_r nat (plus p n) (\lambda (n0: +nat).(eq nat n0 (plus n p))) (sym_eq nat (plus n p) (plus p n) (plus_comm n +p)) (plus m n) H) (plus n m) (plus_comm n m)))))). + +theorem minus_plus_r: + \forall (m: nat).(\forall (n: nat).(eq nat (minus (plus m n) n) m)) +\def + \lambda (m: nat).(\lambda (n: nat).(eq_ind_r nat (plus n m) (\lambda (n0: +nat).(eq nat (minus n0 n) m)) (minus_plus n m) (plus m n) (plus_comm m n))). + +theorem plus_permute_2_in_3: + \forall (x: nat).(\forall (y: nat).(\forall (z: nat).(eq nat (plus (plus x +y) z) (plus (plus x z) y)))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(eq_ind_r nat (plus x +(plus y z)) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) (eq_ind_r nat +(plus z y) (\lambda (n: nat).(eq nat (plus x n) (plus (plus x z) y))) (eq_ind +nat (plus (plus x z) y) (\lambda (n: nat).(eq nat n (plus (plus x z) y))) +(refl_equal nat (plus (plus x z) y)) (plus x (plus z y)) (plus_assoc_reverse +x z y)) (plus y z) (plus_comm y z)) (plus (plus x y) z) (plus_assoc_reverse x +y z)))). + +theorem plus_permute_2_in_3_assoc: + \forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq nat (plus (plus n +h) k) (plus n (plus k h))))) +\def + \lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind_r nat (plus +(plus n k) h) (\lambda (n0: nat).(eq nat n0 (plus n (plus k h)))) (eq_ind_r +nat (plus (plus n k) h) (\lambda (n0: nat).(eq nat (plus (plus n k) h) n0)) +(refl_equal nat (plus (plus n k) h)) (plus n (plus k h)) (plus_assoc n k h)) +(plus (plus n h) k) (plus_permute_2_in_3 n h k)))). + +theorem plus_O: + \forall (x: nat).(\forall (y: nat).((eq nat (plus x y) O) \to (land (eq nat +x O) (eq nat y O)))) +\def + \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((eq nat (plus +n y) O) \to (land (eq nat n O) (eq nat y O))))) (\lambda (y: nat).(\lambda +(H: (eq nat (plus O y) O)).(conj (eq nat O O) (eq nat y O) (refl_equal nat O) +H))) (\lambda (n: nat).(\lambda (_: ((\forall (y: nat).((eq nat (plus n y) O) +\to (land (eq nat n O) (eq nat y O)))))).(\lambda (y: nat).(\lambda (H0: (eq +nat (plus (S n) y) O)).(let H1 \def (match H0 in eq return (\lambda (n0: +nat).(\lambda (_: (eq ? ? n0)).((eq nat n0 O) \to (land (eq nat (S n) O) (eq +nat y O))))) with [refl_equal \Rightarrow (\lambda (H1: (eq nat (plus (S n) +y) O)).(let H2 \def (eq_ind nat (plus (S n) y) (\lambda (e: nat).(match e in +nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) +\Rightarrow True])) I O H1) in (False_ind (land (eq nat (S n) O) (eq nat y +O)) H2)))]) in (H1 (refl_equal nat O))))))) x). + +theorem minus_Sx_SO: + \forall (x: nat).(eq nat (minus (S x) (S O)) x) +\def + \lambda (x: nat).(eq_ind nat x (\lambda (n: nat).(eq nat n x)) (refl_equal +nat x) (minus x O) (minus_n_O x)). + +theorem eq_nat_dec: + \forall (i: nat).(\forall (j: nat).(or (not (eq nat i j)) (eq nat i j))) +\def + \lambda (i: nat).(nat_ind (\lambda (n: nat).(\forall (j: nat).(or (not (eq +nat n j)) (eq nat n j)))) (\lambda (j: nat).(nat_ind (\lambda (n: nat).(or +(not (eq nat O n)) (eq nat O n))) (or_intror (not (eq nat O O)) (eq nat O O) +(refl_equal nat O)) (\lambda (n: nat).(\lambda (_: (or (not (eq nat O n)) (eq +nat O n))).(or_introl (not (eq nat O (S n))) (eq nat O (S n)) (O_S n)))) j)) +(\lambda (n: nat).(\lambda (H: ((\forall (j: nat).(or (not (eq nat n j)) (eq +nat n j))))).(\lambda (j: nat).(nat_ind (\lambda (n0: nat).(or (not (eq nat +(S n) n0)) (eq nat (S n) n0))) (or_introl (not (eq nat (S n) O)) (eq nat (S +n) O) (sym_not_eq nat O (S n) (O_S n))) (\lambda (n0: nat).(\lambda (_: (or +(not (eq nat (S n) n0)) (eq nat (S n) n0))).(or_ind (not (eq nat n n0)) (eq +nat n n0) (or (not (eq nat (S n) (S n0))) (eq nat (S n) (S n0))) (\lambda +(H1: (not (eq nat n n0))).(or_introl (not (eq nat (S n) (S n0))) (eq nat (S +n) (S n0)) (not_eq_S n n0 H1))) (\lambda (H1: (eq nat n n0)).(or_intror (not +(eq nat (S n) (S n0))) (eq nat (S n) (S n0)) (f_equal nat nat S n n0 H1))) (H +n0)))) j)))) i). + +theorem neq_eq_e: + \forall (i: nat).(\forall (j: nat).(\forall (P: Prop).((((not (eq nat i j)) +\to P)) \to ((((eq nat i j) \to P)) \to P)))) +\def + \lambda (i: nat).(\lambda (j: nat).(\lambda (P: Prop).(\lambda (H: (((not +(eq nat i j)) \to P))).(\lambda (H0: (((eq nat i j) \to P))).(let o \def +(eq_nat_dec i j) in (or_ind (not (eq nat i j)) (eq nat i j) P H H0 o)))))). + +theorem le_false: + \forall (m: nat).(\forall (n: nat).(\forall (P: Prop).((le m n) \to ((le (S +n) m) \to P)))) +\def + \lambda (m: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).(\forall (P: +Prop).((le n n0) \to ((le (S n0) n) \to P))))) (\lambda (n: nat).(\lambda (P: +Prop).(\lambda (_: (le O n)).(\lambda (H0: (le (S n) O)).(let H1 \def (match +H0 in le return (\lambda (n0: nat).(\lambda (_: (le ? n0)).((eq nat n0 O) \to +P))) 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 return (\lambda (_: +nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in +(False_ind P H2))) | (le_S m0 H1) \Rightarrow (\lambda (H2: (eq nat (S m0) +O)).((let H3 \def (eq_ind nat (S m0) (\lambda (e: nat).(match e in nat return +(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) +I O H2) in (False_ind ((le (S n) m0) \to P) H3)) H1))]) in (H1 (refl_equal +nat O))))))) (\lambda (n: nat).(\lambda (H: ((\forall (n0: nat).(\forall (P: +Prop).((le n n0) \to ((le (S n0) n) \to P)))))).(\lambda (n0: nat).(nat_ind +(\lambda (n1: nat).(\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) +\to P)))) (\lambda (P: Prop).(\lambda (H0: (le (S n) O)).(\lambda (_: (le (S +O) (S n))).(let H2 \def (match H0 in le return (\lambda (n1: nat).(\lambda +(_: (le ? n1)).((eq nat n1 O) \to P))) with [le_n \Rightarrow (\lambda (H2: +(eq nat (S n) O)).(let H3 \def (eq_ind nat (S n) (\lambda (e: nat).(match e +in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) +\Rightarrow True])) I O H2) in (False_ind P H3))) | (le_S m0 H2) \Rightarrow +(\lambda (H3: (eq nat (S m0) O)).((let H4 \def (eq_ind nat (S m0) (\lambda +(e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow +False | (S _) \Rightarrow True])) I O H3) in (False_ind ((le (S n) m0) \to P) +H4)) H2))]) in (H2 (refl_equal nat O)))))) (\lambda (n1: nat).(\lambda (_: +((\forall (P: Prop).((le (S n) n1) \to ((le (S n1) (S n)) \to P))))).(\lambda +(P: Prop).(\lambda (H1: (le (S n) (S n1))).(\lambda (H2: (le (S (S n1)) (S +n))).(H n1 P (le_S_n n n1 H1) (le_S_n (S n1) n H2))))))) n0)))) m). + +theorem le_Sx_x: + \forall (x: nat).((le (S x) x) \to (\forall (P: Prop).P)) +\def + \lambda (x: nat).(\lambda (H: (le (S x) x)).