X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;ds=inline;f=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_computation%2Fgcp_cr.ma;fp=matita%2Fmatita%2Fcontribs%2Flambdadelta%2Fbasic_2%2Frt_computation%2Fgcp_cr.ma;h=dee244aac86223aafb08e91fb3ae28f50d2b03d4;hb=e9f96fa56226dfd74de214c89d827de0c5018ac7;hp=0000000000000000000000000000000000000000;hpb=ad3ca38634cfae29e8c26d0ab23cb466407eca5e;p=helm.git diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_computation/gcp_cr.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/gcp_cr.ma new file mode 100644 index 000000000..dee244aac --- /dev/null +++ b/matita/matita/contribs/lambdadelta/basic_2/rt_computation/gcp_cr.ma @@ -0,0 +1,169 @@ +(**************************************************************************) +(* ___ *) +(* ||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 *) +(* *) +(**************************************************************************) + +include "basic_2/notation/relations/ineint_5.ma". +include "basic_2/grammar/aarity.ma". +include "basic_2/multiple/mr2_mr2.ma". +include "basic_2/multiple/lifts_lift_vector.ma". +include "basic_2/multiple/drops_drop.ma". +include "basic_2/computation/gcp.ma". + +(* GENERIC COMPUTATION PROPERTIES *******************************************) + +(* Note: this is Girard's CR1 *) +definition S1 ≝ λRP,C:candidate. + ∀G,L,T. C G L T → RP G L T. + +(* Note: this is Tait's iii, or Girard's CR4 *) +definition S2 ≝ λRR:relation4 genv lenv term term. λRS:relation term. λRP,C:candidate. + ∀G,L,Vs. all … (RP G L) Vs → + ∀T. 𝐒⦃T⦄ → NF … (RR G L) RS T → C G L (ⒶVs.T). + +(* Note: this generalizes Tait's ii *) +definition S3 ≝ λC:candidate. + ∀a,G,L,Vs,V,T,W. + C G L (ⒶVs.ⓓ{a}ⓝW.V.T) → C G L (ⒶVs.ⓐV.ⓛ{a}W.T). + +definition S4 ≝ λRP,C:candidate. + ∀G,L,Vs. all … (RP G L) Vs → ∀s. C G L (ⒶVs.⋆s). + +definition S5 ≝ λC:candidate. ∀I,G,L,K,Vs,V1,V2,i. + C G L (ⒶVs.V2) → ⬆[0, i+1] V1 ≡ V2 → + ⬇[i] L ≡ K.ⓑ{I}V1 → C G L (ⒶVs.#i). + +definition S6 ≝ λRP,C:candidate. + ∀G,L,V1c,V2c. ⬆[0, 1] V1c ≡ V2c → + ∀a,V,T. C G (L.ⓓV) (ⒶV2c.T) → RP G L V → C G L (ⒶV1c.ⓓ{a}V.T). + +definition S7 ≝ λC:candidate. + ∀G,L,Vs,T,W. C G L (ⒶVs.T) → C G L (ⒶVs.W) → C G L (ⒶVs.ⓝW.T). + +(* requirements for the generic reducibility candidate *) +record gcr (RR:relation4 genv lenv term term) (RS:relation term) (RP,C:candidate) : Prop ≝ +{ c1: S1 RP C; + c2: S2 RR RS RP C; + c3: S3 C; + c4: S4 RP C; + c5: S5 C; + c6: S6 RP C; + c7: S7 C +}. + +(* the functional construction for candidates *) +definition cfun: candidate → candidate → candidate ≝ + λC1,C2,G,K,T. ∀L,W,U,cs. + ⬇*[Ⓕ, cs] L ≡ K → ⬆*[cs] T ≡ U → C1 G L W → C2 G L (ⓐW.U). + +(* the reducibility candidate associated to an atomic arity *) +rec definition acr (RP:candidate) (A:aarity) on A: candidate ≝ +match A with +[ AAtom ⇒ RP +| APair B A ⇒ cfun (acr RP B) (acr RP A) +]. + +interpretation + "candidate of reducibility of an atomic arity (abstract)" + 'InEInt RP G L T A = (acr RP A G L T). + +(* Basic properties *********************************************************) + +(* Basic 1: was: sc3_lift *) +lemma gcr_lift: ∀RR,RS,RP. gcp RR RS RP → ∀A,G. d_liftable1 (acr RP A G) (Ⓕ). +#RR #RS #RP #H #A elim A -A +/3 width=8 by cp2, drops_cons, lifts_cons/ +qed. + +(* Basic_1: was: sc3_lift1 *) +lemma gcr_lifts: ∀RR,RS,RP. gcp RR RS RP → ∀A,G. d_liftables1 (acr RP A G) (Ⓕ). +#RR #RS #RP #H #A #G @d1_liftable_liftables /2 width=7 by gcr_lift/ +qed. + +(* Basic_1: was: + sc3_sn3 sc3_abst sc3_appl sc3_abbr sc3_bind sc3_cast +*) +lemma acr_gcr: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP → + ∀A. gcr RR RS RP (acr RP A). +#RR #RS #RP #H1RP #H2RP #A elim A -A // +#B #A #IHB #IHA @mk_gcr +[ #G #L #T #H + elim (cp1 … H1RP G L) #s #HK + lapply (H L (⋆s) T (◊) ? ? ?) -H // + [ lapply (c2 … IHB G L (◊) … HK) // + | /3 width=6 by c1, cp3/ + ] +| #G #L #Vs #HVs #T #H1T #H2T #L0 #V0 #X #cs #HL0 #H #HB + elim (lifts_inv_applv1 … H) -H #V0c #T0 #HV0c #HT0 #H destruct + lapply (c1 … IHB … HB) #HV0 + @(c2 … IHA … (V0 @ V0c)) + /3 width=14 by gcp2_lifts_all, gcp2_lifts, gcp0_lifts, lifts_simple_dx, conj/ +| #a #G #L #Vs #U #T #W #HA #L0 #V0 #X #cs #HL0 #H #HB + elim (lifts_inv_applv1 … H) -H #V0c #Y #HV0c #HY #H destruct + elim (lifts_inv_flat1 … HY) -HY #U0 #X #HU0 #HX #H destruct + elim (lifts_inv_bind1 … HX) -HX #W0 #T0 #HW0 #HT0 #H destruct + @(c3 … IHA … (V0 @ V0c)) /5 width=6 by lifts_applv, lifts_flat, lifts_bind/ +| #G #L #Vs #HVs #s #L0 #V0 #X #cs #HL0 #H #HB + elim (lifts_inv_applv1 … H) -H #V0c #Y #HV0c #HY #H destruct + >(lifts_inv_sort1 … HY) -Y + lapply (c1 … IHB … HB) #HV0 + @(c4 … IHA … (V0 @ V0c)) /3 width=7 by gcp2_lifts_all, conj/ +| #I #G #L #K #Vs #V1 #V2 #i #HA #HV12 #HLK #L0 #V0 #X #cs #HL0 #H #HB + elim (lifts_inv_applv1 … H) -H #V0c #Y #HV0c #HY #H destruct + elim (lifts_inv_lref1 … HY) -HY #i0 #Hi0 #H destruct + elim (drops_drop_trans … HL0 … HLK) #X #cs0 #i1 #HL02 #H #Hi1 #Hcs0 + >(at_mono … Hi1 … Hi0) in HL02; -i1 #HL02 + elim (drops_inv_skip2 … Hcs0 … H) -H -cs0 #L2 #W1 #cs0 #Hcs0 #HLK #HVW1 #H destruct + elim (lift_total W1 0 (i0 + 1)) #W2 #HW12 + elim (lifts_lift_trans … Hcs0 … HVW1 … HW12) // -Hcs0 -Hi0 #V3 #HV13 #HVW2 + >(lift_mono … HV13 … HV12) in HVW2; -V3 #HVW2 + @(c5 … IHA … (V0 @ V0c) … HW12 HL02) /3 width=5 by lifts_applv/ +| #G #L #V1c #V2c #HV12c #a #V