From 147770d4605b8163f63331f82e800c19a2021d46 Mon Sep 17 00:00:00 2001 From: Ferruccio Guidi Date: Mon, 18 Sep 2006 14:28:30 +0000 Subject: [PATCH] last problem elegantly resolved! --- .../LAMBDA-TYPES/Level-1/Base/ext/arith.ma | 4 +- .../LAMBDA-TYPES/Level-1/Base/ext/preamble.ma | 2 - .../Level-1/LambdaDelta/aplus/props.ma | 105 ++-- .../Level-1/LambdaDelta/drop/props.ma | 382 ++++++------- .../Level-1/LambdaDelta/gz/props.ma | 4 +- .../Level-1/LambdaDelta/lift/fwd.ma | 537 +++++++++--------- .../Level-1/LambdaDelta/lift/props.ma | 176 +++--- .../Level-1/LambdaDelta/lift/tlt.ma | 118 ++-- .../Level-1/LambdaDelta/pc3/dec.ma | 2 +- .../Level-1/LambdaDelta/pr2/subst1.ma | 166 +++++- .../Level-1/LambdaDelta/subst0/subst0.ma | 269 ++++----- .../Level-1/LambdaDelta/subst1/subst1.ma | 10 +- .../Level-1/LambdaDelta/ty3/arity.ma | 12 +- .../Level-1/LambdaDelta/ty3/subst1.ma | 2 +- .../LAMBDA-TYPES/Level-1/problems-1.ma | 183 ------ 15 files changed, 976 insertions(+), 996 deletions(-) delete mode 100644 matita/contribs/LAMBDA-TYPES/Level-1/problems-1.ma diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma b/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma index fc33cc529..8f4719772 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/arith.ma @@ -529,8 +529,8 @@ nat).(nat_ind (\lambda (n: nat).(\forall (y1: nat).(\forall (y2: nat).((le O nat).(\forall (y3: nat).(\forall (y4: nat).((le O z0) \to ((le O z0) \to ((eq nat (plus n y3) (plus n y4)) \to (eq nat y4 y3))))))) H_y z0 (minus_n_O z0)) in (H2 y1 y2 (le_O_n z0) (le_O_n z0) (H2 (plus z0 y2) (plus z0 y1) (le_O_n -z0) (le_O_n z0) (f_equal nat nat (plus z0) (plus z0 y2) (plus z0 y1) -(sym_equal nat (plus z0 y1) (plus z0 y2) (eq_add_S (plus z0 y1) (plus z0 y2) +z0) (le_O_n z0) (f_equal nat nat (plus z0) (plus z0 y2) (plus z0 y1) (sym_eq +nat (plus z0 y1) (plus z0 y2) (eq_add_S (plus z0 y1) (plus z0 y2) H1)))))))))))) (\lambda (x3: nat).(\lambda (_: ((\forall (y1: nat).(\forall (y2: nat).((le O (S z0)) \to ((le x3 (S z0)) \to ((eq nat (S (plus z0 y1)) (plus (match x3 with [O \Rightarrow (S z0) | (S l) \Rightarrow (minus z0 l)]) diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.ma b/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.ma index 040b878aa..b4c69f299 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.ma @@ -152,8 +152,6 @@ theorem plus_le_reg_l: \forall p,n,m. p + n <= p + m \to n <= m. intros. apply plus_le_reg_l; auto. qed. -definition sym_equal \def sym_eq. - default "equality" cic:/Coq/Init/Logic/eq.ind cic:/matita/LAMBDA-TYPES/Level-1/Base/ext/preamble/sym_eq.con diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/aplus/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/aplus/props.ma index 7907d8b96..1ca629bff 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/aplus/props.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/aplus/props.ma @@ -29,12 +29,12 @@ theorem aplus_reg_r: (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)))).(sym_equal A (asucc g (aplus g a2 (plus n -h2))) (asucc g (aplus g a1 (plus n h1))) (sym_equal A (asucc g (aplus g a1 -(plus n h1))) (asucc g (aplus g a2 (plus n h2))) (sym_equal A (asucc g (aplus -g a2 (plus n h2))) (asucc g (aplus g a1 (plus n h1))) (f_equal2 G A A asucc g -g (aplus g a2 (plus n h2)) (aplus g a1 (plus n h1)) (refl_equal G g) (sym_eq -A (aplus g a1 (plus n h1)) (aplus g a2 (plus n h2)) H0))))))) h))))))). +h1)) (aplus g a2 (plus n h2)))).(sym_eq A (asucc g (aplus g a2 (plus n h2))) +(asucc g (aplus g a1 (plus n h1))) (sym_eq A (asucc g (aplus g a1 (plus n +h1))) (asucc g (aplus g a2 (plus n h2))) (sym_eq A (asucc g (aplus g a2 (plus +n h2))) (asucc g (aplus g a1 (plus n h1))) (f_equal2 G A A asucc g g (aplus g +a2 (plus n h2)) (aplus g a1 (plus n h1)) (refl_equal G g) (sym_eq A (aplus g +a1 (plus n h1)) (aplus g a2 (plus n h2)) H0))))))) h))))))). theorem aplus_assoc: \forall (g: G).(\forall (a: A).(\forall (h1: nat).(\forall (h2: nat).(eq A @@ -51,10 +51,10 @@ g a (plus n h2)))))).(\lambda (h2: nat).(nat_ind (\lambda (n0: nat).(eq A 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)))) (sym_equal A (asucc g (asucc g (aplus g a (plus n n0)))) -(asucc g (aplus g (asucc g (aplus g a n)) n0)) (sym_equal A (asucc g (aplus g +(aplus g a n1)))) (sym_eq A (asucc g (asucc g (aplus g a (plus n n0)))) +(asucc g (aplus g (asucc g (aplus g a n)) n0)) (sym_eq A (asucc g (aplus g (asucc g (aplus g a n)) n0)) (asucc g (asucc g (aplus g a (plus n n0)))) -(sym_equal A (asucc g (asucc g (aplus g a (plus n n0)))) (asucc g (aplus g +(sym_eq A (asucc g (asucc g (aplus g a (plus n n0)))) (asucc g (aplus g (asucc g (aplus g a n)) n0)) (f_equal2 G A A asucc g g (asucc g (aplus g a (plus n n0))) (aplus g (asucc g (aplus g a n)) n0) (refl_equal G g) (sym_eq A (aplus g (asucc g (aplus g a n)) n0) (asucc g (aplus g a (plus n n0))) @@ -178,53 +178,54 @@ h) a) \to (\forall (P: Prop).P)))) \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 h) -\Rightarrow (ASort h n0)]) h) (ASort n n0))).(\lambda (P: Prop).((match n in +(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).((match n in nat return (\lambda (n1: nat).((eq A (aplus g (match n1 with [O \Rightarrow -(ASort O (next g n0)) | (S h) \Rightarrow (ASort h n0)]) h) (ASort n1 n0)) +(ASort O (next g n0)) | (S h0) \Rightarrow (ASort h0 n0)]) h) (ASort n1 n0)) \to P)) with [O \Rightarrow (\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 (a: A).(eq A a (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 +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 _ n) \Rightarrow n | (AHead _ _) \Rightarrow ((let rec next_plus -(g: G) (n: nat) (i: nat) on i: nat \def (match i with [O \Rightarrow n | (S -i0) \Rightarrow (next g (next_plus g n 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 (n: -nat).(eq nat (next_plus g (next g n0) n) 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))))) | (S n1) -\Rightarrow (\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 (a: A).(eq A -a (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 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 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 _ n) \Rightarrow n | (AHead _ _) -\Rightarrow ((let rec next_plus (g: G) (n: nat) (i: nat) on i: nat \def -(match i with [O \Rightarrow n | (S i0) \Rightarrow (next g (next_plus g n -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))))]) 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 (a: A).(eq A a (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 (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) g -(asucc g a1) h) | (AHead _ a) \Rightarrow a])) (AHead a0 (aplus g (asucc g -a1) h)) (AHead a0 a1) H2) in (H0 h H3 P)))))))))) a)). +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))))) | (S n1) \Rightarrow (\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))))]) 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)). theorem aplus_inj: \forall (g: G).(\forall (h1: nat).(\forall (h2: nat).(\forall (a: A).((eq A diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop/props.ma index d9578d9f3..133d10413 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop/props.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/drop/props.ma @@ -65,24 +65,24 @@ T).(\lambda (e: C).(\lambda (u: T).(\lambda (h: nat).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e))) (\lambda (H0: (drop O O (CHead c0 k t) (CHead e (Bind b) u))).(let H1 \def (f_equal C C (\lambda (e0: C).(match e0 in C return -(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c _ _) \Rightarrow -c])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead -e (Bind b) u) H0)) in ((let H2 \def (f_equal C K (\lambda (e0: C).(match e0 -in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | (CHead _ k _) -\Rightarrow k])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead c0 -k t) (CHead e (Bind b) u) H0)) in ((let H3 \def (f_equal C T (\lambda (e0: -C).(match e0 in C return (\lambda (_: C).T) with [(CSort _) \Rightarrow t | -(CHead _ _ t) \Rightarrow t])) (CHead c0 k t) (CHead e (Bind b) u) -(drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) in (\lambda (H4: (eq -K k (Bind b))).(\lambda (H5: (eq C c0 e)).(eq_ind C c0 (\lambda (c1: C).(drop -(S O) O (CHead c0 k t) c1)) (eq_ind_r K (Bind b) (\lambda (k0: K).(drop (S O) -O (CHead c0 k0 t) c0)) (drop_drop (Bind b) O c0 c0 (drop_refl c0) t) k H4) e -H5)))) H2)) H1))) (\lambda (n: nat).(\lambda (_: (((drop n O (CHead c0 k t) -(CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 k t) e)))).(\lambda (H1: -(drop (S n) O (CHead c0 k t) (CHead e (Bind b) u))).(drop_drop k (S n) c0 e -(eq_ind_r nat (S (r k n)) (\lambda (n0: nat).(drop n0 O c0 e)) (H e u (r k n) -(drop_gen_drop k c0 (CHead e (Bind b) u) t n H1)) (r k (S n)) (r_S k n)) -t)))) h)))))))) c)). +(\lambda (_: C).C) with [(CSort _) \Rightarrow c0 | (CHead c1 _ _) +\Rightarrow c1])) (CHead c0 k t) (CHead e (Bind b) u) (drop_gen_refl (CHead +c0 k t) (CHead e (Bind b) u) H0)) in ((let H2 \def (f_equal C K (\lambda (e0: +C).(match e0 in C return (\lambda (_: C).K) with [(CSort _) \Rightarrow k | +(CHead _ k0 _) \Rightarrow k0])) (CHead c0 k t) (CHead e (Bind b) u) +(drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) in ((let H3 \def +(f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda (_: C).T) with +[(CSort _) \Rightarrow t | (CHead _ _ t0) \Rightarrow t0])) (CHead c0 k t) +(CHead e (Bind b) u) (drop_gen_refl (CHead c0 k t) (CHead e (Bind b) u) H0)) +in (\lambda (H4: (eq K k (Bind b))).(\lambda (H5: (eq C c0 e)).(eq_ind C c0 +(\lambda (c1: C).(drop (S O) O (CHead c0 k t) c1)) (eq_ind_r K (Bind b) +(\lambda (k0: K).(drop (S O) O (CHead c0 k0 t) c0)) (drop_drop (Bind b) O c0 +c0 (drop_refl c0) t) k H4) e H5)))) H2)) H1))) (\lambda (n: nat).(\lambda (_: +(((drop n O (CHead c0 k t) (CHead e (Bind b) u)) \to (drop (S n) O (CHead c0 +k t) e)))).(\lambda (H1: (drop (S n) O (CHead c0 k t) (CHead e (Bind b) +u))).(drop_drop k (S n) c0 e (eq_ind_r nat (S (r k n)) (\lambda (n0: +nat).(drop n0 O c0 e)) (H e u (r k n) (drop_gen_drop k c0 (CHead e (Bind b) +u) t n H1)) (r k (S n)) (r_S k n)) t)))) h)))))))) c)). theorem drop_ctail: \forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop @@ -126,17 +126,18 @@ c3)).(\lambda (k0: K).(\lambda (u: T).(ex3_2_ind C T (\lambda (e: C).(\lambda (drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c3)) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H1: (eq C c3 (CHead x0 k x1))).(\lambda (H2: (eq T t (lift h (r k n) x1))).(\lambda (H3: (drop h (r k n) c2 x0)).(let H4 -\def (eq_ind C c3 (\lambda (c: C).(\forall (h: nat).((drop h n (CHead c2 k t) -c) \to (\forall (k0: K).(\forall (u: T).(drop h n (CTail k0 u (CHead c2 k t)) -(CTail k0 u c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k x1) -(\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u c))) -(let H5 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h n (CHead -c2 k t) (CHead x0 k x1)) \to (\forall (k0: K).(\forall (u: T).(drop h n -(CTail k0 u (CHead c2 k t)) (CTail k0 u (CHead x0 k x1)))))))) H4 (lift h (r -k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: T).(drop h (S -n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1)))) (drop_skip k h -n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u) x1) t H2)) c3 -H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d))))))) c1). +\def (eq_ind C c3 (\lambda (c: C).(\forall (h0: nat).((drop h0 n (CHead c2 k +t) c) \to (\forall (k1: K).(\forall (u0: T).(drop h0 n (CTail k1 u0 (CHead c2 +k t)) (CTail k1 u0 c))))))) H (CHead x0 k x1) H1) in (eq_ind_r C (CHead x0 k +x1) (\lambda (c: C).(drop h (S n) (CTail k0 u (CHead c2 k t)) (CTail k0 u +c))) (let H5 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: nat).((drop h0 n +(CHead c2 k t0) (CHead x0 k x1)) \to (\forall (k1: K).(\forall (u0: T).(drop +h0 n (CTail k1 u0 (CHead c2 k t0)) (CTail k1 u0 (CHead x0 k x1)))))))) H4 +(lift h (r k n) x1) H2) in (eq_ind_r T (lift h (r k n) x1) (\lambda (t0: +T).(drop h (S n) (CTail k0 u (CHead c2 k t0)) (CTail k0 u (CHead x0 k x1)))) +(drop_skip k h n (CTail k0 u c2) (CTail k0 u x0) (IHc x0 (r k n) h H3 k0 u) +x1) t H2)) c3 H1))))))) (drop_gen_skip_l c2 c3 t h n k H0)))))))) d))))))) +c1). theorem drop_mono: \forall (c: C).(\forall (x1: C).(\forall (d: nat).(\forall (h: nat).((drop h @@ -152,8 +153,8 @@ n))).(\lambda (H2: (eq nat h O)).(\lambda (H3: (eq nat d O)).(and3_ind (eq C x1 (CSort n)) (eq nat h O) (eq nat d O) (eq C x1 x2) (\lambda (H4: (eq C x1 (CSort n))).(\lambda (H5: (eq nat h O)).(\lambda (H6: (eq nat d O)).(eq_ind_r C (CSort n) (\lambda (c0: C).(eq C x1 c0)) (let H7 \def (eq_ind nat h -(\lambda (n: nat).(eq nat n O)) H2 O H5) in (let H8 \def (eq_ind nat d -(\lambda (n: nat).(eq nat n O)) H3 O H6) in (eq_ind_r C (CSort n) (\lambda +(\lambda (n0: nat).(eq nat n0 O)) H2 O H5) in (let H8 \def (eq_ind nat d +(\lambda (n0: nat).(eq nat n0 O)) H3 O H6) in (eq_ind_r C (CSort n) (\lambda (c0: C).(eq C c0 (CSort n))) (refl_equal C (CSort n)) x1 H4))) x2 H1)))) (drop_gen_sort n h d x1 H))))) (drop_gen_sort n h d x2 H0))))))))) (\lambda (c0: C).(\lambda (H: ((\forall (x1: C).(\forall (d: nat).(\forall (h: @@ -187,23 +188,24 @@ C).(\lambda (_: T).(drop h (r k n) c0 e))) (eq C x1 x2) (\lambda (x4: C).(\lambda (x5: T).(\lambda (H6: (eq C x1 (CHead x4 k x5))).(\lambda (H7: (eq T t (lift h (r k n) x5))).(\lambda (H8: (drop h (r k n) c0 x4)).(eq_ind_r C (CHead x0 k x3) (\lambda (c1: C).(eq C x1 c1)) (let H9 \def (eq_ind C x1 -(\lambda (c: C).(\forall (h: nat).((drop h n (CHead c0 k t) c) \to (\forall -(x2: C).((drop h n (CHead c0 k t) x2) \to (eq C c x2)))))) H0 (CHead x4 k x5) -H6) in (eq_ind_r C (CHead x4 k x5) (\lambda (c1: C).(eq C c1 (CHead x0 k -x3))) (let H10 \def (eq_ind T t (\lambda (t: T).(\forall (h: nat).((drop h n -(CHead c0 k t) (CHead x4 k x5)) \to (\forall (x2: C).((drop h n (CHead c0 k -t) x2) \to (eq C (CHead x4 k x5) x2)))))) H9 (lift h (r k n) x5) H7) in (let -H11 \def (eq_ind T t (\lambda (t: T).(eq T t (lift h (r k n) x3))) H4 (lift h -(r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda (t: T).(\forall (h0: -nat).((drop h0 n (CHead c0 k (lift h (r k n) t)) (CHead x4 k t)) \to (\forall -(x2: C).((drop h0 n (CHead c0 k (lift h (r k n) t)) x2) \to (eq C (CHead x4 k -t) x2)))))) H10 x3 (lift_inj x5 x3 h (r k n) H11)) in (eq_ind_r T x3 (\lambda -(t0: T).(eq C (CHead x4 k t0) (CHead x0 k x3))) (sym_equal C (CHead x0 k x3) -(CHead x4 k x3) (sym_equal C (CHead x4 k x3) (CHead x0 k x3) (sym_equal C -(CHead x0 k x3) (CHead x4 k x3) (f_equal3 C K T C CHead x0 x4 k k x3 x3 (H x0 -(r k n) h H5 x4 H8) (refl_equal K k) (refl_equal T x3))))) x5 (lift_inj x5 x3 -h (r k n) H11))))) x1 H6)) x2 H3)))))) (drop_gen_skip_l c0 x1 t h n k -H1))))))) (drop_gen_skip_l c0 x2 t h n k H2)))))))) d))))))) c). +(\lambda (c1: C).(\forall (h0: nat).((drop h0 n (CHead c0 k t) c1) \to +(\forall (x6: C).((drop h0 n (CHead c0 k t) x6) \to (eq C c1 x6)))))) H0 +(CHead x4 k x5) H6) in (eq_ind_r C (CHead x4 k x5) (\lambda (c1: C).(eq C c1 +(CHead x0 k x3))) (let H10 \def (eq_ind T t (\lambda (t0: T).(\forall (h0: +nat).((drop h0 n (CHead c0 k t0) (CHead x4 k x5)) \to (\forall (x6: C).((drop +h0 n (CHead c0 k t0) x6) \to (eq C (CHead x4 k x5) x6)))))) H9 (lift h (r k +n) x5) H7) in (let H11 \def (eq_ind T t (\lambda (t0: T).(eq T t0 (lift h (r +k n) x3))) H4 (lift h (r k n) x5) H7) in (let H12 \def (eq_ind T x5 (\lambda +(t0: T).(\forall (h0: nat).((drop h0 n (CHead c0 k (lift h (r k n) t0)) +(CHead x4 k t0)) \to (\forall (x6: C).((drop h0 n (CHead c0 k (lift h (r k n) +t0)) x6) \to (eq C (CHead x4 k t0) x6)))))) H10 x3 (lift_inj x5 x3 h (r k n) +H11)) in (eq_ind_r T x3 (\lambda (t0: T).(eq C (CHead x4 k t0) (CHead x0 k +x3))) (sym_eq C (CHead x0 k x3) (CHead x4 k x3) (sym_eq C (CHead x4 k x3) +(CHead x0 k x3) (sym_eq C (CHead x0 k x3) (CHead x4 k x3) (f_equal3 C K T C +CHead x0 x4 k k x3 x3 (H x0 (r k n) h H5 x4 H8) (refl_equal K k) (refl_equal +T x3))))) x5 (lift_inj x5 x3 h (r k n) H11))))) x1 H6)) x2 H3)))))) +(drop_gen_skip_l c0 x1 t h n k H1))))))) (drop_gen_skip_l c0 x2 t h n k +H2)))))))) d))))))) c). theorem drop_conf_lt: \forall (k: K).(\forall (i: nat).(\forall (u: T).(\forall (c0: C).(\forall @@ -221,8 +223,8 @@ T).(\forall (c0: C).(\forall (c: C).((drop n O c (CHead c0 k u)) \to (\forall T).(\lambda (e0: C).(drop h (r k d) c0 e0))))))))))))) (\lambda (u: T).(\lambda (c0: C).(\lambda (c: C).(\lambda (H: (drop O O c (CHead c0 k u))).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop -h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c: C).(drop h (S -(plus O d)) c e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in +h (S (plus O d)) c e)).(let H1 \def (eq_ind C c (\lambda (c1: C).(drop h (S +(plus O d)) c1 e)) H0 (CHead c0 k u) (drop_gen_refl c (CHead c0 k u) H)) in (ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 k v)))) (\lambda (_: C).(\lambda (v: T).(eq T u (lift h (r k (plus O d)) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r k (plus O d)) c0 e0))) (ex3_2 T C @@ -348,115 +350,115 @@ nat).((drop h d c e) \to ((le (plus d h) n) \to (drop (minus n h) O e a)))))))))) (\lambda (a: C).(\lambda (c: C).(\lambda (H: (drop O O c a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H0: (drop h d c e)).(\lambda (H1: (le (plus d h) O)).(let H2 \def (eq_ind C c (\lambda -(c: C).(drop h d c e)) H0 a (drop_gen_refl c a H)) in (let H3 \def (match H1 -in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) \to (drop -(minus O h) O e a)))) with [le_n \Rightarrow (\lambda (H: (eq nat (plus d h) -O)).(let H3 \def (f_equal nat nat (\lambda (e0: nat).e0) (plus d h) O H) in -(eq_ind nat (plus d h) (\lambda (n: nat).(drop (minus n h) n e a)) (eq_ind_r -nat O (\lambda (n: nat).(drop (minus n h) n e a)) (and_ind (eq nat d O) (eq -nat h O) (drop O O e a) (\lambda (H0: (eq nat d O)).(\lambda (H1: (eq nat h -O)).(let H2 \def (eq_ind nat d (\lambda (n: nat).(drop h n a e)) H2 O H0) in -(let H4 \def (eq_ind nat h (\lambda (n: nat).(drop n O a e)) H2 O H1) in -(eq_ind C a (\lambda (c: C).(drop O O c a)) (drop_refl a) e (drop_gen_refl a -e H4)))))) (plus_O d h H3)) (plus d h) H3) O H3))) | (le_S m H) \Rightarrow -(\lambda (H2: (eq nat (S m) O)).((let H0 \def (eq_ind nat (S m) (\lambda (e0: -nat).(match e0 in nat return (\lambda (_: nat).Prop) with [O \Rightarrow -False | (S _) \Rightarrow True])) I O H2) in (False_ind ((le (plus d h) m) -\to (drop (minus O h) O e a)) H0)) H))]) in (H3 (refl_equal nat O)))))))))))) -(\lambda (i0: nat).(\lambda (H: ((\forall (a: C).(\forall (c: C).((drop i0 O -c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c e) -\to ((le (plus d h) i0) \to (drop (minus i0 h) O e a))))))))))).(\lambda (a: -C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop (S i0) O c0 a) \to (\forall -(e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le (plus d -h) (S i0)) \to (drop (minus (S i0) h) O e a)))))))) (\lambda (n: -nat).(\lambda (H0: (drop (S i0) O (CSort n) a)).(\lambda (e: C).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H1: (drop h d (CSort n) e)).(\lambda (H2: -(le (plus d h) (S i0))).(and3_ind (eq C e (CSort n)) (eq nat h O) (eq nat d -O) (drop (minus (S i0) h) O e a) (\lambda (H3: (eq C e (CSort n))).(\lambda -(H4: (eq nat h O)).(\lambda (H5: (eq nat d O)).(and3_ind (eq C a (CSort n)) -(eq nat (S i0) O) (eq nat O O) (drop (minus (S i0) h) O e a) (\lambda (H6: -(eq C a (CSort n))).(\lambda (H7: (eq nat (S i0) O)).(\lambda (_: (eq nat O -O)).(let H9 \def (eq_ind nat d (\lambda (n: nat).(le (plus n h) (S i0))) H2 O -H5) in (let H10 \def (eq_ind nat h (\lambda (n: nat).(le (plus O n) (S i0))) -H9 O H4) in (eq_ind_r nat O (\lambda (n0: nat).(drop (minus (S i0) n0) O e -a)) (eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O c0 a)) -(eq_ind_r C (CSort n) (\lambda (c0: C).(drop (minus (S i0) O) O (CSort n) -c0)) (let H11 \def (eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat -return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow -True])) I O H7) in (False_ind (drop (minus (S i0) O) O (CSort n) (CSort n)) -H11)) a H6) e H3) h H4)))))) (drop_gen_sort n (S i0) O a H0))))) -(drop_gen_sort n h d e H1))))))))) (\lambda (c0: C).(\lambda (H0: (((drop (S -i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h -d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e -a))))))))).(\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).((drop (S -i0) O (CHead c0 k0 t) a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: -nat).((drop h d (CHead c0 k0 t) e) \to ((le (plus d h) (S i0)) \to (drop -(minus (S i0) h) O e a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1: -(drop (S i0) O (CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h: -nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t) +(c0: C).(drop h d c0 e)) H0 a (drop_gen_refl c a H)) in (let H3 \def (match +H1 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) \to +(drop (minus O h) O e a)))) with [le_n \Rightarrow (\lambda (H3: (eq nat +(plus d h) O)).(let H4 \def (f_equal nat nat (\lambda (e0: nat).e0) (plus d +h) O H3) in (eq_ind nat (plus d h) (\lambda (n: nat).(drop (minus n h) n e +a)) (eq_ind_r nat O (\lambda (n: nat).(drop (minus n h) n e a)) (and_ind (eq +nat d O) (eq nat h O) (drop O O e a) (\lambda (H5: (eq nat d O)).(\lambda +(H6: (eq nat h O)).(let H7 \def (eq_ind nat d (\lambda (n: nat).(drop h n a +e)) H2 O H5) in (let H8 \def (eq_ind nat h (\lambda (n: nat).(drop n O a e)) +H7 O H6) in (eq_ind C a (\lambda (c0: C).(drop O O c0 a)) (drop_refl a) e +(drop_gen_refl a e H8)))))) (plus_O d h H4)) (plus d h) H4) O H4))) | (le_S m +H3) \Rightarrow (\lambda (H4: (eq nat (S m) O)).((let H5 \def (eq_ind nat (S +m) (\lambda (e0: nat).