From e8656c819b0b5e7bea7b4da244015b480af5f0f5 Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Sun, 22 Feb 2015 19:31:23 +0000 Subject: [PATCH] components: A asucc aplus leq llt aprem ex0 --- .../contribs/lambdadelta/basic_1/A/defs.ma | 4 +- .../contribs/lambdadelta/basic_1/A/fwd.ma | 32 + .../lambdadelta/basic_1/aplus/defs.ma | 9 +- .../lambdadelta/basic_1/aplus/props.ma | 544 +++++--- .../lambdadelta/basic_1/aprem/defs.ma | 2 +- .../contribs/lambdadelta/basic_1/aprem/fwd.ma | 186 +-- .../lambdadelta/basic_1/aprem/props.ma | 100 +- .../lambdadelta/basic_1/asucc/defs.ma | 15 +- .../contribs/lambdadelta/basic_1/asucc/fwd.ma | 173 +-- .../contribs/lambdadelta/basic_1/ex0/defs.ma | 4 +- .../contribs/lambdadelta/basic_1/ex0/fwd.ma | 29 + .../contribs/lambdadelta/basic_1/ex0/props.ma | 505 +++++--- .../contribs/lambdadelta/basic_1/leq/asucc.ma | 1105 ++++++++++------- .../contribs/lambdadelta/basic_1/leq/defs.ma | 2 +- .../contribs/lambdadelta/basic_1/leq/fwd.ma | 494 +++++--- .../contribs/lambdadelta/basic_1/leq/props.ma | 384 +++--- .../contribs/lambdadelta/basic_1/llt/defs.ma | 14 +- .../contribs/lambdadelta/basic_1/llt/fwd.ma | 51 + .../contribs/lambdadelta/basic_1/llt/props.ma | 106 +- 19 files changed, 2311 insertions(+), 1448 deletions(-) create mode 100644 matita/matita/contribs/lambdadelta/basic_1/A/fwd.ma create mode 100644 matita/matita/contribs/lambdadelta/basic_1/ex0/fwd.ma create mode 100644 matita/matita/contribs/lambdadelta/basic_1/llt/fwd.ma diff --git a/matita/matita/contribs/lambdadelta/basic_1/A/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/A/defs.ma index 2290c3de4..dc435ca87 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/A/defs.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/A/defs.ma @@ -14,9 +14,9 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/preamble.ma". +include "basic_1/preamble.ma". -inductive A: Set \def +inductive A: Type[0] \def | ASort: nat \to (nat \to A) | AHead: A \to (A \to A). diff --git a/matita/matita/contribs/lambdadelta/basic_1/A/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/A/fwd.ma new file mode 100644 index 000000000..0f5f065df --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/A/fwd.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 *********************) + +include "basic_1/A/defs.ma". + +let rec A_rect (P: (A \to Type[0])) (f: (\forall (n: nat).(\forall (n0: +nat).(P (ASort n n0))))) (f0: (\forall (a: A).((P a) \to (\forall (a0: A).((P +a0) \to (P (AHead a a0))))))) (a: A) on a: P a \def match a with [(ASort n +n0) \Rightarrow (f n n0) | (AHead a0 a1) \Rightarrow (let TMP_1 \def ((A_rect +P f f0) a0) in (let TMP_2 \def ((A_rect P f f0) a1) in (f0 a0 TMP_1 a1 +TMP_2)))]. + +theorem A_ind: + \forall (P: ((A \to Prop))).(((\forall (n: nat).(\forall (n0: nat).(P (ASort +n n0))))) \to (((\forall (a: A).((P a) \to (\forall (a0: A).((P a0) \to (P +(AHead a a0))))))) \to (\forall (a: A).(P a)))) +\def + \lambda (P: ((A \to Prop))).(A_rect P). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/aplus/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/aplus/defs.ma index 4095b163b..08c752cb8 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/aplus/defs.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/aplus/defs.ma @@ -14,11 +14,8 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/asucc/defs.ma". +include "basic_1/asucc/defs.ma". -definition aplus: - G \to (A \to (nat \to A)) -\def - let rec aplus (g: G) (a: A) (n: nat) on n: A \def (match n with [O -\Rightarrow a | (S n0) \Rightarrow (asucc g (aplus g a n0))]) in aplus. +let rec aplus (g: G) (a: A) (n: nat) on n: A \def match n with [O \Rightarrow +a | (S n0) \Rightarrow (let TMP_1 \def (aplus g a n0) in (asucc g TMP_1))]. diff --git a/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma b/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma index 94bb9a069..7cd40dae7 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/aplus/props.ma @@ -14,9 +14,11 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/aplus/defs.ma". +include "basic_1/aplus/defs.ma". -include "Basic-1/next_plus/props.ma". +include "basic_1/A/fwd.ma". + +include "basic_1/next_plus/props.ma". theorem aplus_reg_r: \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (h1: nat).(\forall @@ -25,258 +27,390 @@ theorem aplus_reg_r: \def \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (H: (eq A (aplus g a1 h1) (aplus g a2 h2))).(\lambda (h: -nat).(nat_ind (\lambda (n: nat).(eq A (aplus g a1 (plus n h1)) (aplus g a2 -(plus n h2)))) H (\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n -h1)) (aplus g a2 (plus n h2)))).(f_equal2 G A A asucc g g (aplus g a1 (plus n -h1)) (aplus g a2 (plus n h2)) (refl_equal G g) H0))) h))))))). -(* COMMENTS -Initial nodes: 143 -END *) +nat).(let TMP_5 \def (\lambda (n: nat).(let TMP_1 \def (plus n h1) in (let +TMP_2 \def (aplus g a1 TMP_1) in (let TMP_3 \def (plus n h2) in (let TMP_4 +\def (aplus g a2 TMP_3) in (eq A TMP_2 TMP_4)))))) in (let TMP_11 \def +(\lambda (n: nat).(\lambda (H0: (eq A (aplus g a1 (plus n h1)) (aplus g a2 +(plus n h2)))).(let TMP_6 \def (plus n h1) in (let TMP_7 \def (aplus g a1 +TMP_6) in (let TMP_8 \def (plus n h2) in (let TMP_9 \def (aplus g a2 TMP_8) +in (let TMP_10 \def (refl_equal G g) in (f_equal2 G A A asucc g g TMP_7 TMP_9 +TMP_10 H0)))))))) in (nat_ind TMP_5 H TMP_11 h))))))))). theorem aplus_assoc: \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A (aplus g (aplus g a h1) h2) (aplus g a (plus h1 h2)))))) \def - \lambda (g: G).(\lambda (a: A).(\lambda (h1: nat).(nat_ind (\lambda (n: -nat).(\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus g a (plus n -h2))))) (\lambda (h2: nat).(refl_equal A (aplus g a h2))) (\lambda (n: -nat).(\lambda (_: ((\forall (h2: nat).(eq A (aplus g (aplus g a n) h2) (aplus -g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A -(aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0))))) -(eq_ind nat n (\lambda (n0: nat).(eq A (asucc g (aplus g a n)) (asucc g -(aplus g a n0)))) (refl_equal A (asucc g (aplus g a n))) (plus n O) (plus_n_O -n)) (\lambda (n0: nat).(\lambda (H0: (eq A (aplus g (asucc g (aplus g a n)) -n0) (asucc g (aplus g a (plus n n0))))).(eq_ind nat (S (plus n n0)) (\lambda -(n1: nat).(eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g -(aplus g a n1)))) (f_equal2 G A A asucc g g (aplus g (asucc g (aplus g a n)) -n0) (asucc g (aplus g a (plus n n0))) (refl_equal G g) H0) (plus n (S n0)) -(plus_n_Sm n n0)))) h2)))) h1))). -(* COMMENTS -Initial nodes: 361 -END *) + \lambda (g: G).(\lambda (a: A).(\lambda (h1: nat).(let TMP_5 \def (\lambda +(n: nat).(\forall (h2: nat).(let TMP_1 \def (aplus g a n) in (let TMP_2 \def +(aplus g TMP_1 h2) in (let TMP_3 \def (plus n h2) in (let TMP_4 \def (aplus g +a TMP_3) in (eq A TMP_2 TMP_4))))))) in (let TMP_7 \def (\lambda (h2: +nat).(let TMP_6 \def (aplus g a h2) in (refl_equal A TMP_6))) in (let TMP_47 +\def (\lambda (n: nat).(\lambda (_: ((\forall (h2: nat).(eq A (aplus g (aplus +g a n) h2) (aplus g a (plus n h2)))))).(\lambda (h2: nat).(let TMP_14 \def +(\lambda (n0: nat).(let TMP_8 \def (aplus g a n) in (let TMP_9 \def (asucc g +TMP_8) in (let TMP_10 \def (aplus g TMP_9 n0) in (let TMP_11 \def (plus n n0) +in (let TMP_12 \def (aplus g a TMP_11) in (let TMP_13 \def (asucc g TMP_12) +in (eq A TMP_10 TMP_13)))))))) in (let TMP_19 \def (\lambda (n0: nat).(let +TMP_15 \def (aplus g a n) in (let TMP_16 \def (asucc g TMP_15) in (let TMP_17 +\def (aplus g a n0) in (let TMP_18 \def (asucc g TMP_17) in (eq A TMP_16 +TMP_18)))))) in (let TMP_20 \def (aplus g a n) in (let TMP_21 \def (asucc g +TMP_20) in (let TMP_22 \def (refl_equal A TMP_21) in (let TMP_23 \def (plus n +O) in (let TMP_24 \def (plus_n_O n) in (let TMP_25 \def (eq_ind nat n TMP_19 +TMP_22 TMP_23 TMP_24) in (let TMP_46 \def (\lambda (n0: nat).(\lambda (H0: +(eq A (aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n +n0))))).(let TMP_26 \def (plus n n0) in (let TMP_27 \def (S TMP_26) in (let +TMP_34 \def (\lambda (n1: nat).(let TMP_28 \def (aplus g a n) in (let TMP_29 +\def (asucc g TMP_28) in (let TMP_30 \def (aplus g TMP_29 n0) in (let TMP_31 +\def (asucc g TMP_30) in (let TMP_32 \def (aplus g a n1) in (let TMP_33 \def +(asucc g TMP_32) in (eq A TMP_31 TMP_33)))))))) in (let TMP_35 \def (aplus g +a n) in (let TMP_36 \def (asucc g TMP_35) in (let TMP_37 \def (aplus g TMP_36 +n0) in (let TMP_38 \def (plus n n0) in (let TMP_39 \def (aplus g a TMP_38) in +(let TMP_40 \def (asucc g TMP_39) in (let TMP_41 \def (refl_equal G g) in +(let TMP_42 \def (f_equal2 G A A asucc g g TMP_37 TMP_40 TMP_41 H0) in (let +TMP_43 \def (S n0) in (let TMP_44 \def (plus n TMP_43) in (let TMP_45 \def +(plus_n_Sm n n0) in (eq_ind nat TMP_27 TMP_34 TMP_42 TMP_44 +TMP_45))))))))))))))))) in (nat_ind TMP_14 TMP_25 TMP_46 h2))))))))))))) in +(nat_ind TMP_5 TMP_7 TMP_47 h1)))))). theorem aplus_asucc: \forall (g: G).(\forall (h: nat).(\forall (a: A).(eq A (aplus g (asucc g a) h) (asucc g (aplus g a h))))) \def - \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(eq_ind_r A (aplus g a -(plus (S O) h)) (\lambda (a0: A).(eq A a0 (asucc g (aplus g a h)))) -(refl_equal A (asucc g (aplus g a h))) (aplus g (aplus g a (S O)) h) -(aplus_assoc g a (S O) h)))). -(* COMMENTS -Initial nodes: 87 -END *) + \lambda (g: G).(\lambda (h: nat).(\lambda (a: A).(let TMP_1 \def (S O) in +(let TMP_2 \def (plus TMP_1 h) in (let TMP_3 \def (aplus g a TMP_2) in (let +TMP_6 \def (\lambda (a0: A).(let TMP_4 \def (aplus g a h) in (let TMP_5 \def +(asucc g TMP_4) in (eq A a0 TMP_5)))) in (let TMP_7 \def (aplus g a h) in +(let TMP_8 \def (asucc g TMP_7) in (let TMP_9 \def (refl_equal A TMP_8) in +(let TMP_10 \def (S O) in (let TMP_11 \def (aplus g a TMP_10) in (let TMP_12 +\def (aplus g TMP_11 h) in (let TMP_13 \def (S O) in (let TMP_14 \def +(aplus_assoc g a TMP_13 h) in (eq_ind_r A TMP_3 TMP_6 TMP_9 TMP_12 +TMP_14))))))))))))))). theorem aplus_sort_O_S_simpl: \forall (g: G).(\forall (n: nat).(\forall (k: nat).(eq A (aplus g (ASort O n) (S k)) (aplus g (ASort O (next g n)) k)))) \def - \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(eq_ind A (aplus g (asucc -g (ASort O n)) k) (\lambda (a: A).(eq A a (aplus g (ASort O (next g n)) k))) -(refl_equal A (aplus g (ASort O (next g n)) k)) (asucc g (aplus g (ASort O n) -k)) (aplus_asucc g k (ASort O n))))). -(* COMMENTS -Initial nodes: 97 -END *) + \lambda (g: G).(\lambda (n: nat).(\lambda (k: nat).(let TMP_1 \def (ASort O +n) in (let TMP_2 \def (asucc g TMP_1) in (let TMP_3 \def (aplus g TMP_2 k) in +(let TMP_7 \def (\lambda (a: A).(let TMP_4 \def (next g n) in (let TMP_5 \def +(ASort O TMP_4) in (let TMP_6 \def (aplus g TMP_5 k) in (eq A a TMP_6))))) in +(let TMP_8 \def (next g n) in (let TMP_9 \def (ASort O TMP_8) in (let TMP_10 +\def (aplus g TMP_9 k) in (let TMP_11 \def (refl_equal A TMP_10) in (let +TMP_12 \def (ASort O n) in (let TMP_13 \def (aplus g TMP_12 k) in (let TMP_14 +\def (asucc g TMP_13) in (let TMP_15 \def (ASort O n) in (let TMP_16 \def +(aplus_asucc g k TMP_15) in (eq_ind A TMP_3 TMP_7 TMP_11 TMP_14 +TMP_16)))))))))))))))). theorem aplus_sort_S_S_simpl: \forall (g: G).(\forall (n: nat).(\forall (h: nat).(\forall (k: nat).(eq A (aplus g (ASort (S h) n) (S k)) (aplus g (ASort h n) k))))) \def - \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(eq_ind -A (aplus g (asucc g (ASort (S h) n)) k) (\lambda (a: A).(eq A a (aplus g -(ASort h n) k))) (refl_equal A (aplus g (ASort h n) k)) (asucc g (aplus g -(ASort (S h) n) k)) (aplus_asucc g k (ASort (S h) n)))))). -(* COMMENTS -Initial nodes: 97 -END *) + \lambda (g: G).(\lambda (n: nat).(\lambda (h: nat).(\lambda (k: nat).(let +TMP_1 \def (S h) in (let TMP_2 \def (ASort TMP_1 n) in (let TMP_3 \def (asucc +g TMP_2) in (let TMP_4 \def (aplus g TMP_3 k) in (let TMP_7 \def (\lambda (a: +A).(let TMP_5 \def (ASort h n) in (let TMP_6 \def (aplus g TMP_5 k) in (eq A +a TMP_6)))) in (let TMP_8 \def (ASort h n) in (let TMP_9 \def (aplus g TMP_8 +k) in (let TMP_10 \def (refl_equal A TMP_9) in (let TMP_11 \def (S h) in (let +TMP_12 \def (ASort TMP_11 n) in (let TMP_13 \def (aplus g TMP_12 k) in (let +TMP_14 \def (asucc g TMP_13) in (let TMP_15 \def (S h) in (let TMP_16 \def +(ASort TMP_15 n) in (let TMP_17 \def (aplus_asucc g k TMP_16) in (eq_ind A +TMP_4 TMP_7 TMP_10 TMP_14 TMP_17))))))))))))))))))). theorem aplus_asort_O_simpl: \forall (g: G).(\forall (h: nat).(\forall (n: nat).(eq A (aplus g (ASort O n) h) (ASort O (next_plus g n h))))) \def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (n0: -nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 n))))) (\lambda -(n: nat).(refl_equal A (ASort O n))) (\lambda (n: nat).(\lambda (H: ((\forall -(n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O (next_plus g n0 -n)))))).(\lambda (n0: nat).(eq_ind A (aplus g (asucc g (ASort O n0)) n) -(\lambda (a: A).(eq A a (ASort O (next g (next_plus g n0 n))))) (eq_ind nat -(next_plus g (next g n0) n) (\lambda (n1: nat).(eq A (aplus g (ASort O (next -g n0)) n) (ASort O n1))) (H (next g n0)) (next g (next_plus g n0 n)) -(next_plus_next g n0 n)) (asucc g (aplus g (ASort O n0) n)) (aplus_asucc g n -(ASort O n0)))))) h)). -(* COMMENTS -Initial nodes: 229 -END *) + \lambda (g: G).(\lambda (h: nat).(let TMP_5 \def (\lambda (n: nat).(\forall +(n0: nat).(let TMP_1 \def (ASort O n0) in (let TMP_2 \def (aplus g TMP_1 n) +in (let TMP_3 \def (next_plus g n0 n) in (let TMP_4 \def (ASort O TMP_3) in +(eq A TMP_2 TMP_4))))))) in (let TMP_7 \def (\lambda (n: nat).(let TMP_6 \def +(ASort O n) in (refl_equal A TMP_6))) in (let TMP_33 \def (\lambda (n: +nat).(\lambda (H: ((\forall (n0: nat).(eq A (aplus g (ASort O n0) n) (ASort O +(next_plus g n0 n)))))).(\lambda (n0: nat).(let TMP_8 \def (ASort O n0) in +(let TMP_9 \def (asucc g TMP_8) in (let TMP_10 \def (aplus g TMP_9 n) in (let +TMP_14 \def (\lambda (a: A).(let TMP_11 \def (next_plus g n0 n) in (let +TMP_12 \def (next g TMP_11) in (let TMP_13 \def (ASort O TMP_12) in (eq A a +TMP_13))))) in (let TMP_15 \def (next g n0) in (let TMP_16 \def (next_plus g +TMP_15 n) in (let TMP_21 \def (\lambda (n1: nat).(let TMP_17 \def (next g n0) +in (let TMP_18 \def (ASort O TMP_17) in (let TMP_19 \def (aplus g TMP_18 n) +in (let TMP_20 \def (ASort O n1) in (eq A TMP_19 TMP_20)))))) in (let TMP_22 +\def (next g n0) in (let TMP_23 \def (H TMP_22) in (let TMP_24 \def +(next_plus g n0 n) in (let TMP_25 \def (next g TMP_24) in (let TMP_26 \def +(next_plus_next g n0 n) in (let TMP_27 \def (eq_ind nat TMP_16 TMP_21 TMP_23 +TMP_25 TMP_26) in (let TMP_28 \def (ASort O n0) in (let TMP_29 \def (aplus g +TMP_28 n) in (let TMP_30 \def (asucc g TMP_29) in (let TMP_31 \def (ASort O +n0) in (let TMP_32 \def (aplus_asucc g n TMP_31) in (eq_ind A TMP_10 TMP_14 +TMP_27 TMP_30 TMP_32)))))))))))))))))))))) in (nat_ind TMP_5 TMP_7 TMP_33 +h))))). theorem aplus_asort_le_simpl: \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).((le h k) \to (eq A (aplus g (ASort k n) h) (ASort (minus k h) n)))))) \def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (k: -nat).(\forall (n0: nat).((le n k) \to (eq A (aplus g (ASort k n0) n) (ASort -(minus k n) n0)))))) (\lambda (k: nat).(\lambda (n: nat).(\lambda (_: (le O -k)).(eq_ind nat k (\lambda (n0: nat).(eq A (ASort k n) (ASort n0 n))) -(refl_equal A (ASort k n)) (minus k O) (minus_n_O k))))) (\lambda (h0: -nat).(\lambda (H: ((\forall (k: nat).(\forall (n: nat).((le h0 k) \to (eq A -(aplus g (ASort k n) h0) (ASort (minus k h0) n))))))).(\lambda (k: -nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le (S h0) n) \to (eq A -(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0))))) (\lambda -(n: nat).(\lambda (H0: (le (S h0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat -O (S n0))) (\lambda (n0: nat).(le h0 n0)) (eq A (asucc g (aplus g (ASort O n) -h0)) (ASort (minus O (S h0)) n)) (\lambda (x: nat).(\lambda (H1: (eq nat O (S -x))).(\lambda (_: (le h0 x)).(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 x) H1) in (False_ind (eq A (asucc g (aplus -g (ASort O n) h0)) (ASort (minus O (S h0)) n)) H3))))) (le_gen_S h0 O H0)))) -(\lambda (n: nat).(\lambda (_: ((\forall (n0: nat).((le (S h0) n) \to (eq A -(asucc g (aplus g (ASort n n0) h0)) (ASort (minus n (S h0)) n0)))))).(\lambda -(n0: nat).(\lambda (H1: (le (S h0) (S n))).(eq_ind A (aplus g (asucc g (ASort -(S n) n0)) h0) (\lambda (a: A).(eq A a (ASort (minus (S n) (S h0)) n0))) (H n -n0 (le_S_n h0 n H1)) (asucc g (aplus g (ASort (S n) n0) h0)) (aplus_asucc g -h0 (ASort (S n) n0))))))) k)))) h)). -(* COMMENTS -Initial nodes: 484 -END *) + \lambda (g: G).(\lambda (h: nat).(let TMP_5 \def (\lambda (n: nat).(\forall +(k: nat).(\forall (n0: nat).((le n k) \to (let TMP_1 \def (ASort k n0) in +(let TMP_2 \def (aplus g TMP_1 n) in (let TMP_3 \def (minus k n) in (let +TMP_4 \def (ASort TMP_3 n0) in (eq A TMP_2 TMP_4))))))))) in (let TMP_13 \def +(\lambda (k: nat).(\lambda (n: nat).(\lambda (_: (le O k)).(let TMP_8 \def +(\lambda (n0: nat).(let TMP_6 \def (ASort k n) in (let TMP_7 \def (ASort n0 +n) in (eq A TMP_6 TMP_7)))) in (let TMP_9 \def (ASort k n) in (let TMP_10 +\def (refl_equal A TMP_9) in (let TMP_11 \def (minus k O) in (let TMP_12 \def +(minus_n_O k) in (eq_ind nat k TMP_8 TMP_10 TMP_11 TMP_12))))))))) in (let +TMP_62 \def (\lambda (h0: nat).(\lambda (H: ((\forall (k: nat).(\forall (n: +nat).((le h0 k) \to (eq A (aplus g (ASort k n) h0) (ASort (minus k h0) +n))))))).(\lambda (k: nat).(let TMP_20 \def (\lambda (n: nat).(\forall (n0: +nat).((le (S h0) n) \to (let TMP_14 \def (ASort n n0) in (let TMP_15 \def +(aplus g TMP_14 h0) in (let TMP_16 \def (asucc g TMP_15) in (let TMP_17 \def +(S h0) in (let TMP_18 \def (minus n TMP_17) in (let TMP_19 \def (ASort TMP_18 +n0) in (eq A TMP_16 TMP_19)))))))))) in (let TMP_42 \def (\lambda (n: +nat).(\lambda (H0: (le (S h0) O)).(let TMP_22 \def (\lambda (n0: nat).(let +TMP_21 \def (S n0) in (eq nat O TMP_21))) in (let TMP_23 \def (\lambda (n0: +nat).(le h0 n0)) in (let TMP_24 \def (ASort O n) in (let TMP_25 \def (aplus g +TMP_24 h0) in (let TMP_26 \def (asucc g TMP_25) in (let TMP_27 \def (S h0) in +(let TMP_28 \def (minus O TMP_27) in (let TMP_29 \def (ASort TMP_28 n) in +(let TMP_30 \def (eq A TMP_26 TMP_29) in (let TMP_40 \def (\lambda (x: +nat).(\lambda (H1: (eq nat O (S x))).(\lambda (_: (le h0 x)).(let TMP_31 \def +(\lambda (ee: nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow +False])) in (let TMP_32 \def (S x) in (let H3 \def (eq_ind nat O TMP_31 I +TMP_32 H1) in (let TMP_33 \def (ASort O n) in (let TMP_34 \def (aplus g +TMP_33 h0) in (let TMP_35 \def (asucc g TMP_34) in (let TMP_36 \def (S h0) in +(let TMP_37 \def (minus O TMP_36) in (let TMP_38 \def (ASort TMP_37 n) in +(let TMP_39 \def (eq A TMP_35 TMP_38) in (False_ind TMP_39 H3)))))))))))))) +in (let TMP_41 \def (le_gen_S h0 O H0) in (ex2_ind nat TMP_22 TMP_23 TMP_30 +TMP_40 TMP_41)))))))))))))) in (let TMP_61 \def (\lambda (n: nat).(\lambda +(_: ((\forall (n0: nat).((le (S h0) n) \to (eq A (asucc g (aplus g (ASort n +n0) h0)) (ASort (minus n (S h0)) n0)))))).(\lambda (n0: nat).(\lambda (H1: +(le (S h0) (S n))).(let TMP_43 \def (S n) in (let TMP_44 \def (ASort TMP_43 +n0) in (let TMP_45 \def (asucc g TMP_44) in (let TMP_46 \def (aplus g TMP_45 +h0) in (let TMP_51 \def (\lambda (a: A).(let TMP_47 \def (S n) in (let TMP_48 +\def (S h0) in (let TMP_49 \def (minus TMP_47 TMP_48) in (let TMP_50 \def +(ASort TMP_49 n0) in (eq A a TMP_50)))))) in (let TMP_52 \def (le_S_n h0 n +H1) in (let TMP_53 \def (H n n0 TMP_52) in (let TMP_54 \def (S n) in (let +TMP_55 \def (ASort TMP_54 n0) in (let TMP_56 \def (aplus g TMP_55 h0) in (let +TMP_57 \def (asucc g TMP_56) in (let TMP_58 \def (S n) in (let TMP_59 \def +(ASort TMP_58 n0) in (let TMP_60 \def (aplus_asucc g h0 TMP_59) in (eq_ind A +TMP_46 TMP_51 TMP_53 TMP_57 TMP_60))))))))))))))))))) in (nat_ind TMP_20 +TMP_42 TMP_61 k))))))) in (nat_ind TMP_5 TMP_13 TMP_62 h))))). theorem aplus_asort_simpl: \forall (g: G).(\forall (h: nat).(\forall (k: nat).(\forall (n: nat).(eq A (aplus g (ASort k n) h) (ASort (minus k h) (next_plus g n (minus h k))))))) \def - \lambda (g: G).(\lambda (h: nat).(\lambda (k: nat).(\lambda (n: -nat).(lt_le_e k h (eq A (aplus g (ASort k n) h) (ASort (minus k h) (next_plus -g n (minus h k)))) (\lambda (H: (lt k h)).(eq_ind_r nat (plus k (minus h k)) -(\lambda (n0: nat).(eq A (aplus g (ASort k n) n0) (ASort (minus k h) -(next_plus g n (minus h k))))) (eq_ind A (aplus g (aplus g (ASort k n) k) -(minus h k)) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n (minus -h k))))) (eq_ind_r A (ASort (minus k k) n) (\lambda (a: A).(eq A (aplus g a -(minus h k)) (ASort (minus k h) (next_plus g n (minus h k))))) (eq_ind nat O -(\lambda (n0: nat).(eq A (aplus g (ASort n0 n) (minus h k)) (ASort (minus k -h) (next_plus g n (minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A -(aplus g (ASort O n) (minus h k)) (ASort n0 (next_plus g n (minus h k))))) -(aplus_asort_O_simpl g (minus h k) n) (minus k h) (O_minus k h (le_S_n k h -(le_S (S k) h H)))) (minus k k) (minus_n_n k)) (aplus g (ASort k n) k) -(aplus_asort_le_simpl g k k n (le_n k))) (aplus g (ASort k n) (plus k (minus -h k))) (aplus_assoc g (ASort k n) k (minus h k))) h (le_plus_minus k h -(le_S_n k h (le_S (S k) h H))))) (\lambda (H: (le h k)).(eq_ind_r A (ASort -(minus k h) n) (\lambda (a: A).(eq A a (ASort (minus k h) (next_plus g n -(minus h k))))) (eq_ind_r nat O (\lambda (n0: nat).(eq A (ASort (minus k h) -n) (ASort (minus k h) (next_plus g n n0)))) (refl_equal A (ASort (minus k h) -(next_plus g n O))) (minus h k) (O_minus h k H)) (aplus g (ASort k n) h) -(aplus_asort_le_simpl g h k n H))))))). -(* COMMENTS -Initial nodes: 587 -END *) + \lambda (g: G).(\lambda (h: nat).(\lambda (k: nat).(\lambda (n: nat).(let +TMP_1 \def (ASort k n) in (let TMP_2 \def (aplus g TMP_1 h) in (let TMP_3 +\def (minus k h) in (let TMP_4 \def (minus h k) in (let TMP_5 \def (next_plus +g n TMP_4) in (let TMP_6 \def (ASort TMP_3 TMP_5) in (let TMP_7 \def (eq A +TMP_2 TMP_6) in (let TMP_92 \def (\lambda (H: (lt k h)).(let TMP_8 \def +(minus h k) in (let TMP_9 \def (plus k TMP_8) in (let TMP_16 \def (\lambda +(n0: nat).(let TMP_10 \def (ASort k n) in (let TMP_11 \def (aplus g TMP_10 +n0) in (let TMP_12 \def (minus k h) in (let TMP_13 \def (minus h k) in (let +TMP_14 \def (next_plus g n TMP_13) in (let TMP_15 \def (ASort TMP_12 TMP_14) +in (eq A TMP_11 TMP_15)))))))) in (let TMP_17 \def (ASort k n) in (let TMP_18 +\def (aplus g TMP_17 k) in (let TMP_19 \def (minus h k) in (let TMP_20 \def +(aplus g TMP_18 TMP_19) in (let TMP_25 \def (\lambda (a: A).(let TMP_21 \def +(minus k h) in (let TMP_22 \def (minus h k) in (let TMP_23 \def (next_plus g +n TMP_22) in (let TMP_24 \def (ASort TMP_21 TMP_23) in (eq A a TMP_24)))))) +in (let TMP_26 \def (minus k k) in (let TMP_27 \def (ASort TMP_26 n) in (let +TMP_34 \def (\lambda (a: A).(let TMP_28 \def (minus h k) in (let TMP_29 \def +(aplus g a TMP_28) in (let TMP_30 \def (minus k h) in (let TMP_31 \def (minus +h k) in (let TMP_32 \def (next_plus g n TMP_31) in (let TMP_33 \def (ASort +TMP_30 TMP_32) in (eq A TMP_29 TMP_33)))))))) in (let TMP_42 \def (\lambda +(n0: nat).(let TMP_35 \def (ASort n0 n) in (let TMP_36 \def (minus h k) in +(let TMP_37 \def (aplus g TMP_35 TMP_36) in (let TMP_38 \def (minus k h) in +(let TMP_39 \def (minus h k) in (let TMP_40 \def (next_plus g n TMP_39) in +(let TMP_41 \def (ASort TMP_38 TMP_40) in (eq A TMP_37 TMP_41))))))))) in +(let TMP_49 \def (\lambda (n0: nat).(let TMP_43 \def (ASort O n) in (let +TMP_44 \def (minus h k) in (let TMP_45 \def (aplus g TMP_43 TMP_44) in (let +TMP_46 \def (minus h k) in (let TMP_47 \def (next_plus g n TMP_46) in (let +TMP_48 \def (ASort n0 TMP_47) in (eq A TMP_45 TMP_48)))))))) in (let TMP_50 +\def (minus h k) in (let TMP_51 \def (aplus_asort_O_simpl g TMP_50 n) in (let +TMP_52 \def (minus k h) in (let TMP_53 \def (S k) in (let TMP_54 \def (S h) +in (let TMP_55 \def (S k) in (let TMP_56 \def (S TMP_55) in (let TMP_57 \def +(S h) in (let TMP_58 \def (S k) in (let TMP_59 \def (le_n_S TMP_58 h H) in +(let TMP_60 \def (le_S TMP_56 TMP_57 TMP_59) in (let TMP_61 \def (le_S_n +TMP_53 TMP_54 TMP_60) in (let TMP_62 \def (le_S_n k h TMP_61) in (let TMP_63 +\def (O_minus k h TMP_62) in (let TMP_64 \def (eq_ind_r nat O TMP_49 TMP_51 +TMP_52 TMP_63) in (let TMP_65 \def (minus k k) in (let TMP_66 \def (minus_n_n +k) in (let TMP_67 \def (eq_ind nat O TMP_42 TMP_64 TMP_65 TMP_66) in (let +TMP_68 \def (ASort k n) in (let TMP_69 \def (aplus g TMP_68 k) in (let TMP_70 +\def (le_n k) in (let TMP_71 \def (aplus_asort_le_simpl g k k n TMP_70) in +(let TMP_72 \def (eq_ind_r A TMP_27 TMP_34 TMP_67 TMP_69 TMP_71) in (let +TMP_73 \def (ASort k n) in (let TMP_74 \def (minus h k) in (let TMP_75 \def +(plus k TMP_74) in (let TMP_76 \def (aplus g TMP_73 TMP_75) in (let TMP_77 +\def (ASort k n) in (let TMP_78 \def (minus h k) in (let TMP_79 \def +(aplus_assoc g TMP_77 k TMP_78) in (let TMP_80 \def (eq_ind A TMP_20 TMP_25 +TMP_72 TMP_76 TMP_79) in (let TMP_81 \def (S k) in (let TMP_82 \def (S h) in +(let TMP_83 \def (S k) in (let TMP_84 \def (S TMP_83) in (let TMP_85 \def (S +h) in (let TMP_86 \def (S k) in (let TMP_87 \def (le_n_S TMP_86 h H) in (let +TMP_88 \def (le_S TMP_84 TMP_85 TMP_87) in (let TMP_89 \def (le_S_n TMP_81 +TMP_82 TMP_88) in (let TMP_90 \def (le_S_n k h TMP_89) in (let TMP_91 \def +(le_plus_minus k h TMP_90) in (eq_ind_r nat TMP_9 TMP_16 TMP_80 h +TMP_91))))))))))))))))))))))))))))))))))))))))))))))))))))))))) in (let +TMP_116 \def (\lambda (H: (le h k)).(let TMP_93 \def (minus k h) in (let +TMP_94 \def (ASort TMP_93 n) in (let TMP_99 \def (\lambda (a: A).(let TMP_95 +\def (minus k h) in (let TMP_96 \def (minus h k) in (let TMP_97 \def +(next_plus g n TMP_96) in (let TMP_98 \def (ASort TMP_95 TMP_97) in (eq A a +TMP_98)))))) in (let TMP_105 \def (\lambda (n0: nat).(let TMP_100 \def (minus +k h) in (let TMP_101 \def (ASort TMP_100 n) in (let TMP_102 \def (minus k h) +in (let TMP_103 \def (next_plus g n n0) in (let TMP_104 \def (ASort TMP_102 +TMP_103) in (eq A TMP_101 TMP_104))))))) in (let TMP_106 \def (minus k h) in +(let TMP_107 \def (next_plus g n O) in (let TMP_108 \def (ASort TMP_106 +TMP_107) in (let TMP_109 \def (refl_equal A TMP_108) in (let TMP_110 \def +(minus h k) in (let TMP_111 \def (O_minus h k H) in (let TMP_112 \def +(eq_ind_r nat O TMP_105 TMP_109 TMP_110 TMP_111) in (let TMP_113 \def (ASort +k n) in (let TMP_114 \def (aplus g TMP_113 h) in (let TMP_115 \def +(aplus_asort_le_simpl g h k n H) in (eq_ind_r A TMP_94 TMP_99 TMP_112 TMP_114 +TMP_115)))))))))))))))) in (lt_le_e k h TMP_7 TMP_92 TMP_116))))))))))))). theorem aplus_ahead_simpl: \forall (g: G).(\forall (h: nat).(\forall (a1: A).(\forall (a2: A).(eq A (aplus g (AHead a1 a2) h) (AHead a1 (aplus g a2 h)))))) \def - \lambda (g: G).(\lambda (h: nat).(nat_ind (\lambda (n: nat).(\forall (a1: -A).(\forall (a2: A).(eq A (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 -n)))))) (\lambda (a1: A).(\lambda (a2: A).(refl_equal A (AHead a1 a2)))) -(\lambda (n: nat).(\lambda (H: ((\forall (a1: A).(\forall (a2: A).(eq A -(aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 n))))))).(\lambda (a1: -A).(\lambda (a2: A).(eq_ind A (aplus g (asucc g (AHead a1 a2)) n) (\lambda -(a: A).(eq A a (AHead a1 (asucc g (aplus g a2 n))))) (eq_ind A (aplus g -(asucc g a2) n) (\lambda (a: A).(eq A (aplus g (asucc g (AHead a1 a2)) n) -(AHead a1 a))) (H a1 (asucc g a2)) (asucc g (aplus g a2 n)) (aplus_asucc g n -a2)) (asucc g (aplus g (AHead a1 a2) n)) (aplus_asucc g n (AHead a1 a2))))))) -h)). -(* COMMENTS -Initial nodes: 239 -END *) + \lambda (g: G).(\lambda (h: nat).(let TMP_5 \def (\lambda (n: nat).(\forall +(a1: A).(\forall (a2: A).(let TMP_1 \def (AHead a1 a2) in (let TMP_2 \def +(aplus g TMP_1 n) in (let TMP_3 \def (aplus g a2 n) in (let TMP_4 \def (AHead +a1 TMP_3) in (eq A TMP_2 TMP_4)))))))) in (let TMP_7 \def (\lambda (a1: +A).(\lambda (a2: A).(let TMP_6 \def (AHead a1 a2) in (refl_equal A TMP_6)))) +in (let TMP_33 \def (\lambda (n: nat).(\lambda (H: ((\forall (a1: A).(\forall +(a2: A).(eq A (aplus g (AHead a1 a2) n) (AHead a1 (aplus g a2 +n))))))).(\lambda (a1: A).(\lambda (a2: A).(let TMP_8 \def (AHead a1 a2) in +(let TMP_9 \def (asucc g TMP_8) in (let TMP_10 \def (aplus g TMP_9 n) in (let +TMP_14 \def (\lambda (a: A).(let TMP_11 \def (aplus g a2 n) in (let TMP_12 +\def (asucc g TMP_11) in (let TMP_13 \def (AHead a1 TMP_12) in (eq A a +TMP_13))))) in (let TMP_15 \def (asucc g a2) in (let TMP_16 \def (aplus g +TMP_15 n) in (let TMP_21 \def (\lambda (a: A).(let TMP_17 \def (AHead a1 a2) +in (let TMP_18 \def (asucc g TMP_17) in (let TMP_19 \def (aplus g TMP_18 n) +in (let TMP_20 \def (AHead a1 a) in (eq A TMP_19 TMP_20)))))) in (let TMP_22 +\def (asucc g a2) in (let TMP_23 \def (H a1 TMP_22) in (let TMP_24 \def +(aplus g a2 n) in (let TMP_25 \def (asucc g TMP_24) in (let TMP_26 \def +(aplus_asucc g n a2) in (let TMP_27 \def (eq_ind A TMP_16 TMP_21 TMP_23 +TMP_25 TMP_26) in (let TMP_28 \def (AHead a1 a2) in (let TMP_29 \def (aplus g +TMP_28 n) in (let TMP_30 \def (asucc g TMP_29) in (let TMP_31 \def (AHead a1 +a2) in (let TMP_32 \def (aplus_asucc g n TMP_31) in (eq_ind A TMP_10 TMP_14 +TMP_27 TMP_30 TMP_32))))))))))))))))))))))) in (nat_ind TMP_5 TMP_7 TMP_33 +h))))). theorem aplus_asucc_false: \forall (g: G).(\forall (a: A).(\forall (h: nat).((eq A (aplus g (asucc g a) h) a) \to (\forall (P: Prop).P)))) \def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(\forall (h: -nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P)))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda (H: (eq A -(aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h0) -\Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).(nat_ind -(\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow (ASort O -(next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to P)) -(\lambda (H0: (eq A (aplus g (ASort O (next g n0)) h) (ASort O n0))).(let H1 -\def (eq_ind A (aplus g (ASort O (next g n0)) h) (\lambda (a0: A).(eq A a0 -(ASort O n0))) H0 (ASort (minus O h) (next_plus g (next g n0) (minus h O))) -(aplus_asort_simpl g h O (next g n0))) in (let H2 \def (f_equal A nat -(\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ n1) -\Rightarrow n1 | (AHead _ _) \Rightarrow ((let rec next_plus (g0: G) (n1: -nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n1 | (S i0) -\Rightarrow (next g0 (next_plus g0 n1 i0))]) in next_plus) g (next g n0) -(minus h O))])) (ASort (minus O h) (next_plus g (next g n0) (minus h O))) -(ASort O n0) H1) in (let H3 \def (eq_ind_r nat (minus h O) (\lambda (n1: -nat).(eq nat (next_plus g (next g n0) n1) n0)) H2 h (minus_n_O h)) in -(le_lt_false (next_plus g (next g n0) h) n0 (eq_ind nat (next_plus g (next g -n0) h) (\lambda (n1: nat).(le (next_plus g (next g n0) h) n1)) (le_n -(next_plus g (next g n0) h)) n0 H3) (next_plus_lt g h n0) P))))) (\lambda -(n1: nat).(\lambda (_: (((eq A (aplus g (match n1 with [O \Rightarrow (ASort -O (next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to + \lambda (g: G).(\lambda (a: A).(let TMP_1 \def (\lambda (a0: A).(\forall (h: +nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: Prop).P)))) in (let +TMP_70 \def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (h: nat).(\lambda +(H: (eq A (aplus g (match n with [O \Rightarrow (ASort O (next g n0)) | (S +h0) \Rightarrow (ASort h0 n0)]) h) (ASort n n0))).(\lambda (P: Prop).(let +TMP_2 \def (\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow +(ASort O (next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) +\to P)) in (let TMP_36 \def (\lambda (H0: (eq A (aplus g (ASort O (next g +n0)) h) (ASort O n0))).(let TMP_3 \def (next g n0) in (let TMP_4 \def (ASort +O TMP_3) in (let TMP_5 \def (aplus g TMP_4 h) in (let TMP_7 \def (\lambda +(a0: A).(let TMP_6 \def (ASort O n0) in (eq A a0 TMP_6))) in (let TMP_8 \def +(minus O h) in (let TMP_9 \def (next g n0) in (let TMP_10 \def (minus h O) in +(let TMP_11 \def (next_plus g TMP_9 TMP_10) in (let TMP_12 \def (ASort TMP_8 +TMP_11) in (let TMP_13 \def (next g n0) in (let TMP_14 \def +(aplus_asort_simpl g h O TMP_13) in (let H1 \def (eq_ind A TMP_5 TMP_7 H0 +TMP_12 TMP_14) in (let TMP_18 \def (\lambda (e: A).(match e with [(ASort _ +n1) \Rightarrow n1 | (AHead _ _) \Rightarrow (let TMP_16 \def (next g n0) in +(let TMP_17 \def (minus h O) in (next_plus g TMP_16 TMP_17)))])) in (let +TMP_19 \def (minus O h) in (let TMP_20 \def (next g n0) in (let TMP_21 \def +(minus h O) in (let TMP_22 \def (next_plus g TMP_20 TMP_21) in (let TMP_23 +\def (ASort TMP_19 TMP_22) in (let TMP_24 \def (ASort O n0) in (let H2 \def +(f_equal A nat TMP_18 TMP_23 TMP_24 H1) in (let TMP_25 \def (minus h O) in +(let TMP_28 \def (\lambda (n1: nat).(let TMP_26 \def (next g n0) in (let +TMP_27 \def (next_plus g TMP_26 n1) in (eq nat TMP_27 n0)))) in (let TMP_29 +\def (minus_n_O h) in (let H3 \def (eq_ind_r nat TMP_25 TMP_28 H2 h TMP_29) +in (let TMP_30 \def (le_n n0) in (let TMP_31 \def (next g n0) in (let TMP_32 +\def (next_plus g TMP_31 h) in (let TMP_33 \def (\lambda (n1: nat).(lt n0 +n1)) in (let TMP_34 \def (next_plus_lt g h n0) in (let TMP_35 \def (eq_ind +nat TMP_32 TMP_33 TMP_34 n0 H3) in (le_lt_false n0 n0 TMP_30 TMP_35 +P)))))))))))))))))))))))))))))))) in (let TMP_69 \def (\lambda (n1: +nat).(\lambda (_: (((eq A (aplus g (match n1 with [O \Rightarrow (ASort O +(next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to P))).(\lambda (H0: (eq A (aplus g (ASort n1 n0) h) (ASort (S n1) n0))).(let -H1 \def (eq_ind A (aplus g (ASort n1 n0) h) (\lambda (a0: A).(eq A a0 (ASort -(S n1) n0))) H0 (ASort (minus n1 h) (next_plus g n0 (minus h n1))) -(aplus_asort_simpl g h n1 n0)) in (let H2 \def (f_equal A nat (\lambda (e: -A).(match e in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow -n2 | (AHead _ _) \Rightarrow ((let rec minus (n2: nat) on n2: (nat \to nat) -\def (\lambda (m: nat).(match n2 with [O \Rightarrow O | (S k) \Rightarrow -(match m with [O \Rightarrow (S k) | (S l) \Rightarrow (minus k l)])])) in -minus) n1 h)])) (ASort (minus n1 h) (next_plus g n0 (minus h n1))) (ASort (S -n1) n0) H1) in ((let H3 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort _ n2) \Rightarrow n2 | (AHead _ _) -\Rightarrow ((let rec next_plus (g0: G) (n2: nat) (i: nat) on i: nat \def -(match i with [O \Rightarrow n2 | (S i0) \Rightarrow (next g0 (next_plus g0 -n2 i0))]) in next_plus) g n0 (minus h n1))])) (ASort (minus n1 h) (next_plus -g n0 (minus h n1))) (ASort (S n1) n0) H1) in (\lambda (H4: (eq nat (minus n1 -h) (S n1))).