X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=sidebyside;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FLAMBDA-TYPES%2FLambdaDelta-1%2Fnf2%2Ffwd.ma;h=849e860904d603e79a610557eb7d561fde347984;hb=fdda444a05fe4c68c925cd94e4e3a38c93d0c35f;hp=27a629724bea9fe9b1eacd717c1467f55304f06a;hpb=9376f52b7f5890d924ae7d93bcae2af9e516126d;p=helm.git diff --git a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/fwd.ma b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/fwd.ma index 27a629724..849e86090 100644 --- a/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/fwd.ma +++ b/helm/software/matita/contribs/LAMBDA-TYPES/LambdaDelta-1/nf2/fwd.ma @@ -20,6 +20,8 @@ include "nf2/defs.ma". include "pr2/clen.ma". +include "subst0/dec.ma". + include "T/props.ma". theorem nf2_gen_lref: @@ -65,6 +67,22 @@ t)) \to (\forall (P: Prop).P)))) (pr2_free c (THead (Flat Cast) u t) t (pr0_epsilon t t (pr0_refl t) u))) P))))). +theorem nf2_gen_beta: + \forall (c: C).(\forall (u: T).(\forall (v: T).(\forall (t: T).((nf2 c +(THead (Flat Appl) u (THead (Bind Abst) v t))) \to (\forall (P: Prop).P))))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (v: T).(\lambda (t: T).(\lambda (H: +((\forall (t2: T).((pr2 c (THead (Flat Appl) u (THead (Bind Abst) v t)) t2) +\to (eq T (THead (Flat Appl) u (THead (Bind Abst) v t)) t2))))).(\lambda (P: +Prop).(let H0 \def (eq_ind T (THead (Flat Appl) u (THead (Bind Abst) v t)) +(\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) with [(TSort _) +\Rightarrow False | (TLRef _) \Rightarrow False | (THead k _ _) \Rightarrow +(match k in K return (\lambda (_: K).Prop) with [(Bind _) \Rightarrow False | +(Flat _) \Rightarrow True])])) I (THead (Bind Abbr) u t) (H (THead (Bind +Abbr) u t) (pr2_free c (THead (Flat Appl) u (THead (Bind Abst) v t)) (THead +(Bind Abbr) u t) (pr0_beta v u u (pr0_refl u) t t (pr0_refl t))))) in +(False_ind P H0))))))). + theorem nf2_gen_flat: \forall (f: F).(\forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Flat f) u t)) \to (land (nf2 c u) (nf2 c t)))))) @@ -83,3 +101,98 @@ u t2)).(let H1 \def (f_equal T T (\lambda (e: T).(match e in T return (THead (Flat f) u t) (THead (Flat f) u t2) (H (THead (Flat f) u t2) (pr2_head_2 c u t t2 (Flat f) (pr2_cflat c t t2 H0 f u)))) in H1)))))))). +theorem nf2_gen__aux: + \forall (b: B).(\forall (x: T).(\forall (u: T).(\forall (d: nat).((eq T +(THead (Bind b) u (lift (S O) d x)) x) \to (\forall (P: Prop).P))))) +\def + \lambda (b: B).(\lambda (x: T).(T_ind (\lambda (t: T).(\forall (u: +T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to +(\forall (P: Prop).P))))) (\lambda (n: nat).(\lambda (u: T).(\lambda (d: +nat).(\lambda (H: (eq T (THead (Bind b) u (lift (S O) d (TSort n))) (TSort +n))).(\lambda (P: Prop).(let H0 \def (eq_ind T (THead (Bind b) u (lift (S O) +d (TSort n))) (\lambda (ee: T).(match ee in T return (\lambda (_: T).Prop) +with [(TSort _) \Rightarrow False | (TLRef _) \Rightarrow False | (THead _ _ +_) \Rightarrow True])) I (TSort n) H) in (False_ind P H0))))))) (\lambda (n: +nat).(\lambda (u: T).(\lambda (d: nat).(\lambda (H: (eq T (THead (Bind b) u +(lift (S O) d (TLRef n))) (TLRef n))).(\lambda (P: Prop).(let H0 \def (eq_ind +T (THead (Bind b) u (lift (S O) d (TLRef n))) (\lambda (ee: T).(match ee in T +return (\lambda (_: T).Prop) with [(TSort _) \Rightarrow False | (TLRef _) +\Rightarrow False | (THead _ _ _) \Rightarrow True])) I (TLRef n) H) in +(False_ind P H0))))))) (\lambda (k: K).(\lambda (t: T).(\lambda (_: ((\forall +(u: T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t)) t) \to +(\forall (P: Prop).P)))))).(\lambda (t0: T).(\lambda (H0: ((\forall (u: +T).(\forall (d: nat).((eq T (THead (Bind b) u (lift (S O) d t0)) t0) \to +(\forall (P: Prop).P)))))).(\lambda (u: T).(\lambda (d: nat).(\lambda (H1: +(eq T (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t +t0))).(\lambda (P: Prop).(let H2 \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 +(lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let H3 \def (f_equal T +T (\lambda (e: T).