(match e0 in nat return (\lambda (_: nat).Prop) with [O +\Rightarrow False | (S _) \Rightarrow True])) I O H4) in (False_ind ((le +(plus d h) m) \to (drop (minus O h) O e a)) H5)) H3))]) in (H3 (refl_equal +nat O)))))))))))) (\lambda (i0: nat).(\lambda (H: ((\forall (a: C).(\forall +(c: C).((drop i0 O c a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: +nat).((drop h d c e) \to ((le (plus d h) i0) \to (drop (minus i0 h) O e +a))))))))))).(\lambda (a: C).(\lambda (c: C).(C_ind (\lambda (c0: C).((drop +(S i0) O c0 a) \to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop +h d c0 e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e +a)))))))) (\lambda (n: nat).(\lambda (H0: (drop (S i0) O (CSort n) +a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (drop h +d (CSort n) e)).(\lambda (H2: (le (plus d h) (S i0))).(and3_ind (eq C e +(CSort n)) (eq nat h O) (eq nat d O) (drop (minus (S i0) h) O e a) (\lambda +(H3: (eq C e (CSort n))).(\lambda (H4: (eq nat h O)).(\lambda (H5: (eq nat d +O)).(and3_ind (eq C a (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop (minus +(S i0) h) O e a) (\lambda (H6: (eq C a (CSort n))).(\lambda (H7: (eq nat (S +i0) O)).(\lambda (_: (eq nat O O)).(let H9 \def (eq_ind nat d (\lambda (n0: +nat).(le (plus n0 h) (S i0))) H2 O H5) in (let H10 \def (eq_ind nat h +(\lambda (n0: nat).(le (plus O n0) (S i0))) H9 O H4) in (eq_ind_r nat O +(\lambda (n0: nat).(drop (minus (S i0) n0) O e a)) (eq_ind_r C (CSort n) +(\lambda (c0: C).(drop (minus (S i0) O) O c0 a)) (eq_ind_r C (CSort n) +(\lambda (c0: C).(drop (minus (S i0) O) O (CSort n) c0)) (let H11 \def +(eq_ind nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: +nat).Prop) with [O \Rightarrow False | (S _) \Rightarrow True])) I O H7) in +(False_ind (drop (minus (S i0) O) O (CSort n) (CSort n)) H11)) a H6) e H3) h +H4)))))) (drop_gen_sort n (S i0) O a H0))))) (drop_gen_sort n h d e +H1))))))))) (\lambda (c0: C).(\lambda (H0: (((drop (S i0) O c0 a) \to +(\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d c0 e) \to ((le +(plus d h) (S i0)) \to (drop (minus (S i0) h) O e a))))))))).(\lambda (k: +K).(K_ind (\lambda (k0: K).(\forall (t: T).((drop (S i0) O (CHead c0 k0 t) a) +\to (\forall (e: C).(\forall (h: nat).(\forall (d: nat).((drop h d (CHead c0 +k0 t) e) \to ((le (plus d h) (S i0)) \to (drop (minus (S i0) h) O e +a))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H1: (drop (S i0) O +(CHead c0 (Bind b) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: +nat).(\lambda (H2: (drop h d (CHead c0 (Bind b) t) e)).(\lambda (H3: (le +(plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Bind b) +t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) +(\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus O +h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 (Bind b) t) e) +\to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda +(H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le (plus O O) (S +i0))).(eq_ind C (CHead c0 (Bind b) t) (\lambda (c1: C).(drop (minus (S i0) O) +O c1 a)) (drop_drop (Bind b) i0 c0 a (drop_gen_drop (Bind b) c0 a t i0 H1) t) +e (drop_gen_refl (CHead c0 (Bind b) t) e H6)))) (\lambda (h0: nat).(\lambda +(_: (((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O h0) (S i0)) \to +(drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 +(Bind b) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H a c0 +(drop_gen_drop (Bind b) c0 a t i0 H1) e h0 O (drop_gen_drop (Bind b) c0 e t +h0 H6) (le_S_n (plus O h0) i0 H7)))))) h H4 H5))) (\lambda (d0: nat).(\lambda +(_: (((drop h d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h) (S i0)) \to +(drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) (CHead c0 +(Bind b) t) e)).(\lambda (H5: (le (plus (S d0) h) (S i0))).(ex3_2_ind C T +(\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Bind b) v)))) (\lambda +(_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0) v)))) (\lambda (e0: +C).(\lambda (_: T).(drop h (r (Bind b) d0) c0 e0))) (drop (minus (S i0) h) O +e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C e (CHead x0 (Bind +b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) d0) x1))).(\lambda (H8: +(drop h (r (Bind b) d0) c0 x0)).(eq_ind_r C (CHead x0 (Bind b) x1) (\lambda +(c1: C).(drop (minus (S i0) h) O c1 a)) (eq_ind nat (S (minus i0 h)) (\lambda +(n: nat).(drop n O (CHead x0 (Bind b) x1) a)) (drop_drop (Bind b) (minus i0 +h) x0 a (H a c0 (drop_gen_drop (Bind b) c0 a t i0 H1) x0 h d0 H8 (le_S_n +(plus d0 h) i0 H5)) x1) (minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 +h i0 (le_S_n (plus d0 h) i0 H5)))) e H6)))))) (drop_gen_skip_l c0 e t h d0 +(Bind b) H4)))))) d H2 H3))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda +(H1: (drop (S i0) O (CHead c0 (Flat f) t) a)).(\lambda (e: C).(\lambda (h: +nat).(\lambda (d: nat).(\lambda (H2: (drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le (plus d h) (S i0))).(nat_ind (\lambda (n: nat).((drop h -n (CHead c0 (Bind b) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S -i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Bind b) t) e)).(\lambda +n (CHead c0 (Flat f) t) e) \to ((le (plus n h) (S i0)) \to (drop (minus (S +i0) h) O e a)))) (\lambda (H4: (drop h O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind (\lambda (n: nat).((drop n O (CHead c0 -(Bind b) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e -a)))) (\lambda (H6: (drop O O (CHead c0 (Bind b) t) e)).(\lambda (_: (le -(plus O O) (S i0))).(eq_ind C (CHead c0 (Bind b) t) (\lambda (c1: C).(drop -(minus (S i0) O) O c1 a)) (drop_drop (Bind b) i0 c0 a (drop_gen_drop (Bind b) -c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Bind b) t) e H6)))) (\lambda -(h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Bind b) t) e) \to ((le (plus O +(Flat f) t) e) \to ((le (plus O n) (S i0)) \to (drop (minus (S i0) n) O e +a)))) (\lambda (H6: (drop O O (CHead c0 (Flat f) t) e)).(\lambda (_: (le +(plus O O) (S i0))).(eq_ind C (CHead c0 (Flat f) t) (\lambda (c1: C).(drop +(minus (S i0) O) O c1 a)) (drop_drop (Flat f) i0 c0 a (drop_gen_drop (Flat f) +c0 a t i0 H1) t) e (drop_gen_refl (CHead c0 (Flat f) t) e H6)))) (\lambda +(h0: nat).(\lambda (_: (((drop h0 O (CHead c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O e a))))).(\lambda (H6: (drop (S h0) -O (CHead c0 (Bind b) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H a -c0 (drop_gen_drop (Bind b) c0 a t i0 H1) e h0 O (drop_gen_drop (Bind b) c0 e -t h0 H6) (le_S_n (plus O h0) i0 H7)))))) h H4 H5))) (\lambda (d0: -nat).(\lambda (_: (((drop h d0 (CHead c0 (Bind b) t) e) \to ((le (plus d0 h) -(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) -(CHead c0 (Bind b) t) e)).(\lambda (H5: (le (plus (S d0) h) (S -i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Bind -b) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Bind b) d0) -v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Bind b) d0) c0 e0))) (drop -(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C -e (CHead x0 (Bind b) x1))).(\lambda (_: (eq T t (lift h (r (Bind b) d0) -x1))).(\lambda (H8: (drop h (r (Bind b) d0) c0 x0)).(eq_ind_r C (CHead x0 -(Bind b) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (eq_ind nat (S -(minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Bind b) x1) a)) -(drop_drop (Bind b) (minus i0 h) x0 a (H a c0 (drop_gen_drop (Bind b) c0 a t -i0 H1) x0 h d0 H8 (le_S_n (plus d0 h) i0 H5)) x1) (minus (S i0) h) -(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) e -H6)))))) (drop_gen_skip_l c0 e t h d0 (Bind b) H4)))))) d H2 H3))))))))) -(\lambda (f: F).(\lambda (t: T).(\lambda (H1: (drop (S i0) O (CHead c0 (Flat -f) t) a)).(\lambda (e: C).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H2: -(drop h d (CHead c0 (Flat f) t) e)).(\lambda (H3: (le (plus d h) (S -i0))).(nat_ind (\lambda (n: nat).((drop h n (CHead c0 (Flat f) t) e) \to ((le -(plus n h) (S i0)) \to (drop (minus (S i0) h) O e a)))) (\lambda (H4: (drop h -O (CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus O h) (S i0))).(nat_ind -(\lambda (n: nat).((drop n O (CHead c0 (Flat f) t) e) \to ((le (plus O n) (S -i0)) \to (drop (minus (S i0) n) O e a)))) (\lambda (H6: (drop O O (CHead c0 -(Flat f) t) e)).(\lambda (_: (le (plus O O) (S i0))).(eq_ind C (CHead c0 -(Flat f) t) (\lambda (c1: C).(drop (minus (S i0) O) O c1 a)) (drop_drop (Flat -f) i0 c0 a (drop_gen_drop (Flat f) c0 a t i0 H1) t) e (drop_gen_refl (CHead -c0 (Flat f) t) e H6)))) (\lambda (h0: nat).(\lambda (_: (((drop h0 O (CHead -c0 (Flat f) t) e) \to ((le (plus O h0) (S i0)) \to (drop (minus (S i0) h0) O -e a))))).(\lambda (H6: (drop (S h0) O (CHead c0 (Flat f) t) e)).(\lambda (H7: -(le (plus O (S h0)) (S i0))).(H0 (drop_gen_drop (Flat f) c0 a t i0 H1) e (S -h0) O (drop_gen_drop (Flat f) c0 e t h0 H6) H7))))) h H4 H5))) (\lambda (d0: -nat).(\lambda (_: (((drop h d0 (CHead c0 (Flat f) t) e) \to ((le (plus d0 h) -(S i0)) \to (drop (minus (S i0) h) O e a))))).(\lambda (H4: (drop h (S d0) -(CHead c0 (Flat f) t) e)).(\lambda (H5: (le (plus (S d0) h) (S -i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda (v: T).(eq C e (CHead e0 (Flat -f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r (Flat f) d0) -v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r (Flat f) d0) c0 e0))) (drop -(minus (S i0) h) O e a) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H6: (eq C -e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t (lift h (r (Flat f) d0) -x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 x0)).(eq_ind_r C (CHead x0 -(Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) h) O c1 a)) (let H9 \def -(eq_ind_r nat (minus (S i0) h) (\lambda (n: nat).(drop n O x0 a)) (H0 -(drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) H8 H5) (S (minus i0 h)) -(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5)))) in -(eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop n O (CHead x0 (Flat f) -x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) (minus (S i0) h) -(minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 h) i0 H5))))) e -H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 H3))))))))) -k)))) c))))) i). +O (CHead c0 (Flat f) t) e)).(\lambda (H7: (le (plus O (S h0)) (S i0))).(H0 +(drop_gen_drop (Flat f) c0 a t i0 H1) e (S h0) O (drop_gen_drop (Flat f) c0 e +t h0 H6) H7))))) h H4 H5))) (\lambda (d0: nat).(\lambda (_: (((drop h d0 +(CHead c0 (Flat f) t) e) \to ((le (plus d0 h) (S i0)) \to (drop (minus (S i0) +h) O e a))))).(\lambda (H4: (drop h (S d0) (CHead c0 (Flat f) t) e)).(\lambda +(H5: (le (plus (S d0) h) (S i0))).(ex3_2_ind C T (\lambda (e0: C).(\lambda +(v: T).(eq C e (CHead e0 (Flat f) v)))) (\lambda (_: C).(\lambda (v: T).(eq T +t (lift h (r (Flat f) d0) v)))) (\lambda (e0: C).(\lambda (_: T).(drop h (r +(Flat f) d0) c0 e0))) (drop (minus (S i0) h) O e a) (\lambda (x0: C).(\lambda +(x1: T).(\lambda (H6: (eq C e (CHead x0 (Flat f) x1))).(\lambda (_: (eq T t +(lift h (r (Flat f) d0) x1))).(\lambda (H8: (drop h (r (Flat f) d0) c0 +x0)).(eq_ind_r C (CHead x0 (Flat f) x1) (\lambda (c1: C).(drop (minus (S i0) +h) O c1 a)) (let H9 \def (eq_ind_r nat (minus (S i0) h) (\lambda (n: +nat).(drop n O x0 a)) (H0 (drop_gen_drop (Flat f) c0 a t i0 H1) x0 h (S d0) +H8 H5) (S (minus i0 h)) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n +(plus d0 h) i0 H5)))) in (eq_ind nat (S (minus i0 h)) (\lambda (n: nat).(drop +n O (CHead x0 (Flat f) x1) a)) (drop_drop (Flat f) (minus i0 h) x0 a H9 x1) +(minus (S i0) h) (minus_Sn_m i0 h (le_trans_plus_r d0 h i0 (le_S_n (plus d0 +h) i0 H5))))) e H6)))))) (drop_gen_skip_l c0 e t h d0 (Flat f) H4)))))) d H2 +H3))))))))) k)))) c))))) i). theorem drop_conf_rev: \forall (j: nat).(\forall (e1: C).(\forall (e2: C).((drop j O e1 e2) \to @@ -645,14 +647,14 @@ nat).(\lambda (H: (drop h d c1 c2)).(\lambda (e2: C).(\lambda (H0: (drop O O c2 e2)).(\lambda (H1: (le d O)).(eq_ind C c2 (\lambda (c: C).(drop (plus O h) O c1 c)) (let H2 \def (match H1 in le return (\lambda (n: nat).(\lambda (_: (le ? n)).((eq nat n O) \to (drop (plus O h) O c1 c2)))) with [le_n -\Rightarrow (\lambda (H0: (eq nat d O)).(eq_ind nat O (\lambda (_: nat).(drop -(plus O h) O c1 c2)) (let H2 \def (eq_ind nat d (\lambda (n: nat).(le n O)) -H1 O H0) in (let H3 \def (eq_ind nat d (\lambda (n: nat).(drop h n c1 c2)) H -O H0) in H3)) d (sym_eq nat d O H0))) | (le_S m H0) \Rightarrow (\lambda (H2: -(eq nat (S m) O)).((let H1 \def (eq_ind nat (S m) (\lambda (e: nat).(match e +\Rightarrow (\lambda (H2: (eq nat d O)).(eq_ind nat O (\lambda (_: nat).(drop +(plus O h) O c1 c2)) (let H3 \def (eq_ind nat d (\lambda (n: nat).(le n O)) +H1 O H2) in (let H4 \def (eq_ind nat d (\lambda (n: nat).(drop h n c1 c2)) H +O H2) in H4)) d (sym_eq nat d O H2))) | (le_S m H2) \Rightarrow (\lambda (H3: +(eq nat (S m) O)).((let H4 \def (eq_ind nat (S m) (\lambda (e: nat).(match e in nat return (\lambda (_: nat).Prop) with [O \Rightarrow False | (S _) -\Rightarrow True])) I O H2) in (False_ind ((le d m) \to (drop (plus O h) O c1 -c2)) H1)) H0))]) in (H2 (refl_equal nat O))) e2 (drop_gen_refl c2 e2 +\Rightarrow True])) I O H3) in (False_ind ((le d m) \to (drop (plus O h) O c1 +c2)) H4)) H2))]) in (H2 (refl_equal nat O))) e2 (drop_gen_refl c2 e2 H0)))))))))) (\lambda (i0: nat).(\lambda (IHi: ((\forall (c1: C).(\forall (c2: C).(\forall (d: nat).(\forall (h: nat).((drop h d c1 c2) \to (\forall (e2: C).((drop i0 O c2 e2) \to ((le d i0) \to (drop (plus i0 h) O c1 @@ -665,16 +667,16 @@ nat).(\lambda (H: (drop h d (CSort n) c2)).(\lambda (e2: C).(\lambda (H0: n)) (eq nat h O) (eq nat d O) (drop (S (plus i0 h)) O (CSort n) e2) (\lambda (H2: (eq C c2 (CSort n))).(\lambda (H3: (eq nat h O)).(\lambda (H4: (eq nat d O)).(eq_ind_r nat O (\lambda (n0: nat).(drop (S (plus i0 n0)) O (CSort n) -e2)) (let H5 \def (eq_ind nat d (\lambda (n: nat).(le n (S i0))) H1 O H4) in -(let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort n) -H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop (S -(plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda (H8: -(eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) (\lambda -(c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind nat (S i0) -(\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) with [O -\Rightarrow False | (S _) \Rightarrow True])) I O H8) in (False_ind (drop (S -(plus i0 O)) O (CSort n) (CSort n)) H10)) e2 H7)))) (drop_gen_sort n (S i0) O -e2 H6)))) h H3)))) (drop_gen_sort n h d c2 H)))))))))) (\lambda (c2: +e2)) (let H5 \def (eq_ind nat d (\lambda (n0: nat).(le n0 (S i0))) H1 O H4) +in (let H6 \def (eq_ind C c2 (\lambda (c: C).(drop (S i0) O c e2)) H0 (CSort +n) H2) in (and3_ind (eq C e2 (CSort n)) (eq nat (S i0) O) (eq nat O O) (drop +(S (plus i0 O)) O (CSort n) e2) (\lambda (H7: (eq C e2 (CSort n))).(\lambda +(H8: (eq nat (S i0) O)).(\lambda (_: (eq nat O O)).(eq_ind_r C (CSort n) +(\lambda (c: C).(drop (S (plus i0 O)) O (CSort n) c)) (let H10 \def (eq_ind +nat (S i0) (\lambda (ee: nat).(match ee in nat return (\lambda (_: nat).Prop) +with [O \Rightarrow False | (S _) \Rightarrow True])) I O H8) in (False_ind +(drop (S (plus i0 O)) O (CSort n) (CSort n)) H10)) e2 H7)))) (drop_gen_sort n +(S i0) O e2 H6)))) h H3)))) (drop_gen_sort n h d c2 H)))))))))) (\lambda (c2: C).(\lambda (IHc: ((\forall (c3: C).(\forall (d: nat).(\forall (h: nat).((drop h d c2 c3) \to (\forall (e2: C).((drop (S i0) O c3 e2) \to ((le d (S i0)) \to (drop (S (plus i0 h)) O c2 e2)))))))))).(\lambda (k: K).(\lambda @@ -710,21 +712,21 @@ v)))) (\lambda (_: C).(\lambda (v: T).(eq T t (lift h (r k d0) v)))) (\lambda (CHead c2 k t) e2) (\lambda (x0: C).(\lambda (x1: T).(\lambda (H2: (eq C c3 (CHead x0 k x1))).(\lambda (H3: (eq T t (lift h (r k d0) x1))).(\lambda (H4: (drop h (r k d0) c2 x0)).(let H5 \def (eq_ind C c3 (\lambda (c: C).(\forall -(h: nat).((drop h d0 (CHead c2 k t) c) \to (\forall (e2: C).((drop (S i0) O c -e2) \to ((le d0 (S i0)) \to (drop (S (plus i0 h)) O (CHead c2 k t) e2))))))) -IHd (CHead x0 k x1) H2) in (let H6 \def (eq_ind C c3 (\lambda (c: C).(drop (S -i0) O c e2)) H0 (CHead x0 k x1) H2) in (let H7 \def (eq_ind T t (\lambda (t: -T).(\forall (h: nat).((drop h d0 (CHead c2 k t) (CHead x0 k x1)) \to (\forall -(e2: C).((drop (S i0) O (CHead x0 k x1) e2) \to ((le d0 (S i0)) \to (drop (S -(plus i0 h)) O (CHead c2 k t) e2))))))) H5 (lift h (r k d0) x1) H3) in -(eq_ind_r T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S (plus i0 h)) O -(CHead c2 k t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind (\lambda (k0: -K).((drop h (r k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to (drop (r k0 -(plus i0 h)) O c2 e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r (Bind b) -d0) c2 x0)).(\lambda (H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 (r (Bind -b) d0) h H8 e2 H9 (le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f: F).(\lambda -(H8: (drop h (r (Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat f) i0) O x0 -e2)).(IHc x0 (r (Flat f) d0) h H8 e2 H9 H1)))) k H4 (drop_gen_drop k x0 e2 x1 -i0 H6)) (lift h (r k d0) x1)) t H3))))))))) (drop_gen_skip_l c2 c3 t h d0 k -H))))))))) d))))))) c1)))) i). +(h0: nat).((drop h0 d0 (CHead c2 k t) c) \to (\forall (e3: C).((drop (S i0) O +c e3) \to ((le d0 (S i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t) +e3))))))) IHd (CHead x0 k x1) H2) in (let H6 \def (eq_ind C c3 (\lambda (c: +C).(drop (S i0) O c e2)) H0 (CHead x0 k x1) H2) in (let H7 \def (eq_ind T t +(\lambda (t0: T).(\forall (h0: nat).((drop h0 d0 (CHead c2 k t0) (CHead x0 k +x1)) \to (\forall (e3: C).((drop (S i0) O (CHead x0 k x1) e3) \to ((le d0 (S +i0)) \to (drop (S (plus i0 h0)) O (CHead c2 k t0) e3))))))) H5 (lift h (r k +d0) x1) H3) in (eq_ind_r T (lift h (r k d0) x1) (\lambda (t0: T).(drop (S +(plus i0 h)) O (CHead c2 k t0) e2)) (drop_drop k (plus i0 h) c2 e2 (K_ind +(\lambda (k0: K).((drop h (r k0 d0) c2 x0) \to ((drop (r k0 i0) O x0 e2) \to +(drop (r k0 (plus i0 h)) O c2 e2)))) (\lambda (b: B).(\lambda (H8: (drop h (r +(Bind b) d0) c2 x0)).(\lambda (H9: (drop (r (Bind b) i0) O x0 e2)).(IHi c2 x0 +(r (Bind b) d0) h H8 e2 H9 (le_S_n (r (Bind b) d0) i0 H1))))) (\lambda (f: +F).(\lambda (H8: (drop h (r (Flat f) d0) c2 x0)).(\lambda (H9: (drop (r (Flat +f) i0) O x0 e2)).(IHc x0 (r (Flat f) d0) h H8 e2 H9 H1)))) k H4 +(drop_gen_drop k x0 e2 x1 i0 H6)) (lift h (r k d0) x1)) t H3))))))))) +(drop_gen_skip_l c2 c3 t h d0 k H))))))))) d))))))) c1)))) i). diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/gz/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/gz/props.ma index 9612d298d..463b3de6a 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/gz/props.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/gz/props.ma @@ -149,8 +149,8 @@ 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_equal 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 +(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 diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd.ma index ba33d1441..44ca12dec 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/fwd.ma @@ -32,7 +32,7 @@ theorem lift_lref_lt: \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (lt n d)).(eq_ind bool true (\lambda (b: bool).(eq T (TLRef (match b with [true \Rightarrow n | false \Rightarrow (plus n h)])) (TLRef n))) (refl_equal T -(TLRef n)) (blt n d) (sym_equal bool (blt n d) true (lt_blt d n H)))))). +(TLRef n)) (blt n d) (sym_eq bool (blt n d) true (lt_blt d n H)))))). theorem lift_lref_ge: \forall (n: nat).(\forall (h: nat).(\forall (d: nat).((le d n) \to (eq T @@ -41,7 +41,7 @@ theorem lift_lref_ge: \lambda (n: nat).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H: (le d n)).(eq_ind bool false (\lambda (b: bool).(eq T (TLRef (match b with [true \Rightarrow n | false \Rightarrow (plus n h)])) (TLRef (plus n h)))) -(refl_equal T (TLRef (plus n h))) (blt n d) (sym_equal bool (blt n d) false +(refl_equal T (TLRef (plus n h))) (blt n d) (sym_eq bool (blt n d) false (le_bge d n H)))))). theorem lift_head: @@ -78,38 +78,38 @@ theorem lift_gen_sort: n0)))).(sym_eq T (TSort n) (TSort n0) H))) (\lambda (n0: nat).(\lambda (H: (eq T (TSort n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TSort n)) (\lambda (H0: (lt n0 d)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) -(\lambda (t: T).(eq T (TSort n) t)) H (TLRef n0) (lift_lref_lt n0 h d H0)) in -(let H2 \def (match H1 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? -t)).((eq T t (TLRef n0)) \to (eq T (TLRef n0) (TSort n))))) with [refl_equal -\Rightarrow (\lambda (H1: (eq T (TSort n) (TLRef n0))).(let H2 \def (eq_ind T -(TSort n) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with -[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) -\Rightarrow False])) I (TLRef n0) H1) in (False_ind (eq T (TLRef n0) (TSort -n)) H2)))]) in (H2 (refl_equal T (TLRef n0)))))) (\lambda (H0: (le d -n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T -(TSort n) t)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d H0)) in (let H2 \def -(match H1 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t -(TLRef (plus n0 h))) \to (eq T (TLRef n0) (TSort n))))) with [refl_equal -\Rightarrow (\lambda (H1: (eq T (TSort n) (TLRef (plus n0 h)))).(let H2 \def -(eq_ind T (TSort n) (\lambda (e: T).(match e in T return (\lambda (_: +(\lambda (t0: T).(eq T (TSort n) t0)) H (TLRef n0) (lift_lref_lt n0 h d H0)) +in (let H2 \def (match H1 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? +t0)).((eq T t0 (TLRef n0)) \to (eq T (TLRef n0) (TSort n))))) with +[refl_equal \Rightarrow (\lambda (H2: (eq T (TSort n) (TLRef n0))).