(le_Sx_x n1 (eq_ind nat (minus n1 h) (\lambda (n2: nat).(le n2 -n1)) (minus_le n1 h) (S n1) H4) P))) H2)))))) n H)))))) (\lambda (a0: -A).(\lambda (_: ((\forall (h: nat).((eq A (aplus g (asucc g a0) h) a0) \to -(\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: -nat).((eq A (aplus g (asucc g a1) h) a1) \to (\forall (P: -Prop).P))))).(\lambda (h: nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc -g a1)) h) (AHead a0 a1))).(\lambda (P: Prop).(let H2 \def (eq_ind A (aplus g -(AHead a0 (asucc g a1)) h) (\lambda (a2: A).(eq A a2 (AHead a0 a1))) H1 -(AHead a0 (aplus g (asucc g a1) h)) (aplus_ahead_simpl g h a0 (asucc g a1))) -in (let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda -(_: A).A) with [(ASort _ _) \Rightarrow ((let rec aplus (g0: G) (a2: A) (n: -nat) on n: A \def (match n with [O \Rightarrow a2 | (S n0) \Rightarrow (asucc -g0 (aplus g0 a2 n0))]) in aplus) g (asucc g a1) h) | (AHead _ a2) \Rightarrow -a2])) (AHead a0 (aplus g (asucc g a1) h)) (AHead a0 a1) H2) in (H0 h H3 -P)))))))))) a)). -(* COMMENTS -Initial nodes: 977 -END *) +TMP_37 \def (ASort n1 n0) in (let TMP_38 \def (aplus g TMP_37 h) in (let +TMP_41 \def (\lambda (a0: A).(let TMP_39 \def (S n1) in (let TMP_40 \def +(ASort TMP_39 n0) in (eq A a0 TMP_40)))) in (let TMP_42 \def (minus n1 h) in +(let TMP_43 \def (minus h n1) in (let TMP_44 \def (next_plus g n0 TMP_43) in +(let TMP_45 \def (ASort TMP_42 TMP_44) in (let TMP_46 \def (aplus_asort_simpl +g h n1 n0) in (let H1 \def (eq_ind A TMP_38 TMP_41 H0 TMP_45 TMP_46) in (let +TMP_47 \def (\lambda (e: A).(match e with [(ASort n2 _) \Rightarrow n2 | +(AHead _ _) \Rightarrow (minus n1 h)])) in (let TMP_48 \def (minus n1 h) in +(let TMP_49 \def (minus h n1) in (let TMP_50 \def (next_plus g n0 TMP_49) in +(let TMP_51 \def (ASort TMP_48 TMP_50) in (let TMP_52 \def (S n1) in (let +TMP_53 \def (ASort TMP_52 n0) in (let H2 \def (f_equal A nat TMP_47 TMP_51 +TMP_53 H1) in (let TMP_56 \def (\lambda (e: A).(match e with [(ASort _ n2) +\Rightarrow n2 | (AHead _ _) \Rightarrow (let TMP_55 \def (minus h n1) in +(next_plus g n0 TMP_55))])) in (let TMP_57 \def (minus n1 h) in (let TMP_58 +\def (minus h n1) in (let TMP_59 \def (next_plus g n0 TMP_58) in (let TMP_60 +\def (ASort TMP_57 TMP_59) in (let TMP_61 \def (S n1) in (let TMP_62 \def +(ASort TMP_61 n0) in (let H3 \def (f_equal A nat TMP_56 TMP_60 TMP_62 H1) in +(let TMP_68 \def (\lambda (H4: (eq nat (minus n1 h) (S n1))).(let TMP_63 \def +(minus n1 h) in (let TMP_64 \def (\lambda (n2: nat).(le n2 n1)) in (let +TMP_65 \def (minus_le n1 h) in (let TMP_66 \def (S n1) in (let TMP_67 \def +(eq_ind nat TMP_63 TMP_64 TMP_65 TMP_66 H4) in (le_Sx_x n1 TMP_67 P))))))) in +(TMP_68 H2)))))))))))))))))))))))))))))) in (nat_ind TMP_2 TMP_36 TMP_69 n +H))))))))) in (let TMP_88 \def (\lambda (a0: A).(\lambda (_: ((\forall (h: +nat).((eq A (aplus g (asucc g a0) h) a0) \to (\forall (P: +Prop).P))))).(\lambda (a1: A).(\lambda (H0: ((\forall (h: nat).((eq A (aplus +g (asucc g a1) h) a1) \to (\forall (P: Prop).P))))).(\lambda (h: +nat).(\lambda (H1: (eq A (aplus g (AHead a0 (asucc g a1)) h) (AHead a0 +a1))).(\lambda (P: Prop).(let TMP_71 \def (asucc g a1) in (let TMP_72 \def +(AHead a0 TMP_71) in (let TMP_73 \def (aplus g TMP_72 h) in (let TMP_75 \def +(\lambda (a2: A).(let TMP_74 \def (AHead a0 a1) in (eq A a2 TMP_74))) in (let +TMP_76 \def (asucc g a1) in (let TMP_77 \def (aplus g TMP_76 h) in (let +TMP_78 \def (AHead a0 TMP_77) in (let TMP_79 \def (asucc g a1) in (let TMP_80 +\def (aplus_ahead_simpl g h a0 TMP_79) in (let H2 \def (eq_ind A TMP_73 +TMP_75 H1 TMP_78 TMP_80) in (let TMP_83 \def (\lambda (e: A).(match e with +[(ASort _ _) \Rightarrow (let TMP_82 \def (asucc g a1) in (aplus g TMP_82 h)) +| (AHead _ a2) \Rightarrow a2])) in (let TMP_84 \def (asucc g a1) in (let +TMP_85 \def (aplus g TMP_84 h) in (let TMP_86 \def (AHead a0 TMP_85) in (let +TMP_87 \def (AHead a0 a1) in (let H3 \def (f_equal A A TMP_83 TMP_86 TMP_87 +H2) in (H0 h H3 P)))))))))))))))))))))))) in (A_ind TMP_1 TMP_70 TMP_88 +a))))). theorem aplus_inj: \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A (aplus g a h1) (aplus g a h2)) \to (eq nat h1 h2))))) \def - \lambda (g: G).(\lambda (h1: nat).(nat_ind (\lambda (n: nat).(\forall (h2: -nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n -h2))))) (\lambda (h2: nat).(nat_ind (\lambda (n: nat).(\forall (a: A).((eq A -(aplus g a O) (aplus g a n)) \to (eq nat O n)))) (\lambda (a: A).(\lambda (_: -(eq A a a)).(refl_equal nat O))) (\lambda (n: nat).(\lambda (_: ((\forall (a: -A).((eq A a (aplus g a n)) \to (eq nat O n))))).(\lambda (a: A).(\lambda (H0: -(eq A a (asucc g (aplus g a n)))).(let H1 \def (eq_ind_r A (asucc g (aplus g -a n)) (\lambda (a0: A).(eq A a a0)) H0 (aplus g (asucc g a) n) (aplus_asucc g -n a)) in (aplus_asucc_false g a n (sym_eq A a (aplus g (asucc g a) n) H1) (eq -nat O (S n)))))))) h2)) (\lambda (n: nat).(\lambda (H: ((\forall (h2: -nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n -h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(\forall (a: A).((eq -A (aplus g a (S n)) (aplus g a n0)) \to (eq nat (S n) n0)))) (\lambda (a: -A).(\lambda (H0: (eq A (asucc g (aplus g a n)) a)).(let H1 \def (eq_ind_r A -(asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 a)) H0 (aplus g (asucc g a) -n) (aplus_asucc g n a)) in (aplus_asucc_false g a n H1 (eq nat (S n) O))))) -(\lambda (n0: nat).(\lambda (_: ((\forall (a: A).((eq A (asucc g (aplus g a -n)) (aplus g a n0)) \to (eq nat (S n) n0))))).(\lambda (a: A).(\lambda (H1: -(eq A (asucc g (aplus g a n)) (asucc g (aplus g a n0)))).(let H2 \def -(eq_ind_r A (asucc g (aplus g a n)) (\lambda (a0: A).(eq A a0 (asucc g (aplus -g a n0)))) H1 (aplus g (asucc g a) n) (aplus_asucc g n a)) in (let H3 \def -(eq_ind_r A (asucc g (aplus g a n0)) (\lambda (a0: A).(eq A (aplus g (asucc g -a) n) a0)) H2 (aplus g (asucc g a) n0) (aplus_asucc g n0 a)) in (f_equal nat -nat S n n0 (H n0 (asucc g a) H3)))))))) h2)))) h1)). -(* COMMENTS -Initial nodes: 599 -END *) + \lambda (g: G).(\lambda (h1: nat).(let TMP_1 \def (\lambda (n: nat).(\forall +(h2: nat).(\forall (a: A).((eq A (aplus g a n) (aplus g a h2)) \to (eq nat n +h2))))) in (let TMP_16 \def (\lambda (h2: nat).(let TMP_2 \def (\lambda (n: +nat).(\forall (a: A).((eq A (aplus g a O) (aplus g a n)) \to (eq nat O n)))) +in (let TMP_3 \def (\lambda (a: A).(\lambda (_: (eq A a a)).(refl_equal nat +O))) in (let TMP_15 \def (\lambda (n: nat).(\lambda (_: ((\forall (a: A).((eq +A a (aplus g a n)) \to (eq nat O n))))).(\lambda (a: A).(\lambda (H0: (eq A a +(asucc g (aplus g a n)))).(let TMP_4 \def (aplus g a n) in (let TMP_5 \def +(asucc g TMP_4) in (let TMP_6 \def (\lambda (a0: A).(eq A a a0)) in (let +TMP_7 \def (asucc g a) in (let TMP_8 \def (aplus g TMP_7 n) in (let TMP_9 +\def (aplus_asucc g n a) in (let H1 \def (eq_ind_r A TMP_5 TMP_6 H0 TMP_8 +TMP_9) in (let TMP_10 \def (asucc g a) in (let TMP_11 \def (aplus g TMP_10 n) +in (let TMP_12 \def (sym_eq A a TMP_11 H1) in (let TMP_13 \def (S n) in (let +TMP_14 \def (eq nat O TMP_13) in (aplus_asucc_false g a n TMP_12 +TMP_14))))))))))))))))) in (nat_ind TMP_2 TMP_3 TMP_15 h2))))) in (let TMP_47 +\def (\lambda (n: nat).(\lambda (H: ((\forall (h2: nat).(\forall (a: A).((eq +A (aplus g a n) (aplus g a h2)) \to (eq nat n h2)))))).(\lambda (h2: +nat).(let TMP_18 \def (\lambda (n0: nat).(\forall (a: A).((eq A (aplus g a (S +n)) (aplus g a n0)) \to (let TMP_17 \def (S n) in (eq nat TMP_17 n0))))) in +(let TMP_27 \def (\lambda (a: A).(\lambda (H0: (eq A (asucc g (aplus g a n)) +a)).(let TMP_19 \def (aplus g a n) in (let TMP_20 \def (asucc g TMP_19) in +(let TMP_21 \def (\lambda (a0: A).(eq A a0 a)) in (let TMP_22 \def (asucc g +a) in (let TMP_23 \def (aplus g TMP_22 n) in (let TMP_24 \def (aplus_asucc g +n a) in (let H1 \def (eq_ind_r A TMP_20 TMP_21 H0 TMP_23 TMP_24) in (let +TMP_25 \def (S n) in (let TMP_26 \def (eq nat TMP_25 O) in (aplus_asucc_false +g a n H1 TMP_26)))))))))))) in (let TMP_46 \def (\lambda (n0: nat).(\lambda +(_: ((\forall (a: A).((eq A (asucc g (aplus g a n)) (aplus g a n0)) \to (eq +nat (S n) n0))))).(\lambda (a: A).(\lambda (H1: (eq A (asucc g (aplus g a n)) +(asucc g (aplus g a n0)))).(let TMP_28 \def (aplus g a n) in (let TMP_29 \def +(asucc g TMP_28) in (let TMP_32 \def (\lambda (a0: A).(let TMP_30 \def (aplus +g a n0) in (let TMP_31 \def (asucc g TMP_30) in (eq A a0 TMP_31)))) in (let +TMP_33 \def (asucc g a) in (let TMP_34 \def (aplus g TMP_33 n) in (let TMP_35 +\def (aplus_asucc g n a) in (let H2 \def (eq_ind_r A TMP_29 TMP_32 H1 TMP_34 +TMP_35) in (let TMP_36 \def (aplus g a n0) in (let TMP_37 \def (asucc g +TMP_36) in (let TMP_40 \def (\lambda (a0: A).(let TMP_38 \def (asucc g a) in +(let TMP_39 \def (aplus g TMP_38 n) in (eq A TMP_39 a0)))) in (let TMP_41 +\def (asucc g a) in (let TMP_42 \def (aplus g TMP_41 n0) in (let TMP_43 \def +(aplus_asucc g n0 a) in (let H3 \def (eq_ind_r A TMP_37 TMP_40 H2 TMP_42 +TMP_43) in (let TMP_44 \def (asucc g a) in (let TMP_45 \def (H n0 TMP_44 H3) +in (f_equal nat nat S n n0 TMP_45))))))))))))))))))))) in (nat_ind TMP_18 +TMP_27 TMP_46 h2))))))) in (nat_ind TMP_1 TMP_16 TMP_47 h1))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/aprem/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/aprem/defs.ma index 78b49e920..e3ea6cc87 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/aprem/defs.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/aprem/defs.ma @@ -14,7 +14,7 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/A/defs.ma". +include "basic_1/A/defs.ma". inductive aprem: nat \to (A \to (A \to Prop)) \def | aprem_zero: \forall (a1: A).(\forall (a2: A).(aprem O (AHead a1 a2) a1)) diff --git a/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma index ed48846a8..02355ca5e 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/aprem/fwd.ma @@ -14,107 +14,131 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/aprem/defs.ma". +include "basic_1/aprem/defs.ma". + +let rec aprem_ind (P: (nat \to (A \to (A \to Prop)))) (f: (\forall (a1: +A).(\forall (a2: A).(P O (AHead a1 a2) a1)))) (f0: (\forall (a2: A).(\forall +(a: A).(\forall (i: nat).((aprem i a2 a) \to ((P i a2 a) \to (\forall (a1: +A).(P (S i) (AHead a1 a2) a)))))))) (n: nat) (a: A) (a0: A) (a1: aprem n a +a0) on a1: P n a a0 \def match a1 with [(aprem_zero a2 a3) \Rightarrow (f a2 +a3) | (aprem_succ a2 a3 i a4 a5) \Rightarrow (let TMP_1 \def ((aprem_ind P f +f0) i a2 a3 a4) in (f0 a2 a3 i a4 TMP_1 a5))]. theorem aprem_gen_sort: \forall (x: A).(\forall (i: nat).(\forall (h: nat).(\forall (n: nat).((aprem i (ASort h n) x) \to False)))) \def \lambda (x: A).(\lambda (i: nat).(\lambda (h: nat).(\lambda (n: -nat).(\lambda (H: (aprem i (ASort h n) x)).(insert_eq A (ASort h n) (\lambda -(a: A).(aprem i a x)) (\lambda (_: A).False) (\lambda (y: A).(\lambda (H0: -(aprem i y x)).(aprem_ind (\lambda (_: nat).(\lambda (a: A).(\lambda (_: -A).((eq A a (ASort h n)) \to False)))) (\lambda (a1: A).(\lambda (a2: -A).(\lambda (H1: (eq A (AHead a1 a2) (ASort h n))).(let H2 \def (eq_ind A -(AHead a1 a2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) -with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I -(ASort h n) H1) in (False_ind False H2))))) (\lambda (a2: A).(\lambda (a: -A).(\lambda (i0: nat).(\lambda (_: (aprem i0 a2 a)).(\lambda (_: (((eq A a2 -(ASort h n)) \to False))).(\lambda (a1: A).(\lambda (H3: (eq A (AHead a1 a2) -(ASort h n))).(let H4 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee -in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | -(AHead _ _) \Rightarrow True])) I (ASort h n) H3) in (False_ind False -H4))))))))) i y x H0))) H))))). -(* COMMENTS -Initial nodes: 227 -END *) +nat).(\lambda (H: (aprem i (ASort h n) x)).(let TMP_1 \def (ASort h n) in +(let TMP_2 \def (\lambda (a: A).(aprem i a x)) in (let TMP_3 \def (\lambda +(_: A).False) in (let TMP_13 \def (\lambda (y: A).(\lambda (H0: (aprem i y +x)).(let TMP_4 \def (\lambda (_: nat).(\lambda (a: A).(\lambda (_: A).((eq A +a (ASort h n)) \to False)))) in (let TMP_8 \def (\lambda (a1: A).(\lambda +(a2: A).(\lambda (H1: (eq A (AHead a1 a2) (ASort h n))).(let TMP_5 \def +(AHead a1 a2) in (let TMP_6 \def (\lambda (ee: A).(match ee with [(ASort _ _) +\Rightarrow False | (AHead _ _) \Rightarrow True])) in (let TMP_7 \def (ASort +h n) in (let H2 \def (eq_ind A TMP_5 TMP_6 I TMP_7 H1) in (False_ind False +H2)))))))) in (let TMP_12 \def (\lambda (a2: A).(\lambda (a: A).(\lambda (i0: +nat).(\lambda (_: (aprem i0 a2 a)).(\lambda (_: (((eq A a2 (ASort h n)) \to +False))).(\lambda (a1: A).(\lambda (H3: (eq A (AHead a1 a2) (ASort h +n))).(let TMP_9 \def (AHead a1 a2) in (let TMP_10 \def (\lambda (ee: +A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow +True])) in (let TMP_11 \def (ASort h n) in (let H4 \def (eq_ind A TMP_9 +TMP_10 I TMP_11 H3) in (False_ind False H4)))))))))))) in (aprem_ind TMP_4 +TMP_8 TMP_12 i y x H0)))))) in (insert_eq A TMP_1 TMP_2 TMP_3 TMP_13 +H))))))))). theorem aprem_gen_head_O: \forall (a1: A).(\forall (a2: A).(\forall (x: A).((aprem O (AHead a1 a2) x) \to (eq A x a1)))) \def \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (H: (aprem O -(AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: A).(aprem O a x)) -(\lambda (_: A).(eq A x a1)) (\lambda (y: A).(\lambda (H0: (aprem O y -x)).(insert_eq nat O (\lambda (n: nat).(aprem n y x)) (\lambda (_: nat).((eq -A y (AHead a1 a2)) \to (eq A x a1))) (\lambda (y0: nat).(\lambda (H1: (aprem -y0 y x)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq -nat n O) \to ((eq A a (AHead a1 a2)) \to (eq A a0 a1)))))) (\lambda (a0: -A).(\lambda (a3: A).(\lambda (_: (eq nat O O)).(\lambda (H3: (eq A (AHead a0 -a3) (AHead a1 a2))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a _) -\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in ((let H5 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a3 | (AHead _ a) \Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) -in (\lambda (H6: (eq A a0 a1)).H6)) H4)))))) (\lambda (a0: A).(\lambda (a: -A).(\lambda (i: nat).(\lambda (H2: (aprem i a0 a)).(\lambda (H3: (((eq nat i -O) \to ((eq A a0 (AHead a1 a2)) \to (eq A a a1))))).(\lambda (a3: A).(\lambda -(H4: (eq nat (S i) O)).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let -H6 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) -with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) (AHead a3 -a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e -in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ -a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in (\lambda (_: (eq A -a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: A).((eq nat i O) \to ((eq A -a4 (AHead a1 a2)) \to (eq A a a1)))) H3 a2 H7) in (let H10 \def (eq_ind A a0 -(\lambda (a4: A).(aprem i a4 a)) H2 a2 H7) in (let H11 \def (eq_ind nat (S i) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind (eq A a -a1) H11)))))) H6)))))))))) y0 y x H1))) H0))) H)))). -(* COMMENTS -Initial nodes: 500 -END *) +(AHead a1 a2) x)).(let TMP_1 \def (AHead a1 a2) in (let TMP_2 \def (\lambda +(a: A).(aprem O a x)) in (let TMP_3 \def (\lambda (_: A).(eq A x a1)) in (let +TMP_29 \def (\lambda (y: A).(\lambda (H0: (aprem O y x)).(let TMP_4 \def +(\lambda (n: nat).(aprem n y x)) in (let TMP_5 \def (\lambda (_: nat).((eq A +y (AHead a1 a2)) \to (eq A x a1))) in (let TMP_28 \def (\lambda (y0: +nat).(\lambda (H1: (aprem y0 y x)).(let TMP_6 \def (\lambda (n: nat).(\lambda +(a: A).(\lambda (a0: A).((eq nat n O) \to ((eq A a (AHead a1 a2)) \to (eq A +a0 a1)))))) in (let TMP_14 \def (\lambda (a0: A).(\lambda (a3: A).(\lambda +(_: (eq nat O O)).(\lambda (H3: (eq A (AHead a0 a3) (AHead a1 a2))).(let +TMP_7 \def (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 | (AHead +a _) \Rightarrow a])) in (let TMP_8 \def (AHead a0 a3) in (let TMP_9 \def +(AHead a1 a2) in (let H4 \def (f_equal A A TMP_7 TMP_8 TMP_9 H3) in (let +TMP_10 \def (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a3 | +(AHead _ a) \Rightarrow a])) in (let TMP_11 \def (AHead a0 a3) in (let TMP_12 +\def (AHead a1 a2) in (let H5 \def (f_equal A A TMP_10 TMP_11 TMP_12 H3) in +(let TMP_13 \def (\lambda (H6: (eq A a0 a1)).H6) in (TMP_13 H4)))))))))))))) +in (let TMP_27 \def (\lambda (a0: A).(\lambda (a: A).(\lambda (i: +nat).(\lambda (H2: (aprem i a0 a)).(\lambda (H3: (((eq nat i O) \to ((eq A a0 +(AHead a1 a2)) \to (eq A a a1))))).(\lambda (a3: A).(\lambda (H4: (eq nat (S +i) O)).(\lambda (H5: (eq A (AHead a3 a0) (AHead a1 a2))).(let TMP_15 \def +(\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) +\Rightarrow a4])) in (let TMP_16 \def (AHead a3 a0) in (let TMP_17 \def +(AHead a1 a2) in (let H6 \def (f_equal A A TMP_15 TMP_16 TMP_17 H5) in (let +TMP_18 \def (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 | +(AHead _ a4) \Rightarrow a4])) in (let TMP_19 \def (AHead a3 a0) in (let +TMP_20 \def (AHead a1 a2) in (let H7 \def (f_equal A A TMP_18 TMP_19 TMP_20 +H5) in (let TMP_26 \def (\lambda (_: (eq A a3 a1)).(let TMP_21 \def (\lambda +(a4: A).((eq nat i O) \to ((eq A a4 (AHead a1 a2)) \to (eq A a a1)))) in (let +H9 \def (eq_ind A a0 TMP_21 H3 a2 H7) in (let TMP_22 \def (\lambda (a4: +A).(aprem i a4 a)) in (let H10 \def (eq_ind A a0 TMP_22 H2 a2 H7) in (let +TMP_23 \def (S i) in (let TMP_24 \def (\lambda (ee: nat).(match ee with [O +\Rightarrow False | (S _) \Rightarrow True])) in (let H11 \def (eq_ind nat +TMP_23 TMP_24 I O H4) in (let TMP_25 \def (eq A a a1) in (False_ind TMP_25 +H11)))))))))) in (TMP_26 H6)))))))))))))))))) in (aprem_ind TMP_6 TMP_14 +TMP_27 y0 y x H1)))))) in (insert_eq nat O TMP_4 TMP_5 TMP_28 H0)))))) in +(insert_eq A TMP_1 TMP_2 TMP_3 TMP_29 H)))))))). theorem aprem_gen_head_S: \forall (a1: A).(\forall (a2: A).(\forall (x: A).(\forall (i: nat).((aprem (S i) (AHead a1 a2) x) \to (aprem i a2 x))))) \def \lambda (a1: A).(\lambda (a2: A).(\lambda (x: A).(\lambda (i: nat).(\lambda -(H: (aprem (S i) (AHead a1 a2) x)).(insert_eq A (AHead a1 a2) (\lambda (a: -A).(aprem (S i) a x)) (\lambda (_: A).(aprem i a2 x)) (\lambda (y: -A).(\lambda (H0: (aprem (S i) y x)).(insert_eq nat (S i) (\lambda (n: -nat).(aprem n y x)) (\lambda (_: nat).((eq A y (AHead a1 a2)) \to (aprem i a2 -x))) (\lambda (y0: nat).(\lambda (H1: (aprem y0 y x)).(aprem_ind (\lambda (n: +(H: (aprem (S i) (AHead a1 a2) x)).(let TMP_1 \def (AHead a1 a2) in (let +TMP_3 \def (\lambda (a: A).(let TMP_2 \def (S i) in (aprem TMP_2 a x))) in +(let TMP_4 \def (\lambda (_: A).(aprem i a2 x)) in (let TMP_38 \def (\lambda +(y: A).(\lambda (H0: (aprem (S i) y x)).(let TMP_5 \def (S i) in (let TMP_6 +\def (\lambda (n: nat).(aprem n y x)) in (let TMP_7 \def (\lambda (_: +nat).((eq A y (AHead a1 a2)) \to (aprem i a2 x))) in (let TMP_37 \def +(\lambda (y0: nat).(\lambda (H1: (aprem y0 y x)).(let TMP_8 \def (\lambda (n: nat).(\lambda (a: A).(\lambda (a0: A).((eq nat n (S i)) \to ((eq A a (AHead -a1 a2)) \to (aprem i a2 a0)))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda -(H2: (eq nat O (S i))).(\lambda (H3: (eq A (AHead a0 a3) (AHead a1 a2))).(let -H4 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) -with [(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) (AHead a0 a3) -(AHead a1 a2) H3) in ((let H5 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead _ a) -\Rightarrow a])) (AHead a0 a3) (AHead a1 a2) H3) in (\lambda (H6: (eq A a0 -a1)).(eq_ind_r A a1 (\lambda (a: A).(aprem i a2 a)) (let H7 \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 i) H2) in (False_ind -(aprem i a2 a1) H7)) a0 H6))) H4)))))) (\lambda (a0: A).(\lambda (a: +a1 a2)) \to (aprem i a2 a0)))))) in (let TMP_21 \def (\lambda (a0: +A).(\lambda (a3: A).(\lambda (H2: (eq nat O (S i))).(\lambda (H3: (eq A +(AHead a0 a3) (AHead a1 a2))).(let TMP_9 \def (\lambda (e: A).(match e with +[(ASort _ _) \Rightarrow a0 | (AHead a _) \Rightarrow a])) in (let TMP_10 +\def (AHead a0 a3) in (let TMP_11 \def (AHead a1 a2) in (let H4 \def (f_equal +A A TMP_9 TMP_10 TMP_11 H3) in (let TMP_12 \def (\lambda (e: A).(match e with +[(ASort _ _) \Rightarrow a3 | (AHead _ a) \Rightarrow a])) in (let TMP_13 +\def (AHead a0 a3) in (let TMP_14 \def (AHead a1 a2) in (let H5 \def (f_equal +A A TMP_12 TMP_13 TMP_14 H3) in (let TMP_20 \def (\lambda (H6: (eq A a0 +a1)).(let TMP_15 \def (\lambda (a: A).(aprem i a2 a)) in (let TMP_16 \def +(\lambda (ee: nat).(match ee with [O \Rightarrow True | (S _) \Rightarrow +False])) in (let TMP_17 \def (S i) in (let H7 \def (eq_ind nat O TMP_16 I +TMP_17 H2) in (let TMP_18 \def (aprem i a2 a1) in (let TMP_19 \def (False_ind +TMP_18 H7) in (eq_ind_r A a1 TMP_15 TMP_19 a0 H6)))))))) in (TMP_20 +H4)))))))))))))) in (let TMP_36 \def (\lambda (a0: A).(\lambda (a: A).(\lambda (i0: nat).(\lambda (H2: (aprem i0 a0 a)).(\lambda (H3: (((eq nat i0 (S i)) \to ((eq A a0 (AHead a1 a2)) \to (aprem i a2 a))))).(\lambda (a3: A).(\lambda (H4: (eq nat (S i0) (S i))).(\lambda (H5: (eq A (AHead a3 a0) -(AHead a1 a2))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a4 _) -\Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) H5) in ((let H7 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a0 | (AHead _ a4) \Rightarrow a4])) (AHead a3 a0) (AHead a1 a2) -H5) in (\lambda (_: (eq A a3 a1)).(let H9 \def (eq_ind A a0 (\lambda (a4: -A).((eq nat i0 (S i)) \to ((eq A a4 (AHead a1 a2)) \to (aprem i a2 a)))) H3 -a2 H7) in (let H10 \def (eq_ind A a0 (\lambda (a4: A).(aprem i0 a4 a)) H2 a2 -H7) in (let H11 \def (f_equal nat nat (\lambda (e: nat).(match e in nat -return (\lambda (_: nat).nat) with [O \Rightarrow i0 | (S n) \Rightarrow n])) -(S i0) (S i) H4) in (let H12 \def (eq_ind nat i0 (\lambda (n: nat).((eq nat n -(S i)) \to ((eq A a2 (AHead a1 a2)) \to (aprem i a2 a)))) H9 i H11) in (let -H13 \def (eq_ind nat i0 (\lambda (n: nat).(aprem n a2 a)) H10 i H11) in -H13))))))) H6)))))))))) y0 y x H1))) H0))) H))))). -(* COMMENTS -Initial nodes: 631 -END *) +(AHead a1 a2))).(let TMP_22 \def (\lambda (e: A).(match e with [(ASort _ _) +\Rightarrow a3 | (AHead a4 _) \Rightarrow a4])) in (let TMP_23 \def (AHead a3 +a0) in (let TMP_24 \def (AHead a1 a2) in (let H6 \def (f_equal A A TMP_22 +TMP_23 TMP_24 H5) in (let TMP_25 \def (\lambda (e: A).(match e with [(ASort _ +_) \Rightarrow a0 | (AHead _ a4) \Rightarrow a4])) in (let TMP_26 \def (AHead +a3 a0) in (let TMP_27 \def (AHead a1 a2) in (let H7 \def (f_equal A A TMP_25 +TMP_26 TMP_27 H5) in (let TMP_35 \def (\lambda (_: (eq A a3 a1)).(let TMP_28 +\def (\lambda (a4: A).((eq nat i0 (S i)) \to ((eq A a4 (AHead a1 a2)) \to +(aprem i a2 a)))) in (let H9 \def (eq_ind A a0 TMP_28 H3 a2 H7) in (let +TMP_29 \def (\lambda (a4: A).(aprem i0 a4 a)) in (let H10 \def (eq_ind A a0 +TMP_29 H2 a2 H7) in (let TMP_30 \def (\lambda (e: nat).(match e with [O +\Rightarrow i0 | (S n) \Rightarrow n])) in (let TMP_31 \def (S i0) in (let +TMP_32 \def (S i) in (let H11 \def (f_equal nat nat TMP_30 TMP_31 TMP_32 H4) +in (let TMP_33 \def (\lambda (n: nat).((eq nat n (S i)) \to ((eq A a2 (AHead +a1 a2)) \to (aprem i a2 a)))) in (let H12 \def (eq_ind nat i0 TMP_33 H9 i +H11) in (let TMP_34 \def (\lambda (n: nat).(aprem n a2 a)) in (let H13 \def +(eq_ind nat i0 TMP_34 H10 i H11) in H13))))))))))))) in (TMP_35 +H6)))))))))))))))))) in (aprem_ind TMP_8 TMP_21 TMP_36 y0 y x H1)))))) in +(insert_eq nat TMP_5 TMP_6 TMP_7 TMP_37 H0))))))) in (insert_eq A TMP_1 TMP_3 +TMP_4 TMP_38 H))))))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/aprem/props.ma b/matita/matita/contribs/lambdadelta/basic_1/aprem/props.ma index fb8321062..df56d5d91 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/aprem/props.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/aprem/props.ma @@ -14,9 +14,9 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/aprem/fwd.ma". +include "basic_1/aprem/fwd.ma". -include "Basic-1/leq/defs.ma". +include "basic_1/leq/fwd.ma". theorem aprem_repl: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall @@ -24,53 +24,63 @@ theorem aprem_repl: b1 b2)) (\lambda (b1: A).(aprem i a1 b1))))))))) \def \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (i: nat).(\forall -(b2: A).((aprem i a0 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda -(b1: A).(aprem i a b1)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda -(n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g -(ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (i: nat).(\lambda (b2: -A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H_x \def (aprem_gen_sort b2 -i h2 n2 H1) in (let H2 \def H_x in (False_ind (ex2 A (\lambda (b1: A).(leq g -b1 b2)) (\lambda (b1: A).(aprem i (ASort h1 n1) b1))) H2)))))))))))) (\lambda -(a0: A).(\lambda (a3: A).(\lambda (H0: (leq g a0 a3)).(\lambda (_: ((\forall -(i: nat).(\forall (b2: A).((aprem i a3 b2) \to (ex2 A (\lambda (b1: A).(leq g -b1 b2)) (\lambda (b1: A).(aprem i a0 b1)))))))).(\lambda (a4: A).(\lambda -(a5: A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: ((\forall (i: nat).(\forall -(b2: A).((aprem i a5 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda -(b1: A).(aprem i a4 b1)))))))).(\lambda (i: nat).(\lambda (b2: A).(\lambda -(H4: (aprem i (AHead a3 a5) b2)).(nat_ind (\lambda (n: nat).((aprem n (AHead -a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem -n (AHead a0 a4) b1))))) (\lambda (H5: (aprem O (AHead a3 a5) b2)).(let H_y -\def (aprem_gen_head_O a3 a5 b2 H5) in (eq_ind_r A a3 (\lambda (a: A).(ex2 A -(\lambda (b1: A).(leq g b1 a)) (\lambda (b1: A).(aprem O (AHead a0 a4) b1)))) -(ex_intro2 A (\lambda (b1: A).(leq g b1 a3)) (\lambda (b1: A).(aprem O (AHead -a0 a4) b1)) a0 H0 (aprem_zero a0 a4)) b2 H_y))) (\lambda (i0: nat).(\lambda -(_: (((aprem i0 (AHead a3 a5) b2) \to (ex2 A (\lambda (b1: A).(leq g b1 b2)) -(\lambda (b1: A).(aprem i0 (AHead a0 a4) b1)))))).(\lambda (H5: (aprem (S i0) -(AHead a3 a5) b2)).(let H_y \def (aprem_gen_head_S a3 a5 b2 i0 H5) in (let -H_x \def (H3 i0 b2 H_y) in (let H6 \def H_x in (ex2_ind A (\lambda (b1: -A).(leq g b1 b2)) (\lambda (b1: A).(aprem i0 a4 b1)) (ex2 A (\lambda (b1: -A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 a4) b1))) (\lambda -(x: A).(\lambda (H7: (leq g x b2)).(\lambda (H8: (aprem i0 a4 x)).(ex_intro2 -A (\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem (S i0) (AHead a0 -a4) b1)) x H7 (aprem_succ a4 x i0 H8 a0))))) H6))))))) i H4)))))))))))) a1 a2 -H)))). -(* COMMENTS -Initial nodes: 621 -END *) +a2)).(let TMP_3 \def (\lambda (a: A).(\lambda (a0: A).(\forall (i: +nat).(\forall (b2: A).((aprem i a0 b2) \to (let TMP_1 \def (\lambda (b1: +A).(leq g b1 b2)) in (let TMP_2 \def (\lambda (b1: A).(aprem i a b1)) in (ex2 +A TMP_1 TMP_2)))))))) in (let TMP_8 \def (\lambda (h1: nat).(\lambda (h2: +nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq +A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (i: +nat).(\lambda (b2: A).(\lambda (H1: (aprem i (ASort h2 n2) b2)).(let H_x \def +(aprem_gen_sort b2 i h2 n2 H1) in (let H2 \def H_x in (let TMP_4 \def +(\lambda (b1: A).(leq g b1 b2)) in (let TMP_6 \def (\lambda (b1: A).(let +TMP_5 \def (ASort h1 n1) in (aprem i TMP_5 b1))) in (let TMP_7 \def (ex2 A +TMP_4 TMP_6) in (False_ind TMP_7 H2))))))))))))))) in (let TMP_37 \def +(\lambda (a0: A).(\lambda (a3: A).(\lambda (H0: (leq g a0 a3)).(\lambda (_: +((\forall (i: nat).(\forall (b2: A).((aprem i a3 b2) \to (ex2 A (\lambda (b1: +A).(leq g b1 b2)) (\lambda (b1: A).(aprem i a0 b1)))))))).(\lambda (a4: +A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: ((\forall (i: +nat).(\forall (b2: A).((aprem i a5 b2) \to (ex2 A (\lambda (b1: A).(leq g b1 +b2)) (\lambda (b1: A).(aprem i a4 b1)))))))).(\lambda (i: nat).(\lambda (b2: +A).(\lambda (H4: (aprem i (AHead a3 a5) b2)).(let TMP_12 \def (\lambda (n: +nat).((aprem n (AHead a3 a5) b2) \to (let TMP_9 \def (\lambda (b1: A).(leq g +b1 b2)) in (let TMP_11 \def (\lambda (b1: A).(let TMP_10 \def (AHead a0 a4) +in (aprem n TMP_10 b1))) in (ex2 A TMP_9 TMP_11))))) in (let TMP_22 \def +(\lambda (H5: (aprem O (AHead a3 a5) b2)).(let H_y \def (aprem_gen_head_O a3 +a5 b2 H5) in (let TMP_16 \def (\lambda (a: A).(let TMP_13 \def (\lambda (b1: +A).(leq g b1 a)) in (let TMP_15 \def (\lambda (b1: A).(let TMP_14 \def (AHead +a0 a4) in (aprem O TMP_14 b1))) in (ex2 A TMP_13 TMP_15)))) in (let TMP_17 +\def (\lambda (b1: A).(leq g b1 a3)) in (let TMP_19 \def (\lambda (b1: +A).(let TMP_18 \def (AHead a0 a4) in (aprem O TMP_18 b1))) in (let TMP_20 +\def (aprem_zero a0 a4) in (let TMP_21 \def (ex_intro2 A TMP_17 TMP_19 a0 H0 +TMP_20) in (eq_ind_r A a3 TMP_16 TMP_21 b2 H_y)))))))) in (let TMP_36 \def +(\lambda (i0: nat).(\lambda (_: (((aprem i0 (AHead a3 a5) b2) \to (ex2 A +(\lambda (b1: A).(leq g b1 b2)) (\lambda (b1: A).(aprem i0 (AHead a0 a4) +b1)))))).(\lambda (H5: (aprem (S i0) (AHead a3 a5) b2)).(let H_y \def +(aprem_gen_head_S a3 a5 b2 i0 H5) in (let H_x \def (H3 i0 b2 H_y) in (let H6 +\def H_x in (let TMP_23 \def (\lambda (b1: A).(leq g b1 b2)) in (let TMP_24 +\def (\lambda (b1: A).(aprem i0 a4 b1)) in (let TMP_25 \def (\lambda (b1: +A).(leq g b1 b2)) in (let TMP_28 \def (\lambda (b1: A).(let TMP_26 \def (S +i0) in (let TMP_27 \def (AHead a0 a4) in (aprem TMP_26 TMP_27 b1)))) in (let +TMP_29 \def (ex2 A TMP_25 TMP_28) in (let TMP_35 \def (\lambda (x: +A).(\lambda (H7: (leq g x b2)).(\lambda (H8: (aprem i0 a4 x)).(let TMP_30 +\def (\lambda (b1: A).(leq g b1 b2)) in (let TMP_33 \def (\lambda (b1: +A).(let TMP_31 \def (S i0) in (let TMP_32 \def (AHead a0 a4) in (aprem TMP_31 +TMP_32 b1)))) in (let TMP_34 \def (aprem_succ a4 x i0 H8 a0) in (ex_intro2 A +TMP_30 TMP_33 x H7 TMP_34))))))) in (ex2_ind A TMP_23 TMP_24 TMP_29 TMP_35 +H6))))))))))))) in (nat_ind TMP_12 TMP_22 TMP_36 i H4))))))))))))))) in +(leq_ind g TMP_3 TMP_8 TMP_37 a1 a2 H))))))). theorem aprem_asucc: \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (i: nat).((aprem i a1 a2) \to (aprem i (asucc g a1) a2))))) \def \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (i: nat).(\lambda -(H: (aprem i a1 a2)).(aprem_ind (\lambda (n: nat).(\lambda (a: A).(\lambda -(a0: A).(aprem n (asucc g a) a0)))) (\lambda (a0: A).(\lambda (a3: -A).(aprem_zero a0 (asucc g a3)))) (\lambda (a0: A).(\lambda (a: A).(\lambda -(i0: nat).(\lambda (_: (aprem i0 a0 a)).(\lambda (H1: (aprem i0 (asucc g a0) -a)).(\lambda (a3: A).(aprem_succ (asucc g a0) a i0 H1 a3))))))) i a1 a2 -H))))). -(* COMMENTS -Initial nodes: 101 -END *) +(H: (aprem i a1 a2)).(let TMP_2 \def (\lambda (n: nat).(\lambda (a: +A).(\lambda (a0: A).(let TMP_1 \def (asucc g a) in (aprem n TMP_1 a0))))) in +(let TMP_4 \def (\lambda (a0: A).(\lambda (a3: A).(let TMP_3 \def (asucc g +a3) in (aprem_zero a0 TMP_3)))) in (let TMP_6 \def (\lambda (a0: A).(\lambda +(a: A).(\lambda (i0: nat).(\lambda (_: (aprem i0 a0 a)).(\lambda (H1: (aprem +i0 (asucc g a0) a)).(\lambda (a3: A).(let TMP_5 \def (asucc g a0) in +(aprem_succ TMP_5 a i0 H1 a3)))))))) in (aprem_ind TMP_2 TMP_4 TMP_6 i a1 a2 +H)))))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/asucc/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/asucc/defs.ma index 2f1af0279..c4943e118 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/asucc/defs.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/asucc/defs.ma @@ -14,15 +14,12 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/A/defs.ma". +include "basic_1/A/defs.ma". -include "Basic-1/G/defs.ma". +include "basic_1/G/defs.ma". -definition asucc: - G \to (A \to A) -\def - let rec asucc (g: G) (l: A) on l: A \def (match l with [(ASort n0 n) -\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g n)) | (S h) -\Rightarrow (ASort h n)]) | (AHead a1 a2) \Rightarrow (AHead a1 (asucc g -a2))]) in asucc. +let rec asucc (g: G) (l: A) on l: A \def match l with [(ASort n0 n) +\Rightarrow (match n0 with [O \Rightarrow (let TMP_2 \def (next g n) in +(ASort O TMP_2)) | (S h) \Rightarrow (ASort h n)]) | (AHead a1 a2) +\Rightarrow (let TMP_1 \def (asucc g a2) in (AHead a1 TMP_1))]. diff --git a/matita/matita/contribs/lambdadelta/basic_1/asucc/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/asucc/fwd.ma index 61fcb5799..6fc72b866 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/asucc/fwd.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/asucc/fwd.ma @@ -14,86 +14,119 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/asucc/defs.ma". +include "basic_1/asucc/defs.ma". + +include "basic_1/A/fwd.ma". theorem asucc_gen_sort: \forall (g: G).(\forall (h: nat).(\forall (n: nat).(\forall (a: A).((eq A (ASort h n) (asucc g a)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a (ASort h0 n0))))))))) \def - \lambda (g: G).(\lambda (h: nat).(\lambda (n: nat).(\lambda (a: A).(A_ind -(\lambda (a0: A).((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat nat (\lambda -(h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 n0))))))) (\lambda (n0: -nat).(\lambda (n1: nat).(\lambda (H: (eq A (ASort h n) (asucc g (ASort n0 -n1)))).(let H0 \def (f_equal A A (\lambda (e: A).e) (ASort h n) (match n0 -with [O \Rightarrow (ASort O (next g n1)) | (S h0) \Rightarrow (ASort h0 -n1)]) H) in (ex_2_intro nat nat (\lambda (h0: nat).(\lambda (n2: nat).(eq A -(ASort n0 n1) (ASort h0 n2)))) n0 n1 (refl_equal A (ASort n0 n1))))))) -(\lambda (a0: A).(\lambda (_: (((eq A (ASort h n) (asucc g a0)) \to (ex_2 nat -nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a0 (ASort h0 -n0)))))))).(\lambda (a1: A).(\lambda (_: (((eq A (ASort h n) (asucc g a1)) -\to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: nat).(eq A a1 (ASort h0 -n0)))))))).(\lambda (H1: (eq A (ASort h n) (asucc g (AHead a0 a1)))).(let H2 -\def (eq_ind A (ASort h n) (\lambda (ee: A).(match ee in A return (\lambda -(_: A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (asucc g (AHead a0 a1)) H1) in (False_ind (ex_2 nat nat (\lambda -(h0: nat).(\lambda (n0: nat).(eq A (AHead a0 a1) (ASort h0 n0))))) H2))))))) -a)))). -(* COMMENTS -Initial nodes: 317 -END *) + \lambda (g: G).(\lambda (h: nat).(\lambda (n: nat).(\lambda (a: A).(let +TMP_3 \def (\lambda (a0: A).((eq A (ASort h n) (asucc g a0)) \to (let TMP_2 +\def (\lambda (h0: nat).(\lambda (n0: nat).(let TMP_1 \def (ASort h0 n0) in +(eq A a0 TMP_1)))) in (ex_2 nat nat TMP_2)))) in (let TMP_13 \def (\lambda +(n0: nat).(\lambda (n1: nat).(\lambda (H: (eq A (ASort h n) (asucc g (ASort +n0 n1)))).