(\lambda (P: Prop).(let H0 \def +le_Sn_n in (False_ind P (H0 x H))))). + +theorem minus_le: + \forall (x: nat).(\forall (y: nat).(le (minus x y) x)) +\def + \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).(le (minus n +y) n))) (\lambda (_: nat).(le_n O)) (\lambda (n: nat).(\lambda (H: ((\forall +(y: nat).(le (minus n y) n)))).(\lambda (y: nat).(nat_ind (\lambda (n0: +nat).(le (minus (S n) n0) (S n))) (le_n (S n)) (\lambda (n0: nat).(\lambda +(_: (le (match n0 with [O \Rightarrow (S n) | (S l) \Rightarrow (minus n l)]) +(S n))).(le_S (minus n n0) n (H n0)))) y)))) x). + +theorem le_plus_minus_sym: + \forall (n: nat).(\forall (m: nat).((le n m) \to (eq nat m (plus (minus m n) +n)))) +\def + \lambda (n: nat).(\lambda (m: nat).(\lambda (H: (le n m)).(eq_ind_r nat +(plus n (minus m n)) (\lambda (n0: nat).(eq nat m n0)) (le_plus_minus n m H) +(plus (minus m n) n) (plus_comm (minus m n) n)))). + +theorem le_minus_minus: + \forall (x: nat).(\forall (y: nat).((le x y) \to (\forall (z: nat).((le y z) +\to (le (minus y x) (minus z x)))))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (z: +nat).(\lambda (H0: (le y z)).(plus_le_reg_l x (minus y x) (minus z x) +(eq_ind_r nat y (\lambda (n: nat).(le n (plus x (minus z x)))) (eq_ind_r nat +z (\lambda (n: nat).(le y n)) H0 (plus x (minus z x)) (le_plus_minus_r x z +(le_trans x y z H H0))) (plus x (minus y x)) (le_plus_minus_r x y H))))))). + +theorem le_minus_plus: + \forall (z: nat).(\forall (x: nat).((le z x) \to (\forall (y: nat).(eq nat +(minus (plus x y) z) (plus (minus x z) y))))) +\def + \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((le n x) \to +(\forall (y: nat).(eq nat (minus (plus x y) n) (plus (minus x n) y)))))) +(\lambda (x: nat).(\lambda (H: (le O x)).(let H0 \def (match H in le return +(\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x) \to (\forall (y: +nat).(eq nat (minus (plus x y) O) (plus (minus x O) y)))))) with [le_n +\Rightarrow (\lambda (H0: (eq nat O x)).(eq_ind nat O (\lambda (n: +nat).(\forall (y: nat).(eq nat (minus (plus n y) O) (plus (minus n O) y)))) +(\lambda (y: nat).(sym_eq nat (plus (minus O O) y) (minus (plus O y) O) +(minus_n_O (plus O y)))) x H0)) | (le_S m H0) \Rightarrow (\lambda (H1: (eq +nat (S m) x)).(eq_ind nat (S m) (\lambda (n: nat).((le O m) \to (\forall (y: +nat).(eq nat (minus (plus n y) O) (plus (minus n O) y))))) (\lambda (_: (le O +m)).(\lambda (y: nat).(refl_equal nat (plus (minus (S m) O) y)))) x H1 H0))]) +in (H0 (refl_equal nat x))))) (\lambda (z0: nat).(\lambda (H: ((\forall (x: +nat).((le z0 x) \to (\forall (y: nat).(eq nat (minus (plus x y) z0) (plus +(minus x z0) y))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).((le (S +z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n +(S z0)) y))))) (\lambda (H0: (le (S z0) O)).(\lambda (y: nat).(let H1 \def +(match H0 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) +\to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))))) with +[le_n \Rightarrow (\lambda (H1: (eq nat (S z0) O)).(let H2 \def (eq_ind nat +(S z0) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with +[O \Rightarrow False | (S _) \Rightarrow True])) I O H1) in (False_ind (eq +nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y)) 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 return (\lambda (_: nat).Prop) with [O +\Rightarrow False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (S +z0) m) \to (eq nat (minus (plus O y) (S z0)) (plus (minus O (S z0)) y))) H3)) +H1))]) in (H1 (refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: (((le (S +z0) n) \to (\forall (y: nat).(eq nat (minus (plus n y) (S z0)) (plus (minus n +(S z0)) y)))))).(\lambda (H1: (le (S z0) (S n))).(\lambda (y: nat).(H n +(le_S_n z0 n H1) y))))) x)))) z). + +theorem le_minus: + \forall (x: nat).(\forall (z: nat).(\forall (y: nat).((le (plus x y) z) \to +(le x (minus z y))))) +\def + \lambda (x: nat).(\lambda (z: nat).(\lambda (y: nat).(\lambda (H: (le (plus +x y) z)).(eq_ind nat (minus (plus x y) y) (\lambda (n: nat).(le n (minus z +y))) (le_minus_minus y (plus x y) (le_plus_r x y) z H) x (minus_plus_r x +y))))). + +theorem le_trans_plus_r: + \forall (x: nat).(\forall (y: nat).(\forall (z: nat).((le (plus x y) z) \to +(le y z)))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (z: nat).(\lambda (H: (le (plus +x y) z)).(le_trans y (plus x y) z (le_plus_r x y) H)))). + +theorem le_gen_S: + \forall (m: nat).(\forall (x: nat).((le (S m) x) \to (ex2 nat (\lambda (n: +nat).(eq nat x (S n))) (\lambda (n: nat).(le m n))))) +\def + \lambda (m: nat).(\lambda (x: nat).(\lambda (H: (le (S m) x)).(let H0 \def +(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n x) +\to (ex2 nat (\lambda (n0: nat).(eq nat x (S n0))) (\lambda (n0: nat).(le m +n0)))))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S m) x)).(eq_ind nat +(S m) (\lambda (n: nat).(ex2 nat (\lambda (n0: nat).(eq nat n (S n0))) +(\lambda (n0: nat).(le m n0)))) (ex_intro2 nat (\lambda (n: nat).(eq nat (S +m) (S n))) (\lambda (n: nat).(le m n)) m (refl_equal nat (S m)) (le_n m)) x +H0)) | (le_S m0 H0) \Rightarrow (\lambda (H1: (eq nat (S m0) x)).(eq_ind nat +(S m0) (\lambda (n: nat).((le (S m) m0) \to (ex2 nat (\lambda (n0: nat).(eq +nat n (S n0))) (\lambda (n0: nat).(le m n0))))) (\lambda (H2: (le (S m) +m0)).(ex_intro2 nat (\lambda (n: nat).(eq nat (S m0) (S n))) (\lambda (n: +nat).(le m n)) m0 (refl_equal nat (S m0)) (le_S_n m m0 (le_S (S m) m0 H2)))) +x H1 H0))]) in (H0 (refl_equal nat x))))). + +theorem lt_x_plus_x_Sy: + \forall (x: nat).(\forall (y: nat).(lt x (plus x (S y)))) +\def + \lambda (x: nat).(\lambda (y: nat).(eq_ind_r nat (plus (S y) x) (\lambda (n: +nat).(lt x n)) (le_S_n (S x) (S (plus y x)) (le_n_S (S x) (S (plus y x)) +(le_n_S x (plus y x) (le_plus_r y x)))) (plus x (S y)) (plus_comm x (S y)))). + +theorem simpl_lt_plus_r: + \forall (p: nat).(\forall (n: nat).(\forall (m: nat).((lt (plus n p) (plus m +p)) \to (lt n m)))) +\def + \lambda (p: nat).(\lambda (n: nat).(\lambda (m: nat).(\lambda (H: (lt (plus +n p) (plus m p))).(plus_lt_reg_l n m p (let H0 \def (eq_ind nat (plus n p) +(\lambda (n0: nat).(lt n0 (plus m p))) H (plus p n) (plus_comm n p)) in (let +H1 \def (eq_ind nat (plus m p) (\lambda (n0: nat).(lt (plus p n) n0)) H0 +(plus p m) (plus_comm m p)) in H1)))))). + +theorem minus_x_Sy: + \forall (x: nat).(\forall (y: nat).((lt y x) \to (eq nat (minus x y) (S +(minus x (S y)))))) +\def + \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((lt y n) \to +(eq nat (minus n y) (S (minus n (S y))))))) (\lambda (y: nat).