(let H3 +\def (eq_ind T (TSort n) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow False])) I (TLRef (plus n0 h)) H1) in (False_ind -(eq T (TLRef n0) (TSort n)) H2)))]) in (H2 (refl_equal T (TLRef (plus n0 -h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TSort n) -(lift h d t0)) \to (eq T t0 (TSort n))))).(\lambda (t1: T).(\lambda (_: (((eq -T (TSort n) (lift h d t1)) \to (eq T t1 (TSort n))))).(\lambda (H1: (eq T -(TSort n) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d -(THead k t0 t1)) (\lambda (t: T).(eq T (TSort n) t)) H1 (THead k (lift h d -t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (match H2 -in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k (lift -h d t0) (lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TSort n))))) with -[refl_equal \Rightarrow (\lambda (H2: (eq T (TSort n) (THead k (lift h d t0) -(lift h (s k d) t1)))).(let H3 \def (eq_ind T (TSort n) (\lambda (e: -T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow -True | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow False])) I -(THead k (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (eq T (THead k -t0 t1) (TSort n)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h -(s k d) t1)))))))))))) t)))). +(THead _ _ _) \Rightarrow False])) I (TLRef n0) H2) in (False_ind (eq T +(TLRef n0) (TSort n)) H3)))]) in (H2 (refl_equal T (TLRef n0)))))) (\lambda +(H0: (le d n0)).(let H1 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: +T).(eq T (TSort n) t0)) H (TLRef (plus n0 h)) (lift_lref_ge n0 h d H0)) in +(let H2 \def (match H1 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? +t0)).((eq T t0 (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TSort n))))) with +[refl_equal \Rightarrow (\lambda (H2: (eq T (TSort n) (TLRef (plus n0 +h)))).(let H3 \def (eq_ind T (TSort n) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow True | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow False])) I (TLRef (plus n0 h)) +H2) in (False_ind (eq T (TLRef n0) (TSort n)) H3)))]) in (H2 (refl_equal T +(TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: +(((eq T (TSort n) (lift h d t0)) \to (eq T t0 (TSort n))))).(\lambda (t1: +T).(\lambda (_: (((eq T (TSort n) (lift h d t1)) \to (eq T t1 (TSort +n))))).(\lambda (H1: (eq T (TSort n) (lift h d (THead k t0 t1)))).(let H2 +\def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TSort n) +t2)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) +in (let H3 \def (match H2 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ? +t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq T (THead +k t0 t1) (TSort n))))) with [refl_equal \Rightarrow (\lambda (H3: (eq T +(TSort n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def (eq_ind +T (TSort n) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with +[(TSort _) \Rightarrow True | (TLRef _) \Rightarrow False | (THead _ _ _) +\Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H3) in +(False_ind (eq T (THead k t0 t1) (TSort n)) H4)))]) in (H3 (refl_equal T +(THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t)))). theorem lift_gen_lref: \forall (t: T).(\forall (d: nat).(\forall (h: nat).(\forall (i: nat).((eq T @@ -121,7 +121,7 @@ nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d) (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i h)))))))))) (\lambda (n: nat).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: nat).(\lambda (H: (eq T (TLRef i) (lift h d (TSort n)))).(let H0 \def -(eq_ind T (lift h d (TSort n)) (\lambda (t: T).(eq T (TLRef i) t)) H (TSort +(eq_ind T (lift h d (TSort n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TSort n) (lift_sort n h d)) in (let H1 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I @@ -131,40 +131,40 @@ False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I nat).(\lambda (H: (eq T (TLRef i) (lift h d (TLRef n)))).(lt_le_e n d (or (land (lt i d) (eq T (TLRef n) (TLRef i))) (land (le (plus d h) i) (eq T (TLRef n) (TLRef (minus i h))))) (\lambda (H0: (lt n d)).(let H1 \def (eq_ind -T (lift h d (TLRef n)) (\lambda (t: T).(eq T (TLRef i) t)) H (TLRef n) +T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef i) t0)) H (TLRef n) (lift_lref_lt n h d H0)) in (let H2 \def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow i | -(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef n) -H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) -(TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 +(TLRef n0) \Rightarrow n0 | (THead _ _ _) \Rightarrow i])) (TLRef i) (TLRef +n) H1) in (eq_ind_r nat n (\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef +n) (TLRef n0))) (land (le (plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (or_introl (land (lt n d) (eq T (TLRef n) (TLRef n))) (land (le (plus d h) n) (eq T (TLRef n) (TLRef (minus n h)))) (conj (lt n d) (eq T (TLRef n) (TLRef n)) H0 (refl_equal T (TLRef n)))) i H2)))) (\lambda (H0: (le d -n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t: T).(eq T (TLRef -i) t)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (f_equal -T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort -_) \Rightarrow i | (TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow i])) -(TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h) (\lambda (n0: -nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le (plus d h) -n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n (\lambda (n0: -nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land -(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0))))) (or_intror (land -(lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land (le (plus d h) -(plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h) (plus n h)) (eq -T (TLRef n) (TLRef n)) (plus_le_compat d n h h H0 (le_n h)) (refl_equal T -(TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i H2)))))))))) (\lambda -(k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d: nat).(\forall (h: -nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to (or (land (lt i d) -(eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 (TLRef (minus i -h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d: nat).(\forall (h: -nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to (or (land (lt i d) -(eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1 (TLRef (minus i -h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: nat).(\lambda -(H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T -(lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TLRef i) t)) H1 (THead k -(lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def -(eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | +n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (TLRef +i) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def +(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with +[(TSort _) \Rightarrow i | (TLRef n0) \Rightarrow n0 | (THead _ _ _) +\Rightarrow i])) (TLRef i) (TLRef (plus n h)) H1) in (eq_ind_r nat (plus n h) +(\lambda (n0: nat).(or (land (lt n0 d) (eq T (TLRef n) (TLRef n0))) (land (le +(plus d h) n0) (eq T (TLRef n) (TLRef (minus n0 h)))))) (eq_ind_r nat n +(\lambda (n0: nat).(or (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n +h)))) (land (le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n0))))) +(or_intror (land (lt (plus n h) d) (eq T (TLRef n) (TLRef (plus n h)))) (land +(le (plus d h) (plus n h)) (eq T (TLRef n) (TLRef n))) (conj (le (plus d h) +(plus n h)) (eq T (TLRef n) (TLRef n)) (plus_le_compat d n h h H0 (le_n h)) +(refl_equal T (TLRef n)))) (minus (plus n h) h) (minus_plus_r n h)) i +H2)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (d: +nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t0)) \to +(or (land (lt i d) (eq T t0 (TLRef i))) (land (le (plus d h) i) (eq T t0 +(TLRef (minus i h))))))))))).(\lambda (t1: T).(\lambda (_: ((\forall (d: +nat).(\forall (h: nat).(\forall (i: nat).((eq T (TLRef i) (lift h d t1)) \to +(or (land (lt i d) (eq T t1 (TLRef i))) (land (le (plus d h) i) (eq T t1 +(TLRef (minus i h))))))))))).(\lambda (d: nat).(\lambda (h: nat).(\lambda (i: +nat).(\lambda (H1: (eq T (TLRef i) (lift h d (THead k t0 t1)))).(let H2 \def +(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef i) t2)) H1 +(THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let +H3 \def (eq_ind T (TLRef i) (\lambda (ee: T).(match ee in T return (\lambda +(_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H2) in (False_ind (or (land (lt i d) (eq T (THead k t0 t1) (TLRef i))) (land (le (plus d h) i) (eq T (THead k t0 t1) (TLRef (minus i h))))) @@ -180,35 +180,36 @@ d)).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef n) (lift h d t0)) (lift h d (TSort n0)))).(sym_eq T (TLRef n) (TSort n0) H0))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef n) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0 d)).(let H2 \def (eq_ind T (lift -h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef n) t)) H0 (TLRef n0) +h d (TLRef n0)) (\lambda (t0: T).(eq T (TLRef n) t0)) H0 (TLRef n0) (lift_lref_lt n0 h d H1)) in (sym_eq T (TLRef n) (TLRef n0) H2))) (\lambda -(H1: (le d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: -T).(eq T (TLRef n) t)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in -(let H3 \def (match H2 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? -t)).((eq T t (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with -[refl_equal \Rightarrow (\lambda (H2: (eq T (TLRef n) (TLRef (plus n0 -h)))).(let H3 \def (f_equal T nat (\lambda (e: T).(match e in T return -(\lambda (_: T).nat) with [(TSort _) \Rightarrow n | (TLRef n) \Rightarrow n -| (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef (plus n0 h)) H2) in (eq_ind -nat (plus n0 h) (\lambda (n: nat).(eq T (TLRef n0) (TLRef n))) (let H0 \def -(eq_ind nat n (\lambda (n: nat).(lt n d)) H (plus n0 h) H3) in (le_false d n0 -(eq T (TLRef n0) (TLRef (plus n0 h))) H1 (lt_le_S n0 d (le_lt_trans n0 (plus -n0 h) d (le_plus_l n0 h) H0)))) n (sym_eq nat n (plus n0 h) H3))))]) in (H3 -(refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: -T).(\lambda (_: (((eq T (TLRef n) (lift h d t0)) \to (eq T t0 (TLRef -n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef n) (lift h d t1)) \to (eq -T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef n) (lift h d (THead k t0 -t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq -T (TLRef n) t)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k -t0 t1 h d)) in (let H4 \def (match H3 in eq return (\lambda (t: T).(\lambda -(_: (eq ? ? t)).((eq T t (THead k (lift h d t0) (lift h (s k d) t1))) \to (eq -T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H3: -(eq T (TLRef n) (THead k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def -(eq_ind T (TLRef n) (\lambda (e: T).(match e in T return (\lambda (_: -T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | -(THead _ _ _) \Rightarrow False])) I (THead k (lift h d t0) (lift h (s k d) -t1)) H3) in (False_ind (eq T (THead k t0 t1) (TLRef n)) H4)))]) in (H4 -(refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))) t))))). +(H1: (le d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: +T).(eq T (TLRef n) t0)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in +(let H3 \def (match H2 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? +t0)).((eq T t0 (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with +[refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef n) (TLRef (plus n0 +h)))).(let H4 \def (f_equal T nat (\lambda (e: T).(match e in T return +(\lambda (_: T).nat) with [(TSort _) \Rightarrow n | (TLRef n1) \Rightarrow +n1 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef (plus n0 h)) H3) in +(eq_ind nat (plus n0 h) (\lambda (n1: nat).(eq T (TLRef n0) (TLRef n1))) (let +H5 \def (eq_ind nat n (\lambda (n1: nat).(lt n1 d)) H (plus n0 h) H4) in +(le_false d n0 (eq T (TLRef n0) (TLRef (plus n0 h))) H1 (lt_le_S n0 d +(le_lt_trans n0 (plus n0 h) d (le_plus_l n0 h) H5)))) n (sym_eq nat n (plus +n0 h) H4))))]) in (H3 (refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: +K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) (lift h d t0)) \to (eq T +t0 (TLRef n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef n) (lift h d +t1)) \to (eq T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef n) (lift h d +(THead k t0 t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda +(t2: T).(eq T (TLRef n) t2)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) +(lift_head k t0 t1 h d)) in (let H4 \def (match H3 in eq return (\lambda (t2: +T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) +t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow +(\lambda (H4: (eq T (TLRef n) (THead k (lift h d t0) (lift h (s k d) +t1)))).(let H5 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e in T return +(\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d +t0) (lift h (s k d) t1)) H4) in (False_ind (eq T (THead k t0 t1) (TLRef n)) +H5)))]) in (H4 (refl_equal T (THead k (lift h d t0) (lift h (s k d) +t1)))))))))))) t))))). theorem lift_gen_lref_false: \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to ((lt n @@ -219,8 +220,8 @@ theorem lift_gen_lref_false: n)).(\lambda (H0: (lt n (plus d h))).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef n) (lift h d t0)) \to (\forall (P: Prop).P))) (\lambda (n0: nat).(\lambda (H1: (eq T (TLRef n) (lift h d (TSort n0)))).(\lambda (P: -Prop).(let H2 \def (match H1 in eq return (\lambda (t: T).(\lambda (_: (eq ? -? t)).((eq T t (lift h d (TSort n0))) \to P))) with [refl_equal \Rightarrow +Prop).(let H2 \def (match H1 in eq return (\lambda (t0: T).(\lambda (_: (eq ? +? t0)).((eq T t0 (lift h d (TSort n0))) \to P))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (TLRef n) (lift h d (TSort n0)))).(let H3 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) @@ -228,40 +229,41 @@ Prop).(let H2 \def (match H1 in eq return (\lambda (t: T).(\lambda (_: (eq ? (H2 (refl_equal T (lift h d (TSort n0)))))))) (\lambda (n0: nat).(\lambda (H1: (eq T (TLRef n) (lift h d (TLRef n0)))).(\lambda (P: Prop).(lt_le_e n0 d P (\lambda (H2: (lt n0 d)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) -(\lambda (t: T).(eq T (TLRef n) t)) H1 (TLRef n0) (lift_lref_lt n0 h d H2)) -in (let H4 \def (match H3 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? -t)).((eq T t (TLRef n0)) \to P))) with [refl_equal \Rightarrow (\lambda (H3: -(eq T (TLRef n) (TLRef n0))).(let H4 \def (f_equal T nat (\lambda (e: +(\lambda (t0: T).(eq T (TLRef n) t0)) H1 (TLRef n0) (lift_lref_lt n0 h d H2)) +in (let H4 \def (match H3 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? +t0)).((eq T t0 (TLRef n0)) \to P))) with [refl_equal \Rightarrow (\lambda +(H4: (eq T (TLRef n) (TLRef n0))).(let H5 \def (f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | -(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef n0) -H3) in (eq_ind nat n0 (\lambda (_: nat).P) (let H1 \def (eq_ind_r nat n0 -(\lambda (n: nat).(lt n d)) H2 n H4) in (le_false d n P H H1)) n (sym_eq nat -n n0 H4))))]) in (H4 (refl_equal T (TLRef n0)))))) (\lambda (H2: (le d -n0)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T -(TLRef n) t)) H1 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H2)) in (let H4 -\def (match H3 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t -(TLRef (plus n0 h))) \to P))) with [refl_equal \Rightarrow (\lambda (H3: (eq -T (TLRef n) (TLRef (plus n0 h)))).(let H4 \def (f_equal T nat (\lambda (e: -T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow n | -(TLRef n) \Rightarrow n | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef -(plus n0 h)) H3) in (eq_ind nat (plus n0 h) (\lambda (_: nat).P) (let H1 \def -(eq_ind nat n (\lambda (n: nat).(lt n (plus d h))) H0 (plus n0 h) H4) in -(le_false d n0 P H2 (lt_le_S n0 d (simpl_lt_plus_r h n0 d H1)))) n (sym_eq -nat n (plus n0 h) H4))))]) in (H4 (refl_equal T (TLRef (plus n0 h))))))))))) -(\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) (lift h d -t0)) \to (\forall (P: Prop).P)))).(\lambda (t1: T).(\lambda (_: (((eq T -(TLRef n) (lift h d t1)) \to (\forall (P: Prop).P)))).(\lambda (H3: (eq T -(TLRef n) (lift h d (THead k t0 t1)))).(\lambda (P: Prop).(let H4 \def -(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t: T).(eq T (TLRef n) t)) H3 +(TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n])) (TLRef n) (TLRef +n0) H4) in (eq_ind nat n0 (\lambda (_: nat).P) (let H6 \def (eq_ind_r nat n0 +(\lambda (n1: nat).(lt n1 d)) H2 n H5) in (le_false d n P H H6)) n (sym_eq +nat n n0 H5))))]) in (H4 (refl_equal T (TLRef n0)))))) (\lambda (H2: (le d +n0)).(let H3 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T +(TLRef n) t0)) H1 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H2)) in (let H4 +\def (match H3 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T +t0 (TLRef (plus n0 h))) \to P))) with [refl_equal \Rightarrow (\lambda (H4: +(eq T (TLRef n) (TLRef (plus n0 h)))).(let H5 \def (f_equal T nat (\lambda +(e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) \Rightarrow +n | (TLRef n1) \Rightarrow n1 | (THead _ _ _) \Rightarrow n])) (TLRef n) +(TLRef (plus n0 h)) H4) in (eq_ind nat (plus n0 h) (\lambda (_: nat).P) (let +H6 \def (eq_ind nat n (\lambda (n1: nat).(lt n1 (plus d h))) H0 (plus n0 h) +H5) in (le_false d n0 P H2 (lt_le_S n0 d (simpl_lt_plus_r h n0 d H6)))) n +(sym_eq nat n (plus n0 h) H5))))]) in (H4 (refl_equal T (TLRef (plus n0 +h))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef n) +(lift h d t0)) \to (\forall (P: Prop).P)))).(\lambda (t1: T).(\lambda (_: +(((eq T (TLRef n) (lift h d t1)) \to (\forall (P: Prop).P)))).(\lambda (H3: +(eq T (TLRef n) (lift h d (THead k t0 t1)))).(\lambda (P: Prop).(let H4 \def +(eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (TLRef n) t2)) H3 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let -H5 \def (match H4 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq -T t (THead k (lift h d t0) (lift h (s k d) t1))) \to P))) with [refl_equal -\Rightarrow (\lambda (H4: (eq T (TLRef n) (THead k (lift h d t0) (lift h (s k -d) t1)))).(let H5 \def (eq_ind T (TLRef n) (\lambda (e: T).(match e in T -return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) -\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d -t0) (lift h (s k d) t1)) H4) in (False_ind P H5)))]) in (H5 (refl_equal T -(THead k (lift h d t0) (lift h (s k d) t1))))))))))))) t)))))). +H5 \def (match H4 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ? +t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to P))) with +[refl_equal \Rightarrow (\lambda (H5: (eq T (TLRef n) (THead k (lift h d t0) +(lift h (s k d) t1)))).(let H6 \def (eq_ind T (TLRef n) (\lambda (e: +T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow +False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow False])) I +(THead k (lift h d t0) (lift h (s k d) t1)) H5) in (False_ind P H6)))]) in +(H5 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1))))))))))))) +t)))))). theorem lift_gen_lref_ge: \forall (h: nat).(\forall (d: nat).(\forall (n: nat).((le d n) \to (\forall @@ -271,7 +273,7 @@ theorem lift_gen_lref_ge: n)).(\lambda (t: T).(T_ind (\lambda (t0: T).((eq T (TLRef (plus n h)) (lift h d t0)) \to (eq T t0 (TLRef n)))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d (TSort n0)))).(let H1 \def (match H0 in eq -return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (lift h d (TSort +return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (lift h d (TSort n0))) \to (eq T (TSort n0) (TLRef n))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (TLRef (plus n h)) (lift h d (TSort n0)))).(let H2 \def (eq_ind T (TLRef (plus n h)) (\lambda (e: T).(match e in T return (\lambda @@ -280,52 +282,53 @@ n0))) \to (eq T (TSort n0) (TLRef n))))) with [refl_equal \Rightarrow (eq T (TSort n0) (TLRef n)) H2)))]) in (H1 (refl_equal T (lift h d (TSort n0))))))) (\lambda (n0: nat).(\lambda (H0: (eq T (TLRef (plus n h)) (lift h d (TLRef n0)))).(lt_le_e n0 d (eq T (TLRef n0) (TLRef n)) (\lambda (H1: (lt n0 -d)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef -(plus n h)) t)) H0 (TLRef n0) (lift_lref_lt n0 h d H1)) in (let H3 \def -(match H2 in eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t -(TLRef n0)) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal \Rightarrow -(\lambda (H2: (eq T (TLRef (plus n h)) (TLRef n0))).(let H3 \def (f_equal T -nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) -\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) n h) | (TLRef n) \Rightarrow n | (THead _ _ _) \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) n h)])) (TLRef -(plus n h)) (TLRef n0) H2) in (eq_ind nat (plus n h) (\lambda (n0: nat).(eq T -(TLRef n0) (TLRef n))) (let H0 \def (eq_ind_r nat n0 (\lambda (n: nat).(lt n -d)) H1 (plus n h) H3) in (le_false d n (eq T (TLRef (plus n h)) (TLRef n)) H -(lt_le_S n d (le_lt_trans n (plus n h) d (le_plus_l n h) H0)))) n0 H3)))]) in -(H3 (refl_equal T (TLRef n0)))))) (\lambda (H1: (le d n0)).(let H2 \def -(eq_ind T (lift h d (TLRef n0)) (\lambda (t: T).(eq T (TLRef (plus n h)) t)) -H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in (let H3 \def (match H2 in -eq return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (TLRef (plus n0 -h))) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal \Rightarrow (\lambda -(H2: (eq T (TLRef (plus n h)) (TLRef (plus n0 h)))).(let H3 \def (f_equal T -nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with [(TSort _) -\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) n h) | (TLRef n) \Rightarrow n | (THead _ _ _) \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) n h)])) (TLRef -(plus n h)) (TLRef (plus n0 h)) H2) in (eq_ind nat (plus n h) (\lambda (_: -nat).