(let TMP_4 \def (\lambda (e: A).e) in (let TMP_5 \def (ASort h n) +in (let TMP_7 \def (match n0 with [O \Rightarrow (let TMP_6 \def (next g n1) +in (ASort O TMP_6)) | (S h0) \Rightarrow (ASort h0 n1)]) in (let H0 \def +(f_equal A A TMP_4 TMP_5 TMP_7 H) in (let TMP_10 \def (\lambda (h0: +nat).(\lambda (n2: nat).(let TMP_8 \def (ASort n0 n1) in (let TMP_9 \def +(ASort h0 n2) in (eq A TMP_8 TMP_9))))) in (let TMP_11 \def (ASort n0 n1) in +(let TMP_12 \def (refl_equal A TMP_11) in (ex_2_intro nat nat TMP_10 n0 n1 +TMP_12))))))))))) in (let TMP_22 \def (\lambda (a0: A).(\lambda (_: (((eq A +(ASort h n) (asucc g a0)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: +nat).(eq A a0 (ASort h0 n0)))))))).(\lambda (a1: A).(\lambda (_: (((eq A +(ASort h n) (asucc g a1)) \to (ex_2 nat nat (\lambda (h0: nat).(\lambda (n0: +nat).(eq A a1 (ASort h0 n0)))))))).(\lambda (H1: (eq A (ASort h n) (asucc g +(AHead a0 a1)))).(let TMP_14 \def (ASort h n) in (let TMP_15 \def (\lambda +(ee: A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) +\Rightarrow False])) in (let TMP_16 \def (AHead a0 a1) in (let TMP_17 \def +(asucc g TMP_16) in (let H2 \def (eq_ind A TMP_14 TMP_15 I TMP_17 H1) in (let +TMP_20 \def (\lambda (h0: nat).(\lambda (n0: nat).(let TMP_18 \def (AHead a0 +a1) in (let TMP_19 \def (ASort h0 n0) in (eq A TMP_18 TMP_19))))) in (let +TMP_21 \def (ex_2 nat nat TMP_20) in (False_ind TMP_21 H2))))))))))))) in +(A_ind TMP_3 TMP_13 TMP_22 a))))))). theorem asucc_gen_head: \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((eq A (AHead a1 a2) (asucc g a)) \to (ex2 A (\lambda (a0: A).(eq A a (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))))) \def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(A_ind -(\lambda (a0: A).((eq A (AHead a1 a2) (asucc g a0)) \to (ex2 A (\lambda (a3: -A).(eq A a0 (AHead a1 a3))) (\lambda (a3: A).(eq A a2 (asucc g a3)))))) -(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H: (eq A (AHead a1 a2) (asucc -g (ASort n n0)))).(nat_ind (\lambda (n1: nat).((eq A (AHead a1 a2) (asucc g -(ASort n1 n0))) \to (ex2 A (\lambda (a0: A).(eq A (ASort n1 n0) (AHead a1 -a0))) (\lambda (a0: A).(eq A a2 (asucc g a0)))))) (\lambda (H0: (eq A (AHead -a1 a2) (asucc g (ASort O n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort O (next g n0)) -H0) in (False_ind (ex2 A (\lambda (a0: A).(eq A (ASort O n0) (AHead a1 a0))) -(\lambda (a0: A).(eq A a2 (asucc g a0)))) H1))) (\lambda (n1: nat).(\lambda -(_: (((eq A (AHead a1 a2) (asucc g (ASort n1 n0))) \to (ex2 A (\lambda (a0: -A).(eq A (ASort n1 n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g -a0))))))).(\lambda (H0: (eq A (AHead a1 a2) (asucc g (ASort (S n1) -n0)))).(let H1 \def (eq_ind A (AHead a1 a2) (\lambda (ee: A).(match ee in A -return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ -_) \Rightarrow True])) I (ASort n1 n0) H0) in (False_ind (ex2 A (\lambda (a0: -A).(eq A (ASort (S n1) n0) (AHead a1 a0))) (\lambda (a0: A).(eq A a2 (asucc g -a0)))) H1))))) n H)))) (\lambda (a0: A).(\lambda (H: (((eq A (AHead a1 a2) -(asucc g a0)) \to (ex2 A (\lambda (a3: A).(eq A a0 (AHead a1 a3))) (\lambda -(a3: A).(eq A a2 (asucc g a3))))))).(\lambda (a3: A).(\lambda (H0: (((eq A -(AHead a1 a2) (asucc g a3)) \to (ex2 A (\lambda (a4: A).(eq A a3 (AHead a1 -a4))) (\lambda (a4: A).(eq A a2 (asucc g a4))))))).(\lambda (H1: (eq A (AHead -a1 a2) (asucc g (AHead a0 a3)))).(let H2 \def (f_equal A A (\lambda (e: -A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a1 | -(AHead a4 _) \Rightarrow a4])) (AHead a1 a2) (AHead a0 (asucc g a3)) H1) in -((let H3 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: -A).A) with [(ASort _ _) \Rightarrow a2 | (AHead _ a4) \Rightarrow a4])) -(AHead a1 a2) (AHead a0 (asucc g a3)) H1) in (\lambda (H4: (eq A a1 a0)).(let -H5 \def (eq_ind_r A a0 (\lambda (a4: A).((eq A (AHead a1 a2) (asucc g a4)) -\to (ex2 A (\lambda (a5: A).(eq A a4 (AHead a1 a5))) (\lambda (a5: A).(eq A -a2 (asucc g a5)))))) H a1 H4) in (eq_ind A a1 (\lambda (a4: A).(ex2 A -(\lambda (a5: A).(eq A (AHead a4 a3) (AHead a1 a5))) (\lambda (a5: A).(eq A -a2 (asucc g a5))))) (let H6 \def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead -a1 a4) (asucc g a3)) \to (ex2 A (\lambda (a5: A).(eq A a3 (AHead a1 a5))) -(\lambda (a5: A).(eq A a4 (asucc g a5)))))) H0 (asucc g a3) H3) in (let H7 -\def (eq_ind A a2 (\lambda (a4: A).((eq A (AHead a1 a4) (asucc g a1)) \to -(ex2 A (\lambda (a5: A).(eq A a1 (AHead a1 a5))) (\lambda (a5: A).(eq A a4 -(asucc g a5)))))) H5 (asucc g a3) H3) in (eq_ind_r A (asucc g a3) (\lambda -(a4: A).(ex2 A (\lambda (a5: A).(eq A (AHead a1 a3) (AHead a1 a5))) (\lambda -(a5: A).(eq A a4 (asucc g a5))))) (ex_intro2 A (\lambda (a4: A).(eq A (AHead -a1 a3) (AHead a1 a4))) (\lambda (a4: A).(eq A (asucc g a3) (asucc g a4))) a3 -(refl_equal A (AHead a1 a3)) (refl_equal A (asucc g a3))) a2 H3))) a0 H4)))) -H2))))))) a)))). -(* COMMENTS -Initial nodes: 957 -END *) + \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(let TMP_5 +\def (\lambda (a0: A).((eq A (AHead a1 a2) (asucc g a0)) \to (let TMP_2 \def +(\lambda (a3: A).(let TMP_1 \def (AHead a1 a3) in (eq A a0 TMP_1))) in (let +TMP_4 \def (\lambda (a3: A).(let TMP_3 \def (asucc g a3) in (eq A a2 TMP_3))) +in (ex2 A TMP_2 TMP_4))))) in (let TMP_34 \def (\lambda (n: nat).(\lambda +(n0: nat).(\lambda (H: (eq A (AHead a1 a2) (asucc g (ASort n n0)))).(let +TMP_11 \def (\lambda (n1: nat).((eq A (AHead a1 a2) (asucc g (ASort n1 n0))) +\to (let TMP_8 \def (\lambda (a0: A).(let TMP_6 \def (ASort n1 n0) in (let +TMP_7 \def (AHead a1 a0) in (eq A TMP_6 TMP_7)))) in (let TMP_10 \def +(\lambda (a0: A).(let TMP_9 \def (asucc g a0) in (eq A a2 TMP_9))) in (ex2 A +TMP_8 TMP_10))))) in (let TMP_22 \def (\lambda (H0: (eq A (AHead a1 a2) +(asucc g (ASort O n0)))).(let TMP_12 \def (AHead a1 a2) in (let TMP_13 \def +(\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _) +\Rightarrow True])) in (let TMP_14 \def (next g n0) in (let TMP_15 \def +(ASort O TMP_14) in (let H1 \def (eq_ind A TMP_12 TMP_13 I TMP_15 H0) in (let +TMP_18 \def (\lambda (a0: A).(let TMP_16 \def (ASort O n0) in (let TMP_17 +\def (AHead a1 a0) in (eq A TMP_16 TMP_17)))) in (let TMP_20 \def (\lambda +(a0: A).(let TMP_19 \def (asucc g a0) in (eq A a2 TMP_19))) in (let TMP_21 +\def (ex2 A TMP_18 TMP_20) in (False_ind TMP_21 H1)))))))))) in (let TMP_33 +\def (\lambda (n1: nat).(\lambda (_: (((eq A (AHead a1 a2) (asucc g (ASort n1 +n0))) \to (ex2 A (\lambda (a0: A).(eq A (ASort n1 n0) (AHead a1 a0))) +(\lambda (a0: A).(eq A a2 (asucc g a0))))))).(\lambda (H0: (eq A (AHead a1 +a2) (asucc g (ASort (S n1) n0)))).(let TMP_23 \def (AHead a1 a2) in (let +TMP_24 \def (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False | +(AHead _ _) \Rightarrow True])) in (let TMP_25 \def (ASort n1 n0) in (let H1 +\def (eq_ind A TMP_23 TMP_24 I TMP_25 H0) in (let TMP_29 \def (\lambda (a0: +A).(let TMP_26 \def (S n1) in (let TMP_27 \def (ASort TMP_26 n0) in (let +TMP_28 \def (AHead a1 a0) in (eq A TMP_27 TMP_28))))) in (let TMP_31 \def +(\lambda (a0: A).(let TMP_30 \def (asucc g a0) in (eq A a2 TMP_30))) in (let +TMP_32 \def (ex2 A TMP_29 TMP_31) in (False_ind TMP_32 H1))))))))))) in +(nat_ind TMP_11 TMP_22 TMP_33 n H))))))) in (let TMP_86 \def (\lambda (a0: +A).(\lambda (H: (((eq A (AHead a1 a2) (asucc g a0)) \to (ex2 A (\lambda (a3: +A).(eq A a0 (AHead a1 a3))) (\lambda (a3: A).(eq A a2 (asucc g +a3))))))).(\lambda (a3: A).(\lambda (H0: (((eq A (AHead a1 a2) (asucc g a3)) +\to (ex2 A (\lambda (a4: A).(eq A a3 (AHead a1 a4))) (\lambda (a4: A).(eq A +a2 (asucc g a4))))))).(\lambda (H1: (eq A (AHead a1 a2) (asucc g (AHead a0 +a3)))).(let TMP_35 \def (\lambda (e: A).(match e with [(ASort _ _) +\Rightarrow a1 | (AHead a4 _) \Rightarrow a4])) in (let TMP_36 \def (AHead a1 +a2) in (let TMP_37 \def (asucc g a3) in (let TMP_38 \def (AHead a0 TMP_37) in +(let H2 \def (f_equal A A TMP_35 TMP_36 TMP_38 H1) in (let TMP_39 \def +(\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a2 | (AHead _ a4) +\Rightarrow a4])) in (let TMP_40 \def (AHead a1 a2) in (let TMP_41 \def +(asucc g a3) in (let TMP_42 \def (AHead a0 TMP_41) in (let H3 \def (f_equal A +A TMP_39 TMP_40 TMP_42 H1) in (let TMP_85 \def (\lambda (H4: (eq A a1 +a0)).(let TMP_47 \def (\lambda (a4: A).((eq A (AHead a1 a2) (asucc g a4)) \to +(let TMP_44 \def (\lambda (a5: A).(let TMP_43 \def (AHead a1 a5) in (eq A a4 +TMP_43))) in (let TMP_46 \def (\lambda (a5: A).(let TMP_45 \def (asucc g a5) +in (eq A a2 TMP_45))) in (ex2 A TMP_44 TMP_46))))) in (let H5 \def (eq_ind_r +A a0 TMP_47 H a1 H4) in (let TMP_53 \def (\lambda (a4: A).(let TMP_50 \def +(\lambda (a5: A).(let TMP_48 \def (AHead a4 a3) in (let TMP_49 \def (AHead a1 +a5) in (eq A TMP_48 TMP_49)))) in (let TMP_52 \def (\lambda (a5: A).(let +TMP_51 \def (asucc g a5) in (eq A a2 TMP_51))) in (ex2 A TMP_50 TMP_52)))) in +(let TMP_58 \def (\lambda (a4: A).((eq A (AHead a1 a4) (asucc g a3)) \to (let +TMP_55 \def (\lambda (a5: A).(let TMP_54 \def (AHead a1 a5) in (eq A a3 +TMP_54))) in (let TMP_57 \def (\lambda (a5: A).(let TMP_56 \def (asucc g a5) +in (eq A a4 TMP_56))) in (ex2 A TMP_55 TMP_57))))) in (let TMP_59 \def (asucc +g a3) in (let H6 \def (eq_ind A a2 TMP_58 H0 TMP_59 H3) in (let TMP_64 \def +(\lambda (a4: A).((eq A (AHead a1 a4) (asucc g a1)) \to (let TMP_61 \def +(\lambda (a5: A).(let TMP_60 \def (AHead a1 a5) in (eq A a1 TMP_60))) in (let +TMP_63 \def (\lambda (a5: A).(let TMP_62 \def (asucc g a5) in (eq A a4 +TMP_62))) in (ex2 A TMP_61 TMP_63))))) in (let TMP_65 \def (asucc g a3) in +(let H7 \def (eq_ind A a2 TMP_64 H5 TMP_65 H3) in (let TMP_66 \def (asucc g +a3) in (let TMP_72 \def (\lambda (a4: A).(let TMP_69 \def (\lambda (a5: +A).(let TMP_67 \def (AHead a1 a3) in (let TMP_68 \def (AHead a1 a5) in (eq A +TMP_67 TMP_68)))) in (let TMP_71 \def (\lambda (a5: A).(let TMP_70 \def +(asucc g a5) in (eq A a4 TMP_70))) in (ex2 A TMP_69 TMP_71)))) in (let TMP_75 +\def (\lambda (a4: A).(let TMP_73 \def (AHead a1 a3) in (let TMP_74 \def +(AHead a1 a4) in (eq A TMP_73 TMP_74)))) in (let TMP_78 \def (\lambda (a4: +A).(let TMP_76 \def (asucc g a3) in (let TMP_77 \def (asucc g a4) in (eq A +TMP_76 TMP_77)))) in (let TMP_79 \def (AHead a1 a3) in (let TMP_80 \def +(refl_equal A TMP_79) in (let TMP_81 \def (asucc g a3) in (let TMP_82 \def +(refl_equal A TMP_81) in (let TMP_83 \def (ex_intro2 A TMP_75 TMP_78 a3 +TMP_80 TMP_82) in (let TMP_84 \def (eq_ind_r A TMP_66 TMP_72 TMP_83 a2 H3) in +(eq_ind A a1 TMP_53 TMP_84 a0 H4))))))))))))))))))))) in (TMP_85 +H2))))))))))))))))) in (A_ind TMP_5 TMP_34 TMP_86 a))))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex0/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/ex0/defs.ma index 6ab98b1af..546e03916 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/ex0/defs.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/ex0/defs.ma @@ -14,9 +14,9 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/A/defs.ma". +include "basic_1/A/defs.ma". -include "Basic-1/G/defs.ma". +include "basic_1/G/defs.ma". definition gz: G diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex0/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/ex0/fwd.ma new file mode 100644 index 000000000..f7ef4265b --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/ex0/fwd.ma @@ -0,0 +1,29 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *********************) + +include "basic_1/ex0/defs.ma". + +let rec leqz_ind (P: (A \to (A \to Prop))) (f: (\forall (h1: nat).(\forall +(h2: nat).(\forall (n1: nat).(\forall (n2: nat).((eq nat (plus h1 n2) (plus +h2 n1)) \to (P (ASort h1 n1) (ASort h2 n2)))))))) (f0: (\forall (a1: +A).(\forall (a2: A).((leqz a1 a2) \to ((P a1 a2) \to (\forall (a3: +A).(\forall (a4: A).((leqz a3 a4) \to ((P a3 a4) \to (P (AHead a1 a3) (AHead +a2 a4))))))))))) (a: A) (a0: A) (l: leqz a a0) on l: P a a0 \def match l with +[(leqz_sort h1 h2 n1 n2 e) \Rightarrow (f h1 h2 n1 n2 e) | (leqz_head a1 a2 +l0 a3 a4 l1) \Rightarrow (let TMP_1 \def ((leqz_ind P f f0) a1 a2 l0) in (let +TMP_2 \def ((leqz_ind P f f0) a3 a4 l1) in (f0 a1 a2 l0 TMP_1 a3 a4 l1 +TMP_2)))]. + diff --git a/matita/matita/contribs/lambdadelta/basic_1/ex0/props.ma b/matita/matita/contribs/lambdadelta/basic_1/ex0/props.ma index 96dd77da3..2f2e8e11e 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/ex0/props.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/ex0/props.ma @@ -14,194 +14,371 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/ex0/defs.ma". +include "basic_1/ex0/fwd.ma". -include "Basic-1/leq/defs.ma". +include "basic_1/leq/fwd.ma". -include "Basic-1/aplus/props.ma". +include "basic_1/aplus/props.ma". theorem aplus_gz_le: \forall (k: nat).(\forall (h: nat).(\forall (n: nat).((le h k) \to (eq A (aplus gz (ASort h n) k) (ASort O (plus (minus k h) n)))))) \def - \lambda (k: nat).(nat_ind (\lambda (n: nat).(\forall (h: nat).(\forall (n0: -nat).((le h n) \to (eq A (aplus gz (ASort h n0) n) (ASort O (plus (minus n h) -n0))))))) (\lambda (h: nat).(\lambda (n: nat).(\lambda (H: (le h O)).(let H_y -\def (le_n_O_eq h H) in (eq_ind nat O (\lambda (n0: nat).(eq A (ASort n0 n) -(ASort O n))) (refl_equal A (ASort O n)) h H_y))))) (\lambda (k0: -nat).(\lambda (IH: ((\forall (h: nat).(\forall (n: nat).((le h k0) \to (eq A -(aplus gz (ASort h n) k0) (ASort O (plus (minus k0 h) n)))))))).(\lambda (h: -nat).(nat_ind (\lambda (n: nat).(\forall (n0: nat).((le n (S k0)) \to (eq A -(asucc gz (aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O -\Rightarrow (S k0) | (S l) \Rightarrow (minus k0 l)]) n0)))))) (\lambda (n: -nat).(\lambda (_: (le O (S k0))).(eq_ind A (aplus gz (asucc gz (ASort O n)) -k0) (\lambda (a: A).(eq A a (ASort O (S (plus k0 n))))) (eq_ind_r A (ASort O -(plus (minus k0 O) (S n))) (\lambda (a: A).(eq A a (ASort O (S (plus k0 -n))))) (eq_ind nat k0 (\lambda (n0: nat).(eq A (ASort O (plus n0 (S n))) -(ASort O (S (plus k0 n))))) (eq_ind nat (S (plus k0 n)) (\lambda (n0: -nat).(eq A (ASort O n0) (ASort O (S (plus k0 n))))) (refl_equal A (ASort O (S -(plus k0 n)))) (plus k0 (S n)) (plus_n_Sm k0 n)) (minus k0 O) (minus_n_O k0)) -(aplus gz (ASort O (S n)) k0) (IH O (S n) (le_O_n k0))) (asucc gz (aplus gz -(ASort O n) k0)) (aplus_asucc gz k0 (ASort O n))))) (\lambda (n: -nat).(\lambda (_: ((\forall (n0: nat).((le n (S k0)) \to (eq A (asucc gz + \lambda (k: nat).(let TMP_6 \def (\lambda (n: nat).(\forall (h: +nat).(\forall (n0: nat).((le h n) \to (let TMP_1 \def (ASort h n0) in (let +TMP_2 \def (aplus gz TMP_1 n) in (let TMP_3 \def (minus n h) in (let TMP_4 +\def (plus TMP_3 n0) in (let TMP_5 \def (ASort O TMP_4) in (eq A TMP_2 +TMP_5)))))))))) in (let TMP_12 \def (\lambda (h: nat).(\lambda (n: +nat).(\lambda (H: (le h O)).(let H_y \def (le_n_O_eq h H) in (let TMP_9 \def +(\lambda (n0: nat).(let TMP_7 \def (ASort n0 n) in (let TMP_8 \def (ASort O +n) in (eq A TMP_7 TMP_8)))) in (let TMP_10 \def (ASort O n) in (let TMP_11 +\def (refl_equal A TMP_10) in (eq_ind nat O TMP_9 TMP_11 h H_y)))))))) in +(let TMP_103 \def (\lambda (k0: nat).(\lambda (IH: ((\forall (h: +nat).(\forall (n: nat).((le h k0) \to (eq A (aplus gz (ASort h n) k0) (ASort +O (plus (minus k0 h) n)))))))).(\lambda (h: nat).(let TMP_19 \def (\lambda +(n: nat).(\forall (n0: nat).((le n (S k0)) \to (let TMP_13 \def (ASort n n0) +in (let TMP_14 \def (aplus gz TMP_13 k0) in (let TMP_15 \def (asucc gz +TMP_14) in (let TMP_16 \def (match n with [O \Rightarrow (S k0) | (S l) +\Rightarrow (minus k0 l)]) in (let TMP_17 \def (plus TMP_16 n0) in (let +TMP_18 \def (ASort O TMP_17) in (eq A TMP_15 TMP_18)))))))))) in (let TMP_72 +\def (\lambda (n: nat).(\lambda (_: (le O (S k0))).(let TMP_20 \def (ASort O +n) in (let TMP_21 \def (asucc gz TMP_20) in (let TMP_22 \def (aplus gz TMP_21 +k0) in (let TMP_26 \def (\lambda (a: A).(let TMP_23 \def (plus k0 n) in (let +TMP_24 \def (S TMP_23) in (let TMP_25 \def (ASort O TMP_24) in (eq A a +TMP_25))))) in (let TMP_27 \def (minus k0 O) in (let TMP_28 \def (S n) in +(let TMP_29 \def (plus TMP_27 TMP_28) in (let TMP_30 \def (ASort O TMP_29) in +(let TMP_34 \def (\lambda (a: A).(let TMP_31 \def (plus k0 n) in (let TMP_32 +\def (S TMP_31) in (let TMP_33 \def (ASort O TMP_32) in (eq A a TMP_33))))) +in (let TMP_41 \def (\lambda (n0: nat).(let TMP_35 \def (S n) in (let TMP_36 +\def (plus n0 TMP_35) in (let TMP_37 \def (ASort O TMP_36) in (let TMP_38 +\def (plus k0 n) in (let TMP_39 \def (S TMP_38) in (let TMP_40 \def (ASort O +TMP_39) in (eq A TMP_37 TMP_40)))))))) in (let TMP_42 \def (plus k0 n) in +(let TMP_43 \def (S TMP_42) in (let TMP_48 \def (\lambda (n0: nat).(let +TMP_44 \def (ASort O n0) in (let TMP_45 \def (plus k0 n) in (let TMP_46 \def +(S TMP_45) in (let TMP_47 \def (ASort O TMP_46) in (eq A TMP_44 TMP_47)))))) +in (let TMP_49 \def (plus k0 n) in (let TMP_50 \def (S TMP_49) in (let TMP_51 +\def (ASort O TMP_50) in (let TMP_52 \def (refl_equal A TMP_51) in (let +TMP_53 \def (S n) in (let TMP_54 \def (plus k0 TMP_53) in (let TMP_55 \def +(plus_n_Sm k0 n) in (let TMP_56 \def (eq_ind nat TMP_43 TMP_48 TMP_52 TMP_54 +TMP_55) in (let TMP_57 \def (minus k0 O) in (let TMP_58 \def (minus_n_O k0) +in (let TMP_59 \def (eq_ind nat k0 TMP_41 TMP_56 TMP_57 TMP_58) in (let +TMP_60 \def (S n) in (let TMP_61 \def (ASort O TMP_60) in (let TMP_62 \def +(aplus gz TMP_61 k0) in (let TMP_63 \def (S n) in (let TMP_64 \def (le_O_n +k0) in (let TMP_65 \def (IH O TMP_63 TMP_64) in (let TMP_66 \def (eq_ind_r A +TMP_30 TMP_34 TMP_59 TMP_62 TMP_65) in (let TMP_67 \def (ASort O n) in (let +TMP_68 \def (aplus gz TMP_67 k0) in (let TMP_69 \def (asucc gz TMP_68) in +(let TMP_70 \def (ASort O n) in (let TMP_71 \def (aplus_asucc gz k0 TMP_70) +in (eq_ind A TMP_22 TMP_26 TMP_66 TMP_69 +TMP_71))))))))))))))))))))))))))))))))))))))) in (let TMP_102 \def (\lambda +(n: nat).(\lambda (_: ((\forall (n0: nat).((le n (S k0)) \to (eq A (asucc gz (aplus gz (ASort n n0) k0)) (ASort O (plus (match n with [O \Rightarrow (S k0) | (S l) \Rightarrow (minus k0 l)]) n0))))))).(\lambda (n0: nat).(\lambda -(H0: (le (S n) (S k0))).(let H_y \def (le_S_n n k0 H0) in (eq_ind A (aplus gz -(ASort n n0) k0) (\lambda (a: A).(eq A (asucc gz (aplus gz (ASort (S n) n0) -k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n) n0)) k0) (\lambda (a: -A).(eq A a (aplus gz (ASort n n0) k0))) (refl_equal A (aplus gz (ASort n n0) -k0)) (asucc gz (aplus gz (ASort (S n) n0) k0)) (aplus_asucc gz k0 (ASort (S -n) n0))) (ASort O (plus (minus k0 n) n0)) (IH n n0 H_y))))))) h)))) k). -(* COMMENTS -Initial nodes: 683 -END *) +(H0: (le (S n) (S k0))).(let H_y \def (le_S_n n k0 H0) in (let TMP_73 \def +(ASort n n0) in (let TMP_74 \def (aplus gz TMP_73 k0) in (let TMP_79 \def +(\lambda (a: A).(let TMP_75 \def (S n) in (let TMP_76 \def (ASort TMP_75 n0) +in (let TMP_77 \def (aplus gz TMP_76 k0) in (let TMP_78 \def (asucc gz +TMP_77) in (eq A TMP_78 a)))))) in (let TMP_80 \def (S n) in (let TMP_81 \def +(ASort TMP_80 n0) in (let TMP_82 \def (asucc gz TMP_81) in (let TMP_83 \def +(aplus gz TMP_82 k0) in (let TMP_86 \def (\lambda (a: A).(let TMP_84 \def +(ASort n n0) in (let TMP_85 \def (aplus gz TMP_84 k0) in (eq A a TMP_85)))) +in (let TMP_87 \def (ASort n n0) in (let TMP_88 \def (aplus gz TMP_87 k0) in +(let TMP_89 \def (refl_equal A TMP_88) in (let TMP_90 \def (S n) in (let +TMP_91 \def (ASort TMP_90 n0) in (let TMP_92 \def (aplus gz TMP_91 k0) in +(let TMP_93 \def (asucc gz TMP_92) in (let TMP_94 \def (S n) in (let TMP_95 +\def (ASort TMP_94 n0) in (let TMP_96 \def (aplus_asucc gz k0 TMP_95) in (let +TMP_97 \def (eq_ind A TMP_83 TMP_86 TMP_89 TMP_93 TMP_96) in (let TMP_98 \def +(minus k0 n) in (let TMP_99 \def (plus TMP_98 n0) in (let TMP_100 \def (ASort +O TMP_99) in (let TMP_101 \def (IH n n0 H_y) in (eq_ind A TMP_74 TMP_79 +TMP_97 TMP_100 TMP_101))))))))))))))))))))))))))))) in (nat_ind TMP_19 TMP_72 +TMP_102 h))))))) in (nat_ind TMP_6 TMP_12 TMP_103 k)))). theorem aplus_gz_ge: \forall (n: nat).(\forall (k: nat).(\forall (h: nat).((le k h) \to (eq A (aplus gz (ASort h n) k) (ASort (minus h k) n))))) \def - \lambda (n: nat).(\lambda (k: nat).(nat_ind (\lambda (n0: nat).(\forall (h: -nat).((le n0 h) \to (eq A (aplus gz (ASort h n) n0) (ASort (minus h n0) -n))))) (\lambda (h: nat).(\lambda (_: (le O h)).(eq_ind nat h (\lambda (n0: -nat).(eq A (ASort h n) (ASort n0 n))) (refl_equal A (ASort h n)) (minus h O) -(minus_n_O h)))) (\lambda (k0: nat).(\lambda (IH: ((\forall (h: nat).((le k0 -h) \to (eq A (aplus gz (ASort h n) k0) (ASort (minus h k0) n)))))).(\lambda -(h: nat).(nat_ind (\lambda (n0: nat).((le (S k0) n0) \to (eq A (asucc gz -(aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n)))) (\lambda (H: (le -(S k0) O)).(ex2_ind nat (\lambda (n0: nat).(eq nat O (S n0))) (\lambda (n0: -nat).(le k0 n0)) (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O n)) -(\lambda (x: nat).(\lambda (H0: (eq nat O (S x))).(\lambda (_: (le k0 -x)).(let H2 \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 x) H0) in (False_ind (eq A (asucc gz (aplus gz (ASort O n) k0)) (ASort O -n)) H2))))) (le_gen_S k0 O H))) (\lambda (n0: nat).(\lambda (_: (((le (S k0) -n0) \to (eq A (asucc gz (aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) -n))))).(\lambda (H0: (le (S k0) (S n0))).(let H_y \def (le_S_n k0 n0 H0) in -(eq_ind A (aplus gz (ASort n0 n) k0) (\lambda (a: A).(eq A (asucc gz (aplus -gz (ASort (S n0) n) k0)) a)) (eq_ind A (aplus gz (asucc gz (ASort (S n0) n)) -k0) (\lambda (a: A).(eq A a (aplus gz (ASort n0 n) k0))) (refl_equal A (aplus -gz (ASort n0 n) k0)) (asucc gz (aplus gz (ASort (S n0) n) k0)) (aplus_asucc -gz k0 (ASort (S n0) n))) (ASort (minus n0 k0) n) (IH n0 H_y)))))) h)))) k)). -(* COMMENTS -Initial nodes: 524 -END *) + \lambda (n: nat).(\lambda (k: nat).(let TMP_5 \def (\lambda (n0: +nat).(\forall (h: nat).((le n0 h) \to (let TMP_1 \def (ASort h n) in (let +TMP_2 \def (aplus gz TMP_1 n0) in (let TMP_3 \def (minus h n0) in (let TMP_4 +\def (ASort TMP_3 n) in (eq A TMP_2 TMP_4)))))))) in (let TMP_13 \def +(\lambda (h: nat).(\lambda (_: (le O h)).(let TMP_8 \def (\lambda (n0: +nat).(let TMP_6 \def (ASort h n) in (let TMP_7 \def (ASort n0 n) in (eq A +TMP_6 TMP_7)))) in (let TMP_9 \def (ASort h n) in (let TMP_10 \def +(refl_equal A TMP_9) in (let TMP_11 \def (minus h O) in (let TMP_12 \def +(minus_n_O h) in (eq_ind nat h TMP_8 TMP_10 TMP_11 TMP_12)))))))) in (let +TMP_68 \def (\lambda (k0: nat).(\lambda (IH: ((\forall (h: nat).((le k0 h) +\to (eq A (aplus gz (ASort h n) k0) (ASort (minus h k0) n)))))).(\lambda (h: +nat).(let TMP_20 \def (\lambda (n0: nat).((le (S k0) n0) \to (let TMP_14 \def +(ASort n0 n) in (let TMP_15 \def (aplus gz TMP_14 k0) in (let TMP_16 \def +(asucc gz TMP_15) in (let TMP_17 \def (S k0) in (let TMP_18 \def (minus n0 +TMP_17) in (let TMP_19 \def (ASort TMP_18 n) in (eq A TMP_16 TMP_19))))))))) +in (let TMP_38 \def (\lambda (H: (le (S k0) O)).(let TMP_22 \def (\lambda +(n0: nat).(let TMP_21 \def (S n0) in (eq nat O TMP_21))) in (let TMP_23 \def +(\lambda (n0: nat).(le k0 n0)) in (let TMP_24 \def (ASort O n) in (let TMP_25 +\def (aplus gz TMP_24 k0) in (let TMP_26 \def (asucc gz TMP_25) in (let +TMP_27 \def (ASort O n) in (let TMP_28 \def (eq A TMP_26 TMP_27) in (let +TMP_36 \def (\lambda (x: nat).(\lambda (H0: (eq nat O (S x))).(\lambda (_: +(le k0 x)).(let TMP_29 \def (\lambda (ee: nat).(match ee with [O \Rightarrow +True | (S _) \Rightarrow False])) in (let TMP_30 \def (S x) in (let H2 \def +(eq_ind nat O TMP_29 I TMP_30 H0) in (let TMP_31 \def (ASort O n) in (let +TMP_32 \def (aplus gz TMP_31 k0) in (let TMP_33 \def (asucc gz TMP_32) in +(let TMP_34 \def (ASort O n) in (let TMP_35 \def (eq A TMP_33 TMP_34) in +(False_ind TMP_35 H2)))))))))))) in (let TMP_37 \def (le_gen_S k0 O H) in +(ex2_ind nat TMP_22 TMP_23 TMP_28 TMP_36 TMP_37))))))))))) in (let TMP_67 +\def (\lambda (n0: nat).(\lambda (_: (((le (S k0) n0) \to (eq A (asucc gz +(aplus gz (ASort n0 n) k0)) (ASort (minus n0 (S k0)) n))))).(\lambda (H0: (le +(S k0) (S n0))).(let H_y \def (le_S_n k0 n0 H0) in (let TMP_39 \def (ASort n0 +n) in (let TMP_40 \def (aplus gz TMP_39 k0) in (let TMP_45 \def (\lambda (a: +A).(let TMP_41 \def (S n0) in (let TMP_42 \def (ASort TMP_41 n) in (let +TMP_43 \def (aplus gz TMP_42 k0) in (let TMP_44 \def (asucc gz TMP_43) in (eq +A TMP_44 a)))))) in (let TMP_46 \def (S n0) in (let TMP_47 \def (ASort TMP_46 +n) in (let TMP_48 \def (asucc gz TMP_47) in (let TMP_49 \def (aplus gz TMP_48 +k0) in (let TMP_52 \def (\lambda (a: A).(let TMP_50 \def (ASort n0 n) in (let +TMP_51 \def (aplus gz TMP_50 k0) in (eq A a TMP_51)))) in (let TMP_53 \def +(ASort n0 n) in (let TMP_54 \def (aplus gz TMP_53 k0) in (let TMP_55 \def +(refl_equal A TMP_54) in (let TMP_56 \def (S n0) in (let TMP_57 \def (ASort +TMP_56 n) in (let TMP_58 \def (aplus gz TMP_57 k0) in (let TMP_59 \def (asucc +gz TMP_58) in (let TMP_60 \def (S n0) in (let TMP_61 \def (ASort TMP_60 n) in +(let TMP_62 \def (aplus_asucc gz k0 TMP_61) in (let TMP_63 \def (eq_ind A +TMP_49 TMP_52 TMP_55 TMP_59 TMP_62) in (let TMP_64 \def (minus n0 k0) in (let +TMP_65 \def (ASort TMP_64 n) in (let TMP_66 \def (IH n0 H_y) in (eq_ind A +TMP_40 TMP_45 TMP_63 TMP_65 TMP_66))))))))))))))))))))))))))) in (nat_ind +TMP_20 TMP_38 TMP_67 h))))))) in (nat_ind TMP_5 TMP_13 TMP_68 k))))). theorem next_plus_gz: \forall (n: nat).(\forall (h: nat).(eq nat (next_plus gz n h) (plus h n))) \def - \lambda (n: nat).(\lambda (h: nat).(nat_ind (\lambda (n0: nat).(eq nat -(next_plus gz n n0) (plus n0 n))) (refl_equal nat n) (\lambda (n0: -nat).(\lambda (H: (eq nat (next_plus gz n n0) (plus n0 n))).(f_equal nat nat -S (next_plus gz n n0) (plus n0 n) H))) h)). -(* COMMENTS -Initial nodes: 77 -END *) + \lambda (n: nat).(\lambda (h: nat).(let TMP_3 \def (\lambda (n0: nat).(let +TMP_1 \def (next_plus gz n n0) in (let TMP_2 \def (plus n0 n) in (eq nat +TMP_1 TMP_2)))) in (let TMP_4 \def (refl_equal nat n) in (let TMP_7 \def +(\lambda (n0: nat).(\lambda (H: (eq nat (next_plus gz n n0) (plus n0 +n))).(let TMP_5 \def (next_plus gz n n0) in (let TMP_6 \def (plus n0 n) in +(f_equal nat nat S TMP_5 TMP_6 H))))) in (nat_ind TMP_3 TMP_4 TMP_7 h))))). theorem leqz_leq: \forall (a1: A).(\forall (a2: A).((leq gz a1 a2) \to (leqz a1 a2))) \def - \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq gz a1 a2)).(leq_ind gz -(\lambda (a: A).(\lambda (a0: A).(leqz a a0))) (\lambda (h1: nat).(\lambda -(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda -(H0: (eq A (aplus gz (ASort h1 n1) k) (aplus gz (ASort h2 n2) k))).(lt_le_e k -h1 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H1: (lt k h1)).(lt_le_e k h2 -(leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k h2)).(let H3 \def -(eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort -h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 (le_S_n k h1 -(le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) k) -(\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort (minus h2 k) n2) -(aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in (let H5 \def -(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with -[(ASort n _) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec minus (n: nat) -on n: (nat \to nat) \def (\lambda (m: nat).(match n with [O \Rightarrow O | -(S k0) \Rightarrow (match m with [O \Rightarrow (S k0) | (S l) \Rightarrow -(minus k0 l)])])) in minus) h1 k)])) (ASort (minus h1 k) n1) (ASort (minus h2 -k) n2) H4) in ((let H6 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) -\Rightarrow n1])) (ASort (minus h1 k) n1) (ASort (minus h2 k) n2) H4) in -(\lambda (H7: (eq nat (minus h1 k) (minus h2 k))).(eq_ind nat n1 (\lambda (n: -nat).(leqz (ASort h1 n1) (ASort h2 n))) (eq_ind nat h1 (\lambda (n: -nat).(leqz (ASort h1 n1) (ASort n n1))) (leqz_sort h1 h1 n1 n1 (refl_equal -nat (plus h1 n1))) h2 (minus_minus k h1 h2 (le_S_n k h1 (le_S (S k) h1 H1)) -(le_S_n k h2 (le_S (S k) h2 H2)) H7)) n2 H6))) H5))))) (\lambda (H2: (le h2 -k)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a -(aplus gz (ASort h2 n2) k))) H0 (ASort (minus h1 k) n1) (aplus_gz_ge n1 k h1 -(le_S_n k h1 (le_S (S k) h1 H1)))) in (let H4 \def (eq_ind A (aplus gz (ASort -h2 n2) k) (\lambda (a: A).(eq A (ASort (minus h1 k) n1) a)) H3 (ASort O (plus -(minus k h2) n2)) (aplus_gz_le k h2 n2 H2)) in (let H5 \def (eq_ind nat -(minus h1 k) (\lambda (n: nat).(eq A (ASort n n1) (ASort O (plus (minus k h2) -n2)))) H4 (S (minus h1 (S k))) (minus_x_Sy h1 k H1)) in (let H6 \def (eq_ind -A (ASort (S (minus h1 (S k))) n1) (\lambda (ee: A).(match ee in A return -(\lambda (_: A).Prop) with [(ASort n _) \Rightarrow (match n in nat return -(\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True]) -| (AHead _ _) \Rightarrow False])) I (ASort O (plus (minus k h2) n2)) H5) in -(False_ind (leqz (ASort h1 n1) (ASort h2 n2)) H6)))))))) (\lambda (H1: (le h1 -k)).(lt_le_e k h2 (leqz (ASort h1 n1) (ASort h2 n2)) (\lambda (H2: (lt k -h2)).(let H3 \def (eq_ind A (aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A -a (aplus gz (ASort h2 n2) k))) H0 (ASort O (plus (minus k h1) n1)) -(aplus_gz_le k h1 n1 H1)) in (let H4 \def (eq_ind A (aplus gz (ASort h2 n2) -k) (\lambda (a: A).(eq A (ASort O (plus (minus k h1) n1)) a)) H3 (ASort -(minus h2 k) n2) (aplus_gz_ge n2 k h2 (le_S_n k h2 (le_S (S k) h2 H2)))) in -(let H5 \def (sym_eq A (ASort O (plus (minus k h1) n1)) (ASort (minus h2 k) -n2) H4) in (let H6 \def (eq_ind nat (minus h2 k) (\lambda (n: nat).(eq A -(ASort n n2) (ASort O (plus (minus k h1) n1)))) H5 (S (minus h2 (S k))) -(minus_x_Sy h2 k H2)) in (let H7 \def (eq_ind A (ASort (S (minus h2 (S k))) -n2) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort -n _) \Rightarrow (match n in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True]) | (AHead _ _) \Rightarrow -False])) I (ASort O (plus (minus k h1) n1)) H6) in (False_ind (leqz (ASort h1 -n1) (ASort h2 n2)) H7))))))) (\lambda (H2: (le h2 k)).(let H3 \def (eq_ind A -(aplus gz (ASort h1 n1) k) (\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) -k))) H0 (ASort O (plus (minus k h1) n1)) (aplus_gz_le k h1 n1 H1)) in (let H4 -\def (eq_ind A (aplus gz (ASort h2 n2) k) (\lambda (a: A).(eq A (ASort O -(plus (minus k h1) n1)) a)) H3 (ASort O (plus (minus k h2) n2)) (aplus_gz_le -k h2 n2 H2)) in (let H5 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) -\Rightarrow ((let rec plus (n: nat) on n: (nat \to nat) \def (\lambda (m: -nat).(match n with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in -plus) (minus k h1) n1)])) (ASort O (plus (minus k h1) n1)) (ASort O (plus -(minus k h2) n2)) H4) in (let H_y \def (plus_plus k h1 h2 n1 n2 H1 H2 H5) in -(leqz_sort h1 h2 n1 n2 H_y))))))))))))))) (\lambda (a0: A).(\lambda (a3: -A).(\lambda (_: (leq gz a0 a3)).(\lambda (H1: (leqz a0 a3)).(\lambda (a4: -A).(\lambda (a5: A).(\lambda (_: (leq gz a4 a5)).(\lambda (H3: (leqz a4 -a5)).(leqz_head a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))). -(* COMMENTS -Initial nodes: 1375 -END *) + \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq gz a1 a2)).(let TMP_1 +\def (\lambda (a: A).(\lambda (a0: A).(leqz a a0))) in (let TMP_225 \def +(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: +nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus gz (ASort h1 n1) k) (aplus +gz (ASort h2 n2) k))).(let TMP_2 \def (ASort h1 n1) in (let TMP_3 \def (ASort +h2 n2) in (let TMP_4 \def (leqz TMP_2 TMP_3) in (let TMP_136 \def (\lambda +(H1: (lt k h1)).(let TMP_5 \def (ASort h1 n1) in (let TMP_6 \def (ASort h2 +n2) in (let TMP_7 \def (leqz TMP_5 TMP_6) in (let TMP_86 \def (\lambda (H2: +(lt k h2)).(let TMP_8 \def (ASort h1 n1) in (let TMP_9 \def (aplus gz TMP_8 +k) in (let TMP_12 \def (\lambda (a: A).(let TMP_10 \def (ASort h2 n2) in (let +TMP_11 \def (aplus gz TMP_10 k) in (eq A a TMP_11)))) in (let TMP_13 \def +(minus h1 k) in (let TMP_14 \def (ASort TMP_13 n1) in (let TMP_15 \def (S k) +in (let TMP_16 \def (S h1) in (let TMP_17 \def (S k) in (let TMP_18 \def (S +TMP_17) in (let TMP_19 \def (S h1) in (let TMP_20 \def (S k) in (let TMP_21 +\def (le_n_S TMP_20 h1 H1) in (let TMP_22 \def (le_S TMP_18 TMP_19 TMP_21) in +(let TMP_23 \def (le_S_n TMP_15 TMP_16 TMP_22) in (let TMP_24 \def (le_S_n k +h1 TMP_23) in (let TMP_25 \def (aplus_gz_ge n1 k h1 TMP_24) in (let H3 \def +(eq_ind A TMP_9 TMP_12 H0 TMP_14 TMP_25) in (let TMP_26 \def (ASort h2 n2) in +(let TMP_27 \def (aplus gz TMP_26 k) in (let TMP_30 \def (\lambda (a: A).(let +TMP_28 \def (minus h1 k) in (let TMP_29 \def (ASort TMP_28 n1) in (eq A +TMP_29 a)))) in (let TMP_31 \def (minus h2 k) in (let TMP_32 \def (ASort +TMP_31 n2) in (let TMP_33 \def (S k) in (let TMP_34 \def (S h2) in (let +TMP_35 \def (S k) in (let TMP_36 \def (S TMP_35) in (let TMP_37 \def (S h2) +in (let TMP_38 \def (S k) in (let TMP_39 \def (le_n_S TMP_38 h2 H2) in (let +TMP_40 \def (le_S TMP_36 TMP_37 TMP_39) in (let TMP_41 \def (le_S_n TMP_33 +TMP_34 TMP_40) in (let TMP_42 \def (le_S_n k h2 TMP_41) in (let TMP_43 \def +(aplus_gz_ge n2 k h2 TMP_42) in (let H4 \def (eq_ind A TMP_27 TMP_30 H3 +TMP_32 TMP_43) in (let TMP_44 \def (\lambda (e: A).(match e with [(ASort n _) +\Rightarrow n | (AHead _ _) \Rightarrow (minus h1 k)])) in (let TMP_45 \def +(minus h1 k) in (let TMP_46 \def (ASort TMP_45 n1) in (let TMP_47 \def (minus +h2 k) in (let TMP_48 \def (ASort TMP_47 n2) in (let H5 \def (f_equal A nat +TMP_44 TMP_46 TMP_48 H4) in (let TMP_49 \def (\lambda (e: A).(match e with +[(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow n1])) in (let TMP_50 +\def (minus h1 k) in (let TMP_51 \def (ASort TMP_50 n1) in (let TMP_52 \def +(minus h2 k) in (let TMP_53 \def (ASort TMP_52 n2) in (let H6 \def (f_equal A +nat TMP_49 TMP_51 TMP_53 H4) in (let TMP_85 \def (\lambda (H7: (eq nat (minus +h1 k) (minus h2 k))).(let TMP_56 \def (\lambda (n: nat).(let TMP_54 \def +(ASort h1 n1) in (let TMP_55 \def (ASort h2 n) in (leqz TMP_54 TMP_55)))) in +(let TMP_59 \def (\lambda (n: nat).