(\lambda (H: +(lt y O)).(let H0 \def (match H in le return (\lambda (n: nat).(\lambda (_: +(le ? n)).((eq nat n O) \to (eq nat (minus O y) (S (minus O (S y))))))) 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 return (\lambda (_: nat).Prop) with [O +\Rightarrow False | (S _) \Rightarrow True])) I O H0) in (False_ind (eq nat +(minus O y) (S (minus O (S y)))) 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 return (\lambda (_: nat).Prop) with [O \Rightarrow False +| (S _) \Rightarrow True])) I O H1) in (False_ind ((le (S y) m) \to (eq nat +(minus O y) (S (minus O (S y))))) H2)) H0))]) in (H0 (refl_equal nat O))))) +(\lambda (n: nat).(\lambda (H: ((\forall (y: nat).((lt y n) \to (eq nat +(minus n y) (S (minus n (S y)))))))).(\lambda (y: nat).(nat_ind (\lambda (n0: +nat).((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S n) (S n0)))))) +(\lambda (_: (lt O (S n))).(eq_ind nat n (\lambda (n0: nat).(eq nat (S n) (S +n0))) (refl_equal nat (S n)) (minus n O) (minus_n_O n))) (\lambda (n0: +nat).(\lambda (_: (((lt n0 (S n)) \to (eq nat (minus (S n) n0) (S (minus (S +n) (S n0))))))).(\lambda (H1: (lt (S n0) (S n))).(let H2 \def (le_S_n (S n0) +n H1) in (H n0 H2))))) y)))) x). + +theorem lt_plus_minus: + \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus x (minus +y (S x))))))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_plus_minus (S +x) y H))). + +theorem lt_plus_minus_r: + \forall (x: nat).(\forall (y: nat).((lt x y) \to (eq nat y (S (plus (minus y +(S x)) x))))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(eq_ind_r nat +(plus x (minus y (S x))) (\lambda (n: nat).(eq nat y (S n))) (lt_plus_minus x +y H) (plus (minus y (S x)) x) (plus_comm (minus y (S x)) x)))). + +theorem minus_x_SO: + \forall (x: nat).((lt O x) \to (eq nat x (S (minus x (S O))))) +\def + \lambda (x: nat).(\lambda (H: (lt O x)).(eq_ind nat (minus x O) (\lambda (n: +nat).(eq nat x n)) (eq_ind nat x (\lambda (n: nat).(eq nat x n)) (refl_equal +nat x) (minus x O) (minus_n_O x)) (S (minus x (S O))) (minus_x_Sy x O H))). + +theorem le_x_pred_y: + \forall (y: nat).(\forall (x: nat).((lt x y) \to (le x (pred y)))) +\def + \lambda (y: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).((lt x n) \to +(le x (pred n))))) (\lambda (x: nat).(\lambda (H: (lt x O)).(let H0 \def +(match H in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) +\to (le x O)))) with [le_n \Rightarrow (\lambda (H0: (eq nat (S x) O)).(let +H1 \def (eq_ind nat (S x) (\lambda (e: nat).(match e in nat return (\lambda +(_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H0) +in (False_ind (le x O) 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 +return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow +True])) I O H1) in (False_ind ((le (S x) m) \to (le x O)) H2)) H0))]) in (H0 +(refl_equal nat O))))) (\lambda (n: nat).(\lambda (_: ((\forall (x: nat).((lt +x n) \to (le x (pred n)))))).(\lambda (x: nat).(\lambda (H0: (lt x (S +n))).(le_S_n x n H0))))) y). + +theorem lt_le_minus: + \forall (x: nat).(\forall (y: nat).((lt x y) \to (le x (minus y (S O))))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(le_minus x y (S +O) (eq_ind_r nat (plus (S O) x) (\lambda (n: nat).(le n y)) H (plus x (S O)) +(plus_comm x (S O)))))). + +theorem lt_le_e: + \forall (n: nat).(\forall (d: nat).(\forall (P: Prop).((((lt n d) \to P)) +\to ((((le d n) \to P)) \to P)))) +\def + \lambda (n: nat).(\lambda (d: nat).(\lambda (P: Prop).(\lambda (H: (((lt n +d) \to P))).(\lambda (H0: (((le d n) \to P))).(let H1 \def (le_or_lt d n) in +(or_ind (le d n) (lt n d) P H0 H H1)))))). + +theorem lt_eq_e: + \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) +\to ((((eq nat x y) \to P)) \to ((le x y) \to P))))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x +y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (le x +y)).(or_ind (lt x y) (eq nat x y) P H H0 (le_lt_or_eq x y H1))))))). + +theorem lt_eq_gt_e: + \forall (x: nat).(\forall (y: nat).(\forall (P: Prop).((((lt x y) \to P)) +\to ((((eq nat x y) \to P)) \to ((((lt y x) \to P)) \to P))))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (P: Prop).(\lambda (H: (((lt x +y) \to P))).(\lambda (H0: (((eq nat x y) \to P))).(\lambda (H1: (((lt y x) +\to P))).(lt_le_e x y P H (\lambda (H2: (le y x)).(lt_eq_e y x P H1 (\lambda +(H3: (eq nat y x)).(H0 (sym_eq nat y x H3))) H2)))))))). + +theorem lt_gen_xS: + \forall (x: nat).(\forall (n: nat).((lt x (S n)) \to (or (eq nat x O) (ex2 +nat (\lambda (m: nat).(eq nat x (S m))) (\lambda (m: nat).(lt m n)))))) +\def + \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((lt n (S +n0)) \to (or (eq nat n O) (ex2 nat (\lambda (m: nat).(eq nat n (S m))) +(\lambda (m: nat).(lt m n0))))))) (\lambda (n: nat).(\lambda (_: (lt O (S +n))).(or_introl (eq nat O O) (ex2 nat (\lambda (m: nat).(eq nat O (S m))) +(\lambda (m: nat).(lt m n))) (refl_equal nat O)))) (\lambda (n: nat).(\lambda +(_: ((\forall (n0: nat).((lt n (S n0)) \to (or (eq nat n O) (ex2 nat (\lambda +(m: nat).(eq nat n (S m))) (\lambda (m: nat).(lt m n0)))))))).(\lambda (n0: +nat).(\lambda (H0: (lt (S n) (S n0))).(or_intror (eq nat (S n) O) (ex2 nat +(\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt m n0))) +(ex_intro2 nat (\lambda (m: nat).(eq nat (S n) (S m))) (\lambda (m: nat).(lt +m n0)) n (refl_equal nat (S n)) (le_S_n (S n) n0 H0))))))) x). + +theorem le_lt_false: + \forall (x: nat).(\forall (y: nat).((le x y) \to ((lt y x) \to (\forall (P: +Prop).P)))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (le x y)).(\lambda (H0: (lt +y x)).(\lambda (P: Prop).(False_ind P (le_not_lt x y H H0)))))). + +theorem lt_neq: + \forall (x: nat).(\forall (y: nat).((lt x y) \to (not (eq nat x y)))) +\def + \lambda (x: nat).(\lambda (y: nat).(\lambda (H: (lt x y)).(\lambda (H0: (eq +nat x y)).(let H1 \def (eq_ind nat x (\lambda (n: nat).(lt n y)) H y H0) in +(lt_irrefl y H1))))). + +theorem arith0: + \forall (h2: nat).(\forall (d2: nat).(\forall (n: nat).((le (plus d2 h2) n) +\to (\forall (h1: nat).(le (plus d2 h1) (minus (plus n h1) h2)))))) +\def + \lambda (h2: nat).(\lambda (d2: nat).(\lambda (n: nat).(\lambda (H: (le +(plus d2 h2) n)).(\lambda (h1: nat).(eq_ind nat (minus (plus h2 (plus d2 h1)) +h2) (\lambda (n0: nat).(le n0 (minus (plus n h1) h2))) (le_minus_minus h2 +(plus h2 (plus d2 h1)) (le_plus_l h2 (plus d2 h1)) (plus n h1) (eq_ind_r nat +(plus (plus h2 d2) h1) (\lambda (n0: nat).