(eq T (TLRef n0) (TLRef n))) (f_equal nat T TLRef n0 n (simpl_plus_r h -n0 n (sym_eq nat (plus n h) (plus n0 h) H3))) (plus n0 h) H3)))]) in (H3 -(refl_equal T (TLRef (plus n0 h)))))))))) (\lambda (k: K).(\lambda (t0: -T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d t0)) \to (eq T t0 (TLRef -n))))).(\lambda (t1: T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d -t1)) \to (eq T t1 (TLRef n))))).(\lambda (H2: (eq T (TLRef (plus n h)) (lift -h d (THead k t0 t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) -(\lambda (t: T).(eq T (TLRef (plus n h)) t)) H2 (THead k (lift h d t0) (lift -h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H4 \def (match H3 in eq -return (\lambda (t: T).(\lambda (_: (eq ? ? t)).((eq T t (THead k (lift h d -t0) (lift h (s k d) t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with -[refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (THead k (lift -h d t0) (lift h (s k d) t1)))).(let H4 \def (eq_ind T (TLRef (plus n h)) -(\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) -\Rightarrow False | (TLRef _) \Rightarrow True | (THead _ _ _) \Rightarrow -False])) I (THead k (lift h d t0) (lift h (s k d) t1)) H3) in (False_ind (eq -T (THead k t0 t1) (TLRef n)) H4)))]) in (H4 (refl_equal T (THead k (lift h d -t0) (lift h (s k d) t1)))))))))))) t))))). +d)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T +(TLRef (plus n h)) t0)) H0 (TLRef n0) (lift_lref_lt n0 h d H1)) in (let H3 +\def (match H2 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T +t0 (TLRef n0)) \to (eq T (TLRef n0) (TLRef n))))) with [refl_equal +\Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (TLRef n0))).(let H4 \def +(f_equal T nat (\lambda (e: T).(match e in T return (\lambda (_: T).nat) with +[(TSort _) \Rightarrow ((let rec plus (n1: nat) on n1: (nat \to nat) \def +(\lambda (m: nat).(match n1 with [O \Rightarrow m | (S p) \Rightarrow (S +(plus p m))])) in plus) n h) | (TLRef n1) \Rightarrow n1 | (THead _ _ _) +\Rightarrow ((let rec plus (n1: nat) on n1: (nat \to nat) \def (\lambda (m: +nat).(match n1 with [O \Rightarrow m | (S p) \Rightarrow (S (plus p m))])) in +plus) n h)])) (TLRef (plus n h)) (TLRef n0) H3) in (eq_ind nat (plus n h) +(\lambda (n1: nat).(eq T (TLRef n1) (TLRef n))) (let H5 \def (eq_ind_r nat n0 +(\lambda (n1: nat).(lt n1 d)) H1 (plus n h) H4) in (le_false d n (eq T (TLRef +(plus n h)) (TLRef n)) H (lt_le_S n d (le_lt_trans n (plus n h) d (le_plus_l +n h) H5)))) n0 H4)))]) in (H3 (refl_equal T (TLRef n0)))))) (\lambda (H1: (le +d n0)).(let H2 \def (eq_ind T (lift h d (TLRef n0)) (\lambda (t0: T).(eq T +(TLRef (plus n h)) t0)) H0 (TLRef (plus n0 h)) (lift_lref_ge n0 h d H1)) in +(let H3 \def (match H2 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? +t0)).((eq T t0 (TLRef (plus n0 h))) \to (eq T (TLRef n0) (TLRef n))))) with +[refl_equal \Rightarrow (\lambda (H3: (eq T (TLRef (plus n h)) (TLRef (plus +n0 h)))).(let H4 \def (f_equal T nat (\lambda (e: T).(match e in T return +(\lambda (_: T).nat) with [(TSort _) \Rightarrow ((let rec plus (n1: nat) on +n1: (nat \to nat) \def (\lambda (m: nat).(match n1 with [O \Rightarrow m | (S +p) \Rightarrow (S (plus p m))])) in plus) n h) | (TLRef n1) \Rightarrow n1 | +(THead _ _ _) \Rightarrow ((let rec plus (n1: nat) on n1: (nat \to nat) \def +(\lambda (m: nat).(match n1 with [O \Rightarrow m | (S p) \Rightarrow (S +(plus p m))])) in plus) n h)])) (TLRef (plus n h)) (TLRef (plus n0 h)) H3) in +(eq_ind nat (plus n h) (\lambda (_: nat).(eq T (TLRef n0) (TLRef n))) +(f_equal nat T TLRef n0 n (simpl_plus_r h n0 n (sym_eq nat (plus n h) (plus +n0 h) H4))) (plus n0 h) H4)))]) in (H3 (refl_equal T (TLRef (plus n0 +h)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: (((eq T (TLRef +(plus n h)) (lift h d t0)) \to (eq T t0 (TLRef n))))).(\lambda (t1: +T).(\lambda (_: (((eq T (TLRef (plus n h)) (lift h d t1)) \to (eq T t1 (TLRef +n))))).(\lambda (H2: (eq T (TLRef (plus n h)) (lift h d (THead k t0 +t1)))).(let H3 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda (t2: T).(eq +T (TLRef (plus n h)) t2)) H2 (THead k (lift h d t0) (lift h (s k d) t1)) +(lift_head k t0 t1 h d)) in (let H4 \def (match H3 in eq return (\lambda (t2: +T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) +t1))) \to (eq T (THead k t0 t1) (TLRef n))))) with [refl_equal \Rightarrow +(\lambda (H4: (eq T (TLRef (plus n h)) (THead k (lift h d t0) (lift h (s k d) +t1)))).(let H5 \def (eq_ind T (TLRef (plus n h)) (\lambda (e: T).(match e in +T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow True | (THead _ _ _) \Rightarrow False])) I (THead k (lift h d +t0) (lift h (s k d) t1)) H4) in (False_ind (eq T (THead k t0 t1) (TLRef n)) +H5)))]) in (H4 (refl_equal T (THead k (lift h d t0) (lift h (s k d) +t1)))))))))))) t))))). theorem lift_gen_head: \forall (k: K).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: @@ -363,27 +366,27 @@ eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef n)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))) with [refl_equal -\Rightarrow (\lambda (H1: (eq T (THead k u t) (TLRef n))).(let H2 \def +\Rightarrow (\lambda (H2: (eq T (THead k u t) (TLRef n))).(let H3 \def (eq_ind T (THead k u t) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T +(THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (s k d) z))))) H2)))]) in (H2 (refl_equal T (TLRef n)))))) +T).(eq T t (lift h (s k d) z))))) H3)))]) in (H2 (refl_equal T (TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead k u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 in eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef (plus n h))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift -h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda (H1: (eq T (THead k -u t) (TLRef (plus n h)))).(let H2 \def (eq_ind T (THead k u t) (\lambda (e: +h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda (H2: (eq T (THead k +u t) (TLRef (plus n h)))).(let H3 \def (eq_ind T (THead k u t) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I -(TLRef (plus n h)) H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: +(TLRef (plus n h)) H2) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) -z))))) H2)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda +z))))) H3)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k0: K).(\lambda (t0: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead k u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead k y z)))) (\lambda (y: T).(\lambda (_: @@ -394,42 +397,42 @@ T).(\lambda (z: T).(eq T t1 (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k u t) (lift h d (THead k0 t0 t1)))).(let H2 \def (eq_ind T (lift h d -(THead k0 t0 t1)) (\lambda (t0: T).(eq T (THead k u t) t0)) H1 (THead k0 +(THead k0 t0 t1)) (\lambda (t2: T).(eq T (THead k u t) t2)) H1 (THead k0 (lift h d t0) (lift h (s k0 d) t1)) (lift_head k0 t0 t1 h d)) in (let H3 \def (match H2 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k0 (lift h d t0) (lift h (s k0 d) t1))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k d) z)))))))) with [refl_equal \Rightarrow (\lambda -(H2: (eq T (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)))).(let -H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) -with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t) -\Rightarrow t])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) -H2) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return +(H3: (eq T (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)))).(let +H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow t | (TLRef _) \Rightarrow t | (THead _ _ t2) +\Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift h (s k0 d) t1)) +H3) in ((let H5 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u | (TLRef _) \Rightarrow u | -(THead _ t _) \Rightarrow t])) (THead k u t) (THead k0 (lift h d t0) (lift h -(s k0 d) t1)) H2) in ((let H5 \def (f_equal T K (\lambda (e: T).(match e in T -return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) -\Rightarrow k | (THead k _ _) \Rightarrow k])) (THead k u t) (THead k0 (lift -h d t0) (lift h (s k0 d) t1)) H2) in (eq_ind K k0 (\lambda (k: K).((eq T u -(lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y: -T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k y z)))) (\lambda (y: +(THead _ t2 _) \Rightarrow t2])) (THead k u t) (THead k0 (lift h d t0) (lift +h (s k0 d) t1)) H3) in ((let H6 \def (f_equal T K (\lambda (e: T).(match e in +T return (\lambda (_: T).K) with [(TSort _) \Rightarrow k | (TLRef _) +\Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k u t) (THead k0 +(lift h d t0) (lift h (s k0 d) t1)) H3) in (eq_ind K k0 (\lambda (k1: K).((eq +T u (lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda +(y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k1 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h (s k d) z)))))))) (\lambda (H6: (eq T u (lift h d +T).(eq T t (lift h (s k1 d) z)))))))) (\lambda (H7: (eq T u (lift h d t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda -(_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z))))))) (\lambda (H7: (eq T -t (lift h (s k0 d) t1))).(eq_ind T (lift h (s k0 d) t1) (\lambda (t: +(_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z))))))) (\lambda (H8: (eq T +t (lift h (s k0 d) t1))).(eq_ind T (lift h (s k0 d) t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d -y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (s k0 d) z)))))) +y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift h (s k0 d) z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead k0 y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s k0 d) t1) (lift h (s k0 d) z)))) t0 t1 (refl_equal T (THead k0 t0 t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h (s k0 d) t1))) t (sym_eq T t (lift h (s k0 d) t1) -H7))) u (sym_eq T u (lift h d t0) H6))) k (sym_eq K k k0 H5))) H4)) H3)))]) +H8))) u (sym_eq T u (lift h d t0) H7))) k (sym_eq K k k0 H6))) H5)) H4)))]) in (H3 (refl_equal T (THead k0 (lift h d t0) (lift h (s k0 d) t1)))))))))))))) x)))). @@ -470,13 +473,13 @@ T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal -\Rightarrow (\lambda (H1: (eq T (THead (Bind b) u t) (TLRef n))).(let H2 \def +\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u t) (TLRef n))).(let H3 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False -| (THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T +| (THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H2)))]) in (H2 (refl_equal T +T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H3)))]) in (H2 (refl_equal T (TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 in eq return (\lambda @@ -484,13 +487,13 @@ h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 in eq return (\lambda T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal -\Rightarrow (\lambda (H1: (eq T (THead (Bind b) u t) (TLRef (plus n -h)))).(let H2 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e in +\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u t) (TLRef (plus n +h)))).(let H3 \def (eq_ind T (THead (Bind b) u t) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) -H1) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) +H2) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d -y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H2)))]) in +y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))) H3)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Bind b) u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 @@ -502,45 +505,45 @@ T).(\lambda (z: T).(eq T t1 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead (Bind b) u t) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T -(lift h d (THead k t0 t1)) (\lambda (t0: T).(eq T (THead (Bind b) u t) t0)) +(lift h d (THead k t0 t1)) (\lambda (t2: T).(eq T (THead (Bind b) u t) t2)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (match H2 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))))) with [refl_equal -\Rightarrow (\lambda (H2: (eq T (THead (Bind b) u t) (THead k (lift h d t0) -(lift h (s k d) t1)))).(let H3 \def (f_equal T T (\lambda (e: T).(match e in +\Rightarrow (\lambda (H3: (eq T (THead (Bind b) u t) (THead k (lift h d t0) +(lift h (s k d) t1)))).(let H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | (TLRef _) -\Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead (Bind b) u t) (THead k -(lift h d t0) (lift h (s k d) t1)) H2) in ((let H4 \def (f_equal T T (\lambda -(e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow u -| (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) (THead (Bind b) u -t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let H5 \def (f_equal -T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) | (THead k _ _) -\Rightarrow k])) (THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) -t1)) H2) in (eq_ind K (Bind b) (\lambda (k: K).((eq T u (lift h d t0)) \to -((eq T t (lift h (s k d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: -T).(eq T (THead k t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: -T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S -d) z)))))))) (\lambda (H6: (eq T u (lift h d t0))).(eq_ind T (lift h d t0) -(\lambda (t2: T).((eq T t (lift h (s (Bind b) d) t1)) \to (ex3_2 T T (\lambda -(y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) (THead (Bind b) y z)))) -(\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T t (lift h (S d) z))))))) (\lambda (H7: (eq T t (lift -h (s (Bind b) d) t1))).(eq_ind T (lift h (s (Bind b) d) t1) (\lambda (t: -T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) -(THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) -(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z)))))) -(ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) t0 t1) -(THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t0) -(lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s (Bind b) d) -t1) (lift h (S d) z)))) t0 t1 (refl_equal T (THead (Bind b) t0 t1)) -(refl_equal T (lift h d t0)) (refl_equal T (lift h (S d) t1))) t (sym_eq T t -(lift h (s (Bind b) d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k H5)) -H4)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d) -t1)))))))))))))) x)))). +\Rightarrow t | (THead _ _ t2) \Rightarrow t2])) (THead (Bind b) u t) (THead +k (lift h d t0) (lift h (s k d) t1)) H3) in ((let H5 \def (f_equal T T +(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2])) +(THead (Bind b) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H3) in ((let +H6 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) +with [(TSort _) \Rightarrow (Bind b) | (TLRef _) \Rightarrow (Bind b) | +(THead k0 _ _) \Rightarrow k0])) (THead (Bind b) u t) (THead k (lift h d t0) +(lift h (s k d) t1)) H3) in (eq_ind K (Bind b) (\lambda (k0: K).((eq T u +(lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y: +T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead (Bind b) y z)))) (\lambda +(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: +T).(eq T t (lift h (S d) z)))))))) (\lambda (H7: (eq T u (lift h d +t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s (Bind b) +d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Bind b) +t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift +h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h (S d) z))))))) +(\lambda (H8: (eq T t (lift h (s (Bind b) d) t1))).(eq_ind T (lift h (s (Bind +b) d) t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T +(THead (Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: +T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T +t2 (lift h (S d) z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq +T (THead (Bind b) t0 t1) (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: +T).(eq T (lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T +(lift h (s (Bind b) d) t1) (lift h (S d) z)))) t0 t1 (refl_equal T (THead +(Bind b) t0 t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h (S d) +t1))) t (sym_eq T t (lift h (s (Bind b) d) t1) H8))) u (sym_eq T u (lift h d +t0) H7))) k H6)) H5)) H4)))]) in (H3 (refl_equal T (THead k (lift h d t0) +(lift h (s k d) t1)))))))))))))) x)))). theorem lift_gen_flat: \forall (f: F).(\forall (u: T).(\forall (t: T).(\forall (x: T).(\forall (h: @@ -578,13 +581,13 @@ eq return (\lambda (t0: T).(\lambda (_: (eq ? ? t0)).((eq T t0 (TLRef n)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow -(\lambda (H1: (eq T (THead (Flat f) u t) (TLRef n))).(let H2 \def (eq_ind T +(\lambda (H2: (eq T (THead (Flat f) u t) (TLRef n))).(let H3 \def (eq_ind T (THead (Flat f) u t) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | -(THead _ _ _) \Rightarrow True])) I (TLRef n) H1) in (False_ind (ex3_2 T T +(THead _ _ _) \Rightarrow True])) I (TLRef n) H2) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T t (lift h d z))))) H2)))]) in (H2 (refl_equal T +T).(\lambda (z: T).(eq T t (lift h d z))))) H3)))]) in (H2 (refl_equal T (TLRef n)))))) (\lambda (H0: (le d n)).(let H1 \def (eq_ind T (lift h d (TLRef n)) (\lambda (t0: T).(eq T (THead (Flat f) u t) t0)) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 in eq return (\lambda @@ -592,13 +595,13 @@ h)) (lift_lref_ge n h d H0)) in (let H2 \def (match H1 in eq return (\lambda T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))) with [refl_equal \Rightarrow -(\lambda (H1: (eq T (THead (Flat f) u t) (TLRef (plus n h)))).(let H2 \def +(\lambda (H2: (eq T (THead (Flat f) u t) (TLRef (plus n h)))).(let H3 \def (eq_ind T (THead (Flat f) u t) (\lambda (e: T).(match e in T return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False -| (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) H1) in (False_ind +| (THead _ _ _) \Rightarrow True])) I (TLRef (plus n h)) H2) in (False_ind (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (TLRef n) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: -T).(\lambda (z: T).(eq T t (lift h d z))))) H2)))]) in (H2 (refl_equal T +T).(\lambda (z: T).(eq T t (lift h d z))))) H3)))]) in (H2 (refl_equal T (TLRef (plus n h)))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (_: ((\forall (h: nat).(\forall (d: nat).((eq T (THead (Flat f) u t) (lift h d t0)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T t0 (THead (Flat f) @@ -610,42 +613,42 @@ f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))))).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead (Flat f) u t) (lift h d (THead k t0 t1)))).(let H2 \def (eq_ind T (lift h d (THead k t0 t1)) (\lambda -(t0: T).(eq T (THead (Flat f) u t) t0)) H1 (THead k (lift h d t0) (lift h (s +(t2: T).(eq T (THead (Flat f) u t) t2)) H1 (THead k (lift h d t0) (lift h (s k d) t1)) (lift_head k t0 t1 h d)) in (let H3 \def (match H2 in eq return (\lambda (t2: T).(\lambda (_: (eq ? ? t2)).((eq T t2 (THead k (lift h d t0) (lift h (s k d) t1))) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z)))))))) -with [refl_equal \Rightarrow (\lambda (H2: (eq T (THead (Flat f) u t) (THead -k (lift h d t0) (lift h (s k d) t1)))).(let H3 \def (f_equal T T (\lambda (e: +with [refl_equal \Rightarrow (\lambda (H3: (eq T (THead (Flat f) u t) (THead +k (lift h d t0) (lift h (s k d) t1)))).(let H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) \Rightarrow t | -(TLRef _) \Rightarrow t | (THead _ _ t) \Rightarrow t])) (THead (Flat f) u t) -(THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let H4 \def (f_equal T T -(\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) -\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t _) \Rightarrow t])) -(THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H2) in ((let -H5 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) +(TLRef _) \Rightarrow t | (THead _ _ t2) \Rightarrow t2])) (THead (Flat f) u +t) (THead k (lift h d t0) (lift h (s k d) t1)) H3) in ((let H5 \def (f_equal +T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t2 _) \Rightarrow t2])) +(THead (Flat f) u t) (THead k (lift h d t0) (lift h (s k d) t1)) H3) in ((let +H6 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) \Rightarrow (Flat f) | (TLRef _) \Rightarrow (Flat f) | -(THead k _ _) \Rightarrow k])) (THead (Flat f) u t) (THead k (lift h d t0) -(lift h (s k d) t1)) H2) in (eq_ind K (Flat f) (\lambda (k: K).((eq T u (lift -h d t0)) \to ((eq T t (lift h (s k d) t1)) \to (ex3_2 T T (\lambda (y: -T).(\lambda (z: T).(eq T (THead k t0 t1) (THead (Flat f) y z)))) (\lambda (y: -T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: -T).(eq T t (lift h d z)))))))) (\lambda (H6: (eq T u (lift h d t0))).(eq_ind +(THead k0 _ _) \Rightarrow k0])) (THead (Flat f) u t) (THead k (lift h d t0) +(lift h (s k d) t1)) H3) in (eq_ind K (Flat f) (\lambda (k0: K).((eq T u +(lift h d t0)) \to ((eq T t (lift h (s k0 d) t1)) \to (ex3_2 T T (\lambda (y: +T).(\lambda (z: T).(eq T (THead k0 t0 t1) (THead (Flat f) y z)))) (\lambda +(y: T).(\lambda (_: T).(eq T u (lift h d y)))) (\lambda (_: T).(\lambda (z: +T).(eq T t (lift h d z)))))))) (\lambda (H7: (eq T u (lift h d t0))).(eq_ind T (lift h d t0) (\lambda (t2: T).((eq T t (lift h (s (Flat f) d) t1)) \to (ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T t2 (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h d z))))))) (\lambda -(H7: (eq T t (lift h (s (Flat f) d) t1))).(eq_ind T (lift h (s (Flat f) d) -t1) (\lambda (t: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead +(H8: (eq T t (lift h (s (Flat f) d) t1))).(eq_ind T (lift h (s (Flat f) d) +t1) (\lambda (t2: T).(ex3_2 T T (\lambda (y: T).(\lambda (z: T).(eq T (THead +(Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T +(lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t2 (lift +h d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T -(lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T t (lift h -d z)))))) (ex3_2_intro T T (\lambda (y: T).