(let TMP_57 \def (ASort h1 n1) in (let +TMP_58 \def (ASort n n1) in (leqz TMP_57 TMP_58)))) in (let TMP_60 \def (plus +h1 n1) in (let TMP_61 \def (refl_equal nat TMP_60) in (let TMP_62 \def +(leqz_sort h1 h1 n1 n1 TMP_61) in (let TMP_63 \def (S k) in (let TMP_64 \def +(S h1) in (let TMP_65 \def (S k) in (let TMP_66 \def (S TMP_65) in (let +TMP_67 \def (S h1) in (let TMP_68 \def (S k) in (let TMP_69 \def (le_n_S +TMP_68 h1 H1) in (let TMP_70 \def (le_S TMP_66 TMP_67 TMP_69) in (let TMP_71 +\def (le_S_n TMP_63 TMP_64 TMP_70) in (let TMP_72 \def (le_S_n k h1 TMP_71) +in (let TMP_73 \def (S k) in (let TMP_74 \def (S h2) in (let TMP_75 \def (S +k) in (let TMP_76 \def (S TMP_75) in (let TMP_77 \def (S h2) in (let TMP_78 +\def (S k) in (let TMP_79 \def (le_n_S TMP_78 h2 H2) in (let TMP_80 \def +(le_S TMP_76 TMP_77 TMP_79) in (let TMP_81 \def (le_S_n TMP_73 TMP_74 TMP_80) +in (let TMP_82 \def (le_S_n k h2 TMP_81) in (let TMP_83 \def (minus_minus k +h1 h2 TMP_72 TMP_82 H7) in (let TMP_84 \def (eq_ind nat h1 TMP_59 TMP_62 h2 +TMP_83) in (eq_ind nat n1 TMP_56 TMP_84 n2 H6))))))))))))))))))))))))))))) in +(TMP_85 H5))))))))))))))))))))))))))))))))))))))))))))))))) in (let TMP_135 +\def (\lambda (H2: (le h2 k)).(let TMP_87 \def (ASort h1 n1) in (let TMP_88 +\def (aplus gz TMP_87 k) in (let TMP_91 \def (\lambda (a: A).(let TMP_89 \def +(ASort h2 n2) in (let TMP_90 \def (aplus gz TMP_89 k) in (eq A a TMP_90)))) +in (let TMP_92 \def (minus h1 k) in (let TMP_93 \def (ASort TMP_92 n1) in +(let TMP_94 \def (S k) in (let TMP_95 \def (S h1) in (let TMP_96 \def (S k) +in (let TMP_97 \def (S TMP_96) in (let TMP_98 \def (S h1) in (let TMP_99 \def +(S k) in (let TMP_100 \def (le_n_S TMP_99 h1 H1) in (let TMP_101 \def (le_S +TMP_97 TMP_98 TMP_100) in (let TMP_102 \def (le_S_n TMP_94 TMP_95 TMP_101) in +(let TMP_103 \def (le_S_n k h1 TMP_102) in (let TMP_104 \def (aplus_gz_ge n1 +k h1 TMP_103) in (let H3 \def (eq_ind A TMP_88 TMP_91 H0 TMP_93 TMP_104) in +(let TMP_105 \def (ASort h2 n2) in (let TMP_106 \def (aplus gz TMP_105 k) in +(let TMP_109 \def (\lambda (a: A).(let TMP_107 \def (minus h1 k) in (let +TMP_108 \def (ASort TMP_107 n1) in (eq A TMP_108 a)))) in (let TMP_110 \def +(minus k h2) in (let TMP_111 \def (plus TMP_110 n2) in (let TMP_112 \def +(ASort O TMP_111) in (let TMP_113 \def (aplus_gz_le k h2 n2 H2) in (let H4 +\def (eq_ind A TMP_106 TMP_109 H3 TMP_112 TMP_113) in (let TMP_114 \def +(minus h1 k) in (let TMP_119 \def (\lambda (n: nat).(let TMP_115 \def (ASort +n n1) in (let TMP_116 \def (minus k h2) in (let TMP_117 \def (plus TMP_116 +n2) in (let TMP_118 \def (ASort O TMP_117) in (eq A TMP_115 TMP_118)))))) in +(let TMP_120 \def (S k) in (let TMP_121 \def (minus h1 TMP_120) in (let +TMP_122 \def (S TMP_121) in (let TMP_123 \def (minus_x_Sy h1 k H1) in (let H5 +\def (eq_ind nat TMP_114 TMP_119 H4 TMP_122 TMP_123) in (let TMP_124 \def (S +k) in (let TMP_125 \def (minus h1 TMP_124) in (let TMP_126 \def (S TMP_125) +in (let TMP_127 \def (ASort TMP_126 n1) in (let TMP_128 \def (\lambda (ee: +A).(match ee with [(ASort n _) \Rightarrow (match n with [O \Rightarrow False +| (S _) \Rightarrow True]) | (AHead _ _) \Rightarrow False])) in (let TMP_129 +\def (minus k h2) in (let TMP_130 \def (plus TMP_129 n2) in (let TMP_131 \def +(ASort O TMP_130) in (let H6 \def (eq_ind A TMP_127 TMP_128 I TMP_131 H5) in +(let TMP_132 \def (ASort h1 n1) in (let TMP_133 \def (ASort h2 n2) in (let +TMP_134 \def (leqz TMP_132 TMP_133) in (False_ind TMP_134 +H6)))))))))))))))))))))))))))))))))))))))))))))) in (lt_le_e k h2 TMP_7 +TMP_86 TMP_135))))))) in (let TMP_224 \def (\lambda (H1: (le h1 k)).(let +TMP_137 \def (ASort h1 n1) in (let TMP_138 \def (ASort h2 n2) in (let TMP_139 +\def (leqz TMP_137 TMP_138) in (let TMP_194 \def (\lambda (H2: (lt k +h2)).(let TMP_140 \def (ASort h1 n1) in (let TMP_141 \def (aplus gz TMP_140 +k) in (let TMP_144 \def (\lambda (a: A).(let TMP_142 \def (ASort h2 n2) in +(let TMP_143 \def (aplus gz TMP_142 k) in (eq A a TMP_143)))) in (let TMP_145 +\def (minus k h1) in (let TMP_146 \def (plus TMP_145 n1) in (let TMP_147 \def +(ASort O TMP_146) in (let TMP_148 \def (aplus_gz_le k h1 n1 H1) in (let H3 +\def (eq_ind A TMP_141 TMP_144 H0 TMP_147 TMP_148) in (let TMP_149 \def +(ASort h2 n2) in (let TMP_150 \def (aplus gz TMP_149 k) in (let TMP_154 \def +(\lambda (a: A).(let TMP_151 \def (minus k h1) in (let TMP_152 \def (plus +TMP_151 n1) in (let TMP_153 \def (ASort O TMP_152) in (eq A TMP_153 a))))) in +(let TMP_155 \def (minus h2 k) in (let TMP_156 \def (ASort TMP_155 n2) in +(let TMP_157 \def (S k) in (let TMP_158 \def (S h2) in (let TMP_159 \def (S +k) in (let TMP_160 \def (S TMP_159) in (let TMP_161 \def (S h2) in (let +TMP_162 \def (S k) in (let TMP_163 \def (le_n_S TMP_162 h2 H2) in (let +TMP_164 \def (le_S TMP_160 TMP_161 TMP_163) in (let TMP_165 \def (le_S_n +TMP_157 TMP_158 TMP_164) in (let TMP_166 \def (le_S_n k h2 TMP_165) in (let +TMP_167 \def (aplus_gz_ge n2 k h2 TMP_166) in (let H4 \def (eq_ind A TMP_150 +TMP_154 H3 TMP_156 TMP_167) in (let TMP_168 \def (minus k h1) in (let TMP_169 +\def (plus TMP_168 n1) in (let TMP_170 \def (ASort O TMP_169) in (let TMP_171 +\def (minus h2 k) in (let TMP_172 \def (ASort TMP_171 n2) in (let H5 \def +(sym_eq A TMP_170 TMP_172 H4) in (let TMP_173 \def (minus h2 k) in (let +TMP_178 \def (\lambda (n: nat).(let TMP_174 \def (ASort n n2) in (let TMP_175 +\def (minus k h1) in (let TMP_176 \def (plus TMP_175 n1) in (let TMP_177 \def +(ASort O TMP_176) in (eq A TMP_174 TMP_177)))))) in (let TMP_179 \def (S k) +in (let TMP_180 \def (minus h2 TMP_179) in (let TMP_181 \def (S TMP_180) in +(let TMP_182 \def (minus_x_Sy h2 k H2) in (let H6 \def (eq_ind nat TMP_173 +TMP_178 H5 TMP_181 TMP_182) in (let TMP_183 \def (S k) in (let TMP_184 \def +(minus h2 TMP_183) in (let TMP_185 \def (S TMP_184) in (let TMP_186 \def +(ASort TMP_185 n2) in (let TMP_187 \def (\lambda (ee: A).(match ee with +[(ASort n _) \Rightarrow (match n with [O \Rightarrow False | (S _) +\Rightarrow True]) | (AHead _ _) \Rightarrow False])) in (let TMP_188 \def +(minus k h1) in (let TMP_189 \def (plus TMP_188 n1) in (let TMP_190 \def +(ASort O TMP_189) in (let H7 \def (eq_ind A TMP_186 TMP_187 I TMP_190 H6) in +(let TMP_191 \def (ASort h1 n1) in (let TMP_192 \def (ASort h2 n2) in (let +TMP_193 \def (leqz TMP_191 TMP_192) in (False_ind TMP_193 +H7)))))))))))))))))))))))))))))))))))))))))))))))))))) in (let TMP_223 \def +(\lambda (H2: (le h2 k)).(let TMP_195 \def (ASort h1 n1) in (let TMP_196 \def +(aplus gz TMP_195 k) in (let TMP_199 \def (\lambda (a: A).(let TMP_197 \def +(ASort h2 n2) in (let TMP_198 \def (aplus gz TMP_197 k) in (eq A a +TMP_198)))) in (let TMP_200 \def (minus k h1) in (let TMP_201 \def (plus +TMP_200 n1) in (let TMP_202 \def (ASort O TMP_201) in (let TMP_203 \def +(aplus_gz_le k h1 n1 H1) in (let H3 \def (eq_ind A TMP_196 TMP_199 H0 TMP_202 +TMP_203) in (let TMP_204 \def (ASort h2 n2) in (let TMP_205 \def (aplus gz +TMP_204 k) in (let TMP_209 \def (\lambda (a: A).(let TMP_206 \def (minus k +h1) in (let TMP_207 \def (plus TMP_206 n1) in (let TMP_208 \def (ASort O +TMP_207) in (eq A TMP_208 a))))) in (let TMP_210 \def (minus k h2) in (let +TMP_211 \def (plus TMP_210 n2) in (let TMP_212 \def (ASort O TMP_211) in (let +TMP_213 \def (aplus_gz_le k h2 n2 H2) in (let H4 \def (eq_ind A TMP_205 +TMP_209 H3 TMP_212 TMP_213) in (let TMP_216 \def (\lambda (e: A).(match e +with [(ASort _ n) \Rightarrow n | (AHead _ _) \Rightarrow (let TMP_215 \def +(minus k h1) in (plus TMP_215 n1))])) in (let TMP_217 \def (minus k h1) in +(let TMP_218 \def (plus TMP_217 n1) in (let TMP_219 \def (ASort O TMP_218) in +(let TMP_220 \def (minus k h2) in (let TMP_221 \def (plus TMP_220 n2) in (let +TMP_222 \def (ASort O TMP_221) in (let H5 \def (f_equal A nat TMP_216 TMP_219 +TMP_222 H4) in (let H_y \def (plus_plus k h1 h2 n1 n2 H1 H2 H5) in (leqz_sort +h1 h2 n1 n2 H_y))))))))))))))))))))))))))) in (lt_le_e k h2 TMP_139 TMP_194 +TMP_223))))))) in (lt_le_e k h1 TMP_4 TMP_136 TMP_224)))))))))))) in (let +TMP_226 \def (\lambda (a0: A).(\lambda (a3: A).(\lambda (_: (leq gz a0 +a3)).(\lambda (H1: (leqz a0 a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda +(_: (leq gz a4 a5)).(\lambda (H3: (leqz a4 a5)).(leqz_head a0 a3 H1 a4 a5 +H3))))))))) in (leq_ind gz TMP_1 TMP_225 TMP_226 a1 a2 H)))))). theorem leq_leqz: \forall (a1: A).(\forall (a2: A).((leqz a1 a2) \to (leq gz a1 a2))) \def - \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leqz a1 a2)).(leqz_ind -(\lambda (a: A).(\lambda (a0: A).(leq gz a a0))) (\lambda (h1: nat).(\lambda -(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (H0: (eq nat (plus -h1 n2) (plus h2 n1))).(leq_sort gz h1 h2 n1 n2 (plus h1 h2) (eq_ind_r A -(ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus (plus h1 h2) h1))) -(\lambda (a: A).(eq A a (aplus gz (ASort h2 n2) (plus h1 h2)))) (eq_ind_r A -(ASort (minus h2 (plus h1 h2)) (next_plus gz n2 (minus (plus h1 h2) h2))) -(\lambda (a: A).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 (minus -(plus h1 h2) h1))) a)) (eq_ind_r nat h2 (\lambda (n: nat).(eq A (ASort (minus -h1 (plus h1 h2)) (next_plus gz n1 n)) (ASort (minus h2 (plus h1 h2)) -(next_plus gz n2 (minus (plus h1 h2) h2))))) (eq_ind_r nat h1 (\lambda (n: -nat).(eq A (ASort (minus h1 (plus h1 h2)) (next_plus gz n1 h2)) (ASort (minus -h2 (plus h1 h2)) (next_plus gz n2 n)))) (eq_ind_r nat O (\lambda (n: nat).(eq -A (ASort n (next_plus gz n1 h2)) (ASort (minus h2 (plus h1 h2)) (next_plus gz -n2 h1)))) (eq_ind_r nat O (\lambda (n: nat).(eq A (ASort O (next_plus gz n1 -h2)) (ASort n (next_plus gz n2 h1)))) (eq_ind_r nat (plus h2 n1) (\lambda (n: -nat).(eq A (ASort O n) (ASort O (next_plus gz n2 h1)))) (eq_ind_r nat (plus -h1 n2) (\lambda (n: nat).(eq A (ASort O (plus h2 n1)) (ASort O n))) (f_equal -nat A (ASort O) (plus h2 n1) (plus h1 n2) (sym_eq nat (plus h1 n2) (plus h2 -n1) H0)) (next_plus gz n2 h1) (next_plus_gz n2 h1)) (next_plus gz n1 h2) -(next_plus_gz n1 h2)) (minus h2 (plus h1 h2)) (O_minus h2 (plus h1 h2) -(le_plus_r h1 h2))) (minus h1 (plus h1 h2)) (O_minus h1 (plus h1 h2) -(le_plus_l h1 h2))) (minus (plus h1 h2) h2) (minus_plus_r h1 h2)) (minus -(plus h1 h2) h1) (minus_plus h1 h2)) (aplus gz (ASort h2 n2) (plus h1 h2)) -(aplus_asort_simpl gz (plus h1 h2) h2 n2)) (aplus gz (ASort h1 n1) (plus h1 -h2)) (aplus_asort_simpl gz (plus h1 h2) h1 n1)))))))) (\lambda (a0: -A).(\lambda (a3: A).(\lambda (_: (leqz a0 a3)).(\lambda (H1: (leq gz a0 -a3)).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leqz a4 a5)).(\lambda -(H3: (leq gz a4 a5)).(leq_head gz a0 a3 H1 a4 a5 H3))))))))) a1 a2 H))). -(* COMMENTS -Initial nodes: 717 -END *) + \lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leqz a1 a2)).(let TMP_1 \def +(\lambda (a: A).(\lambda (a0: A).(leq gz a a0))) in (let TMP_113 \def +(\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: +nat).(\lambda (H0: (eq nat (plus h1 n2) (plus h2 n1))).(let TMP_2 \def (plus +h1 h2) in (let TMP_3 \def (plus h1 h2) in (let TMP_4 \def (minus h1 TMP_3) in +(let TMP_5 \def (plus h1 h2) in (let TMP_6 \def (minus TMP_5 h1) in (let +TMP_7 \def (next_plus gz n1 TMP_6) in (let TMP_8 \def (ASort TMP_4 TMP_7) in +(let TMP_12 \def (\lambda (a: A).(let TMP_9 \def (ASort h2 n2) in (let TMP_10 +\def (plus h1 h2) in (let TMP_11 \def (aplus gz TMP_9 TMP_10) in (eq A a +TMP_11))))) in (let TMP_13 \def (plus h1 h2) in (let TMP_14 \def (minus h2 +TMP_13) in (let TMP_15 \def (plus h1 h2) in (let TMP_16 \def (minus TMP_15 +h2) in (let TMP_17 \def (next_plus gz n2 TMP_16) in (let TMP_18 \def (ASort +TMP_14 TMP_17) in (let TMP_25 \def (\lambda (a: A).(let TMP_19 \def (plus h1 +h2) in (let TMP_20 \def (minus h1 TMP_19) in (let TMP_21 \def (plus h1 h2) in +(let TMP_22 \def (minus TMP_21 h1) in (let TMP_23 \def (next_plus gz n1 +TMP_22) in (let TMP_24 \def (ASort TMP_20 TMP_23) in (eq A TMP_24 a)))))))) +in (let TMP_36 \def (\lambda (n: nat).(let TMP_26 \def (plus h1 h2) in (let +TMP_27 \def (minus h1 TMP_26) in (let TMP_28 \def (next_plus gz n1 n) in (let +TMP_29 \def (ASort TMP_27 TMP_28) in (let TMP_30 \def (plus h1 h2) in (let +TMP_31 \def (minus h2 TMP_30) in (let TMP_32 \def (plus h1 h2) in (let TMP_33 +\def (minus TMP_32 h2) in (let TMP_34 \def (next_plus gz n2 TMP_33) in (let +TMP_35 \def (ASort TMP_31 TMP_34) in (eq A TMP_29 TMP_35)))))))))))) in (let +TMP_45 \def (\lambda (n: nat).(let TMP_37 \def (plus h1 h2) in (let TMP_38 +\def (minus h1 TMP_37) in (let TMP_39 \def (next_plus gz n1 h2) in (let +TMP_40 \def (ASort TMP_38 TMP_39) in (let TMP_41 \def (plus h1 h2) in (let +TMP_42 \def (minus h2 TMP_41) in (let TMP_43 \def (next_plus gz n2 n) in (let +TMP_44 \def (ASort TMP_42 TMP_43) in (eq A TMP_40 TMP_44)))))))))) in (let +TMP_52 \def (\lambda (n: nat).(let TMP_46 \def (next_plus gz n1 h2) in (let +TMP_47 \def (ASort n TMP_46) in (let TMP_48 \def (plus h1 h2) in (let TMP_49 +\def (minus h2 TMP_48) in (let TMP_50 \def (next_plus gz n2 h1) in (let +TMP_51 \def (ASort TMP_49 TMP_50) in (eq A TMP_47 TMP_51)))))))) in (let +TMP_57 \def (\lambda (n: nat).(let TMP_53 \def (next_plus gz n1 h2) in (let +TMP_54 \def (ASort O TMP_53) in (let TMP_55 \def (next_plus gz n2 h1) in (let +TMP_56 \def (ASort n TMP_55) in (eq A TMP_54 TMP_56)))))) in (let TMP_58 \def +(plus h2 n1) in (let TMP_62 \def (\lambda (n: nat).(let TMP_59 \def (ASort O +n) in (let TMP_60 \def (next_plus gz n2 h1) in (let TMP_61 \def (ASort O +TMP_60) in (eq A TMP_59 TMP_61))))) in (let TMP_63 \def (plus h1 n2) in (let +TMP_67 \def (\lambda (n: nat).(let TMP_64 \def (plus h2 n1) in (let TMP_65 +\def (ASort O TMP_64) in (let TMP_66 \def (ASort O n) in (eq A TMP_65 +TMP_66))))) in (let TMP_68 \def (ASort O) in (let TMP_69 \def (plus h2 n1) in +(let TMP_70 \def (plus h1 n2) in (let TMP_71 \def (plus h1 n2) in (let TMP_72 +\def (plus h2 n1) in (let TMP_73 \def (sym_eq nat TMP_71 TMP_72 H0) in (let +TMP_74 \def (f_equal nat A TMP_68 TMP_69 TMP_70 TMP_73) in (let TMP_75 \def +(next_plus gz n2 h1) in (let TMP_76 \def (next_plus_gz n2 h1) in (let TMP_77 +\def (eq_ind_r nat TMP_63 TMP_67 TMP_74 TMP_75 TMP_76) in (let TMP_78 \def +(next_plus gz n1 h2) in (let TMP_79 \def (next_plus_gz n1 h2) in (let TMP_80 +\def (eq_ind_r nat TMP_58 TMP_62 TMP_77 TMP_78 TMP_79) in (let TMP_81 \def +(plus h1 h2) in (let TMP_82 \def (minus h2 TMP_81) in (let TMP_83 \def (plus +h1 h2) in (let TMP_84 \def (le_plus_r h1 h2) in (let TMP_85 \def (O_minus h2 +TMP_83 TMP_84) in (let TMP_86 \def (eq_ind_r nat O TMP_57 TMP_80 TMP_82 +TMP_85) in (let TMP_87 \def (plus h1 h2) in (let TMP_88 \def (minus h1 +TMP_87) in (let TMP_89 \def (plus h1 h2) in (let TMP_90 \def (le_plus_l h1 +h2) in (let TMP_91 \def (O_minus h1 TMP_89 TMP_90) in (let TMP_92 \def +(eq_ind_r nat O TMP_52 TMP_86 TMP_88 TMP_91) in (let TMP_93 \def (plus h1 h2) +in (let TMP_94 \def (minus TMP_93 h2) in (let TMP_95 \def (minus_plus_r h1 +h2) in (let TMP_96 \def (eq_ind_r nat h1 TMP_45 TMP_92 TMP_94 TMP_95) in (let +TMP_97 \def (plus h1 h2) in (let TMP_98 \def (minus TMP_97 h1) in (let TMP_99 +\def (minus_plus h1 h2) in (let TMP_100 \def (eq_ind_r nat h2 TMP_36 TMP_96 +TMP_98 TMP_99) in (let TMP_101 \def (ASort h2 n2) in (let TMP_102 \def (plus +h1 h2) in (let TMP_103 \def (aplus gz TMP_101 TMP_102) in (let TMP_104 \def +(plus h1 h2) in (let TMP_105 \def (aplus_asort_simpl gz TMP_104 h2 n2) in +(let TMP_106 \def (eq_ind_r A TMP_18 TMP_25 TMP_100 TMP_103 TMP_105) in (let +TMP_107 \def (ASort h1 n1) in (let TMP_108 \def (plus h1 h2) in (let TMP_109 +\def (aplus gz TMP_107 TMP_108) in (let TMP_110 \def (plus h1 h2) in (let +TMP_111 \def (aplus_asort_simpl gz TMP_110 h1 n1) in (let TMP_112 \def +(eq_ind_r A TMP_8 TMP_12 TMP_106 TMP_109 TMP_111) in (leq_sort gz h1 h2 n1 n2 +TMP_2 +TMP_112))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))))) +))) in (let TMP_114 \def (\lambda (a0: A).(\lambda (a3: A).(\lambda (_: (leqz +a0 a3)).(\lambda (H1: (leq gz a0 a3)).(\lambda (a4: A).(\lambda (a5: +A).(\lambda (_: (leqz a4 a5)).(\lambda (H3: (leq gz a4 a5)).(leq_head gz a0 +a3 H1 a4 a5 H3))))))))) in (leqz_ind TMP_1 TMP_113 TMP_114 a1 a2 H)))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma index fd9e7c1d3..d79f266be 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/leq/asucc.ma @@ -14,466 +14,739 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/leq/props.ma". +include "basic_1/leq/props.ma". theorem asucc_repl: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g (asucc g a1) (asucc g a2))))) \def \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g (asucc g a) (asucc g -a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: -nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g -(ASort h2 n2) k))).(nat_ind (\lambda (n: nat).((eq A (aplus g (ASort n n1) k) -(aplus g (ASort h2 n2) k)) \to (leq g (match n with [O \Rightarrow (ASort O -(next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow -(ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) (\lambda (H1: (eq -A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda (n: -nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort n n2) k)) \to (leq g -(ASort O (next g n1)) (match n with [O \Rightarrow (ASort O (next g n2)) | (S -h) \Rightarrow (ASort h n2)])))) (\lambda (H2: (eq A (aplus g (ASort O n1) k) -(aplus g (ASort O n2) k))).(leq_sort g O O (next g n1) (next g n2) k (eq_ind -A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g (ASort O -(next g n2)) k))) (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq -A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort O n2) k) -(\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O n2) k)))) -(refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort O n1) k) -H2) (aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k)) (aplus g -(ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))) (\lambda (h3: +a2)).(let TMP_3 \def (\lambda (a: A).(\lambda (a0: A).(let TMP_1 \def (asucc +g a) in (let TMP_2 \def (asucc g a0) in (leq g TMP_1 TMP_2))))) in (let +TMP_186 \def (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: +nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort +h1 n1) k) (aplus g (ASort h2 n2) k))).(let TMP_8 \def (\lambda (n: nat).((eq +A (aplus g (ASort n n1) k) (aplus g (ASort h2 n2) k)) \to (let TMP_5 \def +(match n with [O \Rightarrow (let TMP_4 \def (next g n1) in (ASort O TMP_4)) +| (S h) \Rightarrow (ASort h n1)]) in (let TMP_7 \def (match h2 with [O +\Rightarrow (let TMP_6 \def (next g n2) in (ASort O TMP_6)) | (S h) +\Rightarrow (ASort h n2)]) in (leq g TMP_5 TMP_7))))) in (let TMP_97 \def +(\lambda (H1: (eq A (aplus g (ASort O n1) k) (aplus g (ASort h2 n2) k))).(let +TMP_13 \def (\lambda (n: nat).((eq A (aplus g (ASort O n1) k) (aplus g (ASort +n n2) k)) \to (let TMP_9 \def (next g n1) in (let TMP_10 \def (ASort O TMP_9) +in (let TMP_12 \def (match n with [O \Rightarrow (let TMP_11 \def (next g n2) +in (ASort O TMP_11)) | (S h) \Rightarrow (ASort h n2)]) in (leq g TMP_10 +TMP_12)))))) in (let TMP_54 \def (\lambda (H2: (eq A (aplus g (ASort O n1) k) +(aplus g (ASort O n2) k))).(let TMP_14 \def (next g n1) in (let TMP_15 \def +(next g n2) in (let TMP_16 \def (ASort O n1) in (let TMP_17 \def (S k) in +(let TMP_18 \def (aplus g TMP_16 TMP_17) in (let TMP_22 \def (\lambda (a: +A).(let TMP_19 \def (next g n2) in (let TMP_20 \def (ASort O TMP_19) in (let +TMP_21 \def (aplus g TMP_20 k) in (eq A a TMP_21))))) in (let TMP_23 \def +(ASort O n2) in (let TMP_24 \def (S k) in (let TMP_25 \def (aplus g TMP_23 +TMP_24) in (let TMP_29 \def (\lambda (a: A).(let TMP_26 \def (ASort O n1) in +(let TMP_27 \def (S k) in (let TMP_28 \def (aplus g TMP_26 TMP_27) in (eq A +TMP_28 a))))) in (let TMP_30 \def (ASort O n2) in (let TMP_31 \def (aplus g +TMP_30 k) in (let TMP_36 \def (\lambda (a: A).(let TMP_32 \def (asucc g a) in +(let TMP_33 \def (ASort O n2) in (let TMP_34 \def (aplus g TMP_33 k) in (let +TMP_35 \def (asucc g TMP_34) in (eq A TMP_32 TMP_35)))))) in (let TMP_37 \def +(ASort O n2) in (let TMP_38 \def (aplus g TMP_37 k) in (let TMP_39 \def +(asucc g TMP_38) in (let TMP_40 \def (refl_equal A TMP_39) in (let TMP_41 +\def (ASort O n1) in (let TMP_42 \def (aplus g TMP_41 k) in (let TMP_43 \def +(eq_ind_r A TMP_31 TMP_36 TMP_40 TMP_42 H2) in (let TMP_44 \def (next g n2) +in (let TMP_45 \def (ASort O TMP_44) in (let TMP_46 \def (aplus g TMP_45 k) +in (let TMP_47 \def (aplus_sort_O_S_simpl g n2 k) in (let TMP_48 \def (eq_ind +A TMP_25 TMP_29 TMP_43 TMP_46 TMP_47) in (let TMP_49 \def (next g n1) in (let +TMP_50 \def (ASort O TMP_49) in (let TMP_51 \def (aplus g TMP_50 k) in (let +TMP_52 \def (aplus_sort_O_S_simpl g n1 k) in (let TMP_53 \def (eq_ind A +TMP_18 TMP_22 TMP_48 TMP_51 TMP_52) in (leq_sort g O O TMP_14 TMP_15 k +TMP_53)))))))))))))))))))))))))))))))) in (let TMP_96 \def (\lambda (h3: nat).(\lambda (_: (((eq A (aplus g (ASort O n1) k) (aplus g (ASort h3 n2) k)) \to (leq g (ASort O (next g n1)) (match h3 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H2: (eq A (aplus g -(ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(leq_sort g O h3 (next g n1) -n2 k (eq_ind A (aplus g (ASort O n1) (S k)) (\lambda (a: A).(eq A a (aplus g -(ASort h3 n2) k))) (eq_ind A (aplus g (ASort (S h3) n2) (S k)) (\lambda (a: -A).(eq A (aplus g (ASort O n1) (S k)) a)) (eq_ind_r A (aplus g (ASort (S h3) -n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort (S h3) n2) -k)))) (refl_equal A (asucc g (aplus g (ASort (S h3) n2) k))) (aplus g (ASort -O n1) k) H2) (aplus g (ASort h3 n2) k) (aplus_sort_S_S_simpl g n2 h3 k)) -(aplus g (ASort O (next g n1)) k) (aplus_sort_O_S_simpl g n1 k)))))) h2 H1)) -(\lambda (h3: nat).(\lambda (IHh1: (((eq A (aplus g (ASort h3 n1) k) (aplus g -(ASort h2 n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g -n1)) | (S h) \Rightarrow (ASort h n1)]) (match h2 with [O \Rightarrow (ASort -O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))).(\lambda (H1: (eq A -(aplus g (ASort (S h3) n1) k) (aplus g (ASort h2 n2) k))).(nat_ind (\lambda -(n: nat).((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort n n2) k)) \to -((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort n n2) k)) \to (leq g -(match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow -(ASort h n1)]) (match n with [O \Rightarrow (ASort O (next g n2)) | (S h) -\Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match n with [O -\Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)]))))) -(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2) +(ASort O n1) k) (aplus g (ASort (S h3) n2) k))).(let TMP_55 \def (next g n1) +in (let TMP_56 \def (ASort O n1) in (let TMP_57 \def (S k) in (let TMP_58 +\def (aplus g TMP_56 TMP_57) in (let TMP_61 \def (\lambda (a: A).(let TMP_59 +\def (ASort h3 n2) in (let TMP_60 \def (aplus g TMP_59 k) in (eq A a +TMP_60)))) in (let TMP_62 \def (S h3) in (let TMP_63 \def (ASort TMP_62 n2) +in (let TMP_64 \def (S k) in (let TMP_65 \def (aplus g TMP_63 TMP_64) in (let +TMP_69 \def (\lambda (a: A).(let TMP_66 \def (ASort O n1) in (let TMP_67 \def +(S k) in (let TMP_68 \def (aplus g TMP_66 TMP_67) in (eq A TMP_68 a))))) in +(let TMP_70 \def (S h3) in (let TMP_71 \def (ASort TMP_70 n2) in (let TMP_72 +\def (aplus g TMP_71 k) in (let TMP_78 \def (\lambda (a: A).(let TMP_73 \def +(asucc g a) in (let TMP_74 \def (S h3) in (let TMP_75 \def (ASort TMP_74 n2) +in (let TMP_76 \def (aplus g TMP_75 k) in (let TMP_77 \def (asucc g TMP_76) +in (eq A TMP_73 TMP_77))))))) in (let TMP_79 \def (S h3) in (let TMP_80 \def +(ASort TMP_79 n2) in (let TMP_81 \def (aplus g TMP_80 k) in (let TMP_82 \def +(asucc g TMP_81) in (let TMP_83 \def (refl_equal A TMP_82) in (let TMP_84 +\def (ASort O n1) in (let TMP_85 \def (aplus g TMP_84 k) in (let TMP_86 \def +(eq_ind_r A TMP_72 TMP_78 TMP_83 TMP_85 H2) in (let TMP_87 \def (ASort h3 n2) +in (let TMP_88 \def (aplus g TMP_87 k) in (let TMP_89 \def +(aplus_sort_S_S_simpl g n2 h3 k) in (let TMP_90 \def (eq_ind A TMP_65 TMP_69 +TMP_86 TMP_88 TMP_89) in (let TMP_91 \def (next g n1) in (let TMP_92 \def +(ASort O TMP_91) in (let TMP_93 \def (aplus g TMP_92 k) in (let TMP_94 \def +(aplus_sort_O_S_simpl g n1 k) in (let TMP_95 \def (eq_ind A TMP_58 TMP_61 +TMP_90 TMP_93 TMP_94) in (leq_sort g O h3 TMP_55 n2 k +TMP_95))))))))))))))))))))))))))))))))))) in (nat_ind TMP_13 TMP_54 TMP_96 h2 +H1))))) in (let TMP_185 \def (\lambda (h3: nat).(\lambda (IHh1: (((eq A +(aplus g (ASort h3 n1) k) (aplus g (ASort h2 n2) k)) \to (leq g (match h3 +with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow (ASort h n1)]) +(match h2 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow +(ASort h n2)]))))).(\lambda (H1: (eq A (aplus g (ASort (S h3) n1) k) (aplus g +(ASort h2 n2) k))).(let TMP_101 \def (\lambda (n: nat).((eq A (aplus g (ASort +(S h3) n1) k) (aplus g (ASort n n2) k)) \to ((((eq A (aplus g (ASort h3 n1) +k) (aplus g (ASort n n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort +O (next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match n with [O +\Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) \to +(let TMP_98 \def (ASort h3 n1) in (let TMP_100 \def (match n with [O +\Rightarrow (let TMP_99 \def (next g n2) in (ASort O TMP_99)) | (S h) +\Rightarrow (ASort h n2)]) in (leq g TMP_98 TMP_100)))))) in (let TMP_141 +\def (\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort O n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort O n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) -\Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(leq_sort g h3 O n1 -(next g n2) k (eq_ind A (aplus g (ASort O n2) (S k)) (\lambda (a: A).(eq A -(aplus g (ASort h3 n1) k) a)) (eq_ind A (aplus g (ASort (S h3) n1) (S k)) -(\lambda (a: A).(eq A a (aplus g (ASort O n2) (S k)))) (eq_ind_r A (aplus g -(ASort O n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g (aplus g (ASort O -n2) k)))) (refl_equal A (asucc g (aplus g (ASort O n2) k))) (aplus g (ASort -(S h3) n1) k) H2) (aplus g (ASort h3 n1) k) (aplus_sort_S_S_simpl g n1 h3 k)) -(aplus g (ASort O (next g n2)) k) (aplus_sort_O_S_simpl g n2 k))))) (\lambda -(h4: nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort -h4 n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k)) -\to (leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) +\Rightarrow (ASort h n1)]) (ASort O (next g n2)))))).(let TMP_102 \def (next +g n2) in (let TMP_103 \def (ASort O n2) in (let TMP_104 \def (S k) in (let +TMP_105 \def (aplus g TMP_103 TMP_104) in (let TMP_108 \def (\lambda (a: +A).(let TMP_106 \def (ASort h3 n1) in (let TMP_107 \def (aplus g TMP_106 k) +in (eq A TMP_107 a)))) in (let TMP_109 \def (S h3) in (let TMP_110 \def +(ASort TMP_109 n1) in (let TMP_111 \def (S k) in (let TMP_112 \def (aplus g +TMP_110 TMP_111) in (let TMP_116 \def (\lambda (a: A).(let TMP_113 \def +(ASort O n2) in (let TMP_114 \def (S k) in (let TMP_115 \def (aplus g TMP_113 +TMP_114) in (eq A a TMP_115))))) in (let TMP_117 \def (ASort O n2) in (let +TMP_118 \def (aplus g TMP_117 k) in (let TMP_123 \def (\lambda (a: A).(let +TMP_119 \def (asucc g a) in (let TMP_120 \def (ASort O n2) in (let TMP_121 +\def (aplus g TMP_120 k) in (let TMP_122 \def (asucc g TMP_121) in (eq A +TMP_119 TMP_122)))))) in (let TMP_124 \def (ASort O n2) in (let TMP_125 \def +(aplus g TMP_124 k) in (let TMP_126 \def (asucc g TMP_125) in (let TMP_127 +\def (refl_equal A TMP_126) in (let TMP_128 \def (S h3) in (let TMP_129 \def +(ASort TMP_128 n1) in (let TMP_130 \def (aplus g TMP_129 k) in (let TMP_131 +\def (eq_ind_r A TMP_118 TMP_123 TMP_127 TMP_130 H2) in (let TMP_132 \def +(ASort h3 n1) in (let TMP_133 \def (aplus g TMP_132 k) in (let TMP_134 \def +(aplus_sort_S_S_simpl g n1 h3 k) in (let TMP_135 \def (eq_ind A TMP_112 +TMP_116 TMP_131 TMP_133 TMP_134) in (let TMP_136 \def (next g n2) in (let +TMP_137 \def (ASort O TMP_136) in (let TMP_138 \def (aplus g TMP_137 k) in +(let TMP_139 \def (aplus_sort_O_S_simpl g n2 k) in (let TMP_140 \def (eq_ind +A TMP_105 TMP_108 TMP_135 TMP_138 TMP_139) in (leq_sort g h3 O n1 TMP_102 k +TMP_140))))))))))))))))))))))))))))))))) in (let TMP_184 \def (\lambda (h4: +nat).(\lambda (_: (((eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort h4 +n2) k)) \to ((((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort h4 n2) k)) \to +(leq g (match h3 with [O \Rightarrow (ASort O (next g n1)) | (S h) \Rightarrow (ASort h n1)]) (match h4 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))) \to (leq g (ASort h3 n1) (match h4 with [O \Rightarrow (ASort O (next g n2)) | (S h) \Rightarrow (ASort h n2)])))))).(\lambda (H2: (eq A (aplus g (ASort (S h3) n1) k) (aplus g (ASort (S h4) n2) k))).(\lambda (_: (((eq A (aplus g (ASort h3 n1) k) (aplus g (ASort (S h4) n2) k)) \to (leq g (match h3 with [O \Rightarrow (ASort O (next -g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(leq_sort g h3 h4 -n1 n2 k (eq_ind A (aplus g (ASort (S h3) n1) (S k)) (\lambda (a: A).(eq A a -(aplus g (ASort h4 n2) k))) (eq_ind A (aplus g (ASort (S h4) n2) (S k)) -(\lambda (a: A).(eq A (aplus g (ASort (S h3) n1) (S k)) a)) (eq_ind_r A -(aplus g (ASort (S h4) n2) k) (\lambda (a: A).(eq A (asucc g a) (asucc g -(aplus g (ASort (S h4) n2) k)))) (refl_equal A (asucc g (aplus g (ASort (S -h4) n2) k))) (aplus g (ASort (S h3) n1) k) H2) (aplus g (ASort h4 n2) k) -(aplus_sort_S_S_simpl g n2 h4 k)) (aplus g (ASort h3 n1) k) -(aplus_sort_S_S_simpl g n1 h3 k))))))) h2 H1 IHh1)))) h1 H0))))))) (\lambda -(a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 a4)).(\lambda (_: (leq g -(asucc g a3) (asucc g a4))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: -(leq g a5 a6)).(\lambda (H3: (leq g (asucc g a5) (asucc g a6))).(leq_head g -a3 a4 H0 (asucc g a5) (asucc g a6) H3))))))))) a1 a2 H)))). -(* COMMENTS -Initial nodes: 1907 -END *) +g n1)) | (S h) \Rightarrow (ASort h n1)]) (ASort h4 n2))))).(let TMP_142 \def +(S h3) in (let TMP_143 \def (ASort TMP_142 n1) in (let TMP_144 \def (S k) in +(let TMP_145 \def (aplus g TMP_143 TMP_144) in (let TMP_148 \def (\lambda (a: +A).(let TMP_146 \def (ASort h4 n2) in (let TMP_147 \def (aplus g TMP_146 k) +in (eq A a TMP_147)))) in (let TMP_149 \def (S h4) in (let TMP_150 \def +(ASort TMP_149 n2) in (let TMP_151 \def (S k) in (let TMP_152 \def (aplus g +TMP_150 TMP_151) in (let TMP_157 \def (\lambda (a: A).(let TMP_153 \def (S +h3) in (let TMP_154 \def (ASort TMP_153 n1) in (let TMP_155 \def (S k) in +(let TMP_156 \def (aplus g TMP_154 TMP_155) in (eq A TMP_156 a)))))) in (let +TMP_158 \def (S h4) in (let TMP_159 \def (ASort TMP_158 n2) in (let TMP_160 +\def (aplus g TMP_159 k) in (let TMP_166 \def (\lambda (a: A).(let TMP_161 +\def (asucc g a) in (let TMP_162 \def (S h4) in (let TMP_163 \def (ASort +TMP_162 n2) in (let TMP_164 \def (aplus g TMP_163 k) in (let TMP_165 \def +(asucc g TMP_164) in (eq A TMP_161 TMP_165))))))) in (let TMP_167 \def (S h4) +in (let TMP_168 \def (ASort TMP_167 n2) in (let TMP_169 \def (aplus g TMP_168 +k) in (let TMP_170 \def (asucc g TMP_169) in (let TMP_171 \def (refl_equal A +TMP_170) in (let TMP_172 \def (S h3) in (let TMP_173 \def (ASort TMP_172 n1) +in (let TMP_174 \def (aplus g TMP_173 k) in (let TMP_175 \def (eq_ind_r A +TMP_160 TMP_166 TMP_171 TMP_174 H2) in (let TMP_176 \def (ASort h4 n2) in +(let TMP_177 \def (aplus g TMP_176 k) in (let TMP_178 \def +(aplus_sort_S_S_simpl g n2 h4 k) in (let TMP_179 \def (eq_ind A TMP_152 +TMP_157 TMP_175 TMP_177 TMP_178) in (let TMP_180 \def (ASort h3 n1) in (let +TMP_181 \def (aplus g TMP_180 k) in (let TMP_182 \def (aplus_sort_S_S_simpl g +n1 h3 k) in (let TMP_183 \def (eq_ind A TMP_145 TMP_148 TMP_179 TMP_181 +TMP_182) in (leq_sort g h3 h4 n1 n2 k +TMP_183)))))))))))))))))))))))))))))))))))) in (nat_ind TMP_101 TMP_141 +TMP_184 h2 H1 IHh1))))))) in (nat_ind TMP_8 TMP_97 TMP_185 h1 H0)))))))))) in +(let TMP_189 \def (\lambda (a3: A).(\lambda (a4: A).(\lambda (H0: (leq g a3 +a4)).(\lambda (_: (leq g (asucc g a3) (asucc g a4))).(\lambda (a5: +A).(\lambda (a6: A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g (asucc g +a5) (asucc g a6))).(let TMP_187 \def (asucc g a5) in (let TMP_188 \def (asucc +g a6) in (leq_head g a3 a4 H0 TMP_187 TMP_188 H3))))))))))) in (leq_ind g +TMP_3 TMP_186 TMP_189 a1 a2 H))))))). theorem asucc_inj: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (asucc g a1) (asucc g a2)) \to (leq g a1 a2)))) \def - \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: -A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (a2: A).(A_ind (\lambda (a: A).((leq g -(asucc g (ASort n n0)) (asucc g a)) \to (leq g (ASort n n0) a))) (\lambda -(n1: nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0)) -(asucc g (ASort n1 n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort -n3 n0)) (asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2)))) -(\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1 -n2)))).(nat_ind (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g -(ASort n3 n2))) \to (leq g (ASort O n0) (ASort n3 n2)))) (\lambda (H1: (leq g -(asucc g (ASort O n0)) (asucc g (ASort O n2)))).(let H_x \def (leq_gen_sort1 -g O (next g n0) (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind -nat nat nat (\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A -(aplus g (ASort O (next g n0)) k) (aplus g (ASort h2 n3) k))))) (\lambda (n3: -nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort -h2 n3))))) (leq g (ASort O n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: -nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) -x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) -(ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort n3 _) \Rightarrow n3 | (AHead _ _) -\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def -(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with -[(ASort _ n3) \Rightarrow n3 | (AHead _ _) \Rightarrow ((match g with [(mk_G -next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in -(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n3: -nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 x0) x2))) H3 -O H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n3: nat).(eq A (aplus g -(ASort O (next g n0)) x2) (aplus g (ASort O n3) x2))) H8 (next g n2) H6) in -(let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) (\lambda (a: -A).(eq A a (aplus g (ASort O (next g n2)) x2))) H9 (aplus g (ASort O n0) (S -x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g -(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) -a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in -(leq_sort g O O n0 n2 (S x2) H11))))))) H5))))))) H2)))) (\lambda (n3: -nat).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g (ASort n3 n2))) -\to (leq g (ASort O n0) (ASort n3 n2))))).(\lambda (H1: (leq g (asucc g -(ASort O n0)) (asucc g (ASort (S n3) n2)))).(let H_x \def (leq_gen_sort1 g O -(next g n0) (ASort n3 n2) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat -(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -O (next g n0)) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda -(h2: nat).