(le n0 (plus n h1))) (eq_ind_r nat +(plus d2 h2) (\lambda (n0: nat).(le (plus n0 h1) (plus n h1))) (le_S_n (plus +(plus d2 h2) h1) (plus n h1) (lt_le_S (plus (plus d2 h2) h1) (S (plus n h1)) +(le_lt_n_Sm (plus (plus d2 h2) h1) (plus n h1) (plus_le_compat (plus d2 h2) n +h1 h1 H (le_n h1))))) (plus h2 d2) (plus_comm h2 d2)) (plus h2 (plus d2 h1)) +(plus_assoc h2 d2 h1))) (plus d2 h1) (minus_plus h2 (plus d2 h1))))))). + +theorem O_minus: + \forall (x: nat).(\forall (y: nat).((le x y) \to (eq nat (minus x y) O))) +\def + \lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le n y) \to +(eq nat (minus n y) O)))) (\lambda (y: nat).(\lambda (_: (le O +y)).(refl_equal nat O))) (\lambda (x0: nat).(\lambda (H: ((\forall (y: +nat).((le x0 y) \to (eq nat (minus x0 y) O))))).(\lambda (y: nat).(nat_ind +(\lambda (n: nat).((le (S x0) n) \to (eq nat (match n with [O \Rightarrow (S +x0) | (S l) \Rightarrow (minus x0 l)]) O))) (\lambda (H0: (le (S x0) +O)).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le x0 +n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H1: (eq nat O (S +x1))).(\lambda (_: (le x0 x1)).(let H3 \def (eq_ind nat O (\lambda (ee: +nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True +| (S _) \Rightarrow False])) I (S x1) H1) in (False_ind (eq nat (S x0) O) +H3))))) (le_gen_S x0 O H0))) (\lambda (n: nat).(\lambda (_: (((le (S x0) n) +\to (eq nat (match n with [O \Rightarrow (S x0) | (S l) \Rightarrow (minus x0 +l)]) O)))).(\lambda (H1: (le (S x0) (S n))).(H n (le_S_n x0 n H1))))) y)))) +x). + +theorem minus_minus: + \forall (z: nat).(\forall (x: nat).(\forall (y: nat).((le z x) \to ((le z y) +\to ((eq nat (minus x z) (minus y z)) \to (eq nat x y)))))) +\def + \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x: nat).(\forall (y: +nat).((le n x) \to ((le n y) \to ((eq nat (minus x n) (minus y n)) \to (eq +nat x y))))))) (\lambda (x: nat).(\lambda (y: nat).(\lambda (_: (le O +x)).(\lambda (_: (le O y)).(\lambda (H1: (eq nat (minus x O) (minus y +O))).(let H2 \def (eq_ind_r nat (minus x O) (\lambda (n: nat).(eq nat n +(minus y O))) H1 x (minus_n_O x)) in (let H3 \def (eq_ind_r nat (minus y O) +(\lambda (n: nat).(eq nat x n)) H2 y (minus_n_O y)) in H3))))))) (\lambda +(z0: nat).(\lambda (IH: ((\forall (x: nat).(\forall (y: nat).((le z0 x) \to +((le z0 y) \to ((eq nat (minus x z0) (minus y z0)) \to (eq nat x +y)))))))).(\lambda (x: nat).(nat_ind (\lambda (n: nat).(\forall (y: nat).((le +(S z0) n) \to ((le (S z0) y) \to ((eq nat (minus n (S z0)) (minus y (S z0))) +\to (eq nat n y)))))) (\lambda (y: nat).(\lambda (H: (le (S z0) O)).(\lambda +(_: (le (S z0) y)).(\lambda (_: (eq nat (minus O (S z0)) (minus y (S +z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda (n: nat).(le +z0 n)) (eq nat O y) (\lambda (x0: nat).(\lambda (H2: (eq nat O (S +x0))).(\lambda (_: (le z0 x0)).(let H4 \def (eq_ind nat O (\lambda (ee: +nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow True +| (S _) \Rightarrow False])) I (S x0) H2) in (False_ind (eq nat O y) H4))))) +(le_gen_S z0 O H)))))) (\lambda (x0: nat).(\lambda (_: ((\forall (y: +nat).((le (S z0) x0) \to ((le (S z0) y) \to ((eq nat (minus x0 (S z0)) (minus +y (S z0))) \to (eq nat x0 y))))))).(\lambda (y: nat).(nat_ind (\lambda (n: +nat).((le (S z0) (S x0)) \to ((le (S z0) n) \to ((eq nat (minus (S x0) (S +z0)) (minus n (S z0))) \to (eq nat (S x0) n))))) (\lambda (_: (le (S z0) (S +x0))).(\lambda (H0: (le (S z0) O)).(\lambda (_: (eq nat (minus (S x0) (S z0)) +(minus O (S z0)))).(ex2_ind nat (\lambda (n: nat).(eq nat O (S n))) (\lambda +(n: nat).(le z0 n)) (eq nat (S x0) O) (\lambda (x1: nat).(\lambda (H2: (eq +nat O (S x1))).(\lambda (_: (le z0 x1)).(let H4 \def (eq_ind nat O (\lambda +(ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O \Rightarrow +True | (S _) \Rightarrow False])) I (S x1) H2) in (False_ind (eq nat (S x0) +O) H4))))) (le_gen_S z0 O H0))))) (\lambda (y0: nat).(\lambda (_: (((le (S +z0) (S x0)) \to ((le (S z0) y0) \to ((eq nat (minus (S x0) (S z0)) (minus y0 +(S z0))) \to (eq nat (S x0) y0)))))).(\lambda (H: (le (S z0) (S +x0))).(\lambda (H0: (le (S z0) (S y0))).(\lambda (H1: (eq nat (minus (S x0) +(S z0)) (minus (S y0) (S z0)))).(f_equal nat nat S x0 y0 (IH x0 y0 (le_S_n z0 +x0 H) (le_S_n z0 y0 H0) H1))))))) y)))) x)))) z). + +theorem plus_plus: + \forall (z: nat).(\forall (x1: nat).(\forall (x2: nat).(\forall (y1: +nat).(\forall (y2: nat).((le x1 z) \to ((le x2 z) \to ((eq nat (plus (minus z +x1) y1) (plus (minus z x2) y2)) \to (eq nat (plus x1 y2) (plus x2 y1))))))))) +\def + \lambda (z: nat).(nat_ind (\lambda (n: nat).(\forall (x1: nat).(\forall (x2: +nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 n) \to ((le x2 n) \to ((eq +nat (plus (minus n x1) y1) (plus (minus n x2) y2)) \to (eq nat (plus x1 y2) +(plus x2 y1)))))))))) (\lambda (x1: nat).(\lambda (x2: nat).(\lambda (y1: +nat).(\lambda (y2: nat).(\lambda (H: (le x1 O)).(\lambda (H0: (le x2 +O)).(\lambda (H1: (eq nat y1 y2)).(eq_ind nat y1 (\lambda (n: nat).(eq nat +(plus x1 n) (plus x2 y1))) (let H_y \def (le_n_O_eq x2 H0) in (eq_ind nat O +(\lambda (n: nat).(eq nat (plus x1 y1) (plus n y1))) (let H_y0 \def +(le_n_O_eq x1 H) in (eq_ind nat O (\lambda (n: nat).(eq nat (plus n y1) (plus +O y1))) (refl_equal nat (plus O y1)) x1 H_y0)) x2 H_y)) y2 H1)))))))) +(\lambda (z0: nat).(\lambda (IH: ((\forall (x1: nat).(\forall (x2: +nat).(\forall (y1: nat).(\forall (y2: nat).((le x1 z0) \to ((le x2 z0) \to +((eq nat (plus (minus z0 x1) y1) (plus (minus z0 x2) y2)) \to (eq nat (plus +x1 y2) (plus x2 y1))))))))))).(\lambda (x1: nat).(nat_ind (\lambda (n: +nat).(\forall (x2: nat).(\forall (y1: nat).(\forall (y2: nat).((le n (S z0)) +\to ((le x2 (S z0)) \to ((eq nat (plus (minus (S z0) n) y1) (plus (minus (S +z0) x2) y2)) \to (eq nat (plus n y2) (plus x2 y1))))))))) (\lambda (x2: +nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O +(S z0)) \to ((le n (S z0)) \to ((eq nat (plus (minus (S z0) O) y1) (plus +(minus (S z0) n) y2)) \to (eq nat (plus O y2) (plus n y1)))))))) (\lambda +(y1: nat).(\lambda (y2: nat).(\lambda (_: (le O (S z0))).(\lambda (_: (le O +(S z0))).(\lambda (H1: (eq nat (S (plus z0 y1)) (S (plus z0 y2)))).(let H_y +\def (IH O O) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: +nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq +nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0)) +in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (H2 (plus z0 y2) (plus z0 y1) (le_O_n +z0) (le_O_n z0) (f_equal nat nat (plus z0) (plus z0 y2) (plus z0 y1) (sym_eq +nat (plus z0 y1) (plus z0 y2) (eq_add_S (plus z0 y1) (plus z0 y2) +H1)))))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall +(y2: nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1)) +(plus (match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) +y2)) \to (eq nat y2 (plus x3 y1))))))))).