(match e in T return (\lambda (_: T).T) with [(TSort _) +\Rightarrow u | (TLRef _) \Rightarrow u | (THead _ t1 _) \Rightarrow t1])) +(THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t t0) H1) in ((let +H4 \def (f_equal T T (\lambda (e: T).(match e in T return (\lambda (_: T).T) +with [(TSort _) \Rightarrow (THead k ((let rec lref_map (f: ((nat \to nat))) +(d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort +n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i +| false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 +(lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: +nat).(plus x0 (S O))) d t) ((let rec lref_map (f: ((nat \to nat))) (d0: nat) +(t1: T) on t1: T \def (match t1 with [(TSort n) \Rightarrow (TSort n) | +(TLRef i) \Rightarrow (TLRef (match (blt i d0) with [true \Rightarrow i | +false \Rightarrow (f i)])) | (THead k0 u0 t2) \Rightarrow (THead k0 (lref_map +f d0 u0) (lref_map f (s k0 d0) t2))]) in lref_map) (\lambda (x0: nat).(plus +x0 (S O))) (s k d) t0)) | (TLRef _) \Rightarrow (THead k ((let rec lref_map +(f: ((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort +n) \Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) +with [true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2) +\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in +lref_map) (\lambda (x0: nat).(plus x0 (S O))) d t) ((let rec lref_map (f: +((nat \to nat))) (d0: nat) (t1: T) on t1: T \def (match t1 with [(TSort n) +\Rightarrow (TSort n) | (TLRef i) \Rightarrow (TLRef (match (blt i d0) with +[true \Rightarrow i | false \Rightarrow (f i)])) | (THead k0 u0 t2) +\Rightarrow (THead k0 (lref_map f d0 u0) (lref_map f (s k0 d0) t2))]) in +lref_map) (\lambda (x0: nat).(plus x0 (S O))) (s k d) t0)) | (THead _ _ t1) +\Rightarrow t1])) (THead (Bind b) u (lift (S O) d (THead k t t0))) (THead k t +t0) H1) in (\lambda (_: (eq T u t)).(\lambda (H6: (eq K (Bind b) k)).(let H7 +\def (eq_ind_r K k (\lambda (k0: K).(eq T (lift (S O) d (THead k0 t t0)) t0)) +H4 (Bind b) H6) in (let H8 \def (eq_ind T (lift (S O) d (THead (Bind b) t +t0)) (\lambda (t1: T).(eq T t1 t0)) H7 (THead (Bind b) (lift (S O) d t) (lift +(S O) (S d) t0)) (lift_bind b t t0 (S O) d)) in (H0 (lift (S O) d t) (S d) H8 +P)))))) H3)) H2))))))))))) x)). + +theorem nf2_gen_abbr: + \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Abbr) u +t)) \to (\forall (P: Prop).P)))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: +T).((pr2 c (THead (Bind Abbr) u t) t2) \to (eq T (THead (Bind Abbr) u t) +t2))))).(\lambda (P: Prop).(let H_x \def (dnf_dec u t O) in (let H0 \def H_x +in (ex_ind T (\lambda (v: T).(or (subst0 O u t (lift (S O) O v)) (eq T t +(lift (S O) O v)))) P (\lambda (x: T).(\lambda (H1: (or (subst0 O u t (lift +(S O) O x)) (eq T t (lift (S O) O x)))).(or_ind (subst0 O u t (lift (S O) O +x)) (eq T t (lift (S O) O x)) P (\lambda (H2: (subst0 O u t (lift (S O) O +x))).(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 _ +_ t0) \Rightarrow t0])) (THead (Bind Abbr) u t) (THead (Bind Abbr) u (lift (S +O) O x)) (H (THead (Bind Abbr) u (lift (S O) O x)) (pr2_free c (THead (Bind +Abbr) u t) (THead (Bind Abbr) u (lift (S O) O x)) (pr0_delta u u (pr0_refl u) +t t (pr0_refl t) (lift (S O) O x) H2)))) in (let H4 \def (eq_ind T t (\lambda +(t0: T).(subst0 O u t0 (lift (S O) O x))) H2 (lift (S O) O x) H3) in +(subst0_refl u (lift (S O) O x) O H4 P)))) (\lambda (H2: (eq T t (lift (S O) +O x))).(let H3 \def (eq_ind T t (\lambda (t0: T).(\forall (t2: T).((pr2 c +(THead (Bind Abbr) u t0) t2) \to (eq T (THead (Bind Abbr) u t0) t2)))) H +(lift (S O) O x) H2) in (nf2_gen__aux Abbr x u O (H3 x (pr2_free c (THead +(Bind Abbr) u (lift (S O) O x)) x (pr0_zeta Abbr not_abbr_abst x x (pr0_refl +x) u))) P))) H1))) H0))))))). + +theorem nf2_gen_void: + \forall (c: C).(\forall (u: T).(\forall (t: T).((nf2 c (THead (Bind Void) u +(lift (S O) O t))) \to (\forall (P: Prop).P)))) +\def + \lambda (c: C).(\lambda (u: T).(\lambda (t: T).(\lambda (H: ((\forall (t2: +T).((pr2 c (THead (Bind Void) u (lift (S O) O t)) t2) \to (eq T (THead (Bind +Void) u (lift (S O) O t)) t2))))).(\lambda (P: Prop).(nf2_gen__aux Void t u O +(H t (pr2_free c (THead (Bind Void) u (lift (S O) O t)) t (pr0_zeta Void +not_void_abst t t (pr0_refl t) u))) P))))). +