(\lambda (z: T).(eq T (THead (Flat -f) t0 t1) (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift -h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h (s -(Flat f) d) t1) (lift h d z)))) t0 t1 (refl_equal T (THead (Flat f) t0 t1)) -(refl_equal T (lift h d t0)) (refl_equal T (lift h d t1))) t (sym_eq T t -(lift h (s (Flat f) d) t1) H7))) u (sym_eq T u (lift h d t0) H6))) k H5)) -H4)) H3)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d) +(lift h d t0) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h +(s (Flat f) d) t1) (lift h d z)))) t0 t1 (refl_equal T (THead (Flat f) t0 +t1)) (refl_equal T (lift h d t0)) (refl_equal T (lift h d t1))) t (sym_eq T t +(lift h (s (Flat f) d) t1) H8))) u (sym_eq T u (lift h d t0) H7))) k H6)) +H5)) H4)))]) in (H3 (refl_equal T (THead k (lift h d t0) (lift h (s k d) t1)))))))))))))) x)))). diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props.ma index 7c205f57a..3e2301883 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/props.ma @@ -45,40 +45,41 @@ h d t1)) t1) \to (\forall (P: Prop).P))))))).(\lambda (v: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1))).(\lambda (P: Prop).(let H2 \def (f_equal T K (\lambda (e: T).(match e in T return (\lambda (_: T).K) with [(TSort _) -\Rightarrow k | (TLRef _) \Rightarrow k | (THead k _ _) \Rightarrow k])) +\Rightarrow k | (TLRef _) \Rightarrow k | (THead k1 _ _) \Rightarrow k1])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) in ((let H3 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with -[(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t _) -\Rightarrow t])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) H1) -in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda -(_: T).T) with [(TSort _) \Rightarrow (THead k0 ((let rec lref_map (f: ((nat -\to nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow -(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true -\Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow -(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda -(x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat \to nat))) (d: nat) -(t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef -i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false -\Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) -(lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x h)) (s k0 -d) t1)) | (TLRef _) \Rightarrow (THead k0 ((let rec lref_map (f: ((nat \to -nat))) (d: nat) (t: T) on t: T \def (match t with [(TSort n) \Rightarrow -(TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d) with [true -\Rightarrow i | false \Rightarrow (f i)])) | (THead k u t0) \Rightarrow -(THead k (lref_map f d u) (lref_map f (s k d) t0))]) in lref_map) (\lambda -(x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat \to nat))) (d: nat) -(t: T) on t: T \def (match t with [(TSort n) \Rightarrow (TSort n) | (TLRef -i) \Rightarrow (TLRef (match (blt i d) with [true \Rightarrow i | false -\Rightarrow (f i)])) | (THead k u t0) \Rightarrow (THead k (lref_map f d u) -(lref_map f (s k d) t0))]) in lref_map) (\lambda (x: nat).(plus x h)) (s k0 -d) t1)) | (THead _ _ t) \Rightarrow t])) (THead k v (lift h d (THead k0 t0 -t1))) (THead k0 t0 t1) H1) in (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k -k0)).(let H7 \def (eq_ind K k (\lambda (k: K).(\forall (v: T).(\forall (h: -nat).(\forall (d: nat).((eq T (THead k v (lift h d t1)) t1) \to (\forall (P: -Prop).P)))))) H0 k0 H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0 t1)) -(\lambda (t: T).(eq T t t1)) H4 (THead k0 (lift h d t0) (lift h (s k0 d) t1)) -(lift_head k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P)))))) H3)) -H2)))))))))))) t)). +[(TSort _) \Rightarrow v | (TLRef _) \Rightarrow v | (THead _ t2 _) +\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) +H1) in ((let H4 \def (f_equal T T (\lambda (e: T).(match e in T return +(\lambda (_: T).T) with [(TSort _) \Rightarrow (THead k0 ((let rec lref_map +(f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort +n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) +with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) +\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in +lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec lref_map (f: ((nat +\to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) +\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in +lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (TLRef _) \Rightarrow +(THead k0 ((let rec lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T +\def (match t2 with [(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow +(TLRef (match (blt i d0) with [true \Rightarrow i | false \Rightarrow (f +i)])) | (THead k1 u t3) \Rightarrow (THead k1 (lref_map f d0 u) (lref_map f +(s k1 d0) t3))]) in lref_map) (\lambda (x: nat).(plus x h)) d t0) ((let rec +lref_map (f: ((nat \to nat))) (d0: nat) (t2: T) on t2: T \def (match t2 with +[(TSort n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i +d0) with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k1 u t3) +\Rightarrow (THead k1 (lref_map f d0 u) (lref_map f (s k1 d0) t3))]) in +lref_map) (\lambda (x: nat).(plus x h)) (s k0 d) t1)) | (THead _ _ t2) +\Rightarrow t2])) (THead k v (lift h d (THead k0 t0 t1))) (THead k0 t0 t1) +H1) in (\lambda (_: (eq T v t0)).(\lambda (H6: (eq K k k0)).(let H7 \def +(eq_ind K k (\lambda (k1: K).(\forall (v0: T).(\forall (h0: nat).(\forall +(d0: nat).((eq T (THead k1 v0 (lift h0 d0 t1)) t1) \to (\forall (P0: +Prop).P0)))))) H0 k0 H6) in (let H8 \def (eq_ind T (lift h d (THead k0 t0 +t1)) (\lambda (t2: T).(eq T t2 t1)) H4 (THead k0 (lift h d t0) (lift h (s k0 +d) t1)) (lift_head k0 t0 t1 h d)) in (H7 (lift h d t0) h (s k0 d) H8 P)))))) +H3)) H2)))))))))))) t)). theorem lift_r: \forall (t: T).(\forall (d: nat).(eq T (lift O d t) t)) @@ -94,11 +95,11 @@ t0 (TLRef n))) (f_equal nat T TLRef (plus n O) n (sym_eq nat n (plus n O) K).(\lambda (t0: T).(\lambda (H: ((\forall (d: nat).(eq T (lift O d t0) t0)))).(\lambda (t1: T).(\lambda (H0: ((\forall (d: nat).(eq T (lift O d t1) t1)))).(\lambda (d: nat).(eq_ind_r T (THead k (lift O d t0) (lift O (s k d) -t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_equal T (THead k t0 t1) -(THead k (lift O d t0) (lift O (s k d) t1)) (sym_equal T (THead k (lift O d -t0) (lift O (s k d) t1)) (THead k t0 t1) (sym_equal T (THead k t0 t1) (THead -k (lift O d t0) (lift O (s k d) t1)) (f_equal3 K T T T THead k k t0 (lift O d -t0) t1 (lift O (s k d) t1) (refl_equal K k) (sym_eq T (lift O d t0) t0 (H d)) +t1)) (\lambda (t2: T).(eq T t2 (THead k t0 t1))) (sym_eq T (THead k t0 t1) +(THead k (lift O d t0) (lift O (s k d) t1)) (sym_eq T (THead k (lift O d t0) +(lift O (s k d) t1)) (THead k t0 t1) (sym_eq T (THead k t0 t1) (THead k (lift +O d t0) (lift O (s k d) t1)) (f_equal3 K T T T THead k k t0 (lift O d t0) t1 +(lift O (s k d) t1) (refl_equal K k) (sym_eq T (lift O d t0) t0 (H d)) (sym_eq T (lift O (s k d) t1) t1 (H0 (s k d))))))) (lift O d (THead k t0 t1)) (lift_head k t0 t1 O d)))))))) t). @@ -135,17 +136,17 @@ h d (TLRef n)) (\lambda (t0: T).(eq T t0 (lift h d t))) H (TLRef (plus n h)) (lift_lref_ge n h d H0)) in (sym_eq T t (TLRef n) (lift_gen_lref_ge h d n H0 t H1)))))))))) (\lambda (k: K).(K_ind (\lambda (k0: K).(\forall (t: T).(((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) -(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t: -T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to -(eq T t0 t)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d: +(lift h d t0)) \to (eq T t t0)))))) \to (\forall (t0: T).(((\forall (t1: +T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) +\to (eq T t0 t1)))))) \to (\forall (t1: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d (THead k0 t t0)) (lift h d t1)) \to (eq T (THead k0 t t0) t1)))))))))) (\lambda (b: B).(\lambda (t: T).(\lambda (H: ((\forall (t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to -(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t: T).(\forall -(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to (eq T t0 -t))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: +(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall +(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0 +t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: (eq T (lift h d (THead (Bind b) t t0)) (lift h d t1))).(let H2 \def (eq_ind T -(lift h d (THead (Bind b) t t0)) (\lambda (t: T).(eq T t (lift h d t1))) H1 +(lift h d (THead (Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1 (THead (Bind b) (lift h d t) (lift h (S d) t0)) (lift_bind b t t0 h d)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) @@ -154,32 +155,31 @@ z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) (H3: (eq T t1 (THead (Bind b) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h (S d) t0) (lift h (S d) x1))).(eq_ind_r T (THead (Bind b) x0 x1) (\lambda (t2: T).(eq T (THead (Bind b) t t0) t2)) -(sym_equal T (THead (Bind b) x0 x1) (THead (Bind b) t t0) (sym_equal T (THead -(Bind b) t t0) (THead (Bind b) x0 x1) (sym_equal T (THead (Bind b) x0 x1) -(THead (Bind b) t t0) (f_equal3 K T T T THead (Bind b) (Bind b) x0 t x1 t0 -(refl_equal K (Bind b)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1 -h (S d) H5)))))) t1 H3)))))) (lift_gen_bind b (lift h d t) (lift h (S d) t0) -t1 h d H2)))))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H: ((\forall -(t0: T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d -t0)) \to (eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t: -T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t)) \to -(eq T t0 t))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: -nat).(\lambda (H1: (eq T (lift h d (THead (Flat f) t t0)) (lift h d -t1))).(let H2 \def (eq_ind T (lift h d (THead (Flat f) t t0)) (\lambda (t: -T).(eq T t (lift h d t1))) H1 (THead (Flat f) (lift h d t) (lift h d t0)) -(lift_flat f t t0 h d)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq -T t1 (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d -t) (lift h d y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h d t0) (lift -h d z)))) (eq T (THead (Flat f) t t0) t1) (\lambda (x0: T).(\lambda (x1: -T).(\lambda (H3: (eq T t1 (THead (Flat f) x0 x1))).(\lambda (H4: (eq T (lift -h d t) (lift h d x0))).(\lambda (H5: (eq T (lift h d t0) (lift h d -x1))).(eq_ind_r T (THead (Flat f) x0 x1) (\lambda (t2: T).(eq T (THead (Flat -f) t t0) t2)) (sym_equal T (THead (Flat f) x0 x1) (THead (Flat f) t t0) -(sym_equal T (THead (Flat f) t t0) (THead (Flat f) x0 x1) (sym_equal T (THead -(Flat f) x0 x1) (THead (Flat f) t t0) (f_equal3 K T T T THead (Flat f) (Flat -f) x0 t x1 t0 (refl_equal K (Flat f)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T -t0 x1 (H0 x1 h d H5)))))) t1 H3)))))) (lift_gen_flat f (lift h d t) (lift h d -t0) t1 h d H2)))))))))))) k)) x). +(sym_eq T (THead (Bind b) x0 x1) (THead (Bind b) t t0) (sym_eq T (THead (Bind +b) t t0) (THead (Bind b) x0 x1) (sym_eq T (THead (Bind b) x0 x1) (THead (Bind +b) t t0) (f_equal3 K T T T THead (Bind b) (Bind b) x0 t x1 t0 (refl_equal K +(Bind b)) (sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1 h (S d) +H5)))))) t1 H3)))))) (lift_gen_bind b (lift h d t) (lift h (S d) t0) t1 h d +H2)))))))))))) (\lambda (f: F).(\lambda (t: T).(\lambda (H: ((\forall (t0: +T).(\forall (h: nat).(\forall (d: nat).((eq T (lift h d t) (lift h d t0)) \to +(eq T t t0))))))).(\lambda (t0: T).(\lambda (H0: ((\forall (t1: T).(\forall +(h: nat).(\forall (d: nat).((eq T (lift h d t0) (lift h d t1)) \to (eq T t0 +t1))))))).(\lambda (t1: T).(\lambda (h: nat).(\lambda (d: nat).(\lambda (H1: +(eq T (lift h d (THead (Flat f) t t0)) (lift h d t1))).(let H2 \def (eq_ind T +(lift h d (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift h d t1))) H1 +(THead (Flat f) (lift h d t) (lift h d t0)) (lift_flat f t t0 h d)) in +(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T t1 (THead (Flat f) y +z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h d t) (lift h d y)))) +(\lambda (_: T).(\lambda (z: T).(eq T (lift h d t0) (lift h d z)))) (eq T +(THead (Flat f) t t0) t1) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H3: (eq +T t1 (THead (Flat f) x0 x1))).(\lambda (H4: (eq T (lift h d t) (lift h d +x0))).(\lambda (H5: (eq T (lift h d t0) (lift h d x1))).(eq_ind_r T (THead +(Flat f) x0 x1) (\lambda (t2: T).(eq T (THead (Flat f) t t0) t2)) (sym_eq T +(THead (Flat f) x0 x1) (THead (Flat f) t t0) (sym_eq T (THead (Flat f) t t0) +(THead (Flat f) x0 x1) (sym_eq T (THead (Flat f) x0 x1) (THead (Flat f) t t0) +(f_equal3 K T T T THead (Flat f) (Flat f) x0 t x1 t0 (refl_equal K (Flat f)) +(sym_eq T t x0 (H x0 h d H4)) (sym_eq T t0 x1 (H0 x1 h d H5)))))) t1 H3)))))) +(lift_gen_flat f (lift h d t) (lift h d t0) t1 h d H2)))))))))))) k)) x). theorem lift_gen_lift: \forall (t1: T).(\forall (x: T).(\forall (h1: nat).(\forall (h2: @@ -242,8 +242,8 @@ H3 (le_n h1))))) (eq_ind_r nat (plus (plus d2 h2) h1) (\lambda (n0: nat).(lt (plus_lt_compat_r n (plus d2 h2) h1 H4)) (plus (plus d2 h1) h2) (plus_permute_2_in_3 d2 h1 h2)) x H2 (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (TLRef n) (lift h2 d2 t2)))))) (\lambda (H4: -(le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus n h1) (\lambda (n: -nat).(eq T (TLRef n) (lift h2 (plus d2 h1) x))) H2 (plus (minus (plus n h1) +(le (plus d2 h2) n)).(let H5 \def (eq_ind nat (plus n h1) (\lambda (n0: +nat).(eq T (TLRef n0) (lift h2 (plus d2 h1) x))) H2 (plus (minus (plus n h1) h2) h2) (le_plus_minus_sym h2 (plus n h1) (le_plus_trans h2 n h1 (le_trans_plus_r d2 h2 n H4)))) in (eq_ind_r T (TLRef (minus (plus n h1) h2)) (\lambda (t: T).(ex2 T (\lambda (t2: T).(eq T t (lift h1 d1 t2))) (\lambda @@ -283,9 +283,9 @@ t0)) (lift h2 (plus d2 h1) x))).(K_ind (\lambda (k0: K).((eq T (lift h1 d1 x (lift h1 d1 t2))) (\lambda (t2: T).(eq T (THead k0 t t0) (lift h2 d2 t2)))))) (\lambda (b: B).(\lambda (H3: (eq T (lift h1 d1 (THead (Bind b) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind T (lift h1 d1 (THead -(Bind b) t t0)) (\lambda (t: T).(eq T t (lift h2 (plus d2 h1) x))) H3 (THead -(Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1)) in -(ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y +(Bind b) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus d2 h1) x))) H3 +(THead (Bind b) (lift h1 d1 t) (lift h1 (S d1) t0)) (lift_bind b t t0 h1 d1)) +in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Bind b) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift h2 (plus d2 h1) y)))) (\lambda (_: T).(\lambda (z: T).(eq T (lift h1 (S d1) t0) (lift h2 (S (plus d2 h1)) z)))) (ex2 T (\lambda (t2: T).(eq T x (lift h1 d1 t2))) @@ -333,7 +333,7 @@ H12)))) (H0 x1 h1 h2 (S d1) (S d2) (le_S_n (S d1) (S d2) (lt_le_S (S d1) (S x0 h1 h2 d1 d2 H1 H6)) x H5)))))) (lift_gen_bind b (lift h1 d1 t) (lift h1 (S d1) t0) x h2 (plus d2 h1) H4))))) (\lambda (f: F).(\lambda (H3: (eq T (lift h1 d1 (THead (Flat f) t t0)) (lift h2 (plus d2 h1) x))).(let H4 \def (eq_ind -T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t: T).(eq T t (lift h2 (plus +T (lift h1 d1 (THead (Flat f) t t0)) (\lambda (t2: T).(eq T t2 (lift h2 (plus d2 h1) x))) H3 (THead (Flat f) (lift h1 d1 t) (lift h1 d1 t0)) (lift_flat f t t0 h1 d1)) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x (THead (Flat f) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T (lift h1 d1 t) (lift @@ -413,12 +413,12 @@ nat T TLRef (plus (plus n h) k) (plus n (plus k h)) (lift_lref_ge n (plus k h) d H1)) (lift k e (TLRef (plus n h))) (lift_lref_ge (plus n h) k e (le_trans e (plus d h) (plus n h) H (plus_le_compat d n h h H1 (le_n h))))) (lift h d (TLRef n)) (lift_lref_ge n h d H1))))))))))) (\lambda -(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k: +(k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to -(eq T (lift k e (lift h d t0)) (lift (plus k h) d t0)))))))))).(\lambda (t1: -T).(\lambda (H0: ((\forall (h: nat).(\forall (k: nat).(\forall (d: -nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k e -(lift h d t1)) (lift (plus k h) d t1)))))))))).(\lambda (h: nat).(\lambda +(eq T (lift k0 e (lift h d t0)) (lift (plus k0 h) d t0)))))))))).(\lambda +(t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: +nat).(\forall (e: nat).((le e (plus d h)) \to ((le d e) \to (eq T (lift k0 e +(lift h d t1)) (lift (plus k0 h) d t1)))))))))).(\lambda (h: nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e (plus d h))).(\lambda (H2: (le d e)).(eq_ind_r T (THead k (lift h d t0) (lift h (s k d) t1)) (\lambda (t2: T).(eq T (lift k0 e t2) (lift (plus k0 h) d (THead k t0 @@ -489,13 +489,13 @@ n))))) (eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq T (TLRef (plus (plus n k)) (le_lt_n_Sm (plus d k) (plus n k) (plus_le_compat d n k k H1 (le_n k))))))))) (plus k d) (plus_comm k d)) (lift k e (TLRef n)) (lift_lref_ge n k e H0)))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda -(H: ((\forall (h: nat).(\forall (k: nat).(\forall (d: nat).(\forall (e: -nat).((le e d) \to (eq T (lift h (plus k d) (lift k e t0)) (lift k e (lift h -d t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall (k: -nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h (plus k -d) (lift k e t1)) (lift k e (lift h d t1)))))))))).(\lambda (h: nat).(\lambda -(k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le e -d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2: +(H: ((\forall (h: nat).(\forall (k0: nat).(\forall (d: nat).(\forall (e: +nat).((le e d) \to (eq T (lift h (plus k0 d) (lift k0 e t0)) (lift k0 e (lift +h d t0)))))))))).(\lambda (t1: T).(\lambda (H0: ((\forall (h: nat).(\forall +(k0: nat).(\forall (d: nat).(\forall (e: nat).((le e d) \to (eq T (lift h +(plus k0 d) (lift k0 e t1)) (lift k0 e (lift h d t1)))))))))).(\lambda (h: +nat).(\lambda (k0: nat).(\lambda (d: nat).(\lambda (e: nat).(\lambda (H1: (le +e d)).(eq_ind_r T (THead k (lift k0 e t0) (lift k0 (s k e) t1)) (\lambda (t2: T).(eq T (lift h (plus k0 d) t2) (lift k0 e (lift h d (THead k t0 t1))))) (eq_ind_r T (THead k (lift h (plus k0 d) (lift k0 e t0)) (lift h (s k (plus k0 d)) (lift k0 (s k e) t1))) (\lambda (t2: T).(eq T t2 (lift k0 e (lift h d diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/tlt.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/tlt.ma index b67722f58..f03f26618 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/tlt.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/lift/tlt.ma @@ -141,14 +141,14 @@ d) \to (eq nat (g m) (f m)))))).(\lambda (_: (eq nat (g d) w)).(\lambda (H1: (TLRef n)))) (\lambda (H2: (lt n d)).(eq_ind_r T (TLRef n) (\lambda (t0: T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) (eq_ind_r T (TLRef n) (\lambda (t0: T).(eq nat (weight_map f (TLRef n)) -(weight_map g t0))) (sym_equal nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef +(weight_map g t0))) (sym_eq nat (g n) (f n) (H n H2)) (lift (S h) d (TLRef n)) (lift_lref_lt n (S h) d H2)) (lift h d (TLRef n)) (lift_lref_lt n h d H2))) (\lambda (H2: (le d n)).(eq_ind_r T (TLRef (plus n h)) (\lambda (t0: T).(eq nat (weight_map f t0) (weight_map g (lift (S h) d (TLRef n))))) (eq_ind_r T (TLRef (plus n (S h))) (\lambda (t0: T).(eq nat (weight_map f (TLRef (plus n h))) (weight_map g t0))) (eq_ind nat (S (plus n h)) (\lambda -(n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_equal nat (g (S (plus n h))) -(f (plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h)) +(n0: nat).(eq nat (f (plus n h)) (g n0))) (sym_eq nat (g (S (plus n h))) (f +(plus n h)) (H1 (plus n h) (le_plus_trans d n h H2))) (plus n (S h)) (plus_n_Sm n h)) (lift (S h) d (TLRef n)) (lift_lref_ge n (S h) d H2)) (lift h d (TLRef n)) (lift_lref_ge n h d H2)))))))))))) (\lambda (k: K).(\lambda (t0: T).(\lambda (H: ((\forall (h: nat).(\forall (d: nat).(\forall (f: ((nat @@ -195,32 +195,51 @@ d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift (S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (f_equal nat -nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0)) -(sym_equal nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) -(H h d f g H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq -nat m (S m0))) (\lambda (m: nat).(lt m d)))).(ex2_ind nat (\lambda (m0: -nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S -(weight_map g (lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0))) -m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x -d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g -(lift (S h) d t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x -H7) m H6)))) H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: -(le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: -nat).(le d n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m)) -(\lambda (x: nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d -x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g n) (wadd f (S -(weight_map f (lift h d t0))) n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) -(f_equal nat nat S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) -(lift h (S d) t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd -g O) (lift (S h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h -d t0)) (weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) -t1)) (weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h -(S d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S +nat S (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t0)) (sym_eq +nat (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) (H h d f g +H1 H2 H3))) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S +m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat +m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g (S (weight_map g +(lift (S h) d t0))) m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda +(x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r +nat (S x) (\lambda (n: nat).