(\lambda (y1: nat).(\lambda (y2: +nat).(\lambda (_: (le O (S z0))).(\lambda (H0: (le (S x3) (S z0))).(\lambda +(H1: (eq nat (S (plus z0 y1)) (plus (minus z0 x3) y2))).(let H_y \def (IH O +x3 (S y1)) in (let H2 \def (eq_ind_r nat (minus z0 O) (\lambda (n: +nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (plus n (S +y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H_y z0 +(minus_n_O z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y1)) (\lambda (n: +nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat n (plus +(minus z0 x3) y3)) \to (eq nat y3 (plus x3 (S y1)))))))) H2 (S (plus z0 y1)) +(plus_n_Sm z0 y1)) in (let H4 \def (eq_ind_r nat (plus x3 (S y1)) (\lambda +(n: nat).(\forall (y3: nat).((le O z0) \to ((le x3 z0) \to ((eq nat (S (plus +z0 y1)) (plus (minus z0 x3) y3)) \to (eq nat y3 n)))))) H3 (S (plus x3 y1)) +(plus_n_Sm x3 y1)) in (H4 y2 (le_O_n z0) (le_S_n x3 z0 H0) H1)))))))))))) +x2)) (\lambda (x2: nat).(\lambda (_: ((\forall (x3: nat).(\forall (y1: +nat).(\forall (y2: nat).((le x2 (S z0)) \to ((le x3 (S z0)) \to ((eq nat +(plus (minus (S z0) x2) y1) (plus (minus (S z0) x3) y2)) \to (eq nat (plus x2 +y2) (plus x3 y1)))))))))).(\lambda (x3: nat).(nat_ind (\lambda (n: +nat).(\forall (y1: nat).(\forall (y2: nat).((le (S x2) (S z0)) \to ((le n (S +z0)) \to ((eq nat (plus (minus (S z0) (S x2)) y1) (plus (minus (S z0) n) y2)) +\to (eq nat (plus (S x2) y2) (plus n y1)))))))) (\lambda (y1: nat).(\lambda +(y2: nat).(\lambda (H: (le (S x2) (S z0))).(\lambda (_: (le O (S +z0))).(\lambda (H1: (eq nat (plus (minus z0 x2) y1) (S (plus z0 y2)))).(let +H_y \def (IH x2 O y1 (S y2)) in (let H2 \def (eq_ind_r nat (minus z0 O) +(\lambda (n: nat).((le x2 z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) +y1) (plus n (S y2))) \to (eq nat (plus x2 (S y2)) y1))))) H_y z0 (minus_n_O +z0)) in (let H3 \def (eq_ind_r nat (plus z0 (S y2)) (\lambda (n: nat).((le x2 +z0) \to ((le O z0) \to ((eq nat (plus (minus z0 x2) y1) n) \to (eq nat (plus +x2 (S y2)) y1))))) H2 (S (plus z0 y2)) (plus_n_Sm z0 y2)) in (let H4 \def +(eq_ind_r nat (plus x2 (S y2)) (\lambda (n: nat).((le x2 z0) \to ((le O z0) +\to ((eq nat (plus (minus z0 x2) y1) (S (plus z0 y2))) \to (eq nat n y1))))) +H3 (S (plus x2 y2)) (plus_n_Sm x2 y2)) in (H4 (le_S_n x2 z0 H) (le_O_n z0) +H1)))))))))) (\lambda (x4: nat).(\lambda (_: ((\forall (y1: nat).(\forall +(y2: nat).((le (S x2) (S z0)) \to ((le x4 (S z0)) \to ((eq nat (plus (minus +z0 x2) y1) (plus (match x4 with [O \Rightarrow (S z0) | (S l) \Rightarrow +(minus z0 l)]) y2)) \to (eq nat (S (plus x2 y2)) (plus x4 +y1))))))))).(\lambda (y1: nat).(\lambda (y2: nat).(\lambda (H: (le (S x2) (S +z0))).(\lambda (H0: (le (S x4) (S z0))).(\lambda (H1: (eq nat (plus (minus z0 +x2) y1) (plus (minus z0 x4) y2))).(f_equal nat nat S (plus x2 y2) (plus x4 +y1) (IH x2 x4 y1 y2 (le_S_n x2 z0 H) (le_S_n x4 z0 H0) H1))))))))) x3)))) +x1)))) z). + +theorem le_S_minus: + \forall (d: nat).(\forall (h: nat).(\forall (n: nat).((le (plus d h) n) \to +(le d (S (minus n h)))))) +\def + \lambda (d: nat).(\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le (plus +d h) n)).(let H0 \def (le_trans d (plus d h) n (le_plus_l d h) H) in (let H1 +\def (eq_ind nat n (\lambda (n0: nat).(le d n0)) H0 (plus (minus n h) h) +(le_plus_minus_sym h n (le_trans_plus_r d h n H))) in (le_S d (minus n h) +(le_minus d n h H))))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Base-1/Base/ext/tactics.ma b/matita/contribs/LAMBDA-TYPES/Base-1/Base/ext/tactics.ma new file mode 100644 index 000000000..4a7946c68 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Base-1/Base/ext/tactics.ma @@ -0,0 +1,42 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/ext/tactics". + +include "preamble.ma". + +theorem insert_eq: + \forall (S: Set).(\forall (x: S).(\forall (P: ((S \to Prop))).(\forall (G: +Prop).(((\forall (y: S).((P y) \to ((eq S y x) \to G)))) \to ((P x) \to G))))) +\def + \lambda (S: Set).(\lambda (x: S).(\lambda (P: ((S \to Prop))).(\lambda (G: +Prop).(\lambda (H: ((\forall (y: S).((P y) \to ((eq S y x) \to +G))))).(\lambda (H0: (P x)).(H x H0 (refl_equal S x))))))). + +theorem unintro: + \forall (A: Set).(\forall (a: A).(\forall (P: ((A \to Prop))).(((\forall (x: +A).(P x))) \to (P a)))) +\def + \lambda (A: Set).(\lambda (a: A).(\lambda (P: ((A \to Prop))).(\lambda (H: +((\forall (x: A).(P x)))).(H a)))). + +theorem xinduction: + \forall (A: Set).(\forall (t: A).(\forall (P: ((A \to Prop))).(((\forall (x: +A).((eq A t x) \to (P x)))) \to (P t)))) +\def + \lambda (A: Set).(\lambda (t: A).(\lambda (P: ((A \to Prop))).(\lambda (H: +((\forall (x: A).((eq A t x) \to (P x))))).(H t (refl_equal A t))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Base-1/Base/makefile b/matita/contribs/LAMBDA-TYPES/Base-1/Base/makefile new file mode 100644 index 000000000..db1724d0c --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Base-1/Base/makefile @@ -0,0 +1,39 @@ +H=@ + +RT_BASEDIR=../../../../ +OPTIONS=-bench +MMAKE=$(RT_BASEDIR)matitamake $(OPTIONS) +CLEAN=$(RT_BASEDIR)matitaclean $(OPTIONS) +MMAKEO=$(RT_BASEDIR)matitamake.opt $(OPTIONS) +CLEANO=$(RT_BASEDIR)matitaclean.opt $(OPTIONS) + +devel:=$(shell basename `pwd`) + +ifneq "$(SRC)" "" + XXX="SRC=$(SRC)" +endif + +all: preall + $(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) build $(devel) +clean: preall + $(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) clean $(devel) +cleanall: preall + $(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEAN) all + +all.opt opt: preall.opt + $(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) build $(devel) +clean.opt: preall.opt + $(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) clean $(devel) +cleanall.opt: preall.opt + $(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MCLEANO) all + +%.mo: preall + $(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) $@ +%.mo.opt: preall.opt + $(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) $@ + +preall: + $(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKE) init $(devel) + +preall.opt: + $(H)$(XXX) MATITA_FLAGS=$(MATITA_FLAGS) $(MMAKEO) init $(devel) diff --git a/matita/contribs/LAMBDA-TYPES/Base-1/Base/plist/defs.ma b/matita/contribs/LAMBDA-TYPES/Base-1/Base/plist/defs.ma new file mode 100644 index 000000000..1ca1142d9 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Base-1/Base/plist/defs.ma @@ -0,0 +1,45 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/plist/defs". + +include "preamble.ma". + +inductive PList: Set \def +| PNil: PList +| PCons: nat \to (nat \to (PList \to PList)). + +definition PConsTail: + PList \to (nat \to (nat \to PList)) +\def + let rec PConsTail (hds: PList) on hds: (nat \to (nat \to PList)) \def +(\lambda (h0: nat).