(\lambda (_: nat).(eq A (ASort n3 n2) (ASort h2 n4))))) (leq g -(ASort O n0) (ASort (S n3) n2)) (\lambda (x0: nat).(\lambda (x1: -nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) -x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1 -x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return -(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _) -\Rightarrow n3])) (ASort n3 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A -nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ -n4) \Rightarrow n4 | (AHead _ _) \Rightarrow n2])) (ASort n3 n2) (ASort x1 -x0) H4) in (\lambda (H7: (eq nat n3 x1)).(let H8 \def (eq_ind_r nat x1 -(\lambda (n4: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort -n4 x0) x2))) H3 n3 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: -nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort n3 n4) x2))) H8 -n2 H6) in (let H10 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) -(\lambda (a: A).(eq A a (aplus g (ASort n3 n2) x2))) H9 (aplus g (ASort O n0) -(S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H11 \def (eq_ind_r A (aplus g -(ASort n3 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort O n0) (S x2)) a)) H10 -(aplus g (ASort (S n3) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n3 x2)) in -(leq_sort g O (S n3) n0 n2 (S x2) H11))))))) H5))))))) H2)))))) n1 H0)) -(\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) (asucc g -(ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda (H0: (leq -g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(nat_ind (\lambda -(n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 n2))) \to -((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq g (ASort -n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 n2))))) -(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort O -n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort O n2))) -\to (leq g (ASort n3 n0) (ASort O n2))))).(let H_x \def (leq_gen_sort1 g n3 -n0 (ASort O (next g n2)) H1) in (let H2 \def H_x in (ex2_3_ind nat nat nat -(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -n3 n0) k) (aplus g (ASort h2 n4) k))))) (\lambda (n4: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A (ASort O (next g n2)) (ASort h2 n4))))) (leq g -(ASort (S n3) n0) (ASort O n2)) (\lambda (x0: nat).(\lambda (x1: + \lambda (g: G).(\lambda (a1: A).(let TMP_1 \def (\lambda (a: A).(\forall +(a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2)))) in (let TMP_315 +\def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(let TMP_3 \def +(\lambda (a: A).((leq g (asucc g (ASort n n0)) (asucc g a)) \to (let TMP_2 +\def (ASort n n0) in (leq g TMP_2 a)))) in (let TMP_260 \def (\lambda (n1: +nat).(\lambda (n2: nat).(\lambda (H: (leq g (asucc g (ASort n n0)) (asucc g +(ASort n1 n2)))).(let TMP_6 \def (\lambda (n3: nat).((leq g (asucc g (ASort +n3 n0)) (asucc g (ASort n1 n2))) \to (let TMP_4 \def (ASort n3 n0) in (let +TMP_5 \def (ASort n1 n2) in (leq g TMP_4 TMP_5))))) in (let TMP_133 \def +(\lambda (H0: (leq g (asucc g (ASort O n0)) (asucc g (ASort n1 n2)))).(let +TMP_9 \def (\lambda (n3: nat).((leq g (asucc g (ASort O n0)) (asucc g (ASort +n3 n2))) \to (let TMP_7 \def (ASort O n0) in (let TMP_8 \def (ASort n3 n2) in +(leq g TMP_7 TMP_8))))) in (let TMP_73 \def (\lambda (H1: (leq g (asucc g +(ASort O n0)) (asucc g (ASort O n2)))).(let TMP_10 \def (next g n0) in (let +TMP_11 \def (next g n2) in (let TMP_12 \def (ASort O TMP_11) in (let H_x \def +(leq_gen_sort1 g O TMP_10 TMP_12 H1) in (let H2 \def H_x in (let TMP_18 \def +(\lambda (n3: nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_13 \def +(next g n0) in (let TMP_14 \def (ASort O TMP_13) in (let TMP_15 \def (aplus g +TMP_14 k) in (let TMP_16 \def (ASort h2 n3) in (let TMP_17 \def (aplus g +TMP_16 k) in (eq A TMP_15 TMP_17))))))))) in (let TMP_22 \def (\lambda (n3: +nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_19 \def (next g n2) in +(let TMP_20 \def (ASort O TMP_19) in (let TMP_21 \def (ASort h2 n3) in (eq A +TMP_20 TMP_21))))))) in (let TMP_23 \def (ASort O n0) in (let TMP_24 \def +(ASort O n2) in (let TMP_25 \def (leq g TMP_23 TMP_24) in (let TMP_72 \def +(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A +(aplus g (ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: +(eq A (ASort O (next g n2)) (ASort x1 x0))).(let TMP_26 \def (\lambda (e: +A).(match e with [(ASort n3 _) \Rightarrow n3 | (AHead _ _) \Rightarrow O])) +in (let TMP_27 \def (next g n2) in (let TMP_28 \def (ASort O TMP_27) in (let +TMP_29 \def (ASort x1 x0) in (let H5 \def (f_equal A nat TMP_26 TMP_28 TMP_29 +H4) in (let TMP_31 \def (\lambda (e: A).(match e with [(ASort _ n3) +\Rightarrow n3 | (AHead _ _) \Rightarrow (let TMP_30 \def (match g with +[(mk_G next _) \Rightarrow next]) in (TMP_30 n2))])) in (let TMP_32 \def +(next g n2) in (let TMP_33 \def (ASort O TMP_32) in (let TMP_34 \def (ASort +x1 x0) in (let H6 \def (f_equal A nat TMP_31 TMP_33 TMP_34 H4) in (let TMP_71 +\def (\lambda (H7: (eq nat O x1)).(let TMP_40 \def (\lambda (n3: nat).(let +TMP_35 \def (next g n0) in (let TMP_36 \def (ASort O TMP_35) in (let TMP_37 +\def (aplus g TMP_36 x2) in (let TMP_38 \def (ASort n3 x0) in (let TMP_39 +\def (aplus g TMP_38 x2) in (eq A TMP_37 TMP_39))))))) in (let H8 \def +(eq_ind_r nat x1 TMP_40 H3 O H7) in (let TMP_46 \def (\lambda (n3: nat).(let +TMP_41 \def (next g n0) in (let TMP_42 \def (ASort O TMP_41) in (let TMP_43 +\def (aplus g TMP_42 x2) in (let TMP_44 \def (ASort O n3) in (let TMP_45 \def +(aplus g TMP_44 x2) in (eq A TMP_43 TMP_45))))))) in (let TMP_47 \def (next g +n2) in (let H9 \def (eq_ind_r nat x0 TMP_46 H8 TMP_47 H6) in (let TMP_48 \def +(next g n0) in (let TMP_49 \def (ASort O TMP_48) in (let TMP_50 \def (aplus g +TMP_49 x2) in (let TMP_54 \def (\lambda (a: A).(let TMP_51 \def (next g n2) +in (let TMP_52 \def (ASort O TMP_51) in (let TMP_53 \def (aplus g TMP_52 x2) +in (eq A a TMP_53))))) in (let TMP_55 \def (ASort O n0) in (let TMP_56 \def +(S x2) in (let TMP_57 \def (aplus g TMP_55 TMP_56) in (let TMP_58 \def +(aplus_sort_O_S_simpl g n0 x2) in (let H10 \def (eq_ind_r A TMP_50 TMP_54 H9 +TMP_57 TMP_58) in (let TMP_59 \def (next g n2) in (let TMP_60 \def (ASort O +TMP_59) in (let TMP_61 \def (aplus g TMP_60 x2) in (let TMP_65 \def (\lambda +(a: A).(let TMP_62 \def (ASort O n0) in (let TMP_63 \def (S x2) in (let +TMP_64 \def (aplus g TMP_62 TMP_63) in (eq A TMP_64 a))))) in (let TMP_66 +\def (ASort O n2) in (let TMP_67 \def (S x2) in (let TMP_68 \def (aplus g +TMP_66 TMP_67) in (let TMP_69 \def (aplus_sort_O_S_simpl g n2 x2) in (let H11 +\def (eq_ind_r A TMP_61 TMP_65 H10 TMP_68 TMP_69) in (let TMP_70 \def (S x2) +in (leq_sort g O O n0 n2 TMP_70 H11)))))))))))))))))))))))))) in (TMP_71 +H5))))))))))))))))) in (ex2_3_ind nat nat nat TMP_18 TMP_22 TMP_25 TMP_72 +H2))))))))))))) in (let TMP_132 \def (\lambda (n3: nat).(\lambda (_: (((leq g +(asucc g (ASort O n0)) (asucc g (ASort n3 n2))) \to (leq g (ASort O n0) +(ASort n3 n2))))).(\lambda (H1: (leq g (asucc g (ASort O n0)) (asucc g (ASort +(S n3) n2)))).(let TMP_74 \def (next g n0) in (let TMP_75 \def (ASort n3 n2) +in (let H_x \def (leq_gen_sort1 g O TMP_74 TMP_75 H1) in (let H2 \def H_x in +(let TMP_81 \def (\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: nat).(let +TMP_76 \def (next g n0) in (let TMP_77 \def (ASort O TMP_76) in (let TMP_78 +\def (aplus g TMP_77 k) in (let TMP_79 \def (ASort h2 n4) in (let TMP_80 \def +(aplus g TMP_79 k) in (eq A TMP_78 TMP_80))))))))) in (let TMP_84 \def +(\lambda (n4: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_82 \def +(ASort n3 n2) in (let TMP_83 \def (ASort h2 n4) in (eq A TMP_82 TMP_83)))))) +in (let TMP_85 \def (ASort O n0) in (let TMP_86 \def (S n3) in (let TMP_87 +\def (ASort TMP_86 n2) in (let TMP_88 \def (leq g TMP_85 TMP_87) in (let +TMP_131 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: +nat).(\lambda (H3: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort +x1 x0) x2))).(\lambda (H4: (eq A (ASort n3 n2) (ASort x1 x0))).(let TMP_89 +\def (\lambda (e: A).(match e with [(ASort n4 _) \Rightarrow n4 | (AHead _ _) +\Rightarrow n3])) in (let TMP_90 \def (ASort n3 n2) in (let TMP_91 \def +(ASort x1 x0) in (let H5 \def (f_equal A nat TMP_89 TMP_90 TMP_91 H4) in (let +TMP_92 \def (\lambda (e: A).(match e with [(ASort _ n4) \Rightarrow n4 | +(AHead _ _) \Rightarrow n2])) in (let TMP_93 \def (ASort n3 n2) in (let +TMP_94 \def (ASort x1 x0) in (let H6 \def (f_equal A nat TMP_92 TMP_93 TMP_94 +H4) in (let TMP_130 \def (\lambda (H7: (eq nat n3 x1)).(let TMP_100 \def +(\lambda (n4: nat).(let TMP_95 \def (next g n0) in (let TMP_96 \def (ASort O +TMP_95) in (let TMP_97 \def (aplus g TMP_96 x2) in (let TMP_98 \def (ASort n4 +x0) in (let TMP_99 \def (aplus g TMP_98 x2) in (eq A TMP_97 TMP_99))))))) in +(let H8 \def (eq_ind_r nat x1 TMP_100 H3 n3 H7) in (let TMP_106 \def (\lambda +(n4: nat).(let TMP_101 \def (next g n0) in (let TMP_102 \def (ASort O +TMP_101) in (let TMP_103 \def (aplus g TMP_102 x2) in (let TMP_104 \def +(ASort n3 n4) in (let TMP_105 \def (aplus g TMP_104 x2) in (eq A TMP_103 +TMP_105))))))) in (let H9 \def (eq_ind_r nat x0 TMP_106 H8 n2 H6) in (let +TMP_107 \def (next g n0) in (let TMP_108 \def (ASort O TMP_107) in (let +TMP_109 \def (aplus g TMP_108 x2) in (let TMP_112 \def (\lambda (a: A).(let +TMP_110 \def (ASort n3 n2) in (let TMP_111 \def (aplus g TMP_110 x2) in (eq A +a TMP_111)))) in (let TMP_113 \def (ASort O n0) in (let TMP_114 \def (S x2) +in (let TMP_115 \def (aplus g TMP_113 TMP_114) in (let TMP_116 \def +(aplus_sort_O_S_simpl g n0 x2) in (let H10 \def (eq_ind_r A TMP_109 TMP_112 +H9 TMP_115 TMP_116) in (let TMP_117 \def (ASort n3 n2) in (let TMP_118 \def +(aplus g TMP_117 x2) in (let TMP_122 \def (\lambda (a: A).(let TMP_119 \def +(ASort O n0) in (let TMP_120 \def (S x2) in (let TMP_121 \def (aplus g +TMP_119 TMP_120) in (eq A TMP_121 a))))) in (let TMP_123 \def (S n3) in (let +TMP_124 \def (ASort TMP_123 n2) in (let TMP_125 \def (S x2) in (let TMP_126 +\def (aplus g TMP_124 TMP_125) in (let TMP_127 \def (aplus_sort_S_S_simpl g +n2 n3 x2) in (let H11 \def (eq_ind_r A TMP_118 TMP_122 H10 TMP_126 TMP_127) +in (let TMP_128 \def (S n3) in (let TMP_129 \def (S x2) in (leq_sort g O +TMP_128 n0 n2 TMP_129 H11)))))))))))))))))))))))))) in (TMP_130 +H5))))))))))))))) in (ex2_3_ind nat nat nat TMP_81 TMP_84 TMP_88 TMP_131 +H2))))))))))))))) in (nat_ind TMP_9 TMP_73 TMP_132 n1 H0))))) in (let TMP_259 +\def (\lambda (n3: nat).(\lambda (IHn: (((leq g (asucc g (ASort n3 n0)) +(asucc g (ASort n1 n2))) \to (leq g (ASort n3 n0) (ASort n1 n2))))).(\lambda +(H0: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n1 n2)))).(let +TMP_137 \def (\lambda (n4: nat).((leq g (asucc g (ASort (S n3) n0)) (asucc g +(ASort n4 n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 +n2))) \to (leq g (ASort n3 n0) (ASort n4 n2)))) \to (let TMP_134 \def (S n3) +in (let TMP_135 \def (ASort TMP_134 n0) in (let TMP_136 \def (ASort n4 n2) in +(leq g TMP_135 TMP_136))))))) in (let TMP_200 \def (\lambda (H1: (leq g +(asucc g (ASort (S n3) n0)) (asucc g (ASort O n2)))).(\lambda (_: (((leq g +(asucc g (ASort n3 n0)) (asucc g (ASort O n2))) \to (leq g (ASort n3 n0) +(ASort O n2))))).(let TMP_138 \def (next g n2) in (let TMP_139 \def (ASort O +TMP_138) in (let H_x \def (leq_gen_sort1 g n3 n0 TMP_139 H1) in (let H2 \def +H_x in (let TMP_144 \def (\lambda (n4: nat).(\lambda (h2: nat).(\lambda (k: +nat).(let TMP_140 \def (ASort n3 n0) in (let TMP_141 \def (aplus g TMP_140 k) +in (let TMP_142 \def (ASort h2 n4) in (let TMP_143 \def (aplus g TMP_142 k) +in (eq A TMP_141 TMP_143)))))))) in (let TMP_148 \def (\lambda (n4: +nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_145 \def (next g n2) in +(let TMP_146 \def (ASort O TMP_145) in (let TMP_147 \def (ASort h2 n4) in (eq +A TMP_146 TMP_147))))))) in (let TMP_149 \def (S n3) in (let TMP_150 \def +(ASort TMP_149 n0) in (let TMP_151 \def (ASort O n2) in (let TMP_152 \def +(leq g TMP_150 TMP_151) in (let TMP_199 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort O (next g n2)) (ASort x1 -x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A return -(\lambda (_: A).nat) with [(ASort n4 _) \Rightarrow n4 | (AHead _ _) -\Rightarrow O])) (ASort O (next g n2)) (ASort x1 x0) H4) in ((let H6 \def -(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with -[(ASort _ n4) \Rightarrow n4 | (AHead _ _) \Rightarrow ((match g with [(mk_G -next _) \Rightarrow next]) n2)])) (ASort O (next g n2)) (ASort x1 x0) H4) in -(\lambda (H7: (eq nat O x1)).(let H8 \def (eq_ind_r nat x1 (\lambda (n4: -nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n4 x0) x2))) H3 O H7) -in (let H9 \def (eq_ind_r nat x0 (\lambda (n4: nat).(eq A (aplus g (ASort n3 -n0) x2) (aplus g (ASort O n4) x2))) H8 (next g n2) H6) in (let H10 \def -(eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g -(ASort O (next g n2)) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) -(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g -(ASort O (next g n2)) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S -x2)) a)) H10 (aplus g (ASort O n2) (S x2)) (aplus_sort_O_S_simpl g n2 x2)) in -(leq_sort g (S n3) O n0 n2 (S x2) H11))))))) H5))))))) H2))))) (\lambda (n4: -nat).(\lambda (_: (((leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort n4 -n2))) \to ((((leq g (asucc g (ASort n3 n0)) (asucc g (ASort n4 n2))) \to (leq -g (ASort n3 n0) (ASort n4 n2)))) \to (leq g (ASort (S n3) n0) (ASort n4 -n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S n3) n0)) (asucc g (ASort (S -n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort n3 n0)) (asucc g (ASort (S -n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4) n2))))).(let H_x \def -(leq_gen_sort1 g n3 n0 (ASort n4 n2) H1) in (let H2 \def H_x in (ex2_3_ind -nat nat nat (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A -(aplus g (ASort n3 n0) k) (aplus g (ASort h2 n5) k))))) (\lambda (n5: -nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (ASort n4 n2) (ASort h2 -n5))))) (leq g (ASort (S n3) n0) (ASort (S n4) n2)) (\lambda (x0: -nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g -(ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A (ASort n4 -n2) (ASort x1 x0))).(let H5 \def (f_equal A nat (\lambda (e: A).(match e in A -return (\lambda (_: A).nat) with [(ASort n5 _) \Rightarrow n5 | (AHead _ _) -\Rightarrow n4])) (ASort n4 n2) (ASort x1 x0) H4) in ((let H6 \def (f_equal A -nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort _ -n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) (ASort n4 n2) (ASort x1 -x0) H4) in (\lambda (H7: (eq nat n4 x1)).(let H8 \def (eq_ind_r nat x1 -(\lambda (n5: nat).(eq A (aplus g (ASort n3 n0) x2) (aplus g (ASort n5 x0) -x2))) H3 n4 H7) in (let H9 \def (eq_ind_r nat x0 (\lambda (n5: nat).(eq A -(aplus g (ASort n3 n0) x2) (aplus g (ASort n4 n5) x2))) H8 n2 H6) in (let H10 -\def (eq_ind_r A (aplus g (ASort n3 n0) x2) (\lambda (a: A).(eq A a (aplus g -(ASort n4 n2) x2))) H9 (aplus g (ASort (S n3) n0) (S x2)) -(aplus_sort_S_S_simpl g n0 n3 x2)) in (let H11 \def (eq_ind_r A (aplus g -(ASort n4 n2) x2) (\lambda (a: A).(eq A (aplus g (ASort (S n3) n0) (S x2)) -a)) H10 (aplus g (ASort (S n4) n2) (S x2)) (aplus_sort_S_S_simpl g n2 n4 x2)) -in (leq_sort g (S n3) (S n4) n0 n2 (S x2) H11))))))) H5))))))) H2))))))) n1 -H0 IHn)))) n H)))) (\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n -n0)) (asucc g a)) \to (leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda -(H0: (((leq g (asucc g (ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0) -a0)))).(\lambda (H1: (leq g (asucc g (ASort n n0)) (asucc g (AHead a -a0)))).(nat_ind (\lambda (n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g +x0))).(let TMP_153 \def (\lambda (e: A).(match e with [(ASort n4 _) +\Rightarrow n4 | (AHead _ _) \Rightarrow O])) in (let TMP_154 \def (next g +n2) in (let TMP_155 \def (ASort O TMP_154) in (let TMP_156 \def (ASort x1 x0) +in (let H5 \def (f_equal A nat TMP_153 TMP_155 TMP_156 H4) in (let TMP_158 +\def (\lambda (e: A).(match e with [(ASort _ n4) \Rightarrow n4 | (AHead _ _) +\Rightarrow (let TMP_157 \def (match g with [(mk_G next _) \Rightarrow next]) +in (TMP_157 n2))])) in (let TMP_159 \def (next g n2) in (let TMP_160 \def +(ASort O TMP_159) in (let TMP_161 \def (ASort x1 x0) in (let H6 \def (f_equal +A nat TMP_158 TMP_160 TMP_161 H4) in (let TMP_198 \def (\lambda (H7: (eq nat +O x1)).(let TMP_166 \def (\lambda (n4: nat).(let TMP_162 \def (ASort n3 n0) +in (let TMP_163 \def (aplus g TMP_162 x2) in (let TMP_164 \def (ASort n4 x0) +in (let TMP_165 \def (aplus g TMP_164 x2) in (eq A TMP_163 TMP_165)))))) in +(let H8 \def (eq_ind_r nat x1 TMP_166 H3 O H7) in (let TMP_171 \def (\lambda +(n4: nat).(let TMP_167 \def (ASort n3 n0) in (let TMP_168 \def (aplus g +TMP_167 x2) in (let TMP_169 \def (ASort O n4) in (let TMP_170 \def (aplus g +TMP_169 x2) in (eq A TMP_168 TMP_170)))))) in (let TMP_172 \def (next g n2) +in (let H9 \def (eq_ind_r nat x0 TMP_171 H8 TMP_172 H6) in (let TMP_173 \def +(ASort n3 n0) in (let TMP_174 \def (aplus g TMP_173 x2) in (let TMP_178 \def +(\lambda (a: A).(let TMP_175 \def (next g n2) in (let TMP_176 \def (ASort O +TMP_175) in (let TMP_177 \def (aplus g TMP_176 x2) in (eq A a TMP_177))))) in +(let TMP_179 \def (S n3) in (let TMP_180 \def (ASort TMP_179 n0) in (let +TMP_181 \def (S x2) in (let TMP_182 \def (aplus g TMP_180 TMP_181) in (let +TMP_183 \def (aplus_sort_S_S_simpl g n0 n3 x2) in (let H10 \def (eq_ind_r A +TMP_174 TMP_178 H9 TMP_182 TMP_183) in (let TMP_184 \def (next g n2) in (let +TMP_185 \def (ASort O TMP_184) in (let TMP_186 \def (aplus g TMP_185 x2) in +(let TMP_191 \def (\lambda (a: A).(let TMP_187 \def (S n3) in (let TMP_188 +\def (ASort TMP_187 n0) in (let TMP_189 \def (S x2) in (let TMP_190 \def +(aplus g TMP_188 TMP_189) in (eq A TMP_190 a)))))) in (let TMP_192 \def +(ASort O n2) in (let TMP_193 \def (S x2) in (let TMP_194 \def (aplus g +TMP_192 TMP_193) in (let TMP_195 \def (aplus_sort_O_S_simpl g n2 x2) in (let +H11 \def (eq_ind_r A TMP_186 TMP_191 H10 TMP_194 TMP_195) in (let TMP_196 +\def (S n3) in (let TMP_197 \def (S x2) in (leq_sort g TMP_196 O n0 n2 +TMP_197 H11))))))))))))))))))))))))))) in (TMP_198 H5))))))))))))))))) in +(ex2_3_ind nat nat nat TMP_144 TMP_148 TMP_152 TMP_199 H2)))))))))))))) in +(let TMP_258 \def (\lambda (n4: nat).(\lambda (_: (((leq g (asucc g (ASort (S +n3) n0)) (asucc g (ASort n4 n2))) \to ((((leq g (asucc g (ASort n3 n0)) +(asucc g (ASort n4 n2))) \to (leq g (ASort n3 n0) (ASort n4 n2)))) \to (leq g +(ASort (S n3) n0) (ASort n4 n2)))))).(\lambda (H1: (leq g (asucc g (ASort (S +n3) n0)) (asucc g (ASort (S n4) n2)))).(\lambda (_: (((leq g (asucc g (ASort +n3 n0)) (asucc g (ASort (S n4) n2))) \to (leq g (ASort n3 n0) (ASort (S n4) +n2))))).(let TMP_201 \def (ASort n4 n2) in (let H_x \def (leq_gen_sort1 g n3 +n0 TMP_201 H1) in (let H2 \def H_x in (let TMP_206 \def (\lambda (n5: +nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_202 \def (ASort n3 n0) in +(let TMP_203 \def (aplus g TMP_202 k) in (let TMP_204 \def (ASort h2 n5) in +(let TMP_205 \def (aplus g TMP_204 k) in (eq A TMP_203 TMP_205)))))))) in +(let TMP_209 \def (\lambda (n5: nat).(\lambda (h2: nat).(\lambda (_: +nat).(let TMP_207 \def (ASort n4 n2) in (let TMP_208 \def (ASort h2 n5) in +(eq A TMP_207 TMP_208)))))) in (let TMP_210 \def (S n3) in (let TMP_211 \def +(ASort TMP_210 n0) in (let TMP_212 \def (S n4) in (let TMP_213 \def (ASort +TMP_212 n2) in (let TMP_214 \def (leq g TMP_211 TMP_213) in (let TMP_257 \def +(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A +(aplus g (ASort n3 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A +(ASort n4 n2) (ASort x1 x0))).(let TMP_215 \def (\lambda (e: A).(match e with +[(ASort n5 _) \Rightarrow n5 | (AHead _ _) \Rightarrow n4])) in (let TMP_216 +\def (ASort n4 n2) in (let TMP_217 \def (ASort x1 x0) in (let H5 \def +(f_equal A nat TMP_215 TMP_216 TMP_217 H4) in (let TMP_218 \def (\lambda (e: +A).(match e with [(ASort _ n5) \Rightarrow n5 | (AHead _ _) \Rightarrow n2])) +in (let TMP_219 \def (ASort n4 n2) in (let TMP_220 \def (ASort x1 x0) in (let +H6 \def (f_equal A nat TMP_218 TMP_219 TMP_220 H4) in (let TMP_256 \def +(\lambda (H7: (eq nat n4 x1)).(let TMP_225 \def (\lambda (n5: nat).(let +TMP_221 \def (ASort n3 n0) in (let TMP_222 \def (aplus g TMP_221 x2) in (let +TMP_223 \def (ASort n5 x0) in (let TMP_224 \def (aplus g TMP_223 x2) in (eq A +TMP_222 TMP_224)))))) in (let H8 \def (eq_ind_r nat x1 TMP_225 H3 n4 H7) in +(let TMP_230 \def (\lambda (n5: nat).(let TMP_226 \def (ASort n3 n0) in (let +TMP_227 \def (aplus g TMP_226 x2) in (let TMP_228 \def (ASort n4 n5) in (let +TMP_229 \def (aplus g TMP_228 x2) in (eq A TMP_227 TMP_229)))))) in (let H9 +\def (eq_ind_r nat x0 TMP_230 H8 n2 H6) in (let TMP_231 \def (ASort n3 n0) in +(let TMP_232 \def (aplus g TMP_231 x2) in (let TMP_235 \def (\lambda (a: +A).(let TMP_233 \def (ASort n4 n2) in (let TMP_234 \def (aplus g TMP_233 x2) +in (eq A a TMP_234)))) in (let TMP_236 \def (S n3) in (let TMP_237 \def +(ASort TMP_236 n0) in (let TMP_238 \def (S x2) in (let TMP_239 \def (aplus g +TMP_237 TMP_238) in (let TMP_240 \def (aplus_sort_S_S_simpl g n0 n3 x2) in +(let H10 \def (eq_ind_r A TMP_232 TMP_235 H9 TMP_239 TMP_240) in (let TMP_241 +\def (ASort n4 n2) in (let TMP_242 \def (aplus g TMP_241 x2) in (let TMP_247 +\def (\lambda (a: A).(let TMP_243 \def (S n3) in (let TMP_244 \def (ASort +TMP_243 n0) in (let TMP_245 \def (S x2) in (let TMP_246 \def (aplus g TMP_244 +TMP_245) in (eq A TMP_246 a)))))) in (let TMP_248 \def (S n4) in (let TMP_249 +\def (ASort TMP_248 n2) in (let TMP_250 \def (S x2) in (let TMP_251 \def +(aplus g TMP_249 TMP_250) in (let TMP_252 \def (aplus_sort_S_S_simpl g n2 n4 +x2) in (let H11 \def (eq_ind_r A TMP_242 TMP_247 H10 TMP_251 TMP_252) in (let +TMP_253 \def (S n3) in (let TMP_254 \def (S n4) in (let TMP_255 \def (S x2) +in (leq_sort g TMP_253 TMP_254 n0 n2 TMP_255 H11))))))))))))))))))))))))))) +in (TMP_256 H5))))))))))))))) in (ex2_3_ind nat nat nat TMP_206 TMP_209 +TMP_214 TMP_257 H2)))))))))))))))) in (nat_ind TMP_137 TMP_200 TMP_258 n1 H0 +IHn))))))) in (nat_ind TMP_6 TMP_133 TMP_259 n H))))))) in (let TMP_314 \def +(\lambda (a: A).(\lambda (H: (((leq g (asucc g (ASort n n0)) (asucc g a)) \to +(leq g (ASort n n0) a)))).(\lambda (a0: A).(\lambda (H0: (((leq g (asucc g +(ASort n n0)) (asucc g a0)) \to (leq g (ASort n n0) a0)))).(\lambda (H1: (leq +g (asucc g (ASort n n0)) (asucc g (AHead a a0)))).(let TMP_263 \def (\lambda +(n1: nat).((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 +n0) a))) \to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g +(ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a +a0))) \to (let TMP_261 \def (ASort n1 n0) in (let TMP_262 \def (AHead a a0) +in (leq g TMP_261 TMP_262))))))) in (let TMP_288 \def (\lambda (_: (((leq g +(asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O n0) a)))).(\lambda +(_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq g (ASort O n0) +a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g (AHead a +a0)))).(let TMP_264 \def (next g n0) in (let TMP_265 \def (asucc g a0) in +(let TMP_266 \def (AHead a TMP_265) in (let H_x \def (leq_gen_sort1 g O +TMP_264 TMP_266 H4) in (let H5 \def H_x in (let TMP_272 \def (\lambda (n2: +nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_267 \def (next g n0) in +(let TMP_268 \def (ASort O TMP_267) in (let TMP_269 \def (aplus g TMP_268 k) +in (let TMP_270 \def (ASort h2 n2) in (let TMP_271 \def (aplus g TMP_270 k) +in (eq A TMP_269 TMP_271))))))))) in (let TMP_276 \def (\lambda (n2: +nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_273 \def (asucc g a0) in +(let TMP_274 \def (AHead a TMP_273) in (let TMP_275 \def (ASort h2 n2) in (eq +A TMP_274 TMP_275))))))) in (let TMP_277 \def (ASort O n0) in (let TMP_278 +\def (AHead a a0) in (let TMP_279 \def (leq g TMP_277 TMP_278) in (let +TMP_287 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: +nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort x1 +x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 x0))).(let +TMP_280 \def (asucc g a0) in (let TMP_281 \def (AHead a TMP_280) in (let +TMP_282 \def (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False | +(AHead _ _) \Rightarrow True])) in (let TMP_283 \def (ASort x1 x0) in (let H8 +\def (eq_ind A TMP_281 TMP_282 I TMP_283 H7) in (let TMP_284 \def (ASort O +n0) in (let TMP_285 \def (AHead a a0) in (let TMP_286 \def (leq g TMP_284 +TMP_285) in (False_ind TMP_286 H8)))))))))))))) in (ex2_3_ind nat nat nat +TMP_272 TMP_276 TMP_279 TMP_287 H5))))))))))))))) in (let TMP_313 \def +(\lambda (n1: nat).(\lambda (_: (((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a))) \to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0) a0))) \to ((leq g (asucc g (ASort n1 -n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a a0)))))) -(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a)) \to (leq g (ASort O -n0) a)))).(\lambda (_: (((leq g (asucc g (ASort O n0)) (asucc g a0)) \to (leq -g (ASort O n0) a0)))).(\lambda (H4: (leq g (asucc g (ASort O n0)) (asucc g -(AHead a a0)))).(let H_x \def (leq_gen_sort1 g O (next g n0) (AHead a (asucc -g a0)) H4) in (let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort O (next g -n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A (AHead a (asucc g a0)) (ASort h2 n2))))) (leq g -(ASort O n0) (AHead a a0)) (\lambda (x0: nat).(\lambda (x1: nat).(\lambda -(x2: nat).(\lambda (_: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g -(ASort x1 x0) x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 -x0))).(let H8 \def (eq_ind A (AHead a (asucc g a0)) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow False | -(AHead _ _) \Rightarrow True])) I (ASort x1 x0) H7) in (False_ind (leq g -(ASort O n0) (AHead a a0)) H8))))))) H5)))))) (\lambda (n1: nat).(\lambda (_: -(((((leq g (asucc g (ASort n1 n0)) (asucc g a)) \to (leq g (ASort n1 n0) a))) -\to ((((leq g (asucc g (ASort n1 n0)) (asucc g a0)) \to (leq g (ASort n1 n0) -a0))) \to ((leq g (asucc g (ASort n1 n0)) (asucc g (AHead a a0))) \to (leq g -(ASort n1 n0) (AHead a a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1) -n0)) (asucc g a)) \to (leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g -(asucc g (ASort (S n1) n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) -a0)))).(\lambda (H4: (leq g (asucc g (ASort (S n1) n0)) (asucc g (AHead a -a0)))).(let H_x \def (leq_gen_sort1 g n1 n0 (AHead a (asucc g a0)) H4) in -(let H5 \def H_x in (ex2_3_ind nat nat nat (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (k: nat).(eq A (aplus g (ASort n1 n0) k) (aplus g (ASort h2 n2) -k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A (AHead a -(asucc g a0)) (ASort h2 n2))))) (leq g (ASort (S n1) n0) (AHead a a0)) -(\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (_: (eq A -(aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H7: (eq A -(AHead a (asucc g a0)) (ASort x1 x0))).(let H8 \def (eq_ind A (AHead a (asucc -g a0)) (\lambda (ee: A).(match ee in A return (\lambda (_: A).Prop) with -[(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow True])) I (ASort x1 -x0) H7) in (False_ind (leq g (ASort (S n1) n0) (AHead a a0)) H8))))))) -H5)))))))) n H H0 H1)))))) a2)))) (\lambda (a: A).(\lambda (_: ((\forall (a2: -A).((leq g (asucc g a) (asucc g a2)) \to (leq g a a2))))).(\lambda (a0: -A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) (asucc g a2)) \to -(leq g a0 a2))))).(\lambda (a2: A).(A_ind (\lambda (a3: A).((leq g (asucc g -(AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a a0)) (asucc g -(ASort n n0)))).(nat_ind (\lambda (n1: nat).((leq g (asucc g (AHead a a0)) -(asucc g (ASort n1 n0))) \to (leq g (AHead a a0) (ASort n1 n0)))) (\lambda -(H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort O n0)))).(let H_x \def -(leq_gen_head1 g a (asucc g a0) (ASort O (next g n0)) H2) in (let H3 \def H_x -in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a a3))) (\lambda -(_: A).(\lambda (a4: A).(leq g (asucc g a0) a4))) (\lambda (a3: A).(\lambda -(a4: A).(eq A (ASort O (next g n0)) (AHead a3 a4)))) (leq g (AHead a a0) -(ASort O n0)) (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a -x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda (H6: (eq A (ASort O (next -g n0)) (AHead x0 x1))).(let H7 \def (eq_ind A (ASort O (next g n0)) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H6) in -(False_ind (leq g (AHead a a0) (ASort O n0)) H7))))))) H3)))) (\lambda (n1: -nat).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) -\to (leq g (AHead a a0) (ASort n1 n0))))).(\lambda (H2: (leq g (asucc g -(AHead a a0)) (asucc g (ASort (S n1) n0)))).(let H_x \def (leq_gen_head1 g a -(asucc g a0) (ASort n1 n0) H2) in (let H3 \def H_x in (ex3_2_ind A A (\lambda -(a3: A).(\lambda (_: A).(leq g a a3))) (\lambda (_: A).(\lambda (a4: A).(leq -g (asucc g a0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort n1 n0) -(AHead a3 a4)))) (leq g (AHead a a0) (ASort (S n1) n0)) (\lambda (x0: -A).(\lambda (x1: A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g -a0) x1)).(\lambda (H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let H7 \def -(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (AHead x0 x1) H6) in (False_ind (leq g (AHead a a0) (ASort (S n1) -n0)) H7))))))) H3)))))) n H1)))) (\lambda (a3: A).(\lambda (_: (((leq g -(asucc g (AHead a a0)) (asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda -(a4: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g -(AHead a a0) a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g -(AHead a3 a4)))).(let H_x \def (leq_gen_head1 g a (asucc g a0) (AHead a3 -(asucc g a4)) H3) in (let H4 \def H_x in (ex3_2_ind A A (\lambda (a5: -A).(\lambda (_: A).(leq g a a5))) (\lambda (_: A).(\lambda (a6: A).(leq g -(asucc g a0) a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 (asucc g -a4)) (AHead a5 a6)))) (leq g (AHead a a0) (AHead a3 a4)) (\lambda (x0: -A).(\lambda (x1: A).(\lambda (H5: (leq g a x0)).(\lambda (H6: (leq g (asucc g -a0) x1)).(\lambda (H7: (eq A (AHead a3 (asucc g a4)) (AHead x0 x1))).(let H8 -\def (f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) -with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow a5])) (AHead a3 -(asucc g a4)) (AHead x0 x1) H7) in ((let H9 \def (f_equal A A (\lambda (e: -A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow -((let rec asucc (g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) -\Rightarrow (match n0 with [O \Rightarrow (ASort O (next g0 n)) | (S h) -\Rightarrow (ASort h n)]) | (AHead a5 a6) \Rightarrow (AHead a5 (asucc g0 -a6))]) in asucc) g a4) | (AHead _ a5) \Rightarrow a5])) (AHead a3 (asucc g -a4)) (AHead x0 x1) H7) in (\lambda (H10: (eq A a3 x0)).(let H11 \def -(eq_ind_r A x1 (\lambda (a5: A).(leq g (asucc g a0) a5)) H6 (asucc g a4) H9) -in (let H12 \def (eq_ind_r A x0 (\lambda (a5: A).(leq g a a5)) H5 a3 H10) in -(leq_head g a a3 H12 a0 a4 (H0 a4 H11)))))) H8))))))) H4)))))))) a2)))))) -a1)). -(* COMMENTS -Initial nodes: 4697 -END *) +n0)) (asucc g (AHead a a0))) \to (leq g (ASort n1 n0) (AHead a +a0))))))).(\lambda (_: (((leq g (asucc g (ASort (S n1) n0)) (asucc g a)) \to +(leq g (ASort (S n1) n0) a)))).(\lambda (_: (((leq g (asucc g (ASort (S n1) +n0)) (asucc g a0)) \to (leq g (ASort (S n1) n0) a0)))).(\lambda (H4: (leq g +(asucc g (ASort (S n1) n0)) (asucc g (AHead a a0)))).(let TMP_289 \def (asucc +g a0) in (let TMP_290 \def (AHead a TMP_289) in (let H_x \def (leq_gen_sort1 +g n1 n0 TMP_290 H4) in (let H5 \def H_x in (let TMP_295 \def (\lambda (n2: +nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_291 \def (ASort n1 n0) in +(let TMP_292 \def (aplus g TMP_291 k) in (let TMP_293 \def (ASort h2 n2) in +(let TMP_294 \def (aplus g TMP_293 k) in (eq A TMP_292 TMP_294)))))))) in +(let TMP_299 \def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: +nat).(let TMP_296 \def (asucc g a0) in (let TMP_297 \def (AHead a TMP_296) in +(let TMP_298 \def (ASort h2 n2) in (eq A TMP_297 TMP_298))))))) in (let +TMP_300 \def (S n1) in (let TMP_301 \def (ASort TMP_300 n0) in (let TMP_302 +\def (AHead a a0) in (let TMP_303 \def (leq g TMP_301 TMP_302) in (let +TMP_312 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: +nat).(\lambda (_: (eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) +x2))).(\lambda (H7: (eq A (AHead a (asucc g a0)) (ASort x1 x0))).(let TMP_304 +\def (asucc g a0) in (let TMP_305 \def (AHead a TMP_304) in (let TMP_306 \def +(\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _) +\Rightarrow True])) in (let TMP_307 \def (ASort x1 x0) in (let H8 \def +(eq_ind A TMP_305 TMP_306 I TMP_307 H7) in (let TMP_308 \def (S n1) in (let +TMP_309 \def (ASort TMP_308 n0) in (let TMP_310 \def (AHead a a0) in (let +TMP_311 \def (leq g TMP_309 TMP_310) in (False_ind TMP_311 H8))))))))))))))) +in (ex2_3_ind nat nat nat TMP_295 TMP_299 TMP_303 TMP_312 H5))))))))))))))))) +in (nat_ind TMP_263 TMP_288 TMP_313 n H H0 H1))))))))) in (A_ind TMP_3 +TMP_260 TMP_314 a2))))))) in (let TMP_396 \def (\lambda (a: A).(\lambda (_: +((\forall (a2: A).((leq g (asucc g a) (asucc g a2)) \to (leq g a +a2))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a2: A).((leq g (asucc g a0) +(asucc g a2)) \to (leq g a0 a2))))).(\lambda (a2: A).(let TMP_317 \def +(\lambda (a3: A).((leq g (asucc g (AHead a a0)) (asucc g a3)) \to (let +TMP_316 \def (AHead a a0) in (leq g TMP_316 a3)))) in (let TMP_364 \def +(\lambda (n: nat).(\lambda (n0: nat).(\lambda (H1: (leq g (asucc g (AHead a +a0)) (asucc g (ASort n n0)))).(let TMP_320 \def (\lambda (n1: nat).((leq g +(asucc g (AHead a a0)) (asucc g (ASort n1 n0))) \to (let TMP_318 \def (AHead +a a0) in (let TMP_319 \def (ASort n1 n0) in (leq g TMP_318 TMP_319))))) in +(let TMP_342 \def (\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort +O n0)))).(let TMP_321 \def (asucc g a0) in (let TMP_322 \def (next g n0) in +(let TMP_323 \def (ASort O TMP_322) in (let H_x \def (leq_gen_head1 g a +TMP_321 TMP_323 H2) in (let H3 \def H_x in (let TMP_324 \def (\lambda (a3: +A).(\lambda (_: A).(leq g a a3))) in (let TMP_326 \def (\lambda (_: +A).(\lambda (a4: A).