(eq nat (wadd g (S (weight_map g (lift (S h) d +t0))) n) (wadd f (S (weight_map f (lift h d t0))) n))) (H1 x H7) m H6)))) +H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) +m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d +n)) (eq nat (g m) (wadd f (S (weight_map f (lift h d t0))) m)) (\lambda (x: +nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S +x) (\lambda (n: nat).(eq nat (g n) (wadd f (S (weight_map f (lift h d t0))) +n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat S (plus +(weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) t1))) (plus +(weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S h) (S d) +t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) (weight_map g +(lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) (weight_map +(wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O) +(wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O) +(ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d))) +(eq nat (wadd g O m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat +O (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m +H5)) (\lambda (H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda +(m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0))) +(\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x: +nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S +x) (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6)))) +H5)) (lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) +m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d +n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S +x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g +n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) (f_equal nat nat +S (plus (weight_map f (lift h d t0)) (weight_map (wadd f O) (lift h (S d) +t1))) (plus (weight_map g (lift (S h) d t0)) (weight_map (wadd g O) (lift (S +h) (S d) t1))) (f_equal2 nat nat nat plus (weight_map f (lift h d t0)) +(weight_map g (lift (S h) d t0)) (weight_map (wadd f O) (lift h (S d) t1)) +(weight_map (wadd g O) (lift (S h) (S d) t1)) (H h d f g H1 H2 H3) (H0 h (S +d) (wadd f O) (wadd g O) (\lambda (m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O) (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m H5)) (\lambda (H5: (ex2 nat (\lambda (m0: -nat).(eq nat m (S m0))) (\lambda (m: nat).(lt m d)))).(ex2_ind nat (\lambda +nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x: nat).(\lambda (H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (wadd g O n) @@ -229,40 +248,21 @@ nat).(\lambda (H4: (le (S d) m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g n) (wadd f O n))) (H3 x H6) m H5)))) -(le_gen_S d m H4))))))) (f_equal nat nat S (plus (weight_map f (lift h d t0)) -(weight_map (wadd f O) (lift h (S d) t1))) (plus (weight_map g (lift (S h) d -t0)) (weight_map (wadd g O) (lift (S h) (S d) t1))) (f_equal2 nat nat nat -plus (weight_map f (lift h d t0)) (weight_map g (lift (S h) d t0)) -(weight_map (wadd f O) (lift h (S d) t1)) (weight_map (wadd g O) (lift (S h) -(S d) t1)) (H h d f g H1 H2 H3) (H0 h (S d) (wadd f O) (wadd g O) (\lambda -(m: nat).(\lambda (H4: (lt m (S d))).(or_ind (eq nat m O) (ex2 nat (\lambda -(m0: nat).(eq nat m (S m0))) (\lambda (m0: nat).(lt m0 d))) (eq nat (wadd g O -m) (wadd f O m)) (\lambda (H5: (eq nat m O)).(eq_ind_r nat O (\lambda (n: -nat).(eq nat (wadd g O n) (wadd f O n))) (refl_equal nat O) m H5)) (\lambda -(H5: (ex2 nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m: nat).(lt m -d)))).(ex2_ind nat (\lambda (m0: nat).(eq nat m (S m0))) (\lambda (m0: -nat).(lt m0 d)) (eq nat (wadd g O m) (wadd f O m)) (\lambda (x: nat).(\lambda -(H6: (eq nat m (S x))).(\lambda (H7: (lt x d)).(eq_ind_r nat (S x) (\lambda -(n: nat).(eq nat (wadd g O n) (wadd f O n))) (H1 x H7) m H6)))) H5)) -(lt_gen_xS m d H4)))) H2 (\lambda (m: nat).(\lambda (H4: (le (S d) -m)).(ex2_ind nat (\lambda (n: nat).(eq nat m (S n))) (\lambda (n: nat).(le d -n)) (eq nat (g m) (wadd f O m)) (\lambda (x: nat).(\lambda (H5: (eq nat m (S -x))).(\lambda (H6: (le d x)).(eq_ind_r nat (S x) (\lambda (n: nat).(eq nat (g -n) (wadd f O n))) (H3 x H6) m H5)))) (le_gen_S d m H4))))))) b) (lift (S h) d -(THead (Bind b) t0 t1)) (lift_head (Bind b) t0 t1 (S h) d)) (lift h d (THead -(Bind b) t0 t1)) (lift_head (Bind b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r -T (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0) d) t1)) (\lambda (t2: -T).(eq nat (weight_map f t2) (weight_map g (lift (S h) d (THead (Flat f0) t0 -t1))))) (eq_ind_r T (THead (Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat -f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f (THead (Flat f0) (lift h d -t0) (lift h (s (Flat f0) d) t1))) (weight_map g t2))) (f_equal nat nat S -(plus (weight_map f (lift h d t0)) (weight_map f (lift h d t1))) (plus -(weight_map g (lift (S h) d t0)) (weight_map g (lift (S h) d t1))) (f_equal2 -nat nat nat plus (weight_map f (lift h d t0)) (weight_map g (lift (S h) d -t0)) (weight_map f (lift h d t1)) (weight_map g (lift (S h) d t1)) (H h d f g -H1 H2 H3) (H0 h d f g H1 H2 H3))) (lift (S h) d (THead (Flat f0) t0 t1)) -(lift_head (Flat f0) t0 t1 (S h) d)) (lift h d (THead (Flat f0) t0 t1)) -(lift_head (Flat f0) t0 t1 h d))) k))))))))))))) t)). +(le_gen_S d m H4))))))) b) (lift (S h) d (THead (Bind b) t0 t1)) (lift_head +(Bind b) t0 t1 (S h) d)) (lift h d (THead (Bind b) t0 t1)) (lift_head (Bind +b) t0 t1 h d))) (\lambda (f0: F).(eq_ind_r T (THead (Flat f0) (lift h d t0) +(lift h (s (Flat f0) d) t1)) (\lambda (t2: T).(eq nat (weight_map f t2) +(weight_map g (lift (S h) d (THead (Flat f0) t0 t1))))) (eq_ind_r T (THead +(Flat f0) (lift (S h) d t0) (lift (S h) (s (Flat f0) d) t1)) (\lambda (t2: +T).(eq nat (weight_map f (THead (Flat f0) (lift h d t0) (lift h (s (Flat f0) +d) t1))) (weight_map g t2))) (f_equal nat nat S (plus (weight_map f (lift h d +t0)) (weight_map f (lift h d t1))) (plus (weight_map g (lift (S h) d t0)) +(weight_map g (lift (S h) d t1))) (f_equal2 nat nat nat plus (weight_map f +(lift h d t0)) (weight_map g (lift (S h) d t0)) (weight_map f (lift h d t1)) +(weight_map g (lift (S h) d t1)) (H h d f g H1 H2 H3) (H0 h d f g H1 H2 H3))) +(lift (S h) d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 (S h) d)) +(lift h d (THead (Flat f0) t0 t1)) (lift_head (Flat f0) t0 t1 h d))) +k))))))))))))) t)). theorem lift_weight_add_O: \forall (w: nat).(\forall (t: T).(\forall (h: nat).(\forall (f: ((nat \to diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec.ma index 761030b2a..377c4ea39 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pc3/dec.ma @@ -145,7 +145,7 @@ t0)) \to (\forall (P0: Prop).P0)))) H10 (THead (Bind Abst) x5 x6) H24) in x6) H24) in (let H29 \def (nf2_gen_abst c x5 x6 H28) in (and_ind (nf2 c x5) (nf2 (CHead c (Bind Abst) x5) x6) P (\lambda (H30: (nf2 c x5)).(\lambda (_: (nf2 (CHead c (Bind Abst) x5) x6)).(let H32 \def (nf2_pr3_confluence c x0 H8 -x5 H30 u2 H7) in (H27 x6 (sym_equal T (THead (Bind Abst) x0 x6) (THead (Bind +x5 H30 u2 H7) in (H27 x6 (sym_eq T (THead (Bind Abst) x0 x6) (THead (Bind Abst) x5 x6) (f_equal3 K T T T THead (Bind Abst) (Bind Abst) x0 x5 x6 x6 (refl_equal K (Bind Abst)) (H32 (pr3_t x3 u2 c H20 x5 H25)) (refl_equal T x6))) P)))) H29))))))))) H23)))))))) H18))))))) (pr3_confluence c u1 x1 H13 x diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/subst1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/subst1.ma index 012d5b0ee..db6cf7df6 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/subst1.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/pr2/subst1.ma @@ -20,11 +20,15 @@ include "pr2/defs.ma". include "pr0/subst1.ma". -include "csubst1/defs.ma". +include "pr0/fwd.ma". + +include "csubst1/getl.ma". + +include "csubst1/fwd.ma". include "subst1/subst1.ma". -include "getl/props.ma". +include "getl/drop.ma". theorem pr2_delta1: \forall (c: C).(\forall (d: C).(\forall (u: T).(\forall (i: nat).((getl i c @@ -121,12 +125,166 @@ H10 v w1 i H12 v (pr0_refl v)))))))) t (sym_eq T t t2 H8))) t0 (sym_eq T t0 t1 H7))) c0 (sym_eq C c0 c H4) H5 H6 H1 H2 H3))))]) in (H1 (refl_equal C c) (refl_equal T t1) (refl_equal T t2)))))))))). -axiom pr2_gen_cabbr: +theorem pr2_gen_cabbr: \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall (e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) \to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T (\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)))))))))))))))) -. +\def + \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 +t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (e: +C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to +(\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 +a) \to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda +(x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 +x2)))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: +T).(\lambda (H0: (pr0 t3 t4)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: +nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: +C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) +d a0 a)).(\lambda (x1: T).(\lambda (H4: (subst1 d u t3 (lift (S O) d +x1))).(ex2_ind T (\lambda (w2: T).(pr0 (lift (S O) d x1) w2)) (\lambda (w2: +T).(subst1 d u t4 w2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d +x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H5: (pr0 +(lift (S O) d x1) x)).(\lambda (H6: (subst1 d u t4 x)).(ex2_ind T (\lambda +(t5: T).(eq T x (lift (S O) d t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T +(\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a +x1 x2))) (\lambda (x0: T).(\lambda (H7: (eq T x (lift (S O) d x0))).(\lambda +(H8: (pr0 x1 x0)).(let H9 \def (eq_ind T x (\lambda (t: T).(subst1 d u t4 t)) +H6 (lift (S O) d x0) H7) in (ex_intro2 T (\lambda (x2: T).(subst1 d u t4 +(lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)) x0 H9 (pr2_free a x1 x0 +H8)))))) (pr0_gen_lift x1 x (S O) d H5))))) (pr0_subst1 t3 t4 H0 u (lift (S +O) d x1) d H4 u (pr0_refl u))))))))))))))))) (\lambda (c0: C).(\lambda (d: +C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind +Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 +t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (e: +C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e +(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 +a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(\lambda (x1: +T).(\lambda (H6: (subst1 d0 u0 t3 (lift (S O) d0 x1))).(ex2_ind T (\lambda +(w2: T).(pr0 (lift (S O) d0 x1) w2)) (\lambda (w2: T).(subst1 d0 u0 t4 w2)) +(ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: +T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H7: (pr0 (lift (S O) d0 x1) +x)).(\lambda (H8: (subst1 d0 u0 t4 x)).(ex2_ind T (\lambda (t5: T).(eq T x +(lift (S O) d0 t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T (\lambda (x2: +T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) +(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) d0 x0))).(\lambda (H10: +(pr0 x1 x0)).(let H11 \def (eq_ind T x (\lambda (t0: T).(subst1 d0 u0 t4 t0)) +H8 (lift (S O) d0 x0) H9) in (lt_eq_gt_e i d0 (ex2 T (\lambda (x2: T).(subst1 +d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (H12: +(lt i d0)).(ex2_ind T (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: +T).(subst1 i u (lift (S O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 +t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: +T).(\lambda (H13: (subst1 d0 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) +d0 x0) x2)).(ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 i) u0 (CHead d +(Bind Abbr) u) e2)) (\lambda (e2: C).(getl i a0 e2)) (ex2 T (\lambda (x3: +T).(subst1 d0 u0 t (lift (S O) d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) +(\lambda (x3: C).(\lambda (H15: (csubst1 (minus d0 i) u0 (CHead d (Bind Abbr) +u) x3)).(\lambda (H16: (getl i a0 x3)).(let H17 \def (eq_ind nat (minus d0 i) +(\lambda (n: nat).(csubst1 n u0 (CHead d (Bind Abbr) u) x3)) H15 (S (minus d0 +(S i))) (minus_x_Sy d0 i H12)) in (let H18 \def (csubst1_gen_head (Bind Abbr) +d x3 u u0 (minus d0 (S i)) H17) in (ex3_2_ind T C (\lambda (u2: T).(\lambda +(c2: C).(eq C x3 (CHead c2 (Bind Abbr) u2)))) (\lambda (u2: T).(\lambda (_: +C).(subst1 (minus d0 (S i)) u0 u u2))) (\lambda (_: T).(\lambda (c2: +C).(csubst1 (minus d0 (S i)) u0 d c2))) (ex2 T (\lambda (x4: T).(subst1 d0 u0 +t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4))) (\lambda (x4: +T).(\lambda (x5: C).(\lambda (H19: (eq C x3 (CHead x5 (Bind Abbr) +x4))).(\lambda (H20: (subst1 (minus d0 (S i)) u0 u x4)).(\lambda (_: (csubst1 +(minus d0 (S i)) u0 d x5)).(let H22 \def (eq_ind C x3 (\lambda (c1: C).(getl +i a0 c1)) H16 (CHead x5 (Bind Abbr) x4) H19) in (let H23 \def (eq_ind nat d0 +(\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus i (minus d0 (S i)))) +(lt_plus_minus i d0 H12)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: +C).(eq T x4 (lift (S O) (minus d0 (S i)) v)))) (\lambda (v: T).(\lambda (e0: +C).(getl i a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: +C).(drop (S O) (minus d0 (S i)) x5 e0))) (ex2 T (\lambda (x6: T).(subst1 d0 +u0 t (lift (S O) d0 x6))) (\lambda (x6: T).(pr2 a x1 x6))) (\lambda (x6: +T).(\lambda (x7: C).(\lambda (H24: (eq T x4 (lift (S O) (minus d0 (S i)) +x6))).(\lambda (H25: (getl i a (CHead x7 (Bind Abbr) x6))).(\lambda (_: (drop +(S O) (minus d0 (S i)) x5 x7)).(let H27 \def (eq_ind T x4 (\lambda (t0: +T).(subst1 (minus d0 (S i)) u0 u t0)) H20 (lift (S O) (minus d0 (S i)) x6) +H24) in (ex2_ind T (\lambda (t0: T).(subst1 i (lift (S O) (minus d0 (S i)) +x6) (lift (S O) d0 x0) t0)) (\lambda (t0: T).(subst1 (S (plus (minus d0 (S +i)) i)) u0 x2 t0)) (ex2 T (\lambda (x8: T).(subst1 d0 u0 t (lift (S O) d0 +x8))) (\lambda (x8: T).(pr2 a x1 x8))) (\lambda (x8: T).(\lambda (H28: +(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S O) d0 x0) x8)).(\lambda +(H29: (subst1 (S (plus (minus d0 (S i)) i)) u0 x2 x8)).(let H30 \def (eq_ind +nat d0 (\lambda (n: nat).(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S +O) n x0) x8)) H28 (S (plus i (minus d0 (S i)))) (lt_plus_minus i d0 H12)) in +(ex2_ind T (\lambda (t5: T).(eq T x8 (lift (S O) (S (plus i (minus d0 (S +i)))) t5))) (\lambda (t5: T).(subst1 i x6 x0 t5)) (ex2 T (\lambda (x9: +T).(subst1 d0 u0 t (lift (S O) d0 x9))) (\lambda (x9: T).(pr2 a x1 x9))) +(\lambda (x9: T).(\lambda (H31: (eq T x8 (lift (S O) (S (plus i (minus d0 (S +i)))) x9))).(\lambda (H32: (subst1 i x6 x0 x9)).(let H33 \def (eq_ind T x8 +(\lambda (t0: T).(subst1 (S (plus (minus d0 (S i)) i)) u0 x2 t0)) H29 (lift +(S O) (S (plus i (minus d0 (S i)))) x9) H31) in (let H34 \def (eq_ind_r nat +(S (plus i (minus d0 (S i)))) (\lambda (n: nat).(subst1 (S (plus (minus d0 (S +i)) i)) u0 x2 (lift (S O) n x9))) H33 d0 (lt_plus_minus i d0 H12)) in (let +H35 \def (eq_ind_r nat (S (plus (minus d0 (S i)) i)) (\lambda (n: +nat).(subst1 n u0 x2 (lift (S O) d0 x9))) H34 d0 (lt_plus_minus_r i d0 H12)) +in (ex_intro2 T (\lambda (x10: T).(subst1 d0 u0 t (lift (S O) d0 x10))) +(\lambda (x10: T).(pr2 a x1 x10)) x9 (subst1_trans x2 t u0 d0 H13 (lift (S O) +d0 x9) H35) (pr2_delta1 a x7 x6 i H25 x1 x0 H10 x9 H32)))))))) +(subst1_gen_lift_lt x6 x0 x8 i (S O) (minus d0 (S i)) H30)))))) +(subst1_subst1_back (lift (S O) d0 x0) x2 u i H14 (lift (S O) (minus d0 (S +i)) x6) u0 (minus d0 (S i)) H27)))))))) (getl_drop_conf_lt Abbr a0 x5 x4 i +H22 a (S O) (minus d0 (S i)) H23))))))))) H18)))))) (csubst1_getl_lt d0 i H12 +c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0))))) (subst1_confluence_neq t4 t u i +(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (lt_neq i d0 H12)))) +(\lambda (H12: (eq nat i d0)).(let H13 \def (eq_ind_r nat d0 (\lambda (n: +nat).(subst1 n u0 t4 (lift (S O) n x0))) H11 i H12) in (let H14 \def +(eq_ind_r nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 i H12) in (let H15 +\def (eq_ind_r nat d0 (\lambda (n: nat).(csubst1 n u0 c0 a0)) H4 i H12) in +(let H16 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl n c0 (CHead e (Bind +Abbr) u0))) H3 i H12) in (eq_ind nat i (\lambda (n: nat).(ex2 T (\lambda (x2: +T).(subst1 n u0 t (lift (S O) n x2))) (\lambda (x2: T).(pr2 a x1 x2)))) (let +H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) +H0 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead +e (Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match +e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ +_) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) +(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in +((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda +(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) +(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind +Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d +e)).(let H21 \def (eq_ind_r T u0 (\lambda (t0: T).(getl i c0 (CHead e (Bind +Abbr) t0))) H17 u H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t0: +T).(subst1 i t0 t4 (lift (S O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r +T u0 (\lambda (t0: T).(csubst1 i t0 c0 a0)) H15 u H19) in (eq_ind T u +(\lambda (t0: T).(ex2 T (\lambda (x2: T).(subst1 i t0 t (lift (S O) i x2))) +(\lambda (x2: T).(pr2 a x1 x2)))) (let H24 \def (eq_ind_r C e (\lambda (c1: +C).(getl i c0 (CHead c1 (Bind Abbr) u))) H21 d H20) in (ex2_ind T (\lambda +(t0: T).(subst1 i u t t0)) (\lambda (t0: T).(subst1 i u (lift (S O) i x0) +t0)) (ex2 T (\lambda (x2: T).(subst1 i u t (lift (S O) i x2))) (\lambda (x2: +T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H25: (subst1 i u t +x2)).(\lambda (H26: (subst1 i u (lift (S O) i x0) x2)).(let H27 \def (eq_ind +T x2 (\lambda (t0: T).(subst1 i u t t0)) H25 (lift (S O) i x0) +(subst1_gen_lift_eq x0 u x2 (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) +(\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i +(S O))) H26)) in (ex_intro2 T (\lambda (x3: T).(subst1 i u t (lift (S O) i +x3))) (\lambda (x3: T).(pr2 a x1 x3)) x0 H27 (pr2_free a x1 x0 H10)))))) +(subst1_confluence_eq t4 t u i (subst1_single i u t4 t H2) (lift (S O) i x0) +H22))) u0 H19)))))) H18))) d0 H12)))))) (\lambda (H12: (lt d0 i)).(ex2_ind T +(\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: T).(subst1 i u (lift (S +O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) +(\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H13: (subst1 d0 +u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) d0 x0) x2)).(ex2_ind T +(\lambda (t5: T).(eq T x2 (lift (S O) d0 t5))) (\lambda (t5: T).(subst1 +(minus i (S O)) u x0 t5)) (ex2 T (\lambda (x3: T).(subst1 d0 u0 t (lift (S O) +d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) (\lambda (x3: T).(\lambda (H15: (eq +T x2 (lift (S O) d0 x3))).(\lambda (H16: (subst1 (minus i (S O)) u x0 +x3)).(let H17 \def (eq_ind T x2 (\lambda (t0: T).(subst1 d0 u0 t t0)) H13 +(lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda (x4: T).(subst1 d0 u0 t +(lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4)) x3 H17 (pr2_delta1 a d u +(minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0 +(csubst1_getl_ge d0 i (le_S_n d0 i (le_S (S d0) i H12)) c0 a0 u0 H4 (CHead d +(Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n: +nat).(le n i)) H12 (plus d0 (S O)) (plus_comm d0 (S O)))) x1 x0 H10 x3 +H16)))))) (subst1_gen_lift_ge u x0 x2 i (S O) d0 H14 (eq_ind_r nat (plus (S +O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_comm d0 (S +O)))))))) (subst1_confluence_neq t4 t u i (subst1_single i u t4 t H2) (lift +(S O) d0 x0) u0 d0 H11 (sym_not_eq nat d0 i (lt_neq d0 i H12)))))))))) +(pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 t4 H1 u0 (lift (S O) d0 +x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 t2 H)))). diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/subst0.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/subst0.ma index c09ce6c95..8e1e4e5b9 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/subst0.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst0/subst0.ma @@ -38,8 +38,8 @@ T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u t u i0 (S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) (plus i0 (S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 -u0)).(\lambda (H1: ((\forall (u2: T).(\forall (u: T).(\forall (i0: -nat).((subst0 i0 u u2 v) \to (ex2 T (\lambda (t: T).(subst0 i u2 u1 t)) +u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0: +nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u u3 v)).(ex2_ind T (\lambda (t0: T).(subst0 i @@ -68,13 +68,13 @@ u0 x t0)) H5 (s k (S (plus i0 i))) (s_S k (plus i0 i))) in (ex_intro2 T (plus i0 i)) u0 t (THead k u t0))) (THead k u x) (subst0_snd k u1 x t3 i H3 u) (subst0_snd k u0 t0 x (S (plus i0 i)) H6 u))))))) (H1 u1 u0 i0 H2)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda -(i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: ((\forall (u2: -T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u2 v) \to (ex2 T (\lambda -(t: T).(subst0 i u2 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t +(i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: ((\forall (u3: +T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda +(t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u t u0))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: -(subst0 (s k i) v t0 t3)).(\lambda (H3: ((\forall (u1: T).(\forall (u: -T).(\forall (i0: nat).((subst0 i0 u u1 v) \to (ex2 T (\lambda (t: T).