(\lambda (d0: nat).(match hds with [PNil \Rightarrow +(PCons h0 d0 PNil) | (PCons h d hds0) \Rightarrow (PCons h d (PConsTail hds0 +h0 d0))]))) in PConsTail. + +definition Ss: + PList \to PList +\def + let rec Ss (hds: PList) on hds: PList \def (match hds with [PNil \Rightarrow +PNil | (PCons h d hds0) \Rightarrow (PCons h (S d) (Ss hds0))]) in Ss. + +definition papp: + PList \to (PList \to PList) +\def + let rec papp (a: PList) on a: (PList \to PList) \def (\lambda (b: +PList).(match a with [PNil \Rightarrow b | (PCons h d a0) \Rightarrow (PCons +h d (papp a0 b))])) in papp. + diff --git a/matita/contribs/LAMBDA-TYPES/Base-1/Base/plist/props.ma b/matita/contribs/LAMBDA-TYPES/Base-1/Base/plist/props.ma new file mode 100644 index 000000000..7338262f1 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Base-1/Base/plist/props.ma @@ -0,0 +1,33 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/plist/props". + +include "plist/defs.ma". + +theorem papp_ss: + \forall (is1: PList).(\forall (is2: PList).(eq PList (papp (Ss is1) (Ss +is2)) (Ss (papp is1 is2)))) +\def + \lambda (is1: PList).(PList_ind (\lambda (p: PList).(\forall (is2: +PList).(eq PList (papp (Ss p) (Ss is2)) (Ss (papp p is2))))) (\lambda (is2: +PList).(refl_equal PList (Ss is2))) (\lambda (n: nat).(\lambda (n0: +nat).(\lambda (p: PList).(\lambda (H: ((\forall (is2: PList).(eq PList (papp +(Ss p) (Ss is2)) (Ss (papp p is2)))))).(\lambda (is2: PList).(eq_ind_r PList +(Ss (papp p is2)) (\lambda (p0: PList).(eq PList (PCons n (S n0) p0) (PCons n +(S n0) (Ss (papp p is2))))) (refl_equal PList (PCons n (S n0) (Ss (papp p +is2)))) (papp (Ss p) (Ss is2)) (H is2))))))) is1). + diff --git a/matita/contribs/LAMBDA-TYPES/Base-1/Base/preamble.ma b/matita/contribs/LAMBDA-TYPES/Base-1/Base/preamble.ma new file mode 100644 index 000000000..9b2d974f4 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Base-1/Base/preamble.ma @@ -0,0 +1,160 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/preamble". + +include' "../../../../legacy/coq.ma". + +(* FG: This is because "and" is a reserved keyword of the parser *) +alias id "land" = "cic:/Coq/Init/Logic/and.ind#xpointer(1/1)". + +(* FG/CSC: These aliases should disappear: we would like to write something + * like: "disambiguate in cic:/Coq/*" + *) +alias symbol "plus" = "Coq's natural plus". +alias symbol "leq" = "Coq's natural 'less or equal to'". +alias symbol "neq" = "Coq's not equal to (leibnitz)". +alias symbol "eq" = "Coq's leibnitz's equality". + +alias id "bool" = "cic:/Coq/Init/Datatypes/bool.ind#xpointer(1/1)". +alias id "conj" = "cic:/Coq/Init/Logic/and.ind#xpointer(1/1/1)". +alias id "eq_add_S" = "cic:/Coq/Init/Peano/eq_add_S.con". +alias id "eq" = "cic:/Coq/Init/Logic/eq.ind#xpointer(1/1)". +alias id "eq_ind" = "cic:/Coq/Init/Logic/eq_ind.con". +alias id "eq_ind_r" = "cic:/Coq/Init/Logic/eq_ind_r.con". +alias id "ex2" = "cic:/Coq/Init/Logic/ex2.ind#xpointer(1/1)". +alias id "ex2_ind" = "cic:/Coq/Init/Logic/ex2_ind.con". +alias id "ex" = "cic:/Coq/Init/Logic/ex.ind#xpointer(1/1)". +alias id "ex_intro2" = "cic:/Coq/Init/Logic/ex2.ind#xpointer(1/1/1)". +alias id "false" = "cic:/Coq/Init/Datatypes/bool.ind#xpointer(1/1/2)". +alias id "False" = "cic:/Coq/Init/Logic/False.ind#xpointer(1/1)". +alias id "False_ind" = "cic:/Coq/Init/Logic/False_ind.con". +alias id "I" = "cic:/Coq/Init/Logic/True.ind#xpointer(1/1/1)". +alias id "le_antisym" = "cic:/Coq/Arith/Le/le_antisym.con". +alias id "le" = "cic:/Coq/Init/Peano/le.ind#xpointer(1/1)". +alias id "le_lt_n_Sm" = "cic:/Coq/Arith/Lt/le_lt_n_Sm.con". +alias id "le_lt_or_eq" = "cic:/Coq/Arith/Lt/le_lt_or_eq.con". +alias id "le_n" = "cic:/Coq/Init/Peano/le.ind#xpointer(1/1/1)". +alias id "le_n_O_eq" = "cic:/Coq/Arith/Le/le_n_O_eq.con". +alias id "le_not_lt" = "cic:/Coq/Arith/Lt/le_not_lt.con". +alias id "le_n_S" = "cic:/Coq/Arith/Le/le_n_S.con". +alias id "le_O_n" = "cic:/Coq/Arith/Le/le_O_n.con". +alias id "le_or_lt" = "cic:/Coq/Arith/Lt/le_or_lt.con". +alias id "le_plus_l" = "cic:/Coq/Arith/Plus/le_plus_l.con". +alias id "le_plus_minus" = "cic:/Coq/Arith/Minus/le_plus_minus.con". +alias id "le_plus_minus_r" = "cic:/Coq/Arith/Minus/le_plus_minus_r.con". +alias id "le_plus_r" = "cic:/Coq/Arith/Plus/le_plus_r.con". +alias id "le_S" = "cic:/Coq/Init/Peano/le.ind#xpointer(1/1/2)". +alias id "le_S_n" = "cic:/Coq/Arith/Le/le_S_n.con". +alias id "le_Sn_n" = "cic:/Coq/Arith/Le/le_Sn_n.con". +alias id "le_trans" = "cic:/Coq/Arith/Le/le_trans.con". +alias id "lt" = "cic:/Coq/Init/Peano/lt.con". +alias id "lt_irrefl" = "cic:/Coq/Arith/Lt/lt_irrefl.con". +alias id "lt_le_S" = "cic:/Coq/Arith/Lt/lt_le_S.con". +alias id "lt_n_S" = "cic:/Coq/Arith/Lt/lt_n_S.con". +alias id "minus" = "cic:/Coq/Init/Peano/minus.con". +alias id "minus_n_O" = "cic:/Coq/Arith/Minus/minus_n_O.con". +alias id "minus_plus" = "cic:/Coq/Arith/Minus/minus_plus.con". +alias id "nat" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1)". +alias id "nat_ind" = "cic:/Coq/Init/Datatypes/nat_ind.con". +alias id "not" = "cic:/Coq/Init/Logic/not.con". +alias id "not_eq_S" = "cic:/Coq/Init/Peano/not_eq_S.con". +alias id "O" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/1)". +alias id "or" = "cic:/Coq/Init/Logic/or.ind#xpointer(1/1)". +alias id "or_ind" = "cic:/Coq/Init/Logic/or_ind.con". +alias id "or_introl" = "cic:/Coq/Init/Logic/or.ind#xpointer(1/1/1)". +alias id "or_intror" = "cic:/Coq/Init/Logic/or.ind#xpointer(1/1/2)". +alias id "O_S" = "cic:/Coq/Init/Peano/O_S.con". +alias id "plus_assoc" = "cic:/Coq/Arith/Plus/plus_assoc.con". +alias id "plus_assoc_reverse" = "cic:/Coq/Arith/Plus/plus_assoc_reverse.con". +alias id "plus" = "cic:/Coq/Init/Peano/plus.con". +alias id "plus_comm" = "cic:/Coq/Arith/Plus/plus_comm.con". +alias id "plus_le_compat" = "cic:/Coq/Arith/Plus/plus_le_compat.con". +alias id "plus_le_reg_l" = "cic:/Coq/Arith/Plus/plus_le_reg_l.con". +alias id "plus_lt_reg_l" = "cic:/Coq/Arith/Plus/plus_lt_reg_l.con". +alias