X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_1%2Fleq%2Ffwd.ma;h=ccf43169f1012c986779e9173b9bf00384ca91cc;hb=e8656c819b0b5e7bea7b4da244015b480af5f0f5;hp=e259fa19e134290366ff5d14e1aefad6655dae07;hpb=d1ab998b8c8dacdfceee97d6275955675cf8be83;p=helm.git 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)))))))))))))).