(subst0 -(s k i) u1 t0 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t +(subst0 (s k i) v t0 t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: +T).(\forall (i0: nat).((subst0 i0 u u3 v) \to (ex2 T (\lambda (t: T).(subst0 +(s k i) u3 t0 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t t3))))))))).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: (subst0 i0 u u3 v)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t t3)) (ex2 T (\lambda (t: @@ -114,8 +114,8 @@ T).(subst0 i u1 (TLRef i) t)) (\lambda (t: T).(subst0 (plus i0 (S i)) u (lift (S i) H0 O (le_O_n i0))) (S (plus i0 i)) (sym_eq nat (S (plus i0 i)) (plus i0 (S i)) (plus_n_Sm i0 i))))))))) (\lambda (v: T).(\lambda (u0: T).(\lambda (u1: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 u0)).(\lambda (H1: -((\forall (u2: T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u v u2) \to -(ex2 T (\lambda (t: T).(subst0 i u2 u1 t)) (\lambda (t: T).(subst0 (S (plus +((\forall (u3: T).(\forall (u: T).(\forall (i0: nat).((subst0 i0 u v u3) \to +(ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t))))))))).(\lambda (t: T).(\lambda (k: K).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H2: (subst0 i0 u v u3)).(ex2_ind T (\lambda (t0: T).(subst0 i u3 u1 t0)) (\lambda (t0: @@ -144,12 +144,12 @@ t3) t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u0 (THead k u t0) t)) (THead k u x) (subst0_snd k u1 x t3 i H3 u) (subst0_snd k u0 x t0 (S (plus i0 i)) H6 u))))))) (H1 u1 u0 i0 H2)))))))))))))) (\lambda (v: T).(\lambda (u1: T).(\lambda (u0: T).(\lambda (i: nat).(\lambda (_: (subst0 i v u1 -u0)).(\lambda (H1: ((\forall (u2: T).(\forall (u: T).(\forall (i0: -nat).((subst0 i0 u v u2) \to (ex2 T (\lambda (t: T).(subst0 i u2 u1 t)) +u0)).(\lambda (H1: ((\forall (u3: T).(\forall (u: T).(\forall (i0: +nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 i u3 u1 t)) (\lambda (t: T).(subst0 (S (plus i0 i)) u u0 t))))))))).(\lambda (k: K).(\lambda (t0: T).(\lambda (t3: T).(\lambda (_: (subst0 (s k i) v t0 -t3)).(\lambda (H3: ((\forall (u1: T).(\forall (u: T).(\forall (i0: -nat).((subst0 i0 u v u1) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u1 t0 t)) +t3)).(\lambda (H3: ((\forall (u3: T).(\forall (u: T).(\forall (i0: +nat).((subst0 i0 u v u3) \to (ex2 T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) (\lambda (t: T).(subst0 (S (plus i0 (s k i))) u t3 t))))))))).(\lambda (u3: T).(\lambda (u: T).(\lambda (i0: nat).(\lambda (H4: (subst0 i0 u v u3)).(ex2_ind T (\lambda (t: T).(subst0 (s k i) u3 t0 t)) (\lambda (t: @@ -191,21 +191,21 @@ u2 t) t3)).(or3_ind (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) (\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s k i0) v0 t t4)))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda -(H3: (ex2 T (\lambda (u2: T).(eq T t3 (THead k u2 t))) (\lambda (u3: +(H3: (ex2 T (\lambda (u3: T).(eq T t3 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t3 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead k u1 t) t3) (\lambda (x: T).(\lambda (H4: (eq T t3 (THead k x t))).(\lambda (H5: (subst0 i0 v0 u2 x)).(eq_ind_r T (THead k x t) (\lambda (t0: T).(subst0 i0 v0 (THead k u1 t) t0)) (subst0_fst v0 x u1 i0 (H1 x H5) t k) t3 H4)))) H3)) (\lambda -(H3: (ex2 T (\lambda (t2: T).(eq T t3 (THead k u2 t2))) (\lambda (t2: -T).(subst0 (s k i0) v0 t t2)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead k +(H3: (ex2 T (\lambda (t4: T).(eq T t3 (THead k u2 t4))) (\lambda (t4: +T).(subst0 (s k i0) v0 t t4)))).(ex2_ind T (\lambda (t4: T).(eq T t3 (THead k u2 t4))) (\lambda (t4: T).(subst0 (s k i0) v0 t t4)) (subst0 i0 v0 (THead k u1 t) t3) (\lambda (x: T).(\lambda (H4: (eq T t3 (THead k u2 x))).(\lambda (H5: (subst0 (s k i0) v0 t x)).(eq_ind_r T (THead k u2 x) (\lambda (t0: T).(subst0 i0 v0 (THead k u1 t) t0)) (subst0_both v0 u1 u2 i0 H0 k t x H5) t3 -H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T -t3 (THead k u2 t2)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) -(\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 t t2))))).(ex3_2_ind T T +H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t4: T).(eq T +t3 (THead k u3 t4)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) +(\lambda (_: T).(\lambda (t4: T).(subst0 (s k i0) v0 t t4))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t4: T).(eq T t3 (THead k u3 t4)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t4: T).(subst0 (s k i0) v0 t t4))) (subst0 i0 v0 (THead k u1 t) t3) (\lambda (x0: @@ -229,16 +229,16 @@ T t4 (THead k u2 t0))) (\lambda (u2: T).(subst0 i0 v0 u u2)) (subst0 i0 v0 (THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 (THead k x t0))).(\lambda (H5: (subst0 i0 v0 u x)).(eq_ind_r T (THead k x t0) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_both v0 u x i0 H5 k t3 t0 H0) -t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t2: T).(eq T t4 (THead k u -t2))) (\lambda (t2: T).(subst0 (s k i0) v0 t0 t2)))).(ex2_ind T (\lambda (t5: +t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u +t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t0 t5)) (subst0 i0 v0 (THead k u t3) t4) (\lambda (x: T).(\lambda (H4: (eq T t4 (THead k u x))).(\lambda (H5: (subst0 (s k i0) v0 t0 x)).(eq_ind_r T (THead k u x) (\lambda (t: T).(subst0 i0 v0 (THead k u t3) t)) (subst0_snd k v0 x t3 i0 (H1 x H5) u) t4 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u2: -T).(\lambda (t2: T).(eq T t4 (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: -T).(subst0 i0 v0 u u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i0) v0 -t0 t2))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k +T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: +T).(subst0 i0 v0 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 +t0 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v0 u u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t0 t5))) (subst0 i0 v0 (THead k u t3) t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda (H4: (eq T t4 (THead k x0 @@ -257,22 +257,22 @@ v0 t3 t4) \to (subst0 (s k i0) v0 t0 t4))))).(\lambda (t4: T).(\lambda (H4: t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))) (subst0 i0 v0 (THead k u1 -t0) t4) (\lambda (H5: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 t3))) +t0) t4) (\lambda (H5: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) (\lambda (u3: T).(subst0 i0 v0 u2 u3)) (subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k x t3))).(\lambda (H7: (subst0 i0 v0 u2 x)).(eq_ind_r T (THead k x t3) (\lambda (t: T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 x i0 (H1 x H7) k t0 t3 H2) t4 -H6)))) H5)) (\lambda (H5: (ex2 T (\lambda (t2: T).(eq T t4 (THead k u2 t2))) -(\lambda (t2: T).(subst0 (s k i0) v0 t3 t2)))).(ex2_ind T (\lambda (t5: +H6)))) H5)) (\lambda (H5: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u2 t5))) +(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u2 t5))) (\lambda (t5: T).(subst0 (s k i0) v0 t3 t5)) (subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x: T).(\lambda (H6: (eq T t4 (THead k u2 x))).(\lambda (H7: (subst0 (s k i0) v0 t3 x)).(eq_ind_r T (THead k u2 x) (\lambda (t: T).(subst0 i0 v0 (THead k u1 t0) t)) (subst0_both v0 u1 u2 i0 H0 k t0 x (H3 x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda -(u2: T).(\lambda (t2: T).(eq T t4 (THead k u2 t2)))) (\lambda (u3: -T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t2: -T).(subst0 (s k i0) v0 t3 t2))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda +(u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: +T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: +T).(subst0 (s k i0) v0 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v0 u2 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v0 t3 t5))) (subst0 i0 v0 (THead k u1 t0) t4) (\lambda (x0: T).(\lambda (x1: T).(\lambda @@ -318,8 +318,8 @@ nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq T t2 (THead k u4 t))) T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3)))) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (H4: (ex2 T -(\lambda (u2: T).(eq T t2 (THead k u2 t))) (\lambda (u2: T).(subst0 i2 u3 u0 -u2)))).(ex2_ind T (\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: +(\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: T).(subst0 i2 u3 u0 +u4)))).(ex2_ind T (\lambda (u4: T).(eq T t2 (THead k u4 t))) (\lambda (u4: T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq T t2 (THead k x t))).(\lambda (H6: (subst0 i2 u3 u0 x)).(eq_ind_r T (THead k @@ -332,7 +332,7 @@ k x t) t3))) (\lambda (x0: T).(\lambda (H7: (subst0 i2 u3 u2 x0)).(\lambda u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k x t) t3)) (THead k x0 t) (subst0_fst u3 x0 u2 i2 H7 t k) (subst0_fst v x0 x i H8 t k))))) (H1 x u3 i2 H6 H3)) t2 H5)))) H4)) (\lambda (H4: (ex2 T (\lambda (t3: T).(eq T t2 (THead -k u0 t3))) (\lambda (t2: T).(subst0 (s k i2) u3 t t2)))).(ex2_ind T (\lambda +k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u0 t3))) (\lambda (t3: T).(subst0 (s k i2) u3 t t3)) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) (\lambda (x: T).(\lambda (H5: (eq T t2 (THead k u0 @@ -341,10 +341,10 @@ x))).(\lambda (H6: (subst0 (s k i2) u3 t x)).(eq_ind_r T (THead k u0 x) (\lambda (t4: T).(subst0 i v t3 t4)))) (ex_intro2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v (THead k u0 x) t3)) (THead k u2 x) (subst0_snd k u3 x t i2 H6 u2) (subst0_fst v u2 u0 i H0 x k)) -t2 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u2: T).(\lambda (t3: T).(eq -T t2 (THead k u2 t3)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i2 u3 u0 -u2))) (\lambda (_: T).(\lambda (t2: T).(subst0 (s k i2) u3 t -t2))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 +t2 H5)))) H4)) (\lambda (H4: (ex3_2 T T (\lambda (u4: T).(\lambda (t3: T).(eq +T t2 (THead k u4 t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 +u4))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t +t3))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t3: T).(eq T t2 (THead k u4 t3)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t t3))) (ex2 T (\lambda (t3: T).(subst0 i2 u3 (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i v t2 t3))) @@ -373,7 +373,7 @@ t5))) (ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5)))) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) -(\lambda (H4: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 t3))) (\lambda +(\lambda (H4: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k u3 t3))) (\lambda (u3: T).(subst0 i2 u2 u u3)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: @@ -383,8 +383,8 @@ T).(subst0 i2 u2 (THead k u t2) t5)) (\lambda (t5: T).(subst0 i v t t5)))) (ex_intro2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k x t3) t)) (THead k x t2) (subst0_fst u2 x u i2 H6 t2 k) (subst0_snd k v t2 t3 i H0 x)) t4 H5)))) H4)) (\lambda (H4: (ex2 T -(\lambda (t2: T).(eq T t4 (THead k u t2))) (\lambda (t2: T).(subst0 (s k i2) -u2 t3 t2)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda +(\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) +u2 t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u t5))) (\lambda (t5: T).(subst0 (s k i2) u2 t3 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: T).(\lambda (H5: (eq T t4 (THead k u x))).(\lambda (H6: (subst0 (s k i2) u2 @@ -398,9 +398,9 @@ T (\lambda (t: T).(subst0 i2 u2 (THead k u t2) t)) (\lambda (t: T).(subst0 i v (THead k u x) t)) (THead k u x0) (subst0_snd k u2 x0 t2 i2 H7 u) (subst0_snd k v x0 x i H8 u))))) (H1 x u2 (s k i2) H6 (\lambda (H7: (eq nat (s k i) (s k i2))).(H3 (s_inj k i i2 H7))))) t4 H5)))) H4)) (\lambda (H4: -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead k u2 t2)))) +(ex3_2 T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s k i2) u2 t3 t2))))).(ex3_2_ind T T (\lambda +T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i2 u2 u u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) u2 t3 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u2 (THead k @@ -423,8 +423,8 @@ T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 u0 t2) \to ((not (eq nat i i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 u2 t)) (\lambda (t: T).(subst0 i v t2 t)))))))))).(\lambda (k: K).(\lambda (t2: T).(\lambda (t3: T).(\lambda (H2: (subst0 (s k i) v t2 t3)).(\lambda (H3: ((\forall (t4: -T).(\forall (u2: T).(\forall (i2: nat).((subst0 i2 u2 t2 t4) \to ((not (eq -nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u2 t3 t)) (\lambda (t: +T).(\forall (u3: T).(\forall (i2: nat).((subst0 i2 u3 t2 t4) \to ((not (eq +nat (s k i) i2)) \to (ex2 T (\lambda (t: T).(subst0 i2 u3 t3 t)) (\lambda (t: T).(subst0 (s k i) v t4 t)))))))))).(\lambda (t4: T).(\lambda (u3: T).(\lambda (i2: nat).(\lambda (H4: (subst0 i2 u3 (THead k u0 t2) t4)).(\lambda (H5: (not (eq nat i i2))).(or3_ind (ex2 T (\lambda (u4: T).(eq @@ -434,8 +434,8 @@ u3 t2 t5))) (ex3_2 T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5)))) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) -(\lambda (H6: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 t2))) (\lambda -(u2: T).(subst0 i2 u3 u0 u2)))).(ex2_ind T (\lambda (u4: T).(eq T t4 (THead k +(\lambda (H6: (ex2 T (\lambda (u4: T).(eq T t4 (THead k u4 t2))) (\lambda +(u4: T).(subst0 i2 u3 u0 u4)))).(ex2_ind T (\lambda (u4: T).(eq T t4 (THead k u4 t2))) (\lambda (u4: T).(subst0 i2 u3 u0 u4)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: T).(\lambda (H7: (eq T t4 (THead k x t2))).(\lambda (H8: (subst0 @@ -448,8 +448,8 @@ u3 u2 x0)).(\lambda (H10: (subst0 i v x x0)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k x t2) t)) (THead k x0 t3) (subst0_fst u3 x0 u2 i2 H9 t3 k) (subst0_both v x x0 i H10 k t2 t3 H2))))) (H1 x u3 i2 H8 H5)) t4 H7)))) H6)) (\lambda (H6: (ex2 T -(\lambda (t2: T).(eq T t4 (THead k u0 t2))) (\lambda (t3: T).(subst0 (s k i2) -u3 t2 t3)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda +(\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2) +u3 t2 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i2) u3 t2 t5)) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v t4 t))) (\lambda (x: T).(\lambda (H7: (eq T t4 (THead k u0 x))).(\lambda (H8: (subst0 (s k i2) u3 @@ -463,9 +463,9 @@ x0)).(ex_intro2 T (\lambda (t: T).(subst0 i2 u3 (THead k u2 t3) t)) (\lambda (t: T).(subst0 i v (THead k u0 x) t)) (THead k u2 x0) (subst0_snd k u3 x0 t3 i2 H9 u2) (subst0_both v u0 u2 i H0 k x x0 H10))))) (H3 x u3 (s k i2) H8 (\lambda (H9: (eq nat (s k i) (s k i2))).(H5 (s_inj k i i2 H9))))) t4 H7)))) -H6)) (\lambda (H6: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 -(THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i2 u3 u0 u2))) -(\lambda (_: T).(\lambda (t3: T).(subst0 (s k i2) u3 t2 t3))))).(ex3_2_ind T +H6)) (\lambda (H6: (ex3_2 T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 +(THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) +(\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5))))).(ex3_2_ind T T (\lambda (u4: T).(\lambda (t5: T).(eq T t4 (THead k u4 t5)))) (\lambda (u4: T).(\lambda (_: T).(subst0 i2 u3 u0 u4))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i2) u3 t2 t5))) (ex2 T (\lambda (t: T).(subst0 i2 u3 (THead k @@ -521,8 +521,8 @@ t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i0) v t t3)))) (or4 (eq T (THead k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 -(THead k u2 t))) (\lambda (H3: (ex2 T (\lambda (u2: T).(eq T t2 (THead k u2 -t))) (\lambda (u2: T).(subst0 i0 v u1 u2)))).(ex2_ind T (\lambda (u3: T).(eq +(THead k u2 t))) (\lambda (H3: (ex2 T (\lambda (u3: T).(eq T t2 (THead k u3 +t))) (\lambda (u3: T).(subst0 i0 v u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t2 (THead k u3 t))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v t2 t3))) (subst0 i0 v (THead k u2 t) t2) (subst0 i0 v t2 @@ -543,77 +543,78 @@ t)) (subst0 i0 v (THead k x t) (THead k t3 t)))) (or4_intro0 (eq T (THead k x t) (THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k x t) (THead k x t)) (subst0 i0 v (THead k x t) (THead k x t)) (refl_equal T (THead -k x t))) u2 H6)) (\lambda (H6: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v x t)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 v -u2 t3)) (\lambda (t3: T).(subst0 i0 v x t3)) (or4 (eq T (THead k u2 t) (THead -k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda -(t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k -x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (x0: T).(\lambda -(H7: (subst0 i0 v u2 x0)).(\lambda (H8: (subst0 i0 v x x0)).(or4_intro1 (eq T -(THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k -u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v -(THead k u2 t) (THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t)) -(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: -T).(subst0 i0 v (THead k x t) t3)) (THead k x0 t) (subst0_fst v x0 u2 i0 H7 t -k) (subst0_fst v x0 x i0 H8 t k)))))) H6)) (\lambda (H6: (subst0 i0 v u2 -x)).(or4_intro2 (eq T (THead k u2 t) (THead k x t)) (ex2 T (\lambda (t3: +k x t))) u2 H6)) (\lambda (H6: (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) +(\lambda (t3: T).(subst0 i0 v x t3)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 +v u2 t3)) (\lambda (t3: T).(subst0 i0 v x t3)) (or4 (eq T (THead k u2 t) +(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) +(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) +(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t))) (\lambda (x0: +T).(\lambda (H7: (subst0 i0 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(THead k x0 t) +(subst0_fst v x0 u2 i0 H7 t k) (subst0_fst v x0 x i0 H8 t k)))))) H6)) +(\lambda (H6: (subst0 i0 v u2 x)).(or4_intro2 (eq T (THead k u2 t) (THead k x +t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: +T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) (THead k x +t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v x u2 i0 H6 t +k))) (\lambda (H6: (subst0 i0 v x u2)).(or4_intro3 (eq T (THead k u2 t) +(THead k x t)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) +(\lambda (t3: T).(subst0 i0 v (THead k x t) t3))) (subst0 i0 v (THead k u2 t) +(THead k x t)) (subst0 i0 v (THead k x t) (THead k u2 t)) (subst0_fst v u2 x +i0 H6 t k))) (H1 x H5)) t2 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t3: +T).(eq T t2 (THead k u1 t3))) (\lambda (t3: T).(subst0 (s k i0) v t +t3)))).(ex2_ind T (\lambda (t3: T).(eq T t2 (THead k u1 t3))) (\lambda (t3: +T).(subst0 (s k i0) v t t3)) (or4 (eq T (THead k u2 t) t2) (ex2 T (\lambda +(t3: 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t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) (subst0 i0 v +(THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t)) +(ex_intro2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: +T).(subst0 i0 v (THead k u1 x) t3)) (THead k u2 x) (subst0_snd k v x t i0 H5 +u2) (subst0_fst v u2 u1 i0 H0 x k)))) (\lambda (H6: (ex2 T (\lambda (t3: +T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3)))).(ex2_ind T +(\lambda (t3: T).(subst0 i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v u2 t3)) +(or4 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 +v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k u1 x) t3))) +(subst0 i0 v (THead k u2 t) (THead k u1 x)) (subst0 i0 v (THead k u1 x) +(THead k u2 t))) (\lambda (x0: T).(\lambda (_: (subst0 i0 v u2 x0)).(\lambda +(_: (subst0 i0 v u2 x0)).(or4_intro1 (eq T (THead k u2 t) (THead k u1 x)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead 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T).(\lambda (t3: T).(eq T t2 (THead k u3 +(subst0_snd k v x t i0 H5 u2) (subst0_fst v u2 u1 i0 H0 x k)))) (H1 u2 H0)) +t2 H4)))) H3)) (\lambda (H3: (ex3_2 T T (\lambda (u3: T).(\lambda (t3: T).(eq +T t2 (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 +u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i0) v t +t3))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t3: T).(eq T t2 (THead k u3 t3)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i0) v t t3))) (or4 (eq T (THead k u2 t) t2) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: @@ -636,9 +637,9 @@ x0 x1) (THead k t3 t)))) (or4_intro2 (eq T (THead k x0 t) (THead k x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k x0 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k x0 t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k x0 t)) (subst0_snd k v x1 t i0 H6 -x0)) u2 H7)) (\lambda (H7: (ex2 T (\lambda (t: T).(subst0 i0 v u2 t)) -(\lambda (t: T).(subst0 i0 v x0 t)))).(ex2_ind T (\lambda (t3: T).(subst0 i0 -v u2 t3)) (\lambda (t3: T).(subst0 i0 v x0 t3)) (or4 (eq T (THead k u2 t) +x0)) u2 H7)) (\lambda (H7: (ex2 T (\lambda (t3: T).(subst0 i0 v u2 t3)) +(\lambda (t3: T).(subst0 i0 v x0 t3)))).(ex2_ind T (\lambda (t3: T).(subst0 +i0 v u2 t3)) (\lambda (t3: T).(subst0 i0 v x0 t3)) (or4 (eq T (THead k u2 t) (THead k x0 x1)) (ex2 T (\lambda (t3: T).(subst0 i0 v (THead k u2 t) t3)) (\lambda (t3: T).(subst0 i0 v (THead k x0 x1) t3))) (subst0 i0 v (THead k u2 t) (THead k x0 x1)) (subst0 i0 v (THead k x0 x1) (THead k u2 t))) (\lambda @@ -721,8 +722,8 @@ t3) t))) (subst0 i0 v (THead k u0 t2) (THead k x t3)) (subst0 i0 v (THead k x t3) (THead k u0 t2)) (ex_intro2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k x t3) t)) (THead k x t2) (subst0_fst v x u0 i0 H5 t2 k) (subst0_snd k v t2 t3 i0 H0 x)))) (H1 t2 H0)) -t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t2: T).(eq T t4 (THead k u0 -t2))) (\lambda (t2: T).(subst0 (s k i0) v t3 t2)))).(ex2_ind T (\lambda (t5: +t4 H4)))) H3)) (\lambda (H3: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u0 +t5))) (\lambda (t5: T).(subst0 (s k i0) v t3 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u0 t5))) (\lambda (t5: T).