id "plus_n_Sm" = "cic:/Coq/Init/Peano/plus_n_Sm.con". +alias id "plus_reg_l" = "cic:/Coq/Arith/Plus/plus_reg_l.con". +alias id "pred" = "cic:/Coq/Init/Peano/pred.con". +alias id "refl_equal" = "cic:/Coq/Init/Logic/eq.ind#xpointer(1/1/1)". +alias id "S" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/2)". +alias id "true" = "cic:/Coq/Init/Datatypes/bool.ind#xpointer(1/1/1)". +alias id "True" = "cic:/Coq/Init/Logic/True.ind#xpointer(1/1)". +alias id "plus_lt_compat_r" = "cic:/Coq/Arith/Plus/plus_lt_compat_r.con". +alias id "plus_n_O" = "cic:/Coq/Init/Peano/plus_n_O.con". +alias id "plus_le_lt_compat" = "cic:/Coq/Arith/Plus/plus_le_lt_compat.con". +alias id "lt_wf_ind" = "cic:/Coq/Arith/Wf_nat/lt_wf_ind.con". +alias id "minus_Sn_m" = "cic:/Coq/Arith/Minus/minus_Sn_m.con". +alias id "and_ind" = "cic:/Coq/Init/Logic/and_ind.con". +alias id "le_lt_trans" = "cic:/Coq/Arith/Lt/le_lt_trans.con". +alias id "lt_le_trans" = "cic:/Coq/Arith/Lt/lt_le_trans.con". +alias id "le_lt_trans" = "cic:/Coq/Arith/Lt/le_lt_trans.con". +alias id "plus_n_O" = "cic:/Coq/Init/Peano/plus_n_O.con". +alias id "f_equal3" = "cic:/Coq/Init/Logic/f_equal3.con". +alias id "S_pred" = "cic:/Coq/Arith/Lt/S_pred.con". +alias id "lt_le_trans" = "cic:/Coq/Arith/Lt/lt_le_trans.con". +alias id "plus_lt_compat_r" = "cic:/Coq/Arith/Plus/plus_lt_compat_r.con". +alias id "le_plus_trans" = "cic:/Coq/Arith/Plus/le_plus_trans.con". +alias id "f_equal2" = "cic:/Coq/Init/Logic/f_equal2.con". +alias id "le_plus_trans" = "cic:/Coq/Arith/Plus/le_plus_trans.con". +alias id "f_equal2" = "cic:/Coq/Init/Logic/f_equal2.con". +alias id "plus_n_O" = "cic:/Coq/Init/Peano/plus_n_O.con". +alias id "plus_n_O" = "cic:/Coq/Init/Peano/plus_n_O.con". +alias id "lt_trans" = "cic:/Coq/Arith/Lt/lt_trans.con". +alias id "minus_Sn_m" = "cic:/Coq/Arith/Minus/minus_Sn_m.con". +alias id "ex_intro" = "cic:/Coq/Init/Logic/ex.ind#xpointer(1/1/1)". +alias id "lt_trans" = "cic:/Coq/Arith/Lt/lt_trans.con". +alias id "lt_n_Sn" = "cic:/Coq/Arith/Lt/lt_n_Sn.con". +alias id "lt_le_trans" = "cic:/Coq/Arith/Lt/lt_le_trans.con". +alias id "lt_wf_ind" = "cic:/Coq/Arith/Wf_nat/lt_wf_ind.con". +alias id "bool_ind" = "cic:/Coq/Init/Datatypes/bool_ind.con". +alias id "ex_ind" = "cic:/Coq/Init/Logic/ex_ind.con". +alias id "plus_Snm_nSm" = "cic:/Coq/Arith/Plus/plus_Snm_nSm.con". +alias id "plus_lt_le_compat" = "cic:/Coq/Arith/Plus/plus_lt_le_compat.con". +alias id "plus_lt_compat" = "cic:/Coq/Arith/Plus/plus_lt_compat.con". +alias id "lt_S_n" = "cic:/Coq/Arith/Lt/lt_S_n.con". +alias id "minus_n_n" = "cic:/Coq/Arith/Minus/minus_n_n.con". + +theorem f_equal: \forall A,B:Type. \forall f:A \to B. + \forall x,y:A. x = y \to f x = f y. + intros. elim H. reflexivity. +qed. + +theorem sym_eq: \forall A:Type. \forall x,y:A. x = y \to y = x. + intros. rewrite > H. reflexivity. +qed. + +theorem sym_not_eq: \forall A:Type. \forall x,y:A. x \neq y \to y \neq x. + unfold not. intros. apply H. symmetry. assumption. +qed. + +theorem trans_eq : \forall A:Type. \forall x,y,z:A. x=y \to y=z \to x=z. + intros. transitivity y; assumption. +qed. + +theorem plus_reg_l: \forall n,m,p. n + m = n + p \to m = p. + intros. apply plus_reg_l; auto. +qed. + +theorem plus_le_reg_l: \forall p,n,m. p + n <= p + m \to n <= m. + intros. apply plus_le_reg_l; auto. +qed. + +default "equality" + cic:/Coq/Init/Logic/eq.ind + cic:/matita/LAMBDA-TYPES/Base-1/preamble/sym_eq.con + cic:/matita/LAMBDA-TYPES/Base-1/preamble/trans_eq.con + cic:/Coq/Init/Logic/eq_ind.con + cic:/Coq/Init/Logic/eq_ind_r.con + cic:/matita/LAMBDA-TYPES/Base-1/preamble/f_equal.con + cic:/matita/legacy/coq/f_equal1.con. diff --git a/matita/contribs/LAMBDA-TYPES/Base-1/Base/spare.ma b/matita/contribs/LAMBDA-TYPES/Base-1/Base/spare.ma new file mode 100644 index 000000000..f66934f78 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Base-1/Base/spare.ma @@ -0,0 +1,20 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/spare". + +include "theory.ma". + diff --git a/matita/contribs/LAMBDA-TYPES/Base-1/Base/theory.ma b/matita/contribs/LAMBDA-TYPES/Base-1/Base/theory.ma new file mode 100644 index 000000000..d89a21858 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Base-1/Base/theory.ma @@ -0,0 +1,28 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/theory". + +include "ext/tactics.ma". + +include "ext/arith.ma". + +include "types/props.ma". + +include "blt/props.ma". + +include "plist/props.ma". + diff --git a/matita/contribs/LAMBDA-TYPES/Base-1/Base/types/defs.ma b/matita/contribs/LAMBDA-TYPES/Base-1/Base/types/defs.ma new file mode 100644 index 000000000..a60c1ad64 --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Base-1/Base/types/defs.ma @@ -0,0 +1,150 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/types/defs". + +include "preamble.ma". + +inductive and3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def +| and3_intro: P0 \to (P1 \to (P2 \to (and3 P0 P1 P2))). + +inductive or3 (P0: Prop) (P1: Prop) (P2: Prop): Prop \def +| or3_intro0: P0 \to (or3 P0 P1 P2) +| or3_intro1: P1 \to (or3 P0 P1 P2) +| or3_intro2: P2 \to (or3 P0 P1 P2). + +inductive or4 (P0: Prop) (P1: Prop) (P2: Prop) (P3: Prop): Prop \def +| or4_intro0: P0 \to (or4 P0 P1 P2 P3) +| or4_intro1: P1 \to (or4 P0 P1 P2 P3) +| or4_intro2: P2 \to (or4 P0 P1 P2 P3) +| or4_intro3: P3 \to (or4 P0 P1 P2 P3). + +inductive ex3 (A0: Set) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to +Prop): Prop \def +| ex3_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to (ex3 A0 +P0 P1 P2)))). + +inductive ex4 (A0: Set) (P0: A0 \to Prop) (P1: A0 \to Prop) (P2: A0 \to Prop) +(P3: A0 \to Prop): Prop \def +| ex4_intro: \forall (x0: A0).((P0 x0) \to ((P1 x0) \to ((P2 x0) \to ((P3 x0) +\to (ex4 A0 P0 P1 P2 P3))))). + +inductive ex_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)): Prop \def +| ex_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to (ex_2 A0 A1 +P0))). + +inductive ex2_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to +(A1 \to Prop)): Prop \def +| ex2_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) +\to (ex2_2 A0 A1 P0 P1)))). + +inductive ex3_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to +(A1 \to Prop)) (P2: A0 \to (A1 \to Prop)): Prop \def +| ex3_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) +\to ((P2 x0 x1) \to (ex3_2 A0 A1 P0 P1 P2))))). + +inductive ex4_2 (A0: Set) (A1: Set) (P0: A0 \to (A1 \to Prop)) (P1: A0 \to +(A1 \to Prop)) (P2: A0 \to (A1 \to Prop)) (P3: A0 \to (A1 \to Prop)): Prop +\def +| ex4_2_intro: \forall (x0: A0).