(let TMP_325 \def (asucc g a0) in (leq g TMP_325 a4)))) +in (let TMP_330 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_327 \def +(next g n0) in (let TMP_328 \def (ASort O TMP_327) in (let TMP_329 \def +(AHead a3 a4) in (eq A TMP_328 TMP_329)))))) in (let TMP_331 \def (AHead a +a0) in (let TMP_332 \def (ASort O n0) in (let TMP_333 \def (leq g TMP_331 +TMP_332) in (let TMP_341 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: +(leq g a x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda (H6: (eq A +(ASort O (next g n0)) (AHead x0 x1))).(let TMP_334 \def (next g n0) in (let +TMP_335 \def (ASort O TMP_334) in (let TMP_336 \def (\lambda (ee: A).(match +ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow False])) in +(let TMP_337 \def (AHead x0 x1) in (let H7 \def (eq_ind A TMP_335 TMP_336 I +TMP_337 H6) in (let TMP_338 \def (AHead a a0) in (let TMP_339 \def (ASort O +n0) in (let TMP_340 \def (leq g TMP_338 TMP_339) in (False_ind TMP_340 +H7)))))))))))))) in (ex3_2_ind A A TMP_324 TMP_326 TMP_330 TMP_333 TMP_341 +H3)))))))))))))) in (let TMP_363 \def (\lambda (n1: nat).(\lambda (_: (((leq +g (asucc g (AHead a a0)) (asucc g (ASort n1 n0))) \to (leq g (AHead a a0) +(ASort n1 n0))))).(\lambda (H2: (leq g (asucc g (AHead a a0)) (asucc g (ASort +(S n1) n0)))).(let TMP_343 \def (asucc g a0) in (let TMP_344 \def (ASort n1 +n0) in (let H_x \def (leq_gen_head1 g a TMP_343 TMP_344 H2) in (let H3 \def +H_x in (let TMP_345 \def (\lambda (a3: A).(\lambda (_: A).(leq g a a3))) in +(let TMP_347 \def (\lambda (_: A).(\lambda (a4: A).(let TMP_346 \def (asucc g +a0) in (leq g TMP_346 a4)))) in (let TMP_350 \def (\lambda (a3: A).(\lambda +(a4: A).(let TMP_348 \def (ASort n1 n0) in (let TMP_349 \def (AHead a3 a4) in +(eq A TMP_348 TMP_349))))) in (let TMP_351 \def (AHead a a0) in (let TMP_352 +\def (S n1) in (let TMP_353 \def (ASort TMP_352 n0) in (let TMP_354 \def (leq +g TMP_351 TMP_353) in (let TMP_362 \def (\lambda (x0: A).(\lambda (x1: +A).(\lambda (_: (leq g a x0)).(\lambda (_: (leq g (asucc g a0) x1)).(\lambda +(H6: (eq A (ASort n1 n0) (AHead x0 x1))).(let TMP_355 \def (ASort n1 n0) in +(let TMP_356 \def (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow +True | (AHead _ _) \Rightarrow False])) in (let TMP_357 \def (AHead x0 x1) in +(let H7 \def (eq_ind A TMP_355 TMP_356 I TMP_357 H6) in (let TMP_358 \def +(AHead a a0) in (let TMP_359 \def (S n1) in (let TMP_360 \def (ASort TMP_359 +n0) in (let TMP_361 \def (leq g TMP_358 TMP_360) in (False_ind TMP_361 +H7)))))))))))))) in (ex3_2_ind A A TMP_345 TMP_347 TMP_350 TMP_354 TMP_362 +H3)))))))))))))))) in (nat_ind TMP_320 TMP_342 TMP_363 n H1))))))) in (let +TMP_395 \def (\lambda (a3: A).(\lambda (_: (((leq g (asucc g (AHead a a0)) +(asucc g a3)) \to (leq g (AHead a a0) a3)))).(\lambda (a4: A).(\lambda (_: +(((leq g (asucc g (AHead a a0)) (asucc g a4)) \to (leq g (AHead a a0) +a4)))).(\lambda (H3: (leq g (asucc g (AHead a a0)) (asucc g (AHead a3 +a4)))).(let TMP_365 \def (asucc g a0) in (let TMP_366 \def (asucc g a4) in +(let TMP_367 \def (AHead a3 TMP_366) in (let H_x \def (leq_gen_head1 g a +TMP_365 TMP_367 H3) in (let H4 \def H_x in (let TMP_368 \def (\lambda (a5: +A).(\lambda (_: A).(leq g a a5))) in (let TMP_370 \def (\lambda (_: +A).(\lambda (a6: A).(let TMP_369 \def (asucc g a0) in (leq g TMP_369 a6)))) +in (let TMP_374 \def (\lambda (a5: A).(\lambda (a6: A).(let TMP_371 \def +(asucc g a4) in (let TMP_372 \def (AHead a3 TMP_371) in (let TMP_373 \def +(AHead a5 a6) in (eq A TMP_372 TMP_373)))))) in (let TMP_375 \def (AHead a +a0) in (let TMP_376 \def (AHead a3 a4) in (let TMP_377 \def (leq g TMP_375 +TMP_376) in (let TMP_394 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (H5: +(leq g a x0)).(\lambda (H6: (leq g (asucc g a0) x1)).(\lambda (H7: (eq A +(AHead a3 (asucc g a4)) (AHead x0 x1))).(let TMP_378 \def (\lambda (e: +A).(match e with [(ASort _ _) \Rightarrow a3 | (AHead a5 _) \Rightarrow a5])) +in (let TMP_379 \def (asucc g a4) in (let TMP_380 \def (AHead a3 TMP_379) in +(let TMP_381 \def (AHead x0 x1) in (let H8 \def (f_equal A A TMP_378 TMP_380 +TMP_381 H7) in (let TMP_384 \def (\lambda (e: A).(match e with [(ASort _ _) +\Rightarrow (asucc g a4) | (AHead _ a5) \Rightarrow a5])) in (let TMP_385 +\def (asucc g a4) in (let TMP_386 \def (AHead a3 TMP_385) in (let TMP_387 +\def (AHead x0 x1) in (let H9 \def (f_equal A A TMP_384 TMP_386 TMP_387 H7) +in (let TMP_393 \def (\lambda (H10: (eq A a3 x0)).(let TMP_389 \def (\lambda +(a5: A).(let TMP_388 \def (asucc g a0) in (leq g TMP_388 a5))) in (let +TMP_390 \def (asucc g a4) in (let H11 \def (eq_ind_r A x1 TMP_389 H6 TMP_390 +H9) in (let TMP_391 \def (\lambda (a5: A).(leq g a a5)) in (let H12 \def +(eq_ind_r A x0 TMP_391 H5 a3 H10) in (let TMP_392 \def (H0 a4 H11) in +(leq_head g a a3 H12 a0 a4 TMP_392)))))))) in (TMP_393 H8))))))))))))))))) in +(ex3_2_ind A A TMP_368 TMP_370 TMP_374 TMP_377 TMP_394 H4)))))))))))))))))) +in (A_ind TMP_317 TMP_364 TMP_395 a2))))))))) in (A_ind TMP_1 TMP_315 TMP_396 +a1))))). theorem leq_asucc: \forall (g: G).(\forall (a: A).(ex A (\lambda (a0: A).(leq g a (asucc g a0))))) \def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(ex A (\lambda (a1: -A).(leq g a0 (asucc g a1))))) (\lambda (n: nat).(\lambda (n0: nat).(ex_intro -A (\lambda (a0: A).(leq g (ASort n n0) (asucc g a0))) (ASort (S n) n0) -(leq_refl g (ASort n n0))))) (\lambda (a0: A).(\lambda (_: (ex A (\lambda -(a1: A).(leq g a0 (asucc g a1))))).(\lambda (a1: A).(\lambda (H0: (ex A -(\lambda (a2: A).(leq g a1 (asucc g a2))))).(let H1 \def H0 in (ex_ind A -(\lambda (a2: A).(leq g a1 (asucc g a2))) (ex A (\lambda (a2: A).(leq g -(AHead a0 a1) (asucc g a2)))) (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc -g x))).(ex_intro A (\lambda (a2: A).(leq g (AHead a0 a1) (asucc g a2))) -(AHead a0 x) (leq_head g a0 a0 (leq_refl g a0) a1 (asucc g x) H2)))) H1)))))) -a)). -(* COMMENTS -Initial nodes: 221 -END *) + \lambda (g: G).(\lambda (a: A).(let TMP_3 \def (\lambda (a0: A).(let TMP_2 +\def (\lambda (a1: A).(let TMP_1 \def (asucc g a1) in (leq g a0 TMP_1))) in +(ex A TMP_2))) in (let TMP_11 \def (\lambda (n: nat).(\lambda (n0: nat).(let +TMP_6 \def (\lambda (a0: A).(let TMP_4 \def (ASort n n0) in (let TMP_5 \def +(asucc g a0) in (leq g TMP_4 TMP_5)))) in (let TMP_7 \def (S n) in (let TMP_8 +\def (ASort TMP_7 n0) in (let TMP_9 \def (ASort n n0) in (let TMP_10 \def +(leq_refl g TMP_9) in (ex_intro A TMP_6 TMP_8 TMP_10)))))))) in (let TMP_26 +\def (\lambda (a0: A).(\lambda (_: (ex A (\lambda (a1: A).(leq g a0 (asucc g +a1))))).(\lambda (a1: A).(\lambda (H0: (ex A (\lambda (a2: A).(leq g a1 +(asucc g a2))))).(let H1 \def H0 in (let TMP_13 \def (\lambda (a2: A).(let +TMP_12 \def (asucc g a2) in (leq g a1 TMP_12))) in (let TMP_16 \def (\lambda +(a2: A).(let TMP_14 \def (AHead a0 a1) in (let TMP_15 \def (asucc g a2) in +(leq g TMP_14 TMP_15)))) in (let TMP_17 \def (ex A TMP_16) in (let TMP_25 +\def (\lambda (x: A).(\lambda (H2: (leq g a1 (asucc g x))).(let TMP_20 \def +(\lambda (a2: A).(let TMP_18 \def (AHead a0 a1) in (let TMP_19 \def (asucc g +a2) in (leq g TMP_18 TMP_19)))) in (let TMP_21 \def (AHead a0 x) in (let +TMP_22 \def (leq_refl g a0) in (let TMP_23 \def (asucc g x) in (let TMP_24 +\def (leq_head g a0 a0 TMP_22 a1 TMP_23 H2) in (ex_intro A TMP_20 TMP_21 +TMP_24)))))))) in (ex_ind A TMP_13 TMP_17 TMP_25 H1)))))))))) in (A_ind TMP_3 +TMP_11 TMP_26 a))))). theorem leq_ahead_asucc_false: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) (asucc g a1)) \to (\forall (P: Prop).P)))) \def - \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: -A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) (\lambda -(n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead -(ASort n n0) a2) (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) -\Rightarrow (ASort h n0)]))).(\lambda (P: Prop).(nat_ind (\lambda (n1: -nat).((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O -(next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P)) (\lambda (H0: (leq g -(AHead (ASort O n0) a2) (ASort O (next g n0)))).(let H_x \def (leq_gen_head1 -g (ASort O n0) a2 (ASort O (next g n0)) H0) in (let H1 \def H_x in (ex3_2_ind -A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: -A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A -(ASort O (next g n0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: + \lambda (g: G).(\lambda (a1: A).(let TMP_1 \def (\lambda (a: A).(\forall +(a2: A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P)))) in +(let TMP_34 \def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a2: +A).(\lambda (H: (leq g (AHead (ASort n n0) a2) (match n with [O \Rightarrow +(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]))).(\lambda (P: +Prop).(let TMP_2 \def (\lambda (n1: nat).((leq g (AHead (ASort n1 n0) a2) +(match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow +(ASort h n0)])) \to P)) in (let TMP_18 \def (\lambda (H0: (leq g (AHead +(ASort O n0) a2) (ASort O (next g n0)))).(let TMP_3 \def (ASort O n0) in (let +TMP_4 \def (next g n0) in (let TMP_5 \def (ASort O TMP_4) in (let H_x \def +(leq_gen_head1 g TMP_3 a2 TMP_5 H0) in (let H1 \def H_x in (let TMP_7 \def +(\lambda (a3: A).(\lambda (_: A).(let TMP_6 \def (ASort O n0) in (leq g TMP_6 +a3)))) in (let TMP_8 \def (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) in +(let TMP_12 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_9 \def (next g +n0) in (let TMP_10 \def (ASort O TMP_9) in (let TMP_11 \def (AHead a3 a4) in +(eq A TMP_10 TMP_11)))))) in (let TMP_17 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda -(H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let H5 \def (eq_ind A -(ASort O (next g n0)) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))) (\lambda (n1: +(H4: (eq A (ASort O (next g n0)) (AHead x0 x1))).(let TMP_13 \def (next g n0) +in (let TMP_14 \def (ASort O TMP_13) in (let TMP_15 \def (\lambda (ee: +A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow +False])) in (let TMP_16 \def (AHead x0 x1) in (let H5 \def (eq_ind A TMP_14 +TMP_15 I TMP_16 H4) in (False_ind P H5))))))))))) in (ex3_2_ind A A TMP_7 +TMP_8 TMP_12 P TMP_17 H1))))))))))) in (let TMP_33 \def (\lambda (n1: nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (match n1 with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)])) \to P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort n1 n0))).(let -H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 (ASort n1 n0) H0) in (let H1 -\def H_x in (ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (ASort (S -n1) n0) a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A (ASort n1 n0) (AHead a3 a4)))) P (\lambda (x0: -A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) n0) x0)).(\lambda (_: -(leq g a2 x1)).(\lambda (H4: (eq A (ASort n1 n0) (AHead x0 x1))).(let H5 \def -(eq_ind A (ASort n1 n0) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow -False])) I (AHead x0 x1) H4) in (False_ind P H5))))))) H1)))))) n H)))))) -(\lambda (a: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a a2) (asucc g -a)) \to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall -(a2: A).((leq g (AHead a0 a2) (asucc g a0)) \to (\forall (P: -Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) -(AHead a (asucc g a0)))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g -(AHead a a0) a2 (AHead a (asucc g a0)) H1) in (let H2 \def H_x in (ex3_2_ind -A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) (\lambda (_: -A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A -(AHead a (asucc g a0)) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 -x1)).(\lambda (H5: (eq A (AHead a (asucc g a0)) (AHead x0 x1))).(let H6 \def -(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with -[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a (asucc g -a0)) (AHead x0 x1) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e -in A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow ((let rec asucc -(g0: G) (l: A) on l: A \def (match l with [(ASort n0 n) \Rightarrow (match n0 -with [O \Rightarrow (ASort O (next g0 n)) | (S h) \Rightarrow (ASort h n)]) | -(AHead a3 a4) \Rightarrow (AHead a3 (asucc g0 a4))]) in asucc) g a0) | (AHead -_ a3) \Rightarrow a3])) (AHead a (asucc g a0)) (AHead x0 x1) H5) in (\lambda -(H8: (eq A a x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) -H4 (asucc g a0) H7) in (let H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g -(AHead a a0) a3)) H3 a H8) in (leq_ahead_false_1 g a a0 H10 P))))) H6))))))) -H2)))))))))) a1)). -(* COMMENTS -Initial nodes: 927 -END *) +TMP_19 \def (S n1) in (let TMP_20 \def (ASort TMP_19 n0) in (let TMP_21 \def +(ASort n1 n0) in (let H_x \def (leq_gen_head1 g TMP_20 a2 TMP_21 H0) in (let +H1 \def H_x in (let TMP_24 \def (\lambda (a3: A).(\lambda (_: A).(let TMP_22 +\def (S n1) in (let TMP_23 \def (ASort TMP_22 n0) in (leq g TMP_23 a3))))) in +(let TMP_25 \def (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) in (let +TMP_28 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_26 \def (ASort n1 n0) +in (let TMP_27 \def (AHead a3 a4) in (eq A TMP_26 TMP_27))))) in (let TMP_32 +\def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S n1) n0) +x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort n1 n0) (AHead x0 +x1))).(let TMP_29 \def (ASort n1 n0) in (let TMP_30 \def (\lambda (ee: +A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) \Rightarrow +False])) in (let TMP_31 \def (AHead x0 x1) in (let H5 \def (eq_ind A TMP_29 +TMP_30 I TMP_31 H4) in (False_ind P H5)))))))))) in (ex3_2_ind A A TMP_24 +TMP_25 TMP_28 P TMP_32 H1))))))))))))) in (nat_ind TMP_2 TMP_18 TMP_33 n +H))))))))) in (let TMP_61 \def (\lambda (a: A).(\lambda (_: ((\forall (a2: +A).((leq g (AHead a a2) (asucc g a)) \to (\forall (P: Prop).P))))).(\lambda +(a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead a0 a2) (asucc g a0)) +\to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq g (AHead +(AHead a a0) a2) (AHead a (asucc g a0)))).(\lambda (P: Prop).(let TMP_35 \def +(AHead a a0) in (let TMP_36 \def (asucc g a0) in (let TMP_37 \def (AHead a +TMP_36) in (let H_x \def (leq_gen_head1 g TMP_35 a2 TMP_37 H1) in (let H2 +\def H_x in (let TMP_39 \def (\lambda (a3: A).(\lambda (_: A).(let TMP_38 +\def (AHead a a0) in (leq g TMP_38 a3)))) in (let TMP_40 \def (\lambda (_: +A).(\lambda (a4: A).(leq g a2 a4))) in (let TMP_44 \def (\lambda (a3: +A).(\lambda (a4: A).(let TMP_41 \def (asucc g a0) in (let TMP_42 \def (AHead +a TMP_41) in (let TMP_43 \def (AHead a3 a4) in (eq A TMP_42 TMP_43)))))) in +(let TMP_60 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g +(AHead a a0) x0)).(\lambda (H4: (leq g a2 x1)).(\lambda (H5: (eq A (AHead a +(asucc g a0)) (AHead x0 x1))).(let TMP_45 \def (\lambda (e: A).(match e with +[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) in (let TMP_46 +\def (asucc g a0) in (let TMP_47 \def (AHead a TMP_46) in (let TMP_48 \def +(AHead x0 x1) in (let H6 \def (f_equal A A TMP_45 TMP_47 TMP_48 H5) in (let +TMP_51 \def (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow (asucc g +a0) | (AHead _ a3) \Rightarrow a3])) in (let TMP_52 \def (asucc g a0) in (let +TMP_53 \def (AHead a TMP_52) in (let TMP_54 \def (AHead x0 x1) in (let H7 +\def (f_equal A A TMP_51 TMP_53 TMP_54 H5) in (let TMP_59 \def (\lambda (H8: +(eq A a x0)).(let TMP_55 \def (\lambda (a3: A).(leq g a2 a3)) in (let TMP_56 +\def (asucc g a0) in (let H9 \def (eq_ind_r A x1 TMP_55 H4 TMP_56 H7) in (let +TMP_58 \def (\lambda (a3: A).(let TMP_57 \def (AHead a a0) in (leq g TMP_57 +a3))) in (let H10 \def (eq_ind_r A x0 TMP_58 H3 a H8) in (leq_ahead_false_1 g +a a0 H10 P))))))) in (TMP_59 H6))))))))))))))))) in (ex3_2_ind A A TMP_39 +TMP_40 TMP_44 P TMP_60 H2))))))))))))))))) in (A_ind TMP_1 TMP_34 TMP_61 +a1))))). theorem leq_asucc_false: \forall (g: G).(\forall (a: A).((leq g (asucc g a) a) \to (\forall (P: Prop).P))) \def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).((leq g (asucc g a0) -a0) \to (\forall (P: Prop).P))) (\lambda (n: nat).(\lambda (n0: nat).(\lambda -(H: (leq g (match n with [O \Rightarrow (ASort O (next g n0)) | (S h) -\Rightarrow (ASort h n0)]) (ASort n n0))).(\lambda (P: Prop).(nat_ind -(\lambda (n1: nat).((leq g (match n1 with [O \Rightarrow (ASort O (next g -n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P)) (\lambda (H0: -(leq g (ASort O (next g n0)) (ASort O n0))).(let H_x \def (leq_gen_sort1 g O -(next g n0) (ASort O n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat -(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -O (next g n0)) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda -(h2: nat).(\lambda (_: nat).(eq A (ASort O n0) (ASort h2 n2))))) P (\lambda -(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g -(ASort O (next g n0)) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A -(ASort O n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: -A).(match e in A return (\lambda (_: A).nat) with [(ASort n1 _) \Rightarrow -n1 | (AHead _ _) \Rightarrow O])) (ASort O n0) (ASort x1 x0) H3) in ((let H5 -\def (f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) -with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) \Rightarrow n0])) (ASort O -n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat O x1)).(let H7 \def (eq_ind_r -nat x1 (\lambda (n1: nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g -(ASort n1 x0) x2))) H2 O H6) in (let H8 \def (eq_ind_r nat x0 (\lambda (n1: -nat).(eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort O n1) x2))) H7 -n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort O (next g n0)) x2) -(\lambda (a0: A).(eq A a0 (aplus g (ASort O n0) x2))) H8 (aplus g (ASort O -n0) (S x2)) (aplus_sort_O_S_simpl g n0 x2)) in (let H_y \def (aplus_inj g (S -x2) x2 (ASort O n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n1: -nat).(le n1 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))) (\lambda -(n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow (ASort O (next -g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to P))).(\lambda -(H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let H_x \def (leq_gen_sort1 g -n1 n0 (ASort (S n1) n0) H0) in (let H1 \def H_x in (ex2_3_ind nat nat nat -(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -n1 n0) k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A (ASort (S n1) n0) (ASort h2 n2))))) P (\lambda -(x0: nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g + \lambda (g: G).(\lambda (a: A).(let TMP_1 \def (\lambda (a0: A).((leq g +(asucc g a0) a0) \to (\forall (P: Prop).P))) in (let TMP_103 \def (\lambda +(n: nat).(\lambda (n0: nat).(\lambda (H: (leq g (match n with [O \Rightarrow +(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n +n0))).(\lambda (P: Prop).(let TMP_2 \def (\lambda (n1: nat).((leq g (match n1 +with [O \Rightarrow (ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) +(ASort n1 n0)) \to P)) in (let TMP_50 \def (\lambda (H0: (leq g (ASort O +(next g n0)) (ASort O n0))).(let TMP_3 \def (next g n0) in (let TMP_4 \def +(ASort O n0) in (let H_x \def (leq_gen_sort1 g O TMP_3 TMP_4 H0) in (let H1 +\def H_x in (let TMP_10 \def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda +(k: nat).(let TMP_5 \def (next g n0) in (let TMP_6 \def (ASort O TMP_5) in +(let TMP_7 \def (aplus g TMP_6 k) in (let TMP_8 \def (ASort h2 n2) in (let +TMP_9 \def (aplus g TMP_8 k) in (eq A TMP_7 TMP_9))))))))) in (let TMP_13 +\def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_11 \def +(ASort O n0) in (let TMP_12 \def (ASort h2 n2) in (eq A TMP_11 TMP_12)))))) +in (let TMP_49 \def (\lambda (x0: nat).(\lambda (x1: nat).(\lambda (x2: +nat).(\lambda (H2: (eq A (aplus g (ASort O (next g n0)) x2) (aplus g (ASort +x1 x0) x2))).(\lambda (H3: (eq A (ASort O n0) (ASort x1 x0))).(let TMP_14 +\def (\lambda (e: A).(match e with [(ASort n1 _) \Rightarrow n1 | (AHead _ _) +\Rightarrow O])) in (let TMP_15 \def (ASort O n0) in (let TMP_16 \def (ASort +x1 x0) in (let H4 \def (f_equal A nat TMP_14 TMP_15 TMP_16 H3) in (let TMP_17 +\def (\lambda (e: A).(match e with [(ASort _ n1) \Rightarrow n1 | (AHead _ _) +\Rightarrow n0])) in (let TMP_18 \def (ASort O n0) in (let TMP_19 \def (ASort +x1 x0) in (let H5 \def (f_equal A nat TMP_17 TMP_18 TMP_19 H3) in (let TMP_48 +\def (\lambda (H6: (eq nat O x1)).(let TMP_25 \def (\lambda (n1: nat).(let +TMP_20 \def (next g n0) in (let TMP_21 \def (ASort O TMP_20) in (let TMP_22 +\def (aplus g TMP_21 x2) in (let TMP_23 \def (ASort n1 x0) in (let TMP_24 +\def (aplus g TMP_23 x2) in (eq A TMP_22 TMP_24))))))) in (let H7 \def +(eq_ind_r nat x1 TMP_25 H2 O H6) in (let TMP_31 \def (\lambda (n1: nat).(let +TMP_26 \def (next g n0) in (let TMP_27 \def (ASort O TMP_26) in (let TMP_28 +\def (aplus g TMP_27 x2) in (let TMP_29 \def (ASort O n1) in (let TMP_30 \def +(aplus g TMP_29 x2) in (eq A TMP_28 TMP_30))))))) in (let H8 \def (eq_ind_r +nat x0 TMP_31 H7 n0 H5) in (let TMP_32 \def (next g n0) in (let TMP_33 \def +(ASort O TMP_32) in (let TMP_34 \def (aplus g TMP_33 x2) in (let TMP_37 \def +(\lambda (a0: A).(let TMP_35 \def (ASort O n0) in (let TMP_36 \def (aplus g +TMP_35 x2) in (eq A a0 TMP_36)))) in (let TMP_38 \def (ASort O n0) in (let +TMP_39 \def (S x2) in (let TMP_40 \def (aplus g TMP_38 TMP_39) in (let TMP_41 +\def (aplus_sort_O_S_simpl g n0 x2) in (let H9 \def (eq_ind_r A TMP_34 TMP_37 +H8 TMP_40 TMP_41) in (let TMP_42 \def (S x2) in (let TMP_43 \def (ASort O n0) +in (let H_y \def (aplus_inj g TMP_42 x2 TMP_43 H9) in (let TMP_44 \def +(\lambda (n1: nat).(le n1 x2)) in (let TMP_45 \def (le_n x2) in (let TMP_46 +\def (S x2) in (let TMP_47 \def (eq_ind_r nat x2 TMP_44 TMP_45 TMP_46 H_y) in +(le_Sx_x x2 TMP_47 P)))))))))))))))))))))) in (TMP_48 H4))))))))))))))) in +(ex2_3_ind nat nat nat TMP_10 TMP_13 P TMP_49 H1))))))))) in (let TMP_102 +\def (\lambda (n1: nat).(\lambda (_: (((leq g (match n1 with [O \Rightarrow +(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) (ASort n1 n0)) \to +P))).(\lambda (H0: (leq g (ASort n1 n0) (ASort (S n1) n0))).(let TMP_51 \def +(S n1) in (let TMP_52 \def (ASort TMP_51 n0) in (let H_x \def (leq_gen_sort1 +g n1 n0 TMP_52 H0) in (let H1 \def H_x in (let TMP_57 \def (\lambda (n2: +nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_53 \def (ASort n1 n0) in +(let TMP_54 \def (aplus g TMP_53 k) in (let TMP_55 \def (ASort h2 n2) in (let +TMP_56 \def (aplus g TMP_55 k) in (eq A TMP_54 TMP_56)))))))) in (let TMP_61 +\def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_58 \def +(S n1) in (let TMP_59 \def (ASort TMP_58 n0) in (let TMP_60 \def (ASort h2 +n2) in (eq A TMP_59 TMP_60))))))) in (let TMP_101 \def (\lambda (x0: +nat).(\lambda (x1: nat).(\lambda (x2: nat).(\lambda (H2: (eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort x1 x0) x2))).(\lambda (H3: (eq A (ASort (S -n1) n0) (ASort x1 x0))).(let H4 \def (f_equal A nat (\lambda (e: A).(match e -in A return (\lambda (_: A).nat) with [(ASort n2 _) \Rightarrow n2 | (AHead _ -_) \Rightarrow (S n1)])) (ASort (S n1) n0) (ASort x1 x0) H3) in ((let H5 \def -(f_equal A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with -[(ASort _ n2) \Rightarrow n2 | (AHead _ _) \Rightarrow n0])) (ASort (S n1) -n0) (ASort x1 x0) H3) in (\lambda (H6: (eq nat (S n1) x1)).(let H7 \def -(eq_ind_r nat x1 (\lambda (n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g -(ASort n2 x0) x2))) H2 (S n1) H6) in (let H8 \def (eq_ind_r nat x0 (\lambda -(n2: nat).(eq A (aplus g (ASort n1 n0) x2) (aplus g (ASort (S n1) n2) x2))) -H7 n0 H5) in (let H9 \def (eq_ind_r A (aplus g (ASort n1 n0) x2) (\lambda -(a0: A).(eq A a0 (aplus g (ASort (S n1) n0) x2))) H8 (aplus g (ASort (S n1) -n0) (S x2)) (aplus_sort_S_S_simpl g n0 n1 x2)) in (let H_y \def (aplus_inj g -(S x2) x2 (ASort (S n1) n0) H9) in (le_Sx_x x2 (eq_ind_r nat x2 (\lambda (n2: -nat).(le n2 x2)) (le_n x2) (S x2) H_y) P))))))) H4))))))) H1)))))) n H))))) -(\lambda (a0: A).(\lambda (_: (((leq g (asucc g a0) a0) \to (\forall (P: -Prop).P)))).(\lambda (a1: A).(\lambda (H0: (((leq g (asucc g a1) a1) \to -(\forall (P: Prop).P)))).(\lambda (H1: (leq g (AHead a0 (asucc g a1)) (AHead -a0 a1))).(\lambda (P: Prop).(let H_x \def (leq_gen_head1 g a0 (asucc g a1) -(AHead a0 a1) H1) in (let H2 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g a0 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g -(asucc g a1) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (AHead a0 a1) -(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (H3: (leq g a0 -x0)).(\lambda (H4: (leq g (asucc g a1) x1)).(\lambda (H5: (eq A (AHead a0 a1) -(AHead x0 x1))).(let H6 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead a2 _) -\Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) H5) in ((let H7 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a1 | (AHead _ a2) \Rightarrow a2])) (AHead a0 a1) (AHead x0 x1) -H5) in (\lambda (H8: (eq A a0 x0)).(let H9 \def (eq_ind_r A x1 (\lambda (a2: -A).(leq g (asucc g a1) a2)) H4 a1 H7) in (let H10 \def (eq_ind_r A x0 -(\lambda (a2: A).(leq g a0 a2)) H3 a0 H8) in (H0 H9 P))))) H6))))))) -H2))))))))) a)). -(* COMMENTS -Initial nodes: 1327 -END *) +n1) n0) (ASort x1 x0))).(let TMP_62 \def (\lambda (e: A).(match e with +[(ASort n2 _) \Rightarrow n2 | (AHead _ _) \Rightarrow (S n1)])) in (let +TMP_63 \def (S n1) in (let TMP_64 \def (ASort TMP_63 n0) in (let TMP_65 \def +(ASort x1 x0) in (let H4 \def (f_equal A nat TMP_62 TMP_64 TMP_65 H3) in (let +TMP_66 \def (\lambda (e: A).(match e with [(ASort _ n2) \Rightarrow n2 | +(AHead _ _) \Rightarrow n0])) in (let TMP_67 \def (S n1) in (let TMP_68 \def +(ASort TMP_67 n0) in (let TMP_69 \def (ASort x1 x0) in (let H5 \def (f_equal +A nat TMP_66 TMP_68 TMP_69 H3) in (let TMP_100 \def (\lambda (H6: (eq nat (S +n1) x1)).(let TMP_74 \def (\lambda (n2: nat).(let TMP_70 \def (ASort n1 n0) +in (let TMP_71 \def (aplus g TMP_70 x2) in (let TMP_72 \def (ASort n2 x0) in +(let TMP_73 \def (aplus g TMP_72 x2) in (eq A TMP_71 TMP_73)))))) in (let +TMP_75 \def (S n1) in (let H7 \def (eq_ind_r nat x1 TMP_74 H2 TMP_75 H6) in +(let TMP_81 \def (\lambda (n2: nat).(let TMP_76 \def (ASort n1 n0) in (let +TMP_77 \def (aplus g TMP_76 x2) in (let TMP_78 \def (S n1) in (let TMP_79 +\def (ASort TMP_78 n2) in (let TMP_80 \def (aplus g TMP_79 x2) in (eq A +TMP_77 TMP_80))))))) in (let H8 \def (eq_ind_r nat x0 TMP_81 H7 n0 H5) in +(let TMP_82 \def (ASort n1 n0) in (let TMP_83 \def (aplus g TMP_82 x2) in +(let TMP_87 \def (\lambda (a0: A).(let TMP_84 \def (S n1) in (let TMP_85 \def +(ASort TMP_84 n0) in (let TMP_86 \def (aplus g TMP_85 x2) in (eq A a0 +TMP_86))))) in (let TMP_88 \def (S n1) in (let TMP_89 \def (ASort TMP_88 n0) +in (let TMP_90 \def (S x2) in (let TMP_91 \def (aplus g TMP_89 TMP_90) in +(let TMP_92 \def (aplus_sort_S_S_simpl g n0 n1 x2) in (let H9 \def (eq_ind_r +A TMP_83 TMP_87 H8 TMP_91 TMP_92) in (let TMP_93 \def (S x2) in (let TMP_94 +\def (S n1) in (let TMP_95 \def (ASort TMP_94 n0) in (let H_y \def (aplus_inj +g TMP_93 x2 TMP_95 H9) in (let TMP_96 \def (\lambda (n2: nat).(le n2 x2)) in +(let TMP_97 \def (le_n x2) in (let TMP_98 \def (S x2) in (let TMP_99 \def +(eq_ind_r nat x2 TMP_96 TMP_97 TMP_98 H_y) in (le_Sx_x x2 TMP_99 +P)))))))))))))))))))))))) in (TMP_100 H4))))))))))))))))) in (ex2_3_ind nat +nat nat TMP_57 TMP_61 P TMP_101 H1))))))))))) in (nat_ind TMP_2 TMP_50 +TMP_102 n H)))))))) in (let TMP_123 \def (\lambda (a0: A).(\lambda (_: (((leq +g (asucc g a0) a0) \to (\forall (P: Prop).P)))).(\lambda (a1: A).(\lambda +(H0: (((leq g (asucc g a1) a1) \to (\forall (P: Prop).P)))).(\lambda (H1: +(leq g (AHead a0 (asucc g a1)) (AHead a0 a1))).(\lambda (P: Prop).(let +TMP_104 \def (asucc g a1) in (let TMP_105 \def (AHead a0 a1) in (let H_x \def +(leq_gen_head1 g a0 TMP_104 TMP_105 H1) in (let H2 \def H_x in (let TMP_106 +\def (\lambda (a3: A).(\lambda (_: A).(leq g a0 a3))) in (let TMP_108 \def +(\lambda (_: A).(\lambda (a4: A).(let TMP_107 \def (asucc g a1) in (leq g +TMP_107 a4)))) in (let TMP_111 \def (\lambda (a3: A).(\lambda (a4: A).(let +TMP_109 \def (AHead a0 a1) in (let TMP_110 \def (AHead a3 a4) in (eq A +TMP_109 TMP_110))))) in (let TMP_122 \def (\lambda (x0: A).(\lambda (x1: +A).(\lambda (H3: (leq g a0 x0)).(\lambda (H4: (leq g (asucc g a1) +x1)).(\lambda (H5: (eq A (AHead a0 a1) (AHead x0 x1))).(let TMP_112 \def +(\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a0 | (AHead a2 _) +\Rightarrow a2])) in (let TMP_113 \def (AHead a0 a1) in (let TMP_114 \def +(AHead x0 x1) in (let H6 \def (f_equal A A TMP_112 TMP_113 TMP_114 H5) in +(let TMP_115 \def (\lambda (e: A).(match e with [(ASort _ _) \Rightarrow a1 | +(AHead _ a2) \Rightarrow a2])) in (let TMP_116 \def (AHead a0 a1) in (let +TMP_117 \def (AHead x0 x1) in (let H7 \def (f_equal A A TMP_115 TMP_116 +TMP_117 H5) in (let TMP_121 \def (\lambda (H8: (eq A a0 x0)).(let TMP_119 +\def (\lambda (a2: A).(let TMP_118 \def (asucc g a1) in (leq g TMP_118 a2))) +in (let H9 \def (eq_ind_r A x1 TMP_119 H4 a1 H7) in (let TMP_120 \def +(\lambda (a2: A).(leq g a0 a2)) in (let H10 \def (eq_ind_r A x0 TMP_120 H3 a0 +H8) in (H0 H9 P)))))) in (TMP_121 H6))))))))))))))) in (ex3_2_ind A A TMP_106 +TMP_108 TMP_111 P TMP_122 H2))))))))))))))) in (A_ind TMP_1 TMP_103 TMP_123 +a))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/defs.ma index 5a5308280..45a4288aa 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/leq/defs.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/leq/defs.ma @@ -14,7 +14,7 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/aplus/defs.ma". +include "basic_1/aplus/defs.ma". inductive leq (g: G): A \to (A \to Prop) \def | leq_sort: \forall (h1: nat).(\forall (h2: nat).(\forall (n1: nat).(\forall diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/fwd.ma index e259fa19e..ccf43169f 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/leq/fwd.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/leq/fwd.ma @@ -14,7 +14,18 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/leq/defs.ma". +include "basic_1/leq/defs.ma". + +let rec leq_ind (g: G) (P: (A \to (A \to Prop))) (f: (\forall (h1: +nat).(\forall (h2: nat).(\forall (n1: nat).(\forall (n2: nat).(\forall (k: +nat).((eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k)) \to (P +(ASort h1 n1) (ASort h2 n2))))))))) (f0: (\forall (a1: A).(\forall (a2: +A).((leq g a1 a2) \to ((P a1 a2) \to (\forall (a3: A).(\forall (a4: A).((leq +g a3 a4) \to ((P a3 a4) \to (P (AHead a1 a3) (AHead a2 a4))))))))))) (a: A) +(a0: A) (l: leq g a a0) on l: P a a0 \def match l with [(leq_sort h1 h2 n1 n2 +k e) \Rightarrow (f h1 h2 n1 n2 k e) | (leq_head a1 a2 l0 a3 a4 l1) +\Rightarrow (let TMP_1 \def ((leq_ind g P f f0) a1 a2 l0) in (let TMP_2 \def +((leq_ind g P f f0) a3 a4 l1) in (f0 a1 a2 l0 TMP_1 a3 a4 l1 TMP_2)))]. theorem leq_gen_sort1: \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq @@ -24,57 +35,81 @@ k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2)))))))))) \def \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: -A).(\lambda (H: (leq g (ASort h1 n1) a2)).(insert_eq A (ASort h1 n1) (\lambda -(a: A).(leq g a a2)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a k) (aplus g (ASort -h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A -a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g y a2)).(leq_ind g -(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to (ex2_3 nat nat -nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a -k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A a0 (ASort h2 n2))))))))) (\lambda (h0: -nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k: -nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2) -k))).(\lambda (H2: (eq A (ASort h0 n0) (ASort h1 n1))).(let H3 \def (f_equal -A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort -n _) \Rightarrow n | (AHead _ _) \Rightarrow h0])) (ASort h0 n0) (ASort h1 -n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return -(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) -\Rightarrow n0])) (ASort h0 n0) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h0 -h1)).(let H6 \def (eq_ind nat n0 (\lambda (n: nat).(eq A (aplus g (ASort h0 -n) k) (aplus g (ASort h2 n2) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n: -nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: -nat).(eq A (aplus g (ASort h0 n) k0) (aplus g (ASort h3 n3) k0))))) (\lambda -(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) (ASort h3 -n3))))))) (let H7 \def (eq_ind nat h0 (\lambda (n: nat).(eq A (aplus g (ASort -n n1) k) (aplus g (ASort h2 n2) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda -(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda -(k0: nat).(eq A (aplus g (ASort n n1) k0) (aplus g (ASort h3 n3) k0))))) -(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h2 n2) -(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: -nat).(\lambda (k0: nat).(eq A (aplus g (ASort h1 n1) k0) (aplus g (ASort h3 -n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A -(ASort h2 n2) (ASort h3 n3))))) n2 h2 k H7 (refl_equal A (ASort h2 n2))) h0 -H5)) n0 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_: -(leq g a1 a3)).(\lambda (_: (((eq A a1 (ASort h1 n1)) \to (ex2_3 nat nat nat -(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a1 k) -(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A a3 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: -A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a4 (ASort h1 n1)) \to +A).(\lambda (H: (leq g (ASort h1 n1) a2)).(let TMP_1 \def (ASort h1 n1) in +(let TMP_2 \def (\lambda (a: A).(leq g a a2)) in (let TMP_9 \def (\lambda (a: +A).(let TMP_6 \def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: +nat).(let TMP_3 \def (aplus g a k) in (let TMP_4 \def (ASort h2 n2) in (let +TMP_5 \def (aplus g TMP_4 k) in (eq A TMP_3 TMP_5))))))) in (let TMP_8 \def +(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_7 \def +(ASort h2 n2) in (eq A a2 TMP_7))))) in (ex2_3 nat nat nat TMP_6 TMP_8)))) in +(let TMP_78 \def (\lambda (y: A).(\lambda (H0: (leq g y a2)).(let TMP_16 \def +(\lambda (a: A).(\lambda (a0: A).((eq A a (ASort h1 n1)) \to (let TMP_13 \def +(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_10 \def +(aplus g a k) in (let TMP_11 \def (ASort h2 n2) in (let TMP_12 \def (aplus g +TMP_11 k) in (eq A TMP_10 TMP_12))))))) in (let TMP_15 \def (\lambda (n2: +nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_14 \def (ASort h2 n2) in +(eq A a0 TMP_14))))) in (ex2_3 nat nat nat TMP_13 TMP_15)))))) in (let TMP_64 +\def (\lambda (h0: nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: +nat).