include "basic_1/aplus/props.ma".
-theorem leq_refl:
+lemma leq_refl:
\forall (g: G).(\forall (a: A).(leq g a a))
\def
- \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))))).
+ \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)).
-theorem leq_eq:
+lemma 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)).(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)))))).
+a2)).(eq_ind A a1 (\lambda (a: A).(leq g a1 a)) (leq_refl g a1) a2 H)))).
-theorem leq_sym:
+lemma 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)).(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))))))).
+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)))).
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)).(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:
+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:
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 (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))))))).
+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)))).
-theorem leq_ahead_false_1:
+lemma 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).(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
+ \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
(ASort O n0) x0)).(\lambda (_: (leq g a2 x1)).(\lambda (H4: (eq A (ASort O
-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:
+n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee:
+A).(match ee 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 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 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))))).
+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 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 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)).
-theorem leq_ahead_false_2:
+lemma 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).(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
+ \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
x0)).(\lambda (_: (leq g (ASort O n0) x1)).(\lambda (H4: (eq A (ASort O 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_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
+(AHead x0 x1))).(let H5 \def (eq_ind A (ASort O n0) (\lambda (ee: A).(match
+ee 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 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:
+n1) n0) (AHead x0 x1))).(let H5 \def (eq_ind A (ASort (S n1) n0) (\lambda
+(ee: A).(match ee 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 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))))).
+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 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 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)).
projects / helm.git / blobdiff