(\forall (x1: A1).((P0 x0 x1) \to ((P1 x0 x1) +\to ((P2 x0 x1) \to ((P3 x0 x1) \to (ex4_2 A0 A1 P0 P1 P2 P3)))))). + +inductive ex_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to +Prop))): Prop \def +| ex_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 x1 +x2) \to (ex_3 A0 A1 A2 P0)))). + +inductive ex2_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to +Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))): Prop \def +| ex2_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 +x1 x2) \to ((P1 x0 x1 x2) \to (ex2_3 A0 A1 A2 P0 P1))))). + +inductive ex3_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to +Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to +Prop))): Prop \def +| ex3_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 +x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to (ex3_3 A0 A1 A2 P0 P1 +P2)))))). + +inductive ex4_3 (A0: Set) (A1: Set) (A2: Set) (P0: A0 \to (A1 \to (A2 \to +Prop))) (P1: A0 \to (A1 \to (A2 \to Prop))) (P2: A0 \to (A1 \to (A2 \to +Prop))) (P3: A0 \to (A1 \to (A2 \to Prop))): Prop \def +| ex4_3_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).((P0 x0 +x1 x2) \to ((P1 x0 x1 x2) \to ((P2 x0 x1 x2) \to ((P3 x0 x1 x2) \to (ex4_3 A0 +A1 A2 P0 P1 P2 P3))))))). + +inductive ex3_4 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (P0: A0 \to (A1 \to +(A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P2: A0 +\to (A1 \to (A2 \to (A3 \to Prop)))): Prop \def +| ex3_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall +(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to +(ex3_4 A0 A1 A2 A3 P0 P1 P2))))))). + +inductive ex4_4 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (P0: A0 \to (A1 \to +(A2 \to (A3 \to Prop)))) (P1: A0 \to (A1 \to (A2 \to (A3 \to Prop)))) (P2: A0 +\to (A1 \to (A2 \to (A3 \to Prop)))) (P3: A0 \to (A1 \to (A2 \to (A3 \to +Prop)))): Prop \def +| ex4_4_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall +(x3: A3).((P0 x0 x1 x2 x3) \to ((P1 x0 x1 x2 x3) \to ((P2 x0 x1 x2 x3) \to +((P3 x0 x1 x2 x3) \to (ex4_4 A0 A1 A2 A3 P0 P1 P2 P3)))))))). + +inductive ex4_5 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (P0: A0 \to +(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to +(A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P3: +A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))): Prop \def +| ex4_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall +(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to +((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to (ex4_5 A0 A1 A2 A3 A4 P0 P1 +P2 P3))))))))). + +inductive ex5_5 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (P0: A0 \to +(A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to +(A4 \to Prop))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P3: +A0 \to (A1 \to (A2 \to (A3 \to (A4 \to Prop))))) (P4: A0 \to (A1 \to (A2 \to +(A3 \to (A4 \to Prop))))): Prop \def +| ex5_5_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall +(x3: A3).(\forall (x4: A4).((P0 x0 x1 x2 x3 x4) \to ((P1 x0 x1 x2 x3 x4) \to +((P2 x0 x1 x2 x3 x4) \to ((P3 x0 x1 x2 x3 x4) \to ((P4 x0 x1 x2 x3 x4) \to +(ex5_5 A0 A1 A2 A3 A4 P0 P1 P2 P3 P4)))))))))). + +inductive ex6_6 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (A5: Set) +(P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P1: A0 \to +(A1 \to (A2 \to (A3 \to (A4 \to (A5 \to Prop)))))) (P2: A0 \to (A1 \to (A2 +\to (A3 \to (A4 \to (A5 \to Prop)))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to +(A4 \to (A5 \to Prop)))))) (P4: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 +\to Prop)))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to +Prop)))))): Prop \def +| ex6_6_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall +(x3: A3).(\forall (x4: A4).(\forall (x5: A5).((P0 x0 x1 x2 x3 x4 x5) \to ((P1 +x0 x1 x2 x3 x4 x5) \to ((P2 x0 x1 x2 x3 x4 x5) \to ((P3 x0 x1 x2 x3 x4 x5) +\to ((P4 x0 x1 x2 x3 x4 x5) \to ((P5 x0 x1 x2 x3 x4 x5) \to (ex6_6 A0 A1 A2 +A3 A4 A5 P0 P1 P2 P3 P4 P5)))))))))))). + +inductive ex6_7 (A0: Set) (A1: Set) (A2: Set) (A3: Set) (A4: Set) (A5: Set) +(A6: Set) (P0: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to +Prop))))))) (P1: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to +Prop))))))) (P2: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to +Prop))))))) (P3: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to +Prop))))))) (P4: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to +Prop))))))) (P5: A0 \to (A1 \to (A2 \to (A3 \to (A4 \to (A5 \to (A6 \to +Prop))))))): Prop \def +| ex6_7_intro: \forall (x0: A0).(\forall (x1: A1).(\forall (x2: A2).(\forall +(x3: A3).(\forall (x4: A4).(\forall (x5: A5).(\forall (x6: A6).((P0 x0 x1 x2 +x3 x4 x5 x6) \to ((P1 x0 x1 x2 x3 x4 x5 x6) \to ((P2 x0 x1 x2 x3 x4 x5 x6) +\to ((P3 x0 x1 x2 x3 x4 x5 x6) \to ((P4 x0 x1 x2 x3 x4 x5 x6) \to ((P5 x0 x1 +x2 x3 x4 x5 x6) \to (ex6_7 A0 A1 A2 A3 A4 A5 A6 P0 P1 P2 P3 P4 +P5))))))))))))). + diff --git a/matita/contribs/LAMBDA-TYPES/Base-1/Base/types/props.ma b/matita/contribs/LAMBDA-TYPES/Base-1/Base/types/props.ma new file mode 100644 index 000000000..1c9b499bb --- /dev/null +++ b/matita/contribs/LAMBDA-TYPES/Base-1/Base/types/props.ma @@ -0,0 +1,32 @@ +(**************************************************************************) +(* ___ *) +(* ||M|| *) +(* ||A|| A project by Andrea Asperti *) +(* ||T|| *) +(* ||I|| Developers: *) +(* ||T|| The HELM team. *) +(* ||A|| http://helm.cs.unibo.it *) +(* \ / *) +(* \ / This file is distributed under the terms of the *) +(* v GNU General Public License Version 2 *) +(* *) +(**************************************************************************) + +(* This file was automatically generated: do not edit *********************) + +set "baseuri" "cic:/matita/LAMBDA-TYPES/Base-1/types/props". + +include "types/defs.ma". + +theorem ex2_sym: + \forall (A: Set).(\forall (P: ((A \to Prop))).(\forall (Q: ((A \to +Prop))).((ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x))) \to (ex2 A +(\lambda (x: A).(Q x)) (\lambda (x: A).(P x)))))) +\def + \lambda (A: Set).(\lambda (P: ((A \to Prop))).(\lambda (Q: ((A \to +Prop))).(\lambda (H: (ex2 A (\lambda (x: A).(P x)) (\lambda (x: A).(Q +x)))).(ex2_ind A (\lambda (x: A).(P x)) (\lambda (x: A).(Q x)) (ex2 A +(\lambda (x: A).(Q x)) (\lambda (x: A).(P x))) (\lambda (x: A).(\lambda (H0: +(P x)).(\lambda (H1: (Q x)).(ex_intro2 A (\lambda (x0: A).(Q x0)) (\lambda +(x0: A).(P x0)) x H1 H0)))) H)))). + -- 2.39.2