(subst0 (s k i0) v t3 t5)) (or4 (eq T (THead k u0 t2) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u0 t2) t)) (\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u0 t2) @@ -766,9 +767,9 @@ x t2 i0 H6 u0))) (\lambda (H6: (subst0 (s k i0) v x t2)).(or4_intro3 (eq T u0 t2) t)) (\lambda (t: T).(subst0 i0 v (THead k u0 x) t))) (subst0 i0 v (THead k u0 t2) (THead k u0 x)) (subst0 i0 v (THead k u0 x) (THead k u0 t2)) (subst0_snd k v t2 x i0 H6 u0))) (H1 x H5)) t4 H4)))) H3)) (\lambda (H3: -(ex3_2 T T (\lambda (u2: T).(\lambda (t2: T).(eq T t4 (THead k u2 t2)))) +(ex3_2 T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_: -T).(\lambda (t2: T).(subst0 (s k i0) v t3 t2))))).(ex3_2_ind T T (\lambda +T).(\lambda (t5: T).(subst0 (s k i0) v t3 t5))))).(ex3_2_ind T T (\lambda (u2: T).(\lambda (t5: T).(eq T t4 (THead k u2 t5)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v u0 u2))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v t3 t5))) (or4 (eq T (THead k u0 t2) t4) (ex2 T (\lambda @@ -834,8 +835,8 @@ T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v t2 t5)))) (or4 (eq T (THead k u2 t3) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4 (THead k u2 t3))) -(\lambda (H5: (ex2 T (\lambda (u2: T).(eq T t4 (THead k u2 t2))) (\lambda -(u2: T).(subst0 i0 v u1 u2)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k +(\lambda (H5: (ex2 T (\lambda (u3: T).(eq T t4 (THead k u3 t2))) (\lambda +(u3: T).(subst0 i0 v u1 u3)))).(ex2_ind T (\lambda (u3: T).(eq T t4 (THead k u3 t2))) (\lambda (u3: T).(subst0 i0 v u1 u3)) (or4 (eq T (THead k u2 t3) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 i0 v t4 @@ -1002,8 +1003,8 @@ k v t3 t2 i0 H2 x)))) (\lambda (H9: (subst0 i0 v x u2)).(or4_intro3 (eq T u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k x t2) t))) (subst0 i0 v (THead k u2 t3) (THead k x t2)) (subst0 i0 v (THead k x t2) (THead k u2 t3)) (subst0_both v x u2 i0 H9 k t2 t3 H2))) (H1 x H7))) (H3 t3 H2)) t4 H6)))) -H5)) (\lambda (H5: (ex2 T (\lambda (t2: T).(eq T t4 (THead k u1 t2))) -(\lambda (t3: T).(subst0 (s k i0) v t2 t3)))).(ex2_ind T (\lambda (t5: T).(eq +H5)) (\lambda (H5: (ex2 T (\lambda (t5: T).(eq T t4 (THead k u1 t5))) +(\lambda (t5: T).(subst0 (s k i0) v t2 t5)))).(ex2_ind T (\lambda (t5: T).(eq T t4 (THead k u1 t5))) (\lambda (t5: T).(subst0 (s k i0) v t2 t5)) (or4 (eq T (THead k u2 t3) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v t4 t))) (subst0 i0 v (THead k u2 t3) t4) (subst0 @@ -1159,10 +1160,10 @@ u2)).(or4_intro3 (eq T (THead k u2 t3) (THead k u1 x)) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: T).(subst0 i0 v (THead k u1 x) t))) (subst0 i0 v (THead k u2 t3) (THead k u1 x)) (subst0 i0 v (THead k u1 x) (THead k u2 t3)) (subst0_both v u1 u2 i0 H0 k x t3 H8))) (H1 u2 H0))) (H3 -x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda (u2: T).(\lambda (t2: -T).(eq T t4 (THead k u2 t2)))) (\lambda (u2: T).(\lambda (_: T).(subst0 i0 v -u1 u2))) (\lambda (_: T).(\lambda (t3: T).(subst0 (s k i0) v t2 -t3))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 +x H7)) t4 H6)))) H5)) (\lambda (H5: (ex3_2 T T (\lambda (u3: T).(\lambda (t5: +T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v +u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v t2 +t5))))).(ex3_2_ind T T (\lambda (u3: T).(\lambda (t5: T).(eq T t4 (THead k u3 t5)))) (\lambda (u3: T).(\lambda (_: T).(subst0 i0 v u1 u3))) (\lambda (_: T).(\lambda (t5: T).(subst0 (s k i0) v t2 t5))) (or4 (eq T (THead k u2 t3) t4) (ex2 T (\lambda (t: T).(subst0 i0 v (THead k u2 t3) t)) (\lambda (t: @@ -1353,12 +1354,12 @@ t1)) (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: T).(subst0 i u (lift (S O) i t1) t))) (subst0 i u (lift (S O) i t2) (lift (S O) i t1)) (subst0 i u (lift (S O) i t1) (lift (S O) i t2)) (eq T t1 t2) (\lambda (H1: (eq T (lift (S O) i t2) (lift (S O) i t1))).(let H2 \def -(sym_equal T (lift (S O) i t2) (lift (S O) i t1) H1) in (lift_inj t1 t2 (S O) -i H2))) (\lambda (H1: (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) -t)) (\lambda (t: T).(subst0 i u (lift (S O) i t1) t)))).(ex2_ind T (\lambda -(t: T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: T).(subst0 i u (lift (S -O) i t1) t)) (eq T t1 t2) (\lambda (x: T).(\lambda (_: (subst0 i u (lift (S -O) i t2) x)).(\lambda (H3: (subst0 i u (lift (S O) i t1) +(sym_eq T (lift (S O) i t2) (lift (S O) i t1) H1) in (lift_inj t1 t2 (S O) i +H2))) (\lambda (H1: (ex2 T (\lambda (t: T).(subst0 i u (lift (S O) i t2) t)) +(\lambda (t: T).(subst0 i u (lift (S O) i t1) t)))).(ex2_ind T (\lambda (t: +T).(subst0 i u (lift (S O) i t2) t)) (\lambda (t: T).(subst0 i u (lift (S O) +i t1) t)) (eq T t1 t2) (\lambda (x: T).(\lambda (_: (subst0 i u (lift (S O) i +t2) x)).(\lambda (H3: (subst0 i u (lift (S O) i t1) x)).(subst0_gen_lift_false t1 u x (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i (S O))) H3 (eq T t1 t2))))) H1)) (\lambda (H1: (subst0 i u (lift diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst1/subst1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst1/subst1.ma index 182118963..0c54f9156 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst1/subst1.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/subst1/subst1.ma @@ -176,11 +176,11 @@ y)).(subst1_ind i u t0 (\lambda (t: T).((eq T t (lift (S O) i t1)) \to (\lambda (H1: (eq T t0 (lift (S O) i t1))).(\lambda (t2: T).(\lambda (H2: (subst1 i u t0 (lift (S O) i t2))).(let H3 \def (eq_ind T t0 (\lambda (t: T).(subst1 i u t (lift (S O) i t2))) H2 (lift (S O) i t1) H1) in (let H4 \def -(sym_equal T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u -(lift (S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda -(n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i (S O))) -H3)) in (lift_inj t1 t2 (S O) i H4)))))) (\lambda (t2: T).(\lambda (H1: -(subst0 i u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i t1))).(\lambda (t3: +(sym_eq T (lift (S O) i t2) (lift (S O) i t1) (subst1_gen_lift_eq t1 u (lift +(S O) i t2) (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) (\lambda (n: +nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i (S O))) H3)) +in (lift_inj t1 t2 (S O) i H4)))))) (\lambda (t2: T).(\lambda (H1: (subst0 i +u t0 t2)).(\lambda (H2: (eq T t2 (lift (S O) i t1))).(\lambda (t3: T).(\lambda (H3: (subst1 i u t0 (lift (S O) i t3))).(let H4 \def (eq_ind T t2 (\lambda (t: T).(subst0 i u t0 t)) H1 (lift (S O) i t1) H2) in (insert_eq T (lift (S O) i t3) (\lambda (t: T).(subst1 i u t0 t)) (eq T t1 t3) (\lambda diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity.ma index 23f81bacb..73069bcd5 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/arity.ma @@ -161,12 +161,12 @@ A).(\lambda (a2: A).(arity g (CHead c0 (Bind Abst) u) t a2))) (ex2 A (\lambda (THead (Flat Appl) w (THead (Bind Abst) u t)) (asucc g a1)))) (\lambda (x1: A).(\lambda (x2: A).(\lambda (H11: (eq A (asucc g x0) (AHead x1 x2))).(\lambda (H12: (arity g c0 u (asucc g x1))).(\lambda (H13: (arity g -(CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_equal A (asucc g x0) -(AHead x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in -(ex2_ind A (\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A -x2 (asucc g a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) -a1)) (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u -t)) (asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1 +(CHead c0 (Bind Abst) u) t x2)).(let H14 \def (sym_eq A (asucc g x0) (AHead +x1 x2) H11) in (let H15 \def (asucc_gen_head g x1 x2 x0 H14) in (ex2_ind A +(\lambda (a0: A).(eq A x0 (AHead x1 a0))) (\lambda (a0: A).(eq A x2 (asucc g +a0))) (ex2 A (\lambda (a1: A).(arity g c0 (THead (Flat Appl) w v) a1)) +(\lambda (a1: A).(arity g c0 (THead (Flat Appl) w (THead (Bind Abst) u t)) +(asucc g a1)))) (\lambda (x3: A).(\lambda (H16: (eq A x0 (AHead x1 x3))).(\lambda (H17: (eq A x2 (asucc g x3))).(let H18 \def (eq_ind A x2 (\lambda (a: A).(arity g (CHead c0 (Bind Abst) u) t a)) H13 (asucc g x3) H17) in (let H19 \def (eq_ind A x0 (\lambda (a: A).(arity g c0 v a)) H8 (AHead x1 diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1.ma index eb09bb829..094ce7ecb 100644 --- a/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1.ma +++ b/matita/contribs/LAMBDA-TYPES/Level-1/LambdaDelta/ty3/subst1.ma @@ -498,7 +498,7 @@ Appl) w v) (lift (S O) d y1)))) (\lambda (_: T).(\lambda (y2: T).(subst1 d u0 (y1: T).(\lambda (y2: T).(ty3 g a y1 y2)))) (\lambda (x4: T).(\lambda (x5: T).(\lambda (H15: (eq T (lift (S O) d x1) (THead (Bind Abst) x4 x5))).(\lambda (H16: (subst1 d u0 u x4)).(\lambda (H17: (subst1 (s (Bind -Abst) d) u0 t x5)).(let H18 \def (sym_equal T (lift (S O) d x1) (THead (Bind +Abst) d) u0 t x5)).(let H18 \def (sym_eq T (lift (S O) d x1) (THead (Bind Abst) x4 x5) H15) in (ex3_2_ind T T (\lambda (y: T).(\lambda (z: T).(eq T x1 (THead (Bind Abst) y z)))) (\lambda (y: T).(\lambda (_: T).(eq T x4 (lift (S O) d y)))) (\lambda (_: T).(\lambda (z: T).(eq T x5 (lift (S O) (S d) z)))) diff --git a/matita/contribs/LAMBDA-TYPES/Level-1/problems-1.ma b/matita/contribs/LAMBDA-TYPES/Level-1/problems-1.ma deleted file mode 100644 index 32e613ebb..000000000 --- a/matita/contribs/LAMBDA-TYPES/Level-1/problems-1.ma +++ /dev/null @@ -1,183 +0,0 @@ -(**************************************************************************) -(* ___ *) -(* ||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 *) -(* *) -(**************************************************************************) - -(* Problematic objects for disambiguation/typechecking ********************) - -set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/problems". - -include "LambdaDelta/theory.ma". - -theorem pr2_gen_cabbr: - \forall (c: C).(\forall (t1: T).(\forall (t2: T).((pr2 c t1 t2) \to (\forall -(e: C).(\forall (u: T).(\forall (d: nat).((getl d c (CHead e (Bind Abbr) u)) -\to (\forall (a0: C).((csubst1 d u c a0) \to (\forall (a: C).((drop (S O) d -a0 a) \to (\forall (x1: T).((subst1 d u t1 (lift (S O) d x1)) \to (ex2 T -(\lambda (x2: T).(subst1 d u t2 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a -x1 x2)))))))))))))))) -\def - \lambda (c: C).(\lambda (t1: T).(\lambda (t2: T).(\lambda (H: (pr2 c t1 -t2)).(pr2_ind (\lambda (c0: C).(\lambda (t: T).(\lambda (t0: T).(\forall (e: -C).(\forall (u: T).(\forall (d: nat).((getl d c0 (CHead e (Bind Abbr) u)) \to -(\forall (a0: C).((csubst1 d u c0 a0) \to (\forall (a: C).((drop (S O) d a0 -a) \to (\forall (x1: T).((subst1 d u t (lift (S O) d x1)) \to (ex2 T (\lambda -(x2: T).(subst1 d u t0 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 -x2)))))))))))))))) (\lambda (c0: C).(\lambda (t3: T).(\lambda (t4: -T).(\lambda (H0: (pr0 t3 t4)).(\lambda (e: C).(\lambda (u: T).(\lambda (d: -nat).(\lambda (_: (getl d c0 (CHead e (Bind Abbr) u))).(\lambda (a0: -C).(\lambda (_: (csubst1 d u c0 a0)).(\lambda (a: C).(\lambda (_: (drop (S O) -d a0 a)).(\lambda (x1: T).(\lambda (H4: (subst1 d u t3 (lift (S O) d -x1))).(ex2_ind T (\lambda (w2: T).(pr0 (lift (S O) d x1) w2)) (\lambda (w2: -T).(subst1 d u t4 w2)) (ex2 T (\lambda (x2: T).(subst1 d u t4 (lift (S O) d -x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H5: (pr0 -(lift (S O) d x1) x)).(\lambda (H6: (subst1 d u t4 x)).(ex2_ind T (\lambda -(t5: T).(eq T x (lift (S O) d t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T -(\lambda (x2: T).(subst1 d u t4 (lift (S O) d x2))) (\lambda (x2: T).(pr2 a -x1 x2))) (\lambda (x0: T).(\lambda (H7: (eq T x (lift (S O) d x0))).(\lambda -(H8: (pr0 x1 x0)).(let H9 \def (eq_ind T x (\lambda (t: T).(subst1 d u t4 t)) -H6 (lift (S O) d x0) H7) in (ex_intro2 T (\lambda (x2: T).(subst1 d u t4 -(lift (S O) d x2))) (\lambda (x2: T).(pr2 a x1 x2)) x0 H9 (pr2_free a x1 x0 -H8)))))) (pr0_gen_lift x1 x (S O) d H5))))) (pr0_subst1 t3 t4 H0 u (lift (S -O) d x1) d H4 u (pr0_refl u))))))))))))))))) (\lambda (c0: C).(\lambda (d: -C).(\lambda (u: T).(\lambda (i: nat).(\lambda (H0: (getl i c0 (CHead d (Bind -Abbr) u))).(\lambda (t3: T).(\lambda (t4: T).(\lambda (H1: (pr0 t3 -t4)).(\lambda (t: T).(\lambda (H2: (subst0 i u t4 t)).(\lambda (e: -C).(\lambda (u0: T).(\lambda (d0: nat).(\lambda (H3: (getl d0 c0 (CHead e -(Bind Abbr) u0))).(\lambda (a0: C).(\lambda (H4: (csubst1 d0 u0 c0 -a0)).(\lambda (a: C).(\lambda (H5: (drop (S O) d0 a0 a)).(\lambda (x1: -T).(\lambda (H6: (subst1 d0 u0 t3 (lift (S O) d0 x1))).(ex2_ind T (\lambda -(w2: T).(pr0 (lift (S O) d0 x1) w2)) (\lambda (w2: T).(subst1 d0 u0 t4 w2)) -(ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: -T).(pr2 a x1 x2))) (\lambda (x: T).(\lambda (H7: (pr0 (lift (S O) d0 x1) -x)).(\lambda (H8: (subst1 d0 u0 t4 x)).(ex2_ind T (\lambda (t5: T).(eq T x -(lift (S O) d0 t5))) (\lambda (t5: T).(pr0 x1 t5)) (ex2 T (\lambda (x2: -T).(subst1 d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) -(\lambda (x0: T).(\lambda (H9: (eq T x (lift (S O) d0 x0))).(\lambda (H10: -(pr0 x1 x0)).(let H11 \def (eq_ind T x (\lambda (t0: T).(subst1 d0 u0 t4 t0)) -H8 (lift (S O) d0 x0) H9) in (lt_eq_gt_e i d0 (ex2 T (\lambda (x2: T).(subst1 -d0 u0 t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (H12: -(lt i d0)).(ex2_ind T (\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: -T).(subst1 i u (lift (S O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 -t (lift (S O) d0 x2))) (\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: -T).(\lambda (H13: (subst1 d0 u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) -d0 x0) x2)).(ex2_ind C (\lambda (e2: C).(csubst1 (minus d0 i) u0 (CHead d -(Bind Abbr) u) e2)) (\lambda (e2: C).(getl i a0 e2)) (ex2 T (\lambda (x3: -T).(subst1 d0 u0 t (lift (S O) d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) -(\lambda (x3: C).(\lambda (H15: (csubst1 (minus d0 i) u0 (CHead d (Bind Abbr) -u) x3)).(\lambda (H16: (getl i a0 x3)).(let H17 \def (eq_ind nat (minus d0 i) -(\lambda (n: nat).(csubst1 n u0 (CHead d (Bind Abbr) u) x3)) H15 (S (minus d0 -(S i))) (minus_x_Sy d0 i H12)) in (let H18 \def (csubst1_gen_head (Bind Abbr) -d x3 u u0 (minus d0 (S i)) H17) in (ex3_2_ind T C (\lambda (u2: T).(\lambda -(c2: C).(eq C x3 (CHead c2 (Bind Abbr) u2)))) (\lambda (u2: T).(\lambda (_: -C).(subst1 (minus d0 (S i)) u0 u u2))) (\lambda (_: T).(\lambda (c2: -C).(csubst1 (minus d0 (S i)) u0 d c2))) (ex2 T (\lambda (x4: T).(subst1 d0 u0 -t (lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4))) (\lambda (x4: -T).(\lambda (x5: C).(\lambda (H19: (eq C x3 (CHead x5 (Bind Abbr) -x4))).(\lambda (H20: (subst1 (minus d0 (S i)) u0 u x4)).(\lambda (_: (csubst1 -(minus d0 (S i)) u0 d x5)).(let H22 \def (eq_ind C x3 (\lambda (c1: C).(getl -i a0 c1)) H16 (CHead x5 (Bind Abbr) x4) H19) in (let H23 \def (eq_ind nat d0 -(\lambda (n: nat).(drop (S O) n a0 a)) H5 (S (plus i (minus d0 (S i)))) -(lt_plus_minus i d0 H12)) in (ex3_2_ind T C (\lambda (v: T).(\lambda (_: -C).(eq T x4 (lift (S O) (minus d0 (S i)) v)))) (\lambda (v: T).(\lambda (e0: -C).(getl i a (CHead e0 (Bind Abbr) v)))) (\lambda (_: T).(\lambda (e0: -C).(drop (S O) (minus d0 (S i)) x5 e0))) (ex2 T (\lambda (x6: T).(subst1 d0 -u0 t (lift (S O) d0 x6))) (\lambda (x6: T).(pr2 a x1 x6))) (\lambda (x6: -T).(\lambda (x7: C).(\lambda (H24: (eq T x4 (lift (S O) (minus d0 (S i)) -x6))).(\lambda (H25: (getl i a (CHead x7 (Bind Abbr) x6))).(\lambda (_: (drop -(S O) (minus d0 (S i)) x5 x7)).(let H27 \def (eq_ind T x4 (\lambda (t0: -T).(subst1 (minus d0 (S i)) u0 u t0)) H20 (lift (S O) (minus d0 (S i)) x6) -H24) in (ex2_ind T (\lambda (t0: T).(subst1 i (lift (S O) (minus d0 (S i)) -x6) (lift (S O) d0 x0) t0)) (\lambda (t0: T).(subst1 (S (plus (minus d0 (S -i)) i)) u0 x2 t0)) (ex2 T (\lambda (x8: T).(subst1 d0 u0 t (lift (S O) d0 -x8))) (\lambda (x8: T).(pr2 a x1 x8))) (\lambda (x8: T).(\lambda (H28: -(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S O) d0 x0) x8)).(\lambda -(H29: (subst1 (S (plus (minus d0 (S i)) i)) u0 x2 x8)).(let H30 \def (eq_ind -nat d0 (\lambda (n: nat).(subst1 i (lift (S O) (minus d0 (S i)) x6) (lift (S -O) n x0) x8)) H28 (S (plus i (minus d0 (S i)))) (lt_plus_minus i d0 H12)) in -(ex2_ind T (\lambda (t5: T).(eq T x8 (lift (S O) (S (plus i (minus d0 (S -i)))) t5))) (\lambda (t5: T).(subst1 i x6 x0 t5)) (ex2 T (\lambda (x9: -T).(subst1 d0 u0 t (lift (S O) d0 x9))) (\lambda (x9: T).(pr2 a x1 x9))) -(\lambda (x9: T).(\lambda (H31: (eq T x8 (lift (S O) (S (plus i (minus d0 (S -i)))) x9))).(\lambda (H32: (subst1 i x6 x0 x9)).(let H33 \def (eq_ind T x8 -(\lambda (t0: T).(subst1 (S (plus (minus d0 (S i)) i)) u0 x2 t0)) H29 (lift -(S O) (S (plus i (minus d0 (S i)))) x9) H31) in (let H34 \def (eq_ind_r nat -(S (plus i (minus d0 (S i)))) (\lambda (n: nat).(subst1 (S (plus (minus d0 (S -i)) i)) u0 x2 (lift (S O) n x9))) H33 d0 (lt_plus_minus i d0 H12)) in (let -H35 \def (eq_ind_r nat (S (plus (minus d0 (S i)) i)) (\lambda (n: -nat).(subst1 n u0 x2 (lift (S O) d0 x9))) H34 d0 (lt_plus_minus_r i d0 H12)) -in (ex_intro2 T (\lambda (x10: T).(subst1 d0 u0 t (lift (S O) d0 x10))) -(\lambda (x10: T).(pr2 a x1 x10)) x9 (subst1_trans x2 t u0 d0 H13 (lift (S O) -d0 x9) H35) (pr2_delta1 a x7 x6 i H25 x1 x0 H10 x9 H32)))))))) -(subst1_gen_lift_lt x6 x0 x8 i (S O) (minus d0 (S i)) H30)))))) -(subst1_subst1_back (lift (S O) d0 x0) x2 u i H14 (lift (S O) (minus d0 (S -i)) x6) u0 (minus d0 (S i)) H27)))))))) (getl_drop_conf_lt Abbr a0 x5 x4 i -H22 a (S O) (minus d0 (S i)) H23))))))))) H18)))))) (csubst1_getl_lt d0 i H12 -c0 a0 u0 H4 (CHead d (Bind Abbr) u) H0))))) (subst1_confluence_neq t4 t u i -(subst1_single i u t4 t H2) (lift (S O) d0 x0) u0 d0 H11 (lt_neq i d0 H12)))) -(\lambda (H12: (eq nat i d0)).(let H13 \def (eq_ind_r nat d0 (\lambda (n: -nat).(subst1 n u0 t4 (lift (S O) n x0))) H11 i H12) in (let H14 \def -(eq_ind_r nat d0 (\lambda (n: nat).(drop (S O) n a0 a)) H5 i H12) in (let H15 -\def (eq_ind_r nat d0 (\lambda (n: nat).(csubst1 n u0 c0 a0)) H4 i H12) in -(let H16 \def (eq_ind_r nat d0 (\lambda (n: nat).(getl n c0 (CHead e (Bind -Abbr) u0))) H3 i H12) in (eq_ind nat i (\lambda (n: nat).(ex2 T (\lambda (x2: -T).(subst1 n u0 t (lift (S O) n x2))) (\lambda (x2: T).(pr2 a x1 x2)))) (let -H17 \def (eq_ind C (CHead d (Bind Abbr) u) (\lambda (c1: C).(getl i c0 c1)) -H0 (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead -e (Bind Abbr) u0) H16)) in (let H18 \def (f_equal C C (\lambda (e0: C).(match -e0 in C return (\lambda (_: C).C) with [(CSort _) \Rightarrow d | (CHead c1 _ -_) \Rightarrow c1])) (CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) -(getl_mono c0 (CHead d (Bind Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in -((let H19 \def (f_equal C T (\lambda (e0: C).(match e0 in C return (\lambda -(_: C).T) with [(CSort _) \Rightarrow u | (CHead _ _ t0) \Rightarrow t0])) -(CHead d (Bind Abbr) u) (CHead e (Bind Abbr) u0) (getl_mono c0 (CHead d (Bind -Abbr) u) i H0 (CHead e (Bind Abbr) u0) H16)) in (\lambda (H20: (eq C d -e)).(let H21 \def (eq_ind_r T u0 (\lambda (t0: T).(getl i c0 (CHead e (Bind -Abbr) t0))) H17 u H19) in (let H22 \def (eq_ind_r T u0 (\lambda (t0: -T).(subst1 i t0 t4 (lift (S O) i x0))) H13 u H19) in (let H23 \def (eq_ind_r -T u0 (\lambda (t0: T).(csubst1 i t0 c0 a0)) H15 u H19) in (eq_ind T u -(\lambda (t0: T).(ex2 T (\lambda (x2: T).(subst1 i t0 t (lift (S O) i x2))) -(\lambda (x2: T).(pr2 a x1 x2)))) (let H24 \def (eq_ind_r C e (\lambda (c1: -C).(getl i c0 (CHead c1 (Bind Abbr) u))) H21 d H20) in (ex2_ind T (\lambda -(t0: T).(subst1 i u t t0)) (\lambda (t0: T).(subst1 i u (lift (S O) i x0) -t0)) (ex2 T (\lambda (x2: T).(subst1 i u t (lift (S O) i x2))) (\lambda (x2: -T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H25: (subst1 i u t -x2)).(\lambda (H26: (subst1 i u (lift (S O) i x0) x2)).(let H27 \def (eq_ind -T x2 (\lambda (t0: T).(subst1 i u t t0)) H25 (lift (S O) i x0) -(subst1_gen_lift_eq x0 u x2 (S O) i i (le_n i) (eq_ind_r nat (plus (S O) i) -(\lambda (n: nat).(lt i n)) (le_n (plus (S O) i)) (plus i (S O)) (plus_comm i -(S O))) H26)) in (ex_intro2 T (\lambda (x3: T).(subst1 i u t (lift (S O) i -x3))) (\lambda (x3: T).(pr2 a x1 x3)) x0 H27 (pr2_free a x1 x0 H10)))))) -(subst1_confluence_eq t4 t u i (subst1_single i u t4 t H2) (lift (S O) i x0) -H22))) u0 H19)))))) H18))) d0 H12)))))) (\lambda (H12: (lt d0 i)).(ex2_ind T -(\lambda (t0: T).(subst1 d0 u0 t t0)) (\lambda (t0: T).(subst1 i u (lift (S -O) d0 x0) t0)) (ex2 T (\lambda (x2: T).(subst1 d0 u0 t (lift (S O) d0 x2))) -(\lambda (x2: T).(pr2 a x1 x2))) (\lambda (x2: T).(\lambda (H13: (subst1 d0 -u0 t x2)).(\lambda (H14: (subst1 i u (lift (S O) d0 x0) x2)).(ex2_ind T -(\lambda (t5: T).(eq T x2 (lift (S O) d0 t5))) (\lambda (t5: T).(subst1 -(minus i (S O)) u x0 t5)) (ex2 T (\lambda (x3: T).(subst1 d0 u0 t (lift (S O) -d0 x3))) (\lambda (x3: T).(pr2 a x1 x3))) (\lambda (x3: T).(\lambda (H15: (eq -T x2 (lift (S O) d0 x3))).(\lambda (H16: (subst1 (minus i (S O)) u x0 -x3)).(let H17 \def (eq_ind T x2 (\lambda (t0: T).(subst1 d0 u0 t t0)) H13 -(lift (S O) d0 x3) H15) in (ex_intro2 T (\lambda (x4: T).(subst1 d0 u0 t -(lift (S O) d0 x4))) (\lambda (x4: T).(pr2 a x1 x4)) x3 H17 (pr2_delta1 a d u -(minus i (S O)) (getl_drop_conf_ge i (CHead d (Bind Abbr) u) a0 -(csubst1_getl_ge d0 i (le_S_n d0 i (le_S (S d0) i H12)) c0 a0 u0 H4 (CHead d -(Bind Abbr) u) H0) a (S O) d0 H5 (eq_ind_r nat (plus (S O) d0) (\lambda (n: -nat).(le n i)) H12 (plus d0 (S O)) (plus_comm d0 (S O)))) x1 x0 H10 x3 -H16)))))) (subst1_gen_lift_ge u x0 x2 i (S O) d0 H14 (eq_ind_r nat (plus (S -O) d0) (\lambda (n: nat).(le n i)) H12 (plus d0 (S O)) (plus_comm d0 (S -O)))))))) (subst1_confluence_neq t4 t u i (subst1_single i u t4 t H2) (lift -(S O) d0 x0) u0 d0 H11 (sym_not_equal nat d0 i (lt_neq d0 i H12)))))))))) -(pr0_gen_lift x1 x (S O) d0 H7))))) (pr0_subst1 t3 t4 H1 u0 (lift (S O) d0 -x1) d0 H6 u0 (pr0_refl u0))))))))))))))))))))))) c t1 t2 H)))). - -- 2.39.2