(\lambda (k: nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g +(ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h0 n0) (ASort h1 n1))).(let +TMP_17 \def (\lambda (e: A).(match e with [(ASort n _) \Rightarrow n | (AHead +_ _) \Rightarrow h0])) in (let TMP_18 \def (ASort h0 n0) in (let TMP_19 \def +(ASort h1 n1) in (let H3 \def (f_equal A nat TMP_17 TMP_18 TMP_19 H2) in (let +TMP_20 \def (\lambda (e: A).(match e with [(ASort _ n) \Rightarrow n | (AHead +_ _) \Rightarrow n0])) in (let TMP_21 \def (ASort h0 n0) in (let TMP_22 \def +(ASort h1 n1) in (let H4 \def (f_equal A nat TMP_20 TMP_21 TMP_22 H2) in (let +TMP_63 \def (\lambda (H5: (eq nat h0 h1)).(let TMP_27 \def (\lambda (n: +nat).(let TMP_23 \def (ASort h0 n) in (let TMP_24 \def (aplus g TMP_23 k) in +(let TMP_25 \def (ASort h2 n2) in (let TMP_26 \def (aplus g TMP_25 k) in (eq +A TMP_24 TMP_26)))))) in (let H6 \def (eq_ind nat n0 TMP_27 H1 n1 H4) in (let +TMP_36 \def (\lambda (n: nat).(let TMP_32 \def (\lambda (n3: nat).(\lambda +(h3: nat).(\lambda (k0: nat).(let TMP_28 \def (ASort h0 n) in (let TMP_29 +\def (aplus g TMP_28 k0) in (let TMP_30 \def (ASort h3 n3) in (let TMP_31 +\def (aplus g TMP_30 k0) in (eq A TMP_29 TMP_31)))))))) in (let TMP_35 \def +(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(let TMP_33 \def +(ASort h2 n2) in (let TMP_34 \def (ASort h3 n3) in (eq A TMP_33 TMP_34)))))) +in (ex2_3 nat nat nat TMP_32 TMP_35)))) in (let TMP_41 \def (\lambda (n: +nat).(let TMP_37 \def (ASort n n1) in (let TMP_38 \def (aplus g TMP_37 k) in +(let TMP_39 \def (ASort h2 n2) in (let TMP_40 \def (aplus g TMP_39 k) in (eq +A TMP_38 TMP_40)))))) in (let H7 \def (eq_ind nat h0 TMP_41 H6 h1 H5) in (let +TMP_50 \def (\lambda (n: nat).(let TMP_46 \def (\lambda (n3: nat).(\lambda +(h3: nat).(\lambda (k0: nat).(let TMP_42 \def (ASort n n1) in (let TMP_43 +\def (aplus g TMP_42 k0) in (let TMP_44 \def (ASort h3 n3) in (let TMP_45 +\def (aplus g TMP_44 k0) in (eq A TMP_43 TMP_45)))))))) in (let TMP_49 \def +(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(let TMP_47 \def +(ASort h2 n2) in (let TMP_48 \def (ASort h3 n3) in (eq A TMP_47 TMP_48)))))) +in (ex2_3 nat nat nat TMP_46 TMP_49)))) in (let TMP_55 \def (\lambda (n3: +nat).(\lambda (h3: nat).(\lambda (k0: nat).(let TMP_51 \def (ASort h1 n1) in +(let TMP_52 \def (aplus g TMP_51 k0) in (let TMP_53 \def (ASort h3 n3) in +(let TMP_54 \def (aplus g TMP_53 k0) in (eq A TMP_52 TMP_54)))))))) in (let +TMP_58 \def (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(let +TMP_56 \def (ASort h2 n2) in (let TMP_57 \def (ASort h3 n3) in (eq A TMP_56 +TMP_57)))))) in (let TMP_59 \def (ASort h2 n2) in (let TMP_60 \def +(refl_equal A TMP_59) in (let TMP_61 \def (ex2_3_intro nat nat nat TMP_55 +TMP_58 n2 h2 k H7 TMP_60) in (let TMP_62 \def (eq_ind_r nat h1 TMP_50 TMP_61 +h0 H5) in (eq_ind_r nat n1 TMP_36 TMP_62 n0 H4)))))))))))))) in (TMP_63 +H3))))))))))))))))) in (let TMP_77 \def (\lambda (a1: A).(\lambda (a3: +A).(\lambda (_: (leq g a1 a3)).(\lambda (_: (((eq A a1 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: -nat).(eq A (aplus g a4 k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a5 (ASort h2 -n2))))))))).(\lambda (H5: (eq A (AHead a1 a4) (ASort h1 n1))).(let H6 \def -(eq_ind A (AHead a1 a4) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow -True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (AHead a1 a4) k) -(aplus g (ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A (AHead a3 a5) (ASort h2 n2)))))) H6))))))))))) y a2 H0))) -H))))). -(* COMMENTS -Initial nodes: 913 -END *) +nat).(eq A (aplus g a1 k) (aplus g (ASort h2 n2) k))))) (\lambda (n2: +nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a3 (ASort h2 +n2))))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 +a5)).(\lambda (_: (((eq A a4 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda +(n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g a4 k) (aplus g +(ASort h2 n2) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: +nat).(eq A a5 (ASort h2 n2))))))))).(\lambda (H5: (eq A (AHead a1 a4) (ASort +h1 n1))).(let TMP_65 \def (AHead a1 a4) in (let TMP_66 \def (\lambda (ee: +A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow +True])) in (let TMP_67 \def (ASort h1 n1) in (let H6 \def (eq_ind A TMP_65 +TMP_66 I TMP_67 H5) in (let TMP_72 \def (\lambda (n2: nat).(\lambda (h2: +nat).(\lambda (k: nat).(let TMP_68 \def (AHead a1 a4) in (let TMP_69 \def +(aplus g TMP_68 k) in (let TMP_70 \def (ASort h2 n2) in (let TMP_71 \def +(aplus g TMP_70 k) in (eq A TMP_69 TMP_71)))))))) in (let TMP_75 \def +(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_73 \def +(AHead a3 a5) in (let TMP_74 \def (ASort h2 n2) in (eq A TMP_73 TMP_74)))))) +in (let TMP_76 \def (ex2_3 nat nat nat TMP_72 TMP_75) in (False_ind TMP_76 +H6))))))))))))))))) in (leq_ind g TMP_16 TMP_64 TMP_77 y a2 H0)))))) in +(insert_eq A TMP_1 TMP_2 TMP_9 TMP_78 H))))))))). theorem leq_gen_head1: \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g @@ -83,51 +118,65 @@ a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A a (AHead a3 a4))))))))) \def \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda -(H: (leq g (AHead a1 a2) a)).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq -g a0 a)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g -a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0: -(leq g y a)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a0 (AHead a1 -a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda -(_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq -A a3 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: -nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort -h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h1 n1) -(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h1 n1) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | -(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A -(\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda -(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h2 n2) -(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: -(leq g a0 a3)).(\lambda (H2: (((eq A a0 (AHead a1 a2)) \to (ex3_2 A A -(\lambda (a4: A).(\lambda (_: A).(leq g a1 a4))) (\lambda (_: A).(\lambda -(a5: A).(leq g a2 a5))) (\lambda (a4: A).(\lambda (a5: A).(eq A a3 (AHead a4 -a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4 -a5)).(\lambda (H4: (((eq A a4 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: -A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 -a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A a5 (AHead a6 -a7)))))))).(\lambda (H5: (eq A (AHead a0 a4) (AHead a1 a2))).(let H6 \def -(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with -[(ASort _ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) (AHead a0 a4) -(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a4 | (AHead _ a6) -\Rightarrow a6])) (AHead a0 a4) (AHead a1 a2) H5) in (\lambda (H8: (eq A a0 -a1)).(let H9 \def (eq_ind A a4 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to -(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) (\lambda (_: -A).(\lambda (a8: A).(leq g a2 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A -a5 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a4 (\lambda (a6: -A).(leq g a6 a5)) H3 a2 H7) in (let H11 \def (eq_ind A a0 (\lambda (a6: -A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_: -A).(leq g a1 a7))) (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))) (\lambda -(a7: A).(\lambda (a8: A).(eq A a3 (AHead a7 a8))))))) H2 a1 H8) in (let H12 -\def (eq_ind A a0 (\lambda (a6: A).(leq g a6 a3)) H1 a1 H8) in (ex3_2_intro A -A (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6))) (\lambda (_: A).(\lambda -(a7: A).(leq g a2 a7))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a3 a5) -(AHead a6 a7)))) a3 a5 H12 H10 (refl_equal A (AHead a3 a5))))))))) -H6))))))))))) y a H0))) H))))). -(* COMMENTS -Initial nodes: 797 -END *) +(H: (leq g (AHead a1 a2) a)).(let TMP_1 \def (AHead a1 a2) in (let TMP_2 \def +(\lambda (a0: A).(leq g a0 a)) in (let TMP_7 \def (\lambda (_: A).(let TMP_3 +\def (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) in (let TMP_4 \def +(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) in (let TMP_6 \def (\lambda +(a3: A).(\lambda (a4: A).(let TMP_5 \def (AHead a3 a4) in (eq A a TMP_5)))) +in (ex3_2 A A TMP_3 TMP_4 TMP_6))))) in (let TMP_50 \def (\lambda (y: +A).(\lambda (H0: (leq g y a)).(let TMP_12 \def (\lambda (a0: A).(\lambda (a3: +A).((eq A a0 (AHead a1 a2)) \to (let TMP_8 \def (\lambda (a4: A).(\lambda (_: +A).(leq g a1 a4))) in (let TMP_9 \def (\lambda (_: A).(\lambda (a5: A).(leq g +a2 a5))) in (let TMP_11 \def (\lambda (a4: A).(\lambda (a5: A).(let TMP_10 +\def (AHead a4 a5) in (eq A a3 TMP_10)))) in (ex3_2 A A TMP_8 TMP_9 +TMP_11))))))) in (let TMP_22 \def (\lambda (h1: nat).(\lambda (h2: +nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq +A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A +(ASort h1 n1) (AHead a1 a2))).(let TMP_13 \def (ASort h1 n1) in (let TMP_14 +\def (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ +_) \Rightarrow False])) in (let TMP_15 \def (AHead a1 a2) in (let H3 \def +(eq_ind A TMP_13 TMP_14 I TMP_15 H2) in (let TMP_16 \def (\lambda (a3: +A).(\lambda (_: A).(leq g a1 a3))) in (let TMP_17 \def (\lambda (_: +A).(\lambda (a4: A).(leq g a2 a4))) in (let TMP_20 \def (\lambda (a3: +A).(\lambda (a4: A).(let TMP_18 \def (ASort h2 n2) in (let TMP_19 \def (AHead +a3 a4) in (eq A TMP_18 TMP_19))))) in (let TMP_21 \def (ex3_2 A A TMP_16 +TMP_17 TMP_20) in (False_ind TMP_21 H3)))))))))))))))) in (let TMP_49 \def +(\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: (leq g a0 a3)).(\lambda (H2: +(((eq A a0 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: +A).(leq g a1 a4))) (\lambda (_: A).(\lambda (a5: A).(leq g a2 a5))) (\lambda +(a4: A).(\lambda (a5: A).(eq A a3 (AHead a4 a5)))))))).(\lambda (a4: +A).(\lambda (a5: A).(\lambda (H3: (leq g a4 a5)).(\lambda (H4: (((eq A a4 +(AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: A).(\lambda (_: A).(leq g a1 +a6))) (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) (\lambda (a6: +A).(\lambda (a7: A).(eq A a5 (AHead a6 a7)))))))).(\lambda (H5: (eq A (AHead +a0 a4) (AHead a1 a2))).(let TMP_23 \def (\lambda (e: A).(match e with [(ASort +_ _) \Rightarrow a0 | (AHead a6 _) \Rightarrow a6])) in (let TMP_24 \def +(AHead a0 a4) in (let TMP_25 \def (AHead a1 a2) in (let H6 \def (f_equal A A +TMP_23 TMP_24 TMP_25 H5) in (let TMP_26 \def (\lambda (e: A).(match e with +[(ASort _ _) \Rightarrow a4 | (AHead _ a6) \Rightarrow a6])) in (let TMP_27 +\def (AHead a0 a4) in (let TMP_28 \def (AHead a1 a2) in (let H7 \def (f_equal +A A TMP_26 TMP_27 TMP_28 H5) in (let TMP_48 \def (\lambda (H8: (eq A a0 +a1)).(let TMP_33 \def (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to (let +TMP_29 \def (\lambda (a7: A).(\lambda (_: A).(leq g a1 a7))) in (let TMP_30 +\def (\lambda (_: A).(\lambda (a8: A).(leq g a2 a8))) in (let TMP_32 \def +(\lambda (a7: A).(\lambda (a8: A).(let TMP_31 \def (AHead a7 a8) in (eq A a5 +TMP_31)))) in (ex3_2 A A TMP_29 TMP_30 TMP_32)))))) in (let H9 \def (eq_ind A +a4 TMP_33 H4 a2 H7) in (let TMP_34 \def (\lambda (a6: A).(leq g a6 a5)) in +(let H10 \def (eq_ind A a4 TMP_34 H3 a2 H7) in (let TMP_39 \def (\lambda (a6: +A).((eq A a6 (AHead a1 a2)) \to (let TMP_35 \def (\lambda (a7: A).(\lambda +(_: A).(leq g a1 a7))) in (let TMP_36 \def (\lambda (_: A).(\lambda (a8: +A).(leq g a2 a8))) in (let TMP_38 \def (\lambda (a7: A).(\lambda (a8: A).(let +TMP_37 \def (AHead a7 a8) in (eq A a3 TMP_37)))) in (ex3_2 A A TMP_35 TMP_36 +TMP_38)))))) in (let H11 \def (eq_ind A a0 TMP_39 H2 a1 H8) in (let TMP_40 +\def (\lambda (a6: A).(leq g a6 a3)) in (let H12 \def (eq_ind A a0 TMP_40 H1 +a1 H8) in (let TMP_41 \def (\lambda (a6: A).(\lambda (_: A).(leq g a1 a6))) +in (let TMP_42 \def (\lambda (_: A).(\lambda (a7: A).(leq g a2 a7))) in (let +TMP_45 \def (\lambda (a6: A).(\lambda (a7: A).(let TMP_43 \def (AHead a3 a5) +in (let TMP_44 \def (AHead a6 a7) in (eq A TMP_43 TMP_44))))) in (let TMP_46 +\def (AHead a3 a5) in (let TMP_47 \def (refl_equal A TMP_46) in (ex3_2_intro +A A TMP_41 TMP_42 TMP_45 a3 a5 H12 H10 TMP_47))))))))))))))) in (TMP_48 +H6))))))))))))))))))) in (leq_ind g TMP_12 TMP_22 TMP_49 y a H0)))))) in +(insert_eq A TMP_1 TMP_2 TMP_7 TMP_50 H))))))))). theorem leq_gen_sort2: \forall (g: G).(\forall (h1: nat).(\forall (n1: nat).(\forall (a2: A).((leq @@ -137,57 +186,81 @@ k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a2 (ASort h2 n2)))))))))) \def \lambda (g: G).(\lambda (h1: nat).(\lambda (n1: nat).(\lambda (a2: -A).(\lambda (H: (leq g a2 (ASort h1 n1))).(insert_eq A (ASort h1 n1) (\lambda -(a: A).(leq g a2 a)) (\lambda (a: A).(ex2_3 nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) -(aplus g a k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq -A a2 (ASort h2 n2))))))) (\lambda (y: A).(\lambda (H0: (leq g a2 y)).(leq_ind -g (\lambda (a: A).(\lambda (a0: A).((eq A a0 (ASort h1 n1)) \to (ex2_3 nat -nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus -g (ASort h2 n2) k) (aplus g a0 k))))) (\lambda (n2: nat).(\lambda (h2: -nat).(\lambda (_: nat).(eq A a (ASort h2 n2))))))))) (\lambda (h0: -nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda (n2: nat).(\lambda (k: -nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) (aplus g (ASort h2 n2) -k))).(\lambda (H2: (eq A (ASort h2 n2) (ASort h1 n1))).(let H3 \def (f_equal -A nat (\lambda (e: A).(match e in A return (\lambda (_: A).nat) with [(ASort -n _) \Rightarrow n | (AHead _ _) \Rightarrow h2])) (ASort h2 n2) (ASort h1 -n1) H2) in ((let H4 \def (f_equal A nat (\lambda (e: A).(match e in A return -(\lambda (_: A).nat) with [(ASort _ n) \Rightarrow n | (AHead _ _) -\Rightarrow n2])) (ASort h2 n2) (ASort h1 n1) H2) in (\lambda (H5: (eq nat h2 -h1)).(let H6 \def (eq_ind nat n2 (\lambda (n: nat).(eq A (aplus g (ASort h0 -n0) k) (aplus g (ASort h2 n) k))) H1 n1 H4) in (eq_ind_r nat n1 (\lambda (n: -nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: -nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h2 n) k0))))) (\lambda -(n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) (ASort h3 -n3))))))) (let H7 \def (eq_ind nat h2 (\lambda (n: nat).(eq A (aplus g (ASort -h0 n0) k) (aplus g (ASort n n1) k))) H6 h1 H5) in (eq_ind_r nat h1 (\lambda -(n: nat).(ex2_3 nat nat nat (\lambda (n3: nat).(\lambda (h3: nat).(\lambda -(k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort n n1) k0))))) -(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A (ASort h0 n0) -(ASort h3 n3))))))) (ex2_3_intro nat nat nat (\lambda (n3: nat).(\lambda (h3: -nat).(\lambda (k0: nat).(eq A (aplus g (ASort h3 n3) k0) (aplus g (ASort h1 -n1) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A -(ASort h0 n0) (ASort h3 n3))))) n0 h0 k H7 (refl_equal A (ASort h0 n0))) h2 -H5)) n2 H4)))) H3))))))))) (\lambda (a1: A).(\lambda (a3: A).(\lambda (_: -(leq g a1 a3)).(\lambda (_: (((eq A a3 (ASort h1 n1)) \to (ex2_3 nat nat nat +A).(\lambda (H: (leq g a2 (ASort h1 n1))).(let TMP_1 \def (ASort h1 n1) in +(let TMP_2 \def (\lambda (a: A).(leq g a2 a)) in (let TMP_9 \def (\lambda (a: +A).(let TMP_6 \def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: +nat).(let TMP_3 \def (ASort h2 n2) in (let TMP_4 \def (aplus g TMP_3 k) in +(let TMP_5 \def (aplus g a k) in (eq A TMP_4 TMP_5))))))) in (let TMP_8 \def +(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_7 \def +(ASort h2 n2) in (eq A a2 TMP_7))))) in (ex2_3 nat nat nat TMP_6 TMP_8)))) in +(let TMP_78 \def (\lambda (y: A).(\lambda (H0: (leq g a2 y)).(let TMP_16 \def +(\lambda (a: A).(\lambda (a0: A).((eq A a0 (ASort h1 n1)) \to (let TMP_13 +\def (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(let TMP_10 \def +(ASort h2 n2) in (let TMP_11 \def (aplus g TMP_10 k) in (let TMP_12 \def +(aplus g a0 k) in (eq A TMP_11 TMP_12))))))) in (let TMP_15 \def (\lambda +(n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_14 \def (ASort h2 n2) +in (eq A a TMP_14))))) in (ex2_3 nat nat nat TMP_13 TMP_15)))))) in (let +TMP_64 \def (\lambda (h0: nat).(\lambda (h2: nat).(\lambda (n0: nat).(\lambda +(n2: nat).(\lambda (k: nat).(\lambda (H1: (eq A (aplus g (ASort h0 n0) k) +(aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h2 n2) (ASort h1 +n1))).(let TMP_17 \def (\lambda (e: A).(match e with [(ASort n _) \Rightarrow +n | (AHead _ _) \Rightarrow h2])) in (let TMP_18 \def (ASort h2 n2) in (let +TMP_19 \def (ASort h1 n1) in (let H3 \def (f_equal A nat TMP_17 TMP_18 TMP_19 +H2) in (let TMP_20 \def (\lambda (e: A).(match e with [(ASort _ n) +\Rightarrow n | (AHead _ _) \Rightarrow n2])) in (let TMP_21 \def (ASort h2 +n2) in (let TMP_22 \def (ASort h1 n1) in (let H4 \def (f_equal A nat TMP_20 +TMP_21 TMP_22 H2) in (let TMP_63 \def (\lambda (H5: (eq nat h2 h1)).(let +TMP_27 \def (\lambda (n: nat).(let TMP_23 \def (ASort h0 n0) in (let TMP_24 +\def (aplus g TMP_23 k) in (let TMP_25 \def (ASort h2 n) in (let TMP_26 \def +(aplus g TMP_25 k) in (eq A TMP_24 TMP_26)))))) in (let H6 \def (eq_ind nat +n2 TMP_27 H1 n1 H4) in (let TMP_36 \def (\lambda (n: nat).(let TMP_32 \def +(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: nat).(let TMP_28 \def +(ASort h3 n3) in (let TMP_29 \def (aplus g TMP_28 k0) in (let TMP_30 \def +(ASort h2 n) in (let TMP_31 \def (aplus g TMP_30 k0) in (eq A TMP_29 +TMP_31)))))))) in (let TMP_35 \def (\lambda (n3: nat).(\lambda (h3: +nat).(\lambda (_: nat).(let TMP_33 \def (ASort h0 n0) in (let TMP_34 \def +(ASort h3 n3) in (eq A TMP_33 TMP_34)))))) in (ex2_3 nat nat nat TMP_32 +TMP_35)))) in (let TMP_41 \def (\lambda (n: nat).(let TMP_37 \def (ASort h0 +n0) in (let TMP_38 \def (aplus g TMP_37 k) in (let TMP_39 \def (ASort n n1) +in (let TMP_40 \def (aplus g TMP_39 k) in (eq A TMP_38 TMP_40)))))) in (let +H7 \def (eq_ind nat h2 TMP_41 H6 h1 H5) in (let TMP_50 \def (\lambda (n: +nat).(let TMP_46 \def (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (k0: +nat).(let TMP_42 \def (ASort h3 n3) in (let TMP_43 \def (aplus g TMP_42 k0) +in (let TMP_44 \def (ASort n n1) in (let TMP_45 \def (aplus g TMP_44 k0) in +(eq A TMP_43 TMP_45)))))))) in (let TMP_49 \def (\lambda (n3: nat).(\lambda +(h3: nat).(\lambda (_: nat).(let TMP_47 \def (ASort h0 n0) in (let TMP_48 +\def (ASort h3 n3) in (eq A TMP_47 TMP_48)))))) in (ex2_3 nat nat nat TMP_46 +TMP_49)))) in (let TMP_55 \def (\lambda (n3: nat).(\lambda (h3: nat).(\lambda +(k0: nat).(let TMP_51 \def (ASort h3 n3) in (let TMP_52 \def (aplus g TMP_51 +k0) in (let TMP_53 \def (ASort h1 n1) in (let TMP_54 \def (aplus g TMP_53 k0) +in (eq A TMP_52 TMP_54)))))))) in (let TMP_58 \def (\lambda (n3: +nat).(\lambda (h3: nat).(\lambda (_: nat).(let TMP_56 \def (ASort h0 n0) in +(let TMP_57 \def (ASort h3 n3) in (eq A TMP_56 TMP_57)))))) in (let TMP_59 +\def (ASort h0 n0) in (let TMP_60 \def (refl_equal A TMP_59) in (let TMP_61 +\def (ex2_3_intro nat nat nat TMP_55 TMP_58 n0 h0 k H7 TMP_60) in (let TMP_62 +\def (eq_ind_r nat h1 TMP_50 TMP_61 h2 H5) in (eq_ind_r nat n1 TMP_36 TMP_62 +n2 H4)))))))))))))) in (TMP_63 H3))))))))))))))))) in (let TMP_77 \def +(\lambda (a1: A).(\lambda (a3: A).(\lambda (_: (leq g a1 a3)).(\lambda (_: +(((eq A a3 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda +(h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) (aplus g a3 +k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a1 +(ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g +a4 a5)).(\lambda (_: (((eq A a5 (ASort h1 n1)) \to (ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort -h2 n2) k) (aplus g a3 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A a1 (ASort h2 n2))))))))).(\lambda (a4: A).(\lambda (a5: -A).(\lambda (_: (leq g a4 a5)).(\lambda (_: (((eq A a5 (ASort h1 n1)) \to -(ex2_3 nat nat nat (\lambda (n2: nat).(\lambda (h2: nat).(\lambda (k: -nat).(eq A (aplus g (ASort h2 n2) k) (aplus g a5 k))))) (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (_: nat).(eq A a4 (ASort h2 -n2))))))))).(\lambda (H5: (eq A (AHead a3 a5) (ASort h1 n1))).(let H6 \def -(eq_ind A (AHead a3 a5) (\lambda (ee: A).(match ee in A return (\lambda (_: -A).Prop) with [(ASort _ _) \Rightarrow False | (AHead _ _) \Rightarrow -True])) I (ASort h1 n1) H5) in (False_ind (ex2_3 nat nat nat (\lambda (n2: -nat).(\lambda (h2: nat).(\lambda (k: nat).(eq A (aplus g (ASort h2 n2) k) -(aplus g (AHead a3 a5) k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda -(_: nat).(eq A (AHead a1 a4) (ASort h2 n2)))))) H6))))))))))) a2 y H0))) -H))))). -(* COMMENTS -Initial nodes: 913 -END *) +h2 n2) k) (aplus g a5 k))))) (\lambda (n2: nat).(\lambda (h2: nat).(\lambda +(_: nat).(eq A a4 (ASort h2 n2))))))))).(\lambda (H5: (eq A (AHead a3 a5) +(ASort h1 n1))).(let TMP_65 \def (AHead a3 a5) in (let TMP_66 \def (\lambda +(ee: A).(match ee with [(ASort _ _) \Rightarrow False | (AHead _ _) +\Rightarrow True])) in (let TMP_67 \def (ASort h1 n1) in (let H6 \def (eq_ind +A TMP_65 TMP_66 I TMP_67 H5) in (let TMP_72 \def (\lambda (n2: nat).(\lambda +(h2: nat).(\lambda (k: nat).(let TMP_68 \def (ASort h2 n2) in (let TMP_69 +\def (aplus g TMP_68 k) in (let TMP_70 \def (AHead a3 a5) in (let TMP_71 \def +(aplus g TMP_70 k) in (eq A TMP_69 TMP_71)))))))) in (let TMP_75 \def +(\lambda (n2: nat).(\lambda (h2: nat).(\lambda (_: nat).(let TMP_73 \def +(AHead a1 a4) in (let TMP_74 \def (ASort h2 n2) in (eq A TMP_73 TMP_74)))))) +in (let TMP_76 \def (ex2_3 nat nat nat TMP_72 TMP_75) in (False_ind TMP_76 +H6))))))))))))))))) in (leq_ind g TMP_16 TMP_64 TMP_77 a2 y H0)))))) in +(insert_eq A TMP_1 TMP_2 TMP_9 TMP_78 H))))))))). theorem leq_gen_head2: \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a: A).((leq g a @@ -196,49 +269,88 @@ a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: A).(\lambda (a4: A).(eq A a (AHead a3 a4))))))))) \def \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a: A).(\lambda -(H: (leq g a (AHead a1 a2))).(insert_eq A (AHead a1 a2) (\lambda (a0: A).(leq -g a a0)) (\lambda (_: A).(ex3_2 A A (\lambda (a3: A).(\lambda (_: A).(leq g -a3 a1))) (\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) (\lambda (a3: -A).(\lambda (a4: A).(eq A a (AHead a3 a4)))))) (\lambda (y: A).(\lambda (H0: -(leq g a y)).(leq_ind g (\lambda (a0: A).(\lambda (a3: A).((eq A a3 (AHead a1 -a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda -(_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq -A a0 (AHead a4 a5)))))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: -nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort -h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A (ASort h2 n2) -(AHead a1 a2))).(let H3 \def (eq_ind A (ASort h2 n2) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | -(AHead _ _) \Rightarrow False])) I (AHead a1 a2) H2) in (False_ind (ex3_2 A A -(\lambda (a3: A).(\lambda (_: A).(leq g a3 a1))) (\lambda (_: A).(\lambda -(a4: A).(leq g a4 a2))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort h1 n1) -(AHead a3 a4))))) H3))))))))) (\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: -(leq g a0 a3)).(\lambda (H2: (((eq A a3 (AHead a1 a2)) \to (ex3_2 A A -(\lambda (a4: A).(\lambda (_: A).(leq g a4 a1))) (\lambda (_: A).(\lambda -(a5: A).(leq g a5 a2))) (\lambda (a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 -a5)))))))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (H3: (leq g a4 -a5)).(\lambda (H4: (((eq A a5 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: -A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7 -a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A a4 (AHead a6 -a7)))))))).(\lambda (H5: (eq A (AHead a3 a5) (AHead a1 a2))).(let H6 \def -(f_equal A A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with -[(ASort _ _) \Rightarrow a3 | (AHead a6 _) \Rightarrow a6])) (AHead a3 a5) -(AHead a1 a2) H5) in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A -return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a5 | (AHead _ a6) -\Rightarrow a6])) (AHead a3 a5) (AHead a1 a2) H5) in (\lambda (H8: (eq A a3 -a1)).(let H9 \def (eq_ind A a5 (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to -(ex3_2 A A (\lambda (a7: A).(\lambda (_: A).(leq g a7 a1))) (\lambda (_: -A).(\lambda (a8: A).(leq g a8 a2))) (\lambda (a7: A).(\lambda (a8: A).(eq A -a4 (AHead a7 a8))))))) H4 a2 H7) in (let H10 \def (eq_ind A a5 (\lambda (a6: -A).(leq g a4 a6)) H3 a2 H7) in (let H11 \def (eq_ind A a3 (\lambda (a6: -A).((eq A a6 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a7: A).(\lambda (_: -A).(leq g a7 a1))) (\lambda (_: A).(\lambda (a8: A).(leq g a8 a2))) (\lambda -(a7: A).(\lambda (a8: A).(eq A a0 (AHead a7 a8))))))) H2 a1 H8) in (let H12 -\def (eq_ind A a3 (\lambda (a6: A).(leq g a0 a6)) H1 a1 H8) in (ex3_2_intro A -A (\lambda (a6: A).(\lambda (_: A).(leq g a6 a1))) (\lambda (_: A).(\lambda -(a7: A).(leq g a7 a2))) (\lambda (a6: A).(\lambda (a7: A).(eq A (AHead a0 a4) -(AHead a6 a7)))) a0 a4 H12 H10 (refl_equal A (AHead a0 a4))))))))) -H6))))))))))) a y H0))) H))))). -(* COMMENTS -Initial nodes: 797 -END *) +(H: (leq g a (AHead a1 a2))).(let TMP_1 \def (AHead a1 a2) in (let TMP_2 \def +(\lambda (a0: A).(leq g a a0)) in (let TMP_7 \def (\lambda (_: A).(let TMP_3 +\def (\lambda (a3: A).(\lambda (_: A).(leq g a3 a1))) in (let TMP_4 \def +(\lambda (_: A).(\lambda (a4: A).(leq g a4 a2))) in (let TMP_6 \def (\lambda +(a3: A).(\lambda (a4: A).(let TMP_5 \def (AHead a3 a4) in (eq A a TMP_5)))) +in (ex3_2 A A TMP_3 TMP_4 TMP_6))))) in (let TMP_50 \def (\lambda (y: +A).(\lambda (H0: (leq g a y)).(let TMP_12 \def (\lambda (a0: A).(\lambda (a3: +A).((eq A a3 (AHead a1 a2)) \to (let TMP_8 \def (\lambda (a4: A).(\lambda (_: +A).(leq g a4 a1))) in (let TMP_9 \def (\lambda (_: A).(\lambda (a5: A).(leq g +a5 a2))) in (let TMP_11 \def (\lambda (a4: A).(\lambda (a5: A).(let TMP_10 +\def (AHead a4 a5) in (eq A a0 TMP_10)))) in (ex3_2 A A TMP_8 TMP_9 +TMP_11))))))) in (let TMP_22 \def (\lambda (h1: nat).(\lambda (h2: +nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq +A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (H2: (eq A +(ASort h2 n2) (AHead a1 a2))).(let TMP_13 \def (ASort h2 n2) in (let TMP_14 +\def (\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ +_) \Rightarrow False])) in (let TMP_15 \def (AHead a1 a2) in (let H3 \def +(eq_ind A TMP_13 TMP_14 I TMP_15 H2) in (let TMP_16 \def (\lambda (a3: +A).(\lambda (_: A).(leq g a3 a1))) in (let TMP_17 \def (\lambda (_: +A).(\lambda (a4: A).(leq g a4 a2))) in (let TMP_20 \def (\lambda (a3: +A).(\lambda (a4: A).(let TMP_18 \def (ASort h1 n1) in (let TMP_19 \def (AHead +a3 a4) in (eq A TMP_18 TMP_19))))) in (let TMP_21 \def (ex3_2 A A TMP_16 +TMP_17 TMP_20) in (False_ind TMP_21 H3)))))))))))))))) in (let TMP_49 \def +(\lambda (a0: A).(\lambda (a3: A).(\lambda (H1: (leq g a0 a3)).(\lambda (H2: +(((eq A a3 (AHead a1 a2)) \to (ex3_2 A A (\lambda (a4: A).(\lambda (_: +A).(leq g a4 a1))) (\lambda (_: A).(\lambda (a5: A).(leq g a5 a2))) (\lambda +(a4: A).(\lambda (a5: A).(eq A a0 (AHead a4 a5)))))))).(\lambda (a4: +A).(\lambda (a5: A).(\lambda (H3: (leq g a4 a5)).(\lambda (H4: (((eq A a5 +(AHead a1 a2)) \to (ex3_2 A A (\lambda (a6: A).(\lambda (_: A).(leq g a6 +a1))) (\lambda (_: A).(\lambda (a7: A).(leq g a7 a2))) (\lambda (a6: +A).(\lambda (a7: A).(eq A a4 (AHead a6 a7)))))))).(\lambda (H5: (eq A (AHead +a3 a5) (AHead a1 a2))).(let TMP_23 \def (\lambda (e: A).(match e with [(ASort +_ _) \Rightarrow a3 | (AHead a6 _) \Rightarrow a6])) in (let TMP_24 \def +(AHead a3 a5) in (let TMP_25 \def (AHead a1 a2) in (let H6 \def (f_equal A A +TMP_23 TMP_24 TMP_25 H5) in (let TMP_26 \def (\lambda (e: A).(match e with +[(ASort _ _) \Rightarrow a5 | (AHead _ a6) \Rightarrow a6])) in (let TMP_27 +\def (AHead a3 a5) in (let TMP_28 \def (AHead a1 a2) in (let H7 \def (f_equal +A A TMP_26 TMP_27 TMP_28 H5) in (let TMP_48 \def (\lambda (H8: (eq A a3 +a1)).(let TMP_33 \def (\lambda (a6: A).((eq A a6 (AHead a1 a2)) \to (let +TMP_29 \def (\lambda (a7: A).(\lambda (_: A).(leq g a7 a1))) in (let TMP_30 +\def (\lambda (_: A).(\lambda (a8: A).(leq g a8 a2))) in (let TMP_32 \def +(\lambda (a7: A).(\lambda (a8: A).(let TMP_31 \def (AHead a7 a8) in (eq A a4 +TMP_31)))) in (ex3_2 A A TMP_29 TMP_30 TMP_32)))))) in (let H9 \def (eq_ind A +a5 TMP_33 H4 a2 H7) in (let TMP_34 \def (\lambda (a6: A).(leq g a4 a6)) in +(let H10 \def (eq_ind A a5 TMP_34 H3 a2 H7) in (let TMP_39 \def (\lambda (a6: +A).((eq A a6 (AHead a1 a2)) \to (let TMP_35 \def (\lambda (a7: A).(\lambda +(_: A).(leq g a7 a1))) in (let TMP_36 \def (\lambda (_: A).(\lambda (a8: +A).(leq g a8 a2))) in (let TMP_38 \def (\lambda (a7: A).(\lambda (a8: A).(let +TMP_37 \def (AHead a7 a8) in (eq A a0 TMP_37)))) in (ex3_2 A A TMP_35 TMP_36 +TMP_38)))))) in (let H11 \def (eq_ind A a3 TMP_39 H2 a1 H8) in (let TMP_40 +\def (\lambda (a6: A).(leq g a0 a6)) in (let H12 \def (eq_ind A a3 TMP_40 H1 +a1 H8) in (let TMP_41 \def (\lambda (a6: A).(\lambda (_: A).(leq g a6 a1))) +in (let TMP_42 \def (\lambda (_: A).(\lambda (a7: A).(leq g a7 a2))) in (let +TMP_45 \def (\lambda (a6: A).(\lambda (a7: A).(let TMP_43 \def (AHead a0 a4) +in (let TMP_44 \def (AHead a6 a7) in (eq A TMP_43 TMP_44))))) in (let TMP_46 +\def (AHead a0 a4) in (let TMP_47 \def (refl_equal A TMP_46) in (ex3_2_intro +A A TMP_41 TMP_42 TMP_45 a0 a4 H12 H10 TMP_47))))))))))))))) in (TMP_48 +H6))))))))))))))))))) in (leq_ind g TMP_12 TMP_22 TMP_49 a y H0)))))) in +(insert_eq A TMP_1 TMP_2 TMP_7 TMP_50 H))))))))). + +theorem ahead_inj_snd: + \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall +(a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4)))))) +\def + \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda +(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let TMP_1 \def +(AHead a3 a4) in (let H_x \def (leq_gen_head1 g a1 a2 TMP_1 H) in (let H0 +\def H_x in (let TMP_2 \def (\lambda (a5: A).(\lambda (_: A).(leq g a1 a5))) +in (let TMP_3 \def (\lambda (_: A).(\lambda (a6: A).(leq g a2 a6))) in (let +TMP_6 \def (\lambda (a5: A).(\lambda (a6: A).(let TMP_4 \def (AHead a3 a4) in +(let TMP_5 \def (AHead a5 a6) in (eq A TMP_4 TMP_5))))) in (let TMP_7 \def +(leq g a2 a4) in (let TMP_17 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda +(H1: (leq g a1 x0)).(\lambda (H2: (leq g a2 x1)).(\lambda (H3: (eq A (AHead +a3 a4) (AHead x0 x1))).(let TMP_8 \def (\lambda (e: A).(match e with [(ASort +_ _) \Rightarrow a3 | (AHead a _) \Rightarrow a])) in (let TMP_9 \def (AHead +a3 a4) in (let TMP_10 \def (AHead x0 x1) in (let H4 \def (f_equal A A TMP_8 +TMP_9 TMP_10 H3) in (let TMP_11 \def (\lambda (e: A).(match e with [(ASort _ +_) \Rightarrow a4 | (AHead _ a) \Rightarrow a])) in (let TMP_12 \def (AHead +a3 a4) in (let TMP_13 \def (AHead x0 x1) in (let H5 \def (f_equal A A TMP_11 +TMP_12 TMP_13 H3) in (let TMP_16 \def (\lambda (H6: (eq A a3 x0)).(let TMP_14 +\def (\lambda (a: A).(leq g a2 a)) in (let H7 \def (eq_ind_r A x1 TMP_14 H2 +a4 H5) in (let TMP_15 \def (\lambda (a: A).(leq g a1 a)) in (let H8 \def +(eq_ind_r A x0 TMP_15 H1 a3 H6) in H7))))) in (TMP_16 H4))))))))))))))) in +(ex3_2_ind A A TMP_2 TMP_3 TMP_6 TMP_7 TMP_17 H0)))))))))))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/leq/props.ma b/matita/matita/contribs/lambdadelta/basic_1/leq/props.ma index b83fc503e..48b72e636 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/leq/props.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/leq/props.ma @@ -14,220 +14,250 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/leq/fwd.ma". +include "basic_1/leq/fwd.ma". -include "Basic-1/aplus/props.ma". - -theorem ahead_inj_snd: - \forall (g: G).(\forall (a1: A).(\forall (a2: A).(\forall (a3: A).(\forall -(a4: A).((leq g (AHead a1 a2) (AHead a3 a4)) \to (leq g a2 a4)))))) -\def - \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda -(a4: A).(\lambda (H: (leq g (AHead a1 a2) (AHead a3 a4))).(let H_x \def -(leq_gen_head1 g a1 a2 (AHead a3 a4) H) in (let H0 \def H_x in (ex3_2_ind A A -(\lambda (a5: A).(\lambda (_: A).(leq g a1 a5))) (\lambda (_: A).(\lambda -(a6: A).(leq g a2 a6))) (\lambda (a5: A).(\lambda (a6: A).(eq A (AHead a3 a4) -(AHead a5 a6)))) (leq g a2 a4) (\lambda (x0: A).(\lambda (x1: A).(\lambda -(H1: (leq g a1 x0)).(\lambda (H2: (leq g a2 x1)).(\lambda (H3: (eq A (AHead -a3 a4) (AHead x0 x1))).(let H4 \def (f_equal A A (\lambda (e: A).(match e in -A return (\lambda (_: A).A) with [(ASort _ _) \Rightarrow a3 | (AHead a _) -\Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3) in ((let H5 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a4 | (AHead _ a) \Rightarrow a])) (AHead a3 a4) (AHead x0 x1) H3) -in (\lambda (H6: (eq A a3 x0)).(let H7 \def (eq_ind_r A x1 (\lambda (a: -A).(leq g a2 a)) H2 a4 H5) in (let H8 \def (eq_ind_r A x0 (\lambda (a: -A).(leq g a1 a)) H1 a3 H6) in H7)))) H4))))))) H0)))))))). -(* COMMENTS -Initial nodes: 259 -END *) +include "basic_1/aplus/props.ma". theorem leq_refl: \forall (g: G).(\forall (a: A).(leq g a a)) \def - \lambda (g: G).(\lambda (a: A).(A_ind (\lambda (a0: A).(leq g a0 a0)) -(\lambda (n: nat).(\lambda (n0: nat).(leq_sort g n n n0 n0 O (refl_equal A -(aplus g (ASort n n0) O))))) (\lambda (a0: A).(\lambda (H: (leq g a0 -a0)).(\lambda (a1: A).(\lambda (H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1 -H0))))) a)). -(* COMMENTS -Initial nodes: 87 -END *) + \lambda (g: G).(\lambda (a: A).(let TMP_1 \def (\lambda (a0: A).(leq g a0 +a0)) in (let TMP_5 \def (\lambda (n: nat).(\lambda (n0: nat).(let TMP_2 \def +(ASort n n0) in (let TMP_3 \def (aplus g TMP_2 O) in (let TMP_4 \def +(refl_equal A TMP_3) in (leq_sort g n n n0 n0 O TMP_4)))))) in (let TMP_6 +\def (\lambda (a0: A).(\lambda (H: (leq g a0 a0)).(\lambda (a1: A).(\lambda +(H0: (leq g a1 a1)).(leq_head g a0 a0 H a1 a1 H0))))) in (A_ind TMP_1 TMP_5 +TMP_6 a))))). theorem leq_eq: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((eq A a1 a2) \to (leq g a1 a2)))) \def \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (eq A a1 -a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))). -(* COMMENTS -Initial nodes: 39 -END *) +a2)).(let TMP_1 \def (\lambda (a: A).(leq g a1 a)) in (let TMP_2 \def +(leq_refl g a1) in (eq_ind A a1 TMP_1 TMP_2 a2 H)))))). theorem leq_sym: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (leq g a2 a1)))) \def \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) (\lambda (h1: -nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: -nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) -k))).(leq_sort g h2 h1 n2 n1 k (sym_eq A (aplus g (ASort h1 n1) k) (aplus g -(ASort h2 n2) k) H0)))))))) (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: -(leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: A).(\lambda (a6: -A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 a5)).(leq_head g a4 a3 -H1 a6 a5 H3))))))))) a1 a2 H)))). -(* COMMENTS -Initial nodes: 173 -END *) +a2)).(let TMP_1 \def (\lambda (a: A).(\lambda (a0: A).(leq g a0 a))) in (let +TMP_7 \def (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda +(n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort h1 n1) k) +(aplus g (ASort h2 n2) k))).(let TMP_2 \def (ASort h1 n1) in (let TMP_3 \def +(aplus g TMP_2 k) in (let TMP_4 \def (ASort h2 n2) in (let TMP_5 \def (aplus +g TMP_4 k) in (let TMP_6 \def (sym_eq A TMP_3 TMP_5 H0) in (leq_sort g h2 h1 +n2 n1 k TMP_6)))))))))))) in (let TMP_8 \def (\lambda (a3: A).(\lambda (a4: +A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: (leq g a4 a3)).(\lambda (a5: +A).(\lambda (a6: A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: (leq g a6 +a5)).(leq_head g a4 a3 H1 a6 a5 H3))))))))) in (leq_ind g TMP_1 TMP_7 TMP_8 +a1 a2 H))))))). theorem leq_trans: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3: A).((leq g a2 a3) \to (leq g a1 a3)))))) \def \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq g a0 -a3) \to (leq g a a3))))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: -nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (H0: (eq A (aplus g (ASort -h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda (a3: A).(\lambda (H1: (leq g -(ASort h2 n2) a3)).(let H_x \def (leq_gen_sort1 g h2 n2 a3 H1) in (let H2 -\def H_x in (ex2_3_ind nat nat nat (\lambda (n3: nat).(\lambda (h3: -nat).(\lambda (k0: nat).(eq A (aplus g (ASort h2 n2) k0) (aplus g (ASort h3 -n3) k0))))) (\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(eq A a3 -(ASort h3 n3))))) (leq g (ASort h1 n1) a3) (\lambda (x0: nat).(\lambda (x1: -nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort h2 n2) x2) (aplus -g (ASort x1 x0) x2))).(\lambda (H4: (eq A a3 (ASort x1 x0))).(let H5 \def -(f_equal A A (\lambda (e: A).e) a3 (ASort x1 x0) H4) in (eq_ind_r A (ASort x1 -x0) (\lambda (a: A).(leq g (ASort h1 n1) a)) (lt_le_e k x2 (leq g (ASort h1 -n1) (ASort x1 x0)) (\lambda (H6: (lt k x2)).(let H_y \def (aplus_reg_r g -(ASort h1 n1) (ASort h2 n2) k k H0 (minus x2 k)) in (let H7 \def (eq_ind_r -nat (plus (minus x2 k) k) (\lambda (n: nat).(eq A (aplus g (ASort h1 n1) n) -(aplus g (ASort h2 n2) n))) H_y x2 (le_plus_minus_sym k x2 (le_trans k (S k) -x2 (le_S k k (le_n k)) H6))) in (leq_sort g h1 x1 n1 x0 x2 (trans_eq A (aplus -g (ASort h1 n1) x2) (aplus g (ASort h2 n2) x2) (aplus g (ASort x1 x0) x2) H7 -H3))))) (\lambda (H6: (le x2 k)).(let H_y \def (aplus_reg_r g (ASort h2 n2) -(ASort x1 x0) x2 x2 H3 (minus k x2)) in (let H7 \def (eq_ind_r nat (plus -(minus k x2) x2) (\lambda (n: nat).(eq A (aplus g (ASort h2 n2) n) (aplus g -(ASort x1 x0) n))) H_y k (le_plus_minus_sym x2 k H6)) in (leq_sort g h1 x1 n1 -x0 k (trans_eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k) (aplus g -(ASort x1 x0) k) H0 H7)))))) a3 H5))))))) H2))))))))))) (\lambda (a3: +a2)).(let TMP_1 \def (\lambda (a: A).(\lambda (a0: A).(\forall (a3: A).((leq +g a0 a3) \to (leq g a a3))))) in (let TMP_63 \def (\lambda (h1: nat).(\lambda +(h2: nat).(\lambda (n1: nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda +(H0: (eq A (aplus g (ASort h1 n1) k) (aplus g (ASort h2 n2) k))).(\lambda +(a3: A).(\lambda (H1: (leq g (ASort h2 n2) a3)).(let H_x \def (leq_gen_sort1 +g h2 n2 a3 H1) in (let H2 \def H_x in (let TMP_6 \def (\lambda (n3: +nat).(\lambda (h3: nat).(\lambda (k0: nat).(let TMP_2 \def (ASort h2 n2) in +(let TMP_3 \def (aplus g TMP_2 k0) in (let TMP_4 \def (ASort h3 n3) in (let +TMP_5 \def (aplus g TMP_4 k0) in (eq A TMP_3 TMP_5)))))))) in (let TMP_8 \def +(\lambda (n3: nat).(\lambda (h3: nat).(\lambda (_: nat).(let TMP_7 \def +(ASort h3 n3) in (eq A a3 TMP_7))))) in (let TMP_9 \def (ASort h1 n1) in (let +TMP_10 \def (leq g TMP_9 a3) in (let TMP_62 \def (\lambda (x0: nat).(\lambda +(x1: nat).(\lambda (x2: nat).(\lambda (H3: (eq A (aplus g (ASort h2 n2) x2) +(aplus g (ASort x1 x0) x2))).(\lambda (H4: (eq A a3 (ASort x1 x0))).(let +TMP_11 \def (\lambda (e: A).e) in (let TMP_12 \def (ASort x1 x0) in (let H5 +\def (f_equal A A TMP_11 a3 TMP_12 H4) in (let TMP_13 \def (ASort x1 x0) in +(let TMP_15 \def (\lambda (a: A).(let TMP_14 \def (ASort h1 n1) in (leq g +TMP_14 a))) in (let TMP_16 \def (ASort h1 n1) in (let TMP_17 \def (ASort x1 +x0) in (let TMP_18 \def (leq g TMP_16 TMP_17) in (let TMP_41 \def (\lambda +(H6: (lt k x2)).(let TMP_19 \def (ASort h1 n1) in (let TMP_20 \def (ASort h2 +n2) in (let TMP_21 \def (minus x2 k) in (let H_y \def (aplus_reg_r g TMP_19 +TMP_20 k k H0 TMP_21) in (let TMP_22 \def (minus x2 k) in (let TMP_23 \def +(plus TMP_22 k) in (let TMP_28 \def (\lambda (n: nat).(let TMP_24 \def (ASort +h1 n1) in (let TMP_25 \def (aplus g TMP_24 n) in (let TMP_26 \def (ASort h2 +n2) in (let TMP_27 \def (aplus g TMP_26 n) in (eq A TMP_25 TMP_27)))))) in +(let TMP_29 \def (S k) in (let TMP_30 \def (le_n k) in (let TMP_31 \def (le_S +k k TMP_30) in (let TMP_32 \def (le_trans k TMP_29 x2 TMP_31 H6) in (let +TMP_33 \def (le_plus_minus_sym k x2 TMP_32) in (let H7 \def (eq_ind_r nat +TMP_23 TMP_28 H_y x2 TMP_33) in (let TMP_34 \def (ASort h1 n1) in (let TMP_35 +\def (aplus g TMP_34 x2) in (let TMP_36 \def (ASort h2 n2) in (let TMP_37 +\def (aplus g TMP_36 x2) in (let TMP_38 \def (ASort x1 x0) in (let TMP_39 +\def (aplus g TMP_38 x2) in (let TMP_40 \def (trans_eq A TMP_35 TMP_37 TMP_39 +H7 H3) in (leq_sort g h1 x1 n1 x0 x2 TMP_40)))))))))))))))))))))) in (let +TMP_60 \def (\lambda (H6: (le x2 k)).(let TMP_42 \def (ASort h2 n2) in (let +TMP_43 \def (ASort x1 x0) in (let TMP_44 \def (minus k x2) in (let H_y \def +(aplus_reg_r g TMP_42 TMP_43 x2 x2 H3 TMP_44) in (let TMP_45 \def (minus k +x2) in (let TMP_46 \def (plus TMP_45 x2) in (let TMP_51 \def (\lambda (n: +nat).(let TMP_47 \def (ASort h2 n2) in (let TMP_48 \def (aplus g TMP_47 n) in +(let TMP_49 \def (ASort x1 x0) in (let TMP_50 \def (aplus g TMP_49 n) in (eq +A TMP_48 TMP_50)))))) in (let TMP_52 \def (le_plus_minus_sym x2 k H6) in (let +H7 \def (eq_ind_r nat TMP_46 TMP_51 H_y k TMP_52) in (let TMP_53 \def (ASort +h1 n1) in (let TMP_54 \def (aplus g TMP_53 k) in (let TMP_55 \def (ASort h2 +n2) in (let TMP_56 \def (aplus g TMP_55 k) in (let TMP_57 \def (ASort x1 x0) +in (let TMP_58 \def (aplus g TMP_57 k) in (let TMP_59 \def (trans_eq A TMP_54 +TMP_56 TMP_58 H0 H7) in (leq_sort g h1 x1 n1 x0 k TMP_59)))))))))))))))))) in +(let TMP_61 \def (lt_le_e k x2 TMP_18 TMP_41 TMP_60) in (eq_ind_r A TMP_13 +TMP_15 TMP_61 a3 H5))))))))))))))))) in (ex2_3_ind nat nat nat TMP_6 TMP_8 +TMP_10 TMP_62 H2)))))))))))))))) in (let TMP_79 \def (\lambda (a3: A).(\lambda (a4: A).(\lambda (_: (leq g a3 a4)).(\lambda (H1: ((\forall (a5: A).((leq g a4 a5) \to (leq g a3 a5))))).(\lambda (a5: A).(\lambda (a6: A).(\lambda (_: (leq g a5 a6)).(\lambda (H3: ((\forall (a7: A).((leq g a6 a7) \to (leq g a5 a7))))).(\lambda (a0: A).(\lambda (H4: (leq g (AHead a4 a6) -a0)).(let H_x \def (leq_gen_head1 g a4 a6 a0 H4) in (let H5 \def H_x in -(ex3_2_ind A A (\lambda (a7: A).(\lambda (_: A).(leq g a4 a7))) (\lambda (_: -A).(\lambda (a8: A).(leq g a6 a8))) (\lambda (a7: A).(\lambda (a8: A).(eq A -a0 (AHead a7 a8)))) (leq g (AHead a3 a5) a0) (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H6: (leq g a4 x0)).(\lambda (H7: (leq g a6 x1)).(\lambda (H8: -(eq A a0 (AHead x0 x1))).(let H9 \def (f_equal A A (\lambda (e: A).e) a0 -(AHead x0 x1) H8) in (eq_ind_r A (AHead x0 x1) (\lambda (a: A).(leq g (AHead -a3 a5) a)) (leq_head g a3 x0 (H1 x0 H6) a5 x1 (H3 x1 H7)) a0 H9))))))) -H5))))))))))))) a1 a2 H)))). -(* COMMENTS -Initial nodes: 869 -END *) +a0)).(let H_x \def (leq_gen_head1 g a4 a6 a0 H4) in (let H5 \def H_x in (let +TMP_64 \def (\lambda (a7: A).(\lambda (_: A).(leq g a4 a7))) in (let TMP_65 +\def (\lambda (_: A).(\lambda (a8: A).(leq g a6 a8))) in (let TMP_67 \def +(\lambda (a7: A).(\lambda (a8: A).(let TMP_66 \def (AHead a7 a8) in (eq A a0 +TMP_66)))) in (let TMP_68 \def (AHead a3 a5) in (let TMP_69 \def (leq g +TMP_68 a0) in (let TMP_78 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda +(H6: (leq g a4 x0)).(\lambda (H7: (leq g a6 x1)).(\lambda (H8: (eq A a0 +(AHead x0 x1))).(let TMP_70 \def (\lambda (e: A).e) in (let TMP_71 \def +(AHead x0 x1) in (let H9 \def (f_equal A A TMP_70 a0 TMP_71 H8) in (let +TMP_72 \def (AHead x0 x1) in (let TMP_74 \def (\lambda (a: A).(let TMP_73 +\def (AHead a3 a5) in (leq g TMP_73 a))) in (let TMP_75 \def (H1 x0 H6) in +(let TMP_76 \def (H3 x1 H7) in (let TMP_77 \def (leq_head g a3 x0 TMP_75 a5 +x1 TMP_76) in (eq_ind_r A TMP_72 TMP_74 TMP_77 a0 H9)))))))))))))) in +(ex3_2_ind A A TMP_64 TMP_65 TMP_67 TMP_69 TMP_78 H5))))))))))))))))))) in +(leq_ind g TMP_1 TMP_63 TMP_79 a1 a2 H))))))). theorem leq_ahead_false_1: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g (AHead a1 a2) a1) \to (\forall (P: Prop).P)))) \def - \lambda (g: G).(\lambda (a1: A).(A_ind (\lambda (a: A).(\forall (a2: -A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq g (AHead (ASort n -n0) a2) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g -(AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead -(ASort O n0) a2) (ASort O n0))).(let H_x \def (leq_gen_head1 g (ASort O n0) -a2 (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g (ASort O n0) a3))) (\lambda (_: A).(\lambda (a4: -A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) -(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g + \lambda (g: G).(\lambda (a1: A).(let TMP_1 \def (\lambda (a: A).(\forall +(a2: A).((leq g (AHead a a2) a) \to (\forall (P: Prop).P)))) in (let TMP_34 +\def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a2: A).(\lambda (H: (leq +g (AHead (ASort n n0) a2) (ASort n n0))).(\lambda (P: Prop).(let TMP_2 \def +(\lambda (n1: nat).((leq g (AHead (ASort n1 n0) a2) (ASort n1 n0)) \to P)) in +(let TMP_15 \def (\lambda (H0: (leq g (AHead (ASort O n0) a2) (ASort O +n0))).(let TMP_3 \def (ASort O n0) in (let TMP_4 \def (ASort O n0) in (let +H_x \def (leq_gen_head1 g TMP_3 a2 TMP_4 H0) in (let H1 \def H_x in (let +TMP_6 \def (\lambda (a3: A).(\lambda (_: A).(let TMP_5 \def (ASort O n0) in +(leq g TMP_5 a3)))) in (let TMP_7 \def (\lambda (_: A).(\lambda (a4: A).(leq +g a2 a4))) in (let TMP_10 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_8 +\def (ASort O n0) in (let TMP_9 \def (AHead a3 a4) in (eq A TMP_8 TMP_9))))) +in (let TMP_14 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g (ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort O -n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: -A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow -True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P -H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead (ASort n1 -n0) a2) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) -a2) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g (ASort (S n1) n0) a2 -(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g (ASort (S n1) n0) a3))) (\lambda (_: A).(\lambda -(a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) -n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g -(ASort (S n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort -(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in -(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (H: -((\forall (a2: A).((leq g (AHead a a2) a) \to (\forall (P: -Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: A).((leq g (AHead -a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: A).(\lambda (H1: (leq -g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: Prop).(let H_x \def -(leq_gen_head1 g (AHead a a0) a2 (AHead a a0) H1) in (let H2 \def H_x in -(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g (AHead a a0) a3))) -(\lambda (_: A).(\lambda (a4: A).(leq g a2 a4))) (\lambda (a3: A).(\lambda -(a4: A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 -x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) -in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda -(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) -(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def -(eq_ind_r A x1 (\lambda (a3: A).(leq g a2 a3)) H4 a0 H7) in (let H10 \def -(eq_ind_r A x0 (\lambda (a3: A).(leq g (AHead a a0) a3)) H3 a H8) in (H a0 -H10 P))))) H6))))))) H2)))))))))) a1)). -(* COMMENTS -Initial nodes: 797 -END *) +n0) (AHead x0 x1))).(let TMP_11 \def (ASort O n0) in (let TMP_12 \def +(\lambda (ee: A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) +\Rightarrow False])) in (let TMP_13 \def (AHead x0 x1) in (let H5 \def +(eq_ind A TMP_11 TMP_12 I TMP_13 H4) in (False_ind P H5)))))))))) in +(ex3_2_ind A A TMP_6 TMP_7 TMP_10 P TMP_14 H1)))))))))) in (let TMP_33 \def +(\lambda (n1: nat).(\lambda (_: (((leq g (AHead (ASort n1 n0) a2) (ASort n1 +n0)) \to P))).(\lambda (H0: (leq g (AHead (ASort (S n1) n0) a2) (ASort (S n1) +n0))).(let TMP_16 \def (S n1) in (let TMP_17 \def (ASort TMP_16 n0) in (let +TMP_18 \def (S n1) in (let TMP_19 \def (ASort TMP_18 n0) in (let H_x \def +(leq_gen_head1 g TMP_17 a2 TMP_19 H0) in (let H1 \def H_x in (let TMP_22 \def +(\lambda (a3: A).(\lambda (_: A).(let TMP_20 \def (S n1) in (let TMP_21 \def +(ASort TMP_20 n0) in (leq g TMP_21 a3))))) in (let TMP_23 \def (\lambda (_: +A).(\lambda (a4: A).(leq g a2 a4))) in (let TMP_27 \def (\lambda (a3: +A).(\lambda (a4: A).(let TMP_24 \def (S n1) in (let TMP_25 \def (ASort TMP_24 +n0) in (let TMP_26 \def (AHead a3 a4) in (eq A TMP_25 TMP_26)))))) in (let +TMP_32 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g (ASort (S +n1) n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort (S n1) +n0) (AHead x0 x1))).(let TMP_28 \def (S n1) in (let TMP_29 \def (ASort TMP_28 +n0) in (let TMP_30 \def (\lambda (ee: A).(match ee with [(ASort _ _) +\Rightarrow True | (AHead _ _) \Rightarrow False])) in (let TMP_31 \def +(AHead x0 x1) in (let H5 \def (eq_ind A TMP_29 TMP_30 I TMP_31 H4) in +(False_ind P H5))))))))))) in (ex3_2_ind A A TMP_22 TMP_23 TMP_27 P TMP_32 +H1)))))))))))))) in (nat_ind TMP_2 TMP_15 TMP_33 n H))))))))) in (let TMP_54 +\def (\lambda (a: A).(\lambda (H: ((\forall (a2: A).((leq g (AHead a a2) a) +\to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (_: ((\forall (a2: +A).((leq g (AHead a0 a2) a0) \to (\forall (P: Prop).P))))).(\lambda (a2: +A).(\lambda (H1: (leq g (AHead (AHead a a0) a2) (AHead a a0))).(\lambda (P: +Prop).(let TMP_35 \def (AHead a a0) in (let TMP_36 \def (AHead a a0) in (let +H_x \def (leq_gen_head1 g TMP_35 a2 TMP_36 H1) in (let H2 \def H_x in (let +TMP_38 \def (\lambda (a3: A).(\lambda (_: A).(let TMP_37 \def (AHead a a0) in +(leq g TMP_37 a3)))) in (let TMP_39 \def (\lambda (_: A).(\lambda (a4: +A).(leq g a2 a4))) in (let TMP_42 \def (\lambda (a3: A).(\lambda (a4: A).(let +TMP_40 \def (AHead a a0) in (let TMP_41 \def (AHead a3 a4) in (eq A TMP_40 +TMP_41))))) in (let TMP_53 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda +(H3: (leq g (AHead a a0) x0)).(\lambda (H4: (leq g a2 x1)).(\lambda (H5: (eq +A (AHead a a0) (AHead x0 x1))).(let TMP_43 \def (\lambda (e: A).(match e with +[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) in (let TMP_44 +\def (AHead a a0) in (let TMP_45 \def (AHead x0 x1) in (let H6 \def (f_equal +A A TMP_43 TMP_44 TMP_45 H5) in (let TMP_46 \def (\lambda (e: A).(match e +with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) in (let +TMP_47 \def (AHead a a0) in (let TMP_48 \def (AHead x0 x1) in (let H7 \def +(f_equal A A TMP_46 TMP_47 TMP_48 H5) in (let TMP_52 \def (\lambda (H8: (eq A +a x0)).(let TMP_49 \def (\lambda (a3: A).(leq g a2 a3)) in (let H9 \def +(eq_ind_r A x1 TMP_49 H4 a0 H7) in (let TMP_51 \def (\lambda (a3: A).(let +TMP_50 \def (AHead a a0) in (leq g TMP_50 a3))) in (let H10 \def (eq_ind_r A +x0 TMP_51 H3 a H8) in (H a0 H10 P)))))) in (TMP_52 H6))))))))))))))) in +(ex3_2_ind A A TMP_38 TMP_39 TMP_42 P TMP_53 H2)))))))))))))))) in (A_ind +TMP_1 TMP_34 TMP_54 a1))))). theorem leq_ahead_false_2: \forall (g: G).(\forall (a2: A).(\forall (a1: A).((leq g (AHead a1 a2) a2) \to (\forall (P: Prop).P)))) \def - \lambda (g: G).(\lambda (a2: A).(A_ind (\lambda (a: A).(\forall (a1: -A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) (\lambda (n: -nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq g (AHead a1 (ASort -n n0)) (ASort n n0))).(\lambda (P: Prop).(nat_ind (\lambda (n1: nat).((leq g -(AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) (\lambda (H0: (leq g (AHead -a1 (ASort O n0)) (ASort O n0))).(let H_x \def (leq_gen_head1 g a1 (ASort O -n0) (ASort O n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g -(ASort O n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort O n0) -(AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 + \lambda (g: G).(\lambda (a2: A).(let TMP_1 \def (\lambda (a: A).(\forall +(a1: A).((leq g (AHead a1 a) a) \to (\forall (P: Prop).P)))) in (let TMP_34 +\def (\lambda (n: nat).(\lambda (n0: nat).(\lambda (a1: A).(\lambda (H: (leq +g (AHead a1 (ASort n n0)) (ASort n n0))).(\lambda (P: Prop).(let TMP_2 \def +(\lambda (n1: nat).((leq g (AHead a1 (ASort n1 n0)) (ASort n1 n0)) \to P)) in +(let TMP_15 \def (\lambda (H0: (leq g (AHead a1 (ASort O n0)) (ASort O +n0))).(let TMP_3 \def (ASort O n0) in (let TMP_4 \def (ASort O n0) in (let +H_x \def (leq_gen_head1 g a1 TMP_3 TMP_4 H0) in (let H1 \def H_x in (let +TMP_5 \def (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) in (let TMP_7 +\def (\lambda (_: A).(\lambda (a4: A).(let TMP_6 \def (ASort O n0) in (leq g +TMP_6 a4)))) in (let TMP_10 \def (\lambda (a3: A).(\lambda (a4: A).(let TMP_8 +\def (ASort O n0) in (let TMP_9 \def (AHead a3 a4) in (eq A TMP_8 TMP_9))))) +in (let TMP_14 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 x0)).(\lambda (_: (leq g (ASort O n0) x1)).(\lambda (H4: (eq A (ASort O n0) -(AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: A).(match -ee in A return (\lambda (_: A).Prop) with [(ASort _ _) \Rightarrow True | -(AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in (False_ind P -H5))))))) H1)))) (\lambda (n1: nat).(\lambda (_: (((leq g (AHead a1 (ASort n1 -n0)) (ASort n1 n0)) \to P))).(\lambda (H0: (leq g (AHead a1 (ASort (S n1) -n0)) (ASort (S n1) n0))).(let H_x \def (leq_gen_head1 g a1 (ASort (S n1) n0) -(ASort (S n1) n0) H0) in (let H1 \def H_x in (ex3_2_ind A A (\lambda (a3: -A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: A).(\lambda (a4: A).(leq g -(ASort (S n1) n0) a4))) (\lambda (a3: A).(\lambda (a4: A).(eq A (ASort (S n1) -n0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g -a1 x0)).(\lambda (_: (leq g (ASort (S n1) n0) x1)).(\lambda (H4: (eq A (ASort -(S n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda -(ee: A).(match ee in A return (\lambda (_: A).Prop) with [(ASort _ _) -\Rightarrow True | (AHead _ _) \Rightarrow False])) I (AHead x0 x1) H4) in -(False_ind P H5))))))) H1)))))) n H)))))) (\lambda (a: A).(\lambda (_: -((\forall (a1: A).((leq g (AHead a1 a) a) \to (\forall (P: -Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1: A).((leq g (AHead -a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: A).(\lambda (H1: (leq -g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P: Prop).(let H_x \def -(leq_gen_head1 g a1 (AHead a a0) (AHead a a0) H1) in (let H2 \def H_x in -(ex3_2_ind A A (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) (\lambda (_: -A).(\lambda (a4: A).(leq g (AHead a a0) a4))) (\lambda (a3: A).(\lambda (a4: -A).(eq A (AHead a a0) (AHead a3 a4)))) P (\lambda (x0: A).(\lambda (x1: -A).(\lambda (H3: (leq g a1 x0)).(\lambda (H4: (leq g (AHead a a0) -x1)).(\lambda (H5: (eq A (AHead a a0) (AHead x0 x1))).(let H6 \def (f_equal A -A (\lambda (e: A).(match e in A return (\lambda (_: A).A) with [(ASort _ _) -\Rightarrow a | (AHead a3 _) \Rightarrow a3])) (AHead a a0) (AHead x0 x1) H5) -in ((let H7 \def (f_equal A A (\lambda (e: A).(match e in A return (\lambda -(_: A).A) with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) -(AHead a a0) (AHead x0 x1) H5) in (\lambda (H8: (eq A a x0)).(let H9 \def -(eq_ind_r A x1 (\lambda (a3: A).(leq g (AHead a a0) a3)) H4 a0 H7) in (let -H10 \def (eq_ind_r A x0 (\lambda (a3: A).(leq g a1 a3)) H3 a H8) in (H0 a H9 -P))))) H6))))))) H2)))))))))) a2)). -(* COMMENTS -Initial nodes: 797 -END *) +(AHead x0 x1))).(let TMP_11 \def (ASort O n0) in (let TMP_12 \def (\lambda +(ee: A).(match ee with [(ASort _ _) \Rightarrow True | (AHead _ _) +\Rightarrow False])) in (let TMP_13 \def (AHead x0 x1) in (let H5 \def +(eq_ind A TMP_11 TMP_12 I TMP_13 H4) in (False_ind P H5)))))))))) in +(ex3_2_ind A A TMP_5 TMP_7 TMP_10 P TMP_14 H1)))))))))) in (let TMP_33 \def +(\lambda (n1: nat).(\lambda (_: (((leq g (AHead a1 (ASort n1 n0)) (ASort n1 +n0)) \to P))).(\lambda (H0: (leq g (AHead a1 (ASort (S n1) n0)) (ASort (S n1) +n0))).(let TMP_16 \def (S n1) in (let TMP_17 \def (ASort TMP_16 n0) in (let +TMP_18 \def (S n1) in (let TMP_19 \def (ASort TMP_18 n0) in (let H_x \def +(leq_gen_head1 g a1 TMP_17 TMP_19 H0) in (let H1 \def H_x in (let TMP_20 \def +(\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) in (let TMP_23 \def (\lambda +(_: A).(\lambda (a4: A).(let TMP_21 \def (S n1) in (let TMP_22 \def (ASort +TMP_21 n0) in (leq g TMP_22 a4))))) in (let TMP_27 \def (\lambda (a3: +A).(\lambda (a4: A).(let TMP_24 \def (S n1) in (let TMP_25 \def (ASort TMP_24 +n0) in (let TMP_26 \def (AHead a3 a4) in (eq A TMP_25 TMP_26)))))) in (let +TMP_32 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda (_: (leq g a1 +x0)).(\lambda (_: (leq g (ASort (S n1) n0) x1)).(\lambda (H4: (eq A (ASort (S +n1) n0) (AHead x0 x1))).(let TMP_28 \def (S n1) in (let TMP_29 \def (ASort +TMP_28 n0) in (let TMP_30 \def (\lambda (ee: A).(match ee with [(ASort _ _) +\Rightarrow True | (AHead _ _) \Rightarrow False])) in (let TMP_31 \def +(AHead x0 x1) in (let H5 \def (eq_ind A TMP_29 TMP_30 I TMP_31 H4) in +(False_ind P H5))))))))))) in (ex3_2_ind A A TMP_20 TMP_23 TMP_27 P TMP_32 +H1)))))))))))))) in (nat_ind TMP_2 TMP_15 TMP_33 n H))))))))) in (let TMP_54 +\def (\lambda (a: A).(\lambda (_: ((\forall (a1: A).((leq g (AHead a1 a) a) +\to (\forall (P: Prop).P))))).(\lambda (a0: A).(\lambda (H0: ((\forall (a1: +A).((leq g (AHead a1 a0) a0) \to (\forall (P: Prop).P))))).(\lambda (a1: +A).(\lambda (H1: (leq g (AHead a1 (AHead a a0)) (AHead a a0))).(\lambda (P: +Prop).(let TMP_35 \def (AHead a a0) in (let TMP_36 \def (AHead a a0) in (let +H_x \def (leq_gen_head1 g a1 TMP_35 TMP_36 H1) in (let H2 \def H_x in (let +TMP_37 \def (\lambda (a3: A).(\lambda (_: A).(leq g a1 a3))) in (let TMP_39 +\def (\lambda (_: A).(\lambda (a4: A).(let TMP_38 \def (AHead a a0) in (leq g +TMP_38 a4)))) in (let TMP_42 \def (\lambda (a3: A).(\lambda (a4: A).(let +TMP_40 \def (AHead a a0) in (let TMP_41 \def (AHead a3 a4) in (eq A TMP_40 +TMP_41))))) in (let TMP_53 \def (\lambda (x0: A).(\lambda (x1: A).(\lambda +(H3: (leq g a1 x0)).(\lambda (H4: (leq g (AHead a a0) x1)).(\lambda (H5: (eq +A (AHead a a0) (AHead x0 x1))).(let TMP_43 \def (\lambda (e: A).(match e with +[(ASort _ _) \Rightarrow a | (AHead a3 _) \Rightarrow a3])) in (let TMP_44 +\def (AHead a a0) in (let TMP_45 \def (AHead x0 x1) in (let H6 \def (f_equal +A A TMP_43 TMP_44 TMP_45 H5) in (let TMP_46 \def (\lambda (e: A).(match e +with [(ASort _ _) \Rightarrow a0 | (AHead _ a3) \Rightarrow a3])) in (let +TMP_47 \def (AHead a a0) in (let TMP_48 \def (AHead x0 x1) in (let H7 \def +(f_equal A A TMP_46 TMP_47 TMP_48 H5) in (let TMP_52 \def (\lambda (H8: (eq A +a x0)).(let TMP_50 \def (\lambda (a3: A).(let TMP_49 \def (AHead a a0) in +(leq g TMP_49 a3))) in (let H9 \def (eq_ind_r A x1 TMP_50 H4 a0 H7) in (let +TMP_51 \def (\lambda (a3: A).(leq g a1 a3)) in (let H10 \def (eq_ind_r A x0 +TMP_51 H3 a H8) in (H0 a H9 P)))))) in (TMP_52 H6))))))))))))))) in +(ex3_2_ind A A TMP_37 TMP_39 TMP_42 P TMP_53 H2)))))))))))))))) in (A_ind +TMP_1 TMP_34 TMP_54 a2))))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/llt/defs.ma b/matita/matita/contribs/lambdadelta/basic_1/llt/defs.ma index 96d869935..d56fda894 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/llt/defs.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/llt/defs.ma @@ -14,17 +14,15 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/A/defs.ma". +include "basic_1/A/defs.ma". -definition lweight: - A \to nat -\def - let rec lweight (a: A) on a: nat \def (match a with [(ASort _ _) \Rightarrow -O | (AHead a1 a2) \Rightarrow (S (plus (lweight a1) (lweight a2)))]) in -lweight. +let rec lweight (a: A) on a: nat \def match a with [(ASort _ _) \Rightarrow O +| (AHead a1 a2) \Rightarrow (let TMP_1 \def (lweight a1) in (let TMP_2 \def +(lweight a2) in (let TMP_3 \def (plus TMP_1 TMP_2) in (S TMP_3))))]. definition llt: A \to (A \to Prop) \def - \lambda (a1: A).(\lambda (a2: A).(lt (lweight a1) (lweight a2))). + \lambda (a1: A).(\lambda (a2: A).(let TMP_1 \def (lweight a1) in (let TMP_2 +\def (lweight a2) in (lt TMP_1 TMP_2)))). diff --git a/matita/matita/contribs/lambdadelta/basic_1/llt/fwd.ma b/matita/matita/contribs/lambdadelta/basic_1/llt/fwd.ma new file mode 100644 index 000000000..025bf660f --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_1/llt/fwd.ma @@ -0,0 +1,51 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *********************) + +include "basic_1/llt/defs.ma". + +theorem llt_wf__q_ind: + \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P0: ((A \to +Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P0 +a))))) P n))) \to (\forall (a: A).(P a))) +\def + let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a: +A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to +Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a) +n) \to (P a)))))).(\lambda (a: A).(let TMP_1 \def (lweight a) in (let TMP_2 +\def (lweight a) in (let TMP_3 \def (refl_equal nat TMP_2) in (H TMP_1 a +TMP_3))))))). + +theorem llt_wf_ind: + \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1 +a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a))) +\def + let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a: +A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to +Prop))).(\lambda (H: ((\forall (a2: A).(((\forall (a1: A).((lt (lweight a1) +(lweight a2)) \to (P a1)))) \to (P a2))))).(\lambda (a: A).(let TMP_1 \def +(\lambda (a0: A).(P a0)) in (let TMP_11 \def (\lambda (n: nat).(let TMP_2 +\def (\lambda (a0: A).(P a0)) in (let TMP_3 \def (Q TMP_2) in (let TMP_10 +\def (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) \to (Q +(\lambda (a0: A).(P a0)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat +(lweight a0) n0)).(let TMP_4 \def (\lambda (n1: nat).(\forall (m: nat).((lt m +n1) \to (\forall (a1: A).((eq nat (lweight a1) m) \to (P a1)))))) in (let +TMP_5 \def (lweight a0) in (let H2 \def (eq_ind_r nat n0 TMP_4 H0 TMP_5 H1) +in (let TMP_9 \def (\lambda (a1: A).(\lambda (H3: (lt (lweight a1) (lweight +a0))).(let TMP_6 \def (lweight a1) in (let TMP_7 \def (lweight a1) in (let +TMP_8 \def (refl_equal nat TMP_7) in (H2 TMP_6 H3 a1 TMP_8)))))) in (H a0 +TMP_9))))))))) in (lt_wf_ind n TMP_3 TMP_10))))) in (llt_wf__q_ind TMP_1 +TMP_11 a)))))). + diff --git a/matita/matita/contribs/lambdadelta/basic_1/llt/props.ma b/matita/matita/contribs/lambdadelta/basic_1/llt/props.ma index ef1f15a96..fb70372f9 100644 --- a/matita/matita/contribs/lambdadelta/basic_1/llt/props.ma +++ b/matita/matita/contribs/lambdadelta/basic_1/llt/props.ma @@ -14,101 +14,67 @@ (* This file was automatically generated: do not edit *********************) -include "Basic-1/llt/defs.ma". +include "basic_1/llt/defs.ma". -include "Basic-1/leq/defs.ma". +include "basic_1/leq/fwd.ma". theorem lweight_repl: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (eq nat (lweight a1) (lweight a2))))) \def \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(leq_ind g (\lambda (a: A).(\lambda (a0: A).(eq nat (lweight a) (lweight -a0)))) (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: nat).(\lambda (n2: -nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort h1 n1) k) (aplus g -(ASort h2 n2) k))).(refl_equal nat O))))))) (\lambda (a0: A).(\lambda (a3: -A).(\lambda (_: (leq g a0 a3)).(\lambda (H1: (eq nat (lweight a0) (lweight -a3))).(\lambda (a4: A).(\lambda (a5: A).(\lambda (_: (leq g a4 a5)).(\lambda -(H3: (eq nat (lweight a4) (lweight a5))).(f_equal nat nat S (plus (lweight -a0) (lweight a4)) (plus (lweight a3) (lweight a5)) (f_equal2 nat nat nat plus -(lweight a0) (lweight a3) (lweight a4) (lweight a5) H1 H3)))))))))) a1 a2 -H)))). -(* COMMENTS -Initial nodes: 189 -END *) +a2)).(let TMP_3 \def (\lambda (a: A).(\lambda (a0: A).(let TMP_1 \def +(lweight a) in (let TMP_2 \def (lweight a0) in (eq nat TMP_1 TMP_2))))) in +(let TMP_4 \def (\lambda (h1: nat).(\lambda (h2: nat).(\lambda (n1: +nat).(\lambda (n2: nat).(\lambda (k: nat).(\lambda (_: (eq A (aplus g (ASort +h1 n1) k) (aplus g (ASort h2 n2) k))).(refl_equal nat O))))))) in (let TMP_16 +\def (\lambda (a0: A).(\lambda (a3: A).(\lambda (_: (leq g a0 a3)).(\lambda +(H1: (eq nat (lweight a0) (lweight a3))).(\lambda (a4: A).(\lambda (a5: +A).(\lambda (_: (leq g a4 a5)).(\lambda (H3: (eq nat (lweight a4) (lweight +a5))).(let TMP_5 \def (lweight a0) in (let TMP_6 \def (lweight a4) in (let +TMP_7 \def (plus TMP_5 TMP_6) in (let TMP_8 \def (lweight a3) in (let TMP_9 +\def (lweight a5) in (let TMP_10 \def (plus TMP_8 TMP_9) in (let TMP_11 \def +(lweight a0) in (let TMP_12 \def (lweight a3) in (let TMP_13 \def (lweight +a4) in (let TMP_14 \def (lweight a5) in (let TMP_15 \def (f_equal2 nat nat +nat plus TMP_11 TMP_12 TMP_13 TMP_14 H1 H3) in (f_equal nat nat S TMP_7 +TMP_10 TMP_15)))))))))))))))))))) in (leq_ind g TMP_3 TMP_4 TMP_16 a1 a2 +H))))))). theorem llt_repl: \forall (g: G).(\forall (a1: A).(\forall (a2: A).((leq g a1 a2) \to (\forall (a3: A).((llt a1 a3) \to (llt a2 a3)))))) \def \lambda (g: G).(\lambda (a1: A).(\lambda (a2: A).(\lambda (H: (leq g a1 -a2)).(\lambda (a3: A).(\lambda (H0: (lt (lweight a1) (lweight a3))).(let H1 -\def (eq_ind nat (lweight a1) (\lambda (n: nat).(lt n (lweight a3))) H0 -(lweight a2) (lweight_repl g a1 a2 H)) in H1)))))). -(* COMMENTS -Initial nodes: 61 -END *) +a2)).(\lambda (a3: A).(\lambda (H0: (lt (lweight a1) (lweight a3))).(let +TMP_1 \def (lweight a1) in (let TMP_3 \def (\lambda (n: nat).(let TMP_2 \def +(lweight a3) in (lt n TMP_2))) in (let TMP_4 \def (lweight a2) in (let TMP_5 +\def (lweight_repl g a1 a2 H) in (let H1 \def (eq_ind nat TMP_1 TMP_3 H0 +TMP_4 TMP_5) in H1)))))))))). theorem llt_trans: \forall (a1: A).(\forall (a2: A).(\forall (a3: A).((llt a1 a2) \to ((llt a2 a3) \to (llt a1 a3))))) \def \lambda (a1: A).(\lambda (a2: A).(\lambda (a3: A).(\lambda (H: (lt (lweight -a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(lt_trans -(lweight a1) (lweight a2) (lweight a3) H H0))))). -(* COMMENTS -Initial nodes: 43 -END *) +a1) (lweight a2))).(\lambda (H0: (lt (lweight a2) (lweight a3))).(let TMP_1 +\def (lweight a1) in (let TMP_2 \def (lweight a2) in (let TMP_3 \def (lweight +a3) in (lt_trans TMP_1 TMP_2 TMP_3 H H0)))))))). theorem llt_head_sx: \forall (a1: A).(\forall (a2: A).(llt a1 (AHead a1 a2))) \def - \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a1) (plus (lweight a1) -(lweight a2)) (le_plus_l (lweight a1) (lweight a2)))). -(* COMMENTS -Initial nodes: 29 -END *) + \lambda (a1: A).(\lambda (a2: A).(let TMP_1 \def (lweight a1) in (let TMP_2 +\def (lweight a1) in (let TMP_3 \def (lweight a2) in (let TMP_4 \def (plus +TMP_2 TMP_3) in (let TMP_5 \def (lweight a1) in (let TMP_6 \def (lweight a2) +in (let TMP_7 \def (le_plus_l TMP_5 TMP_6) in (le_n_S TMP_1 TMP_4 +TMP_7))))))))). theorem llt_head_dx: \forall (a1: A).(\forall (a2: A).(llt a2 (AHead a1 a2))) \def - \lambda (a1: A).(\lambda (a2: A).(le_n_S (lweight a2) (plus (lweight a1) -(lweight a2)) (le_plus_r (lweight a1) (lweight a2)))). -(* COMMENTS -Initial nodes: 29 -END *) - -theorem llt_wf__q_ind: - \forall (P: ((A \to Prop))).(((\forall (n: nat).((\lambda (P0: ((A \to -Prop))).(\lambda (n0: nat).(\forall (a: A).((eq nat (lweight a) n0) \to (P0 -a))))) P n))) \to (\forall (a: A).(P a))) -\def - let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a: -A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to -Prop))).(\lambda (H: ((\forall (n: nat).(\forall (a: A).((eq nat (lweight a) -n) \to (P a)))))).(\lambda (a: A).(H (lweight a) a (refl_equal nat (lweight -a)))))). -(* COMMENTS -Initial nodes: 61 -END *) - -theorem llt_wf_ind: - \forall (P: ((A \to Prop))).(((\forall (a2: A).(((\forall (a1: A).((llt a1 -a2) \to (P a1)))) \to (P a2)))) \to (\forall (a: A).(P a))) -\def - let Q \def (\lambda (P: ((A \to Prop))).(\lambda (n: nat).(\forall (a: -A).((eq nat (lweight a) n) \to (P a))))) in (\lambda (P: ((A \to -Prop))).(\lambda (H: ((\forall (a2: A).(((\forall (a1: A).((lt (lweight a1) -(lweight a2)) \to (P a1)))) \to (P a2))))).(\lambda (a: A).(llt_wf__q_ind -(\lambda (a0: A).(P a0)) (\lambda (n: nat).(lt_wf_ind n (Q (\lambda (a0: -A).(P a0))) (\lambda (n0: nat).(\lambda (H0: ((\forall (m: nat).((lt m n0) -\to (Q (\lambda (a0: A).(P a0)) m))))).(\lambda (a0: A).(\lambda (H1: (eq nat -(lweight a0) n0)).(let H2 \def (eq_ind_r nat n0 (\lambda (n1: nat).(\forall -(m: nat).((lt m n1) \to (\forall (a1: A).((eq nat (lweight a1) m) \to (P -a1)))))) H0 (lweight a0) H1) in (H a0 (\lambda (a1: A).(\lambda (H3: (lt -(lweight a1) (lweight a0))).(H2 (lweight a1) H3 a1 (refl_equal nat (lweight -a1))))))))))))) a)))). -(* COMMENTS -Initial nodes: 179 -END *) + \lambda (a1: A).(\lambda (a2: A).(let TMP_1 \def (lweight a2) in (let TMP_2 +\def (lweight a1) in (let TMP_3 \def (lweight a2) in (let TMP_4 \def (plus +TMP_2 TMP_3) in (let TMP_5 \def (lweight a1) in (let TMP_6 \def (lweight a2) +in (let TMP_7 \def (le_plus_r TMP_5 TMP_6) in (le_n_S TMP_1 TMP_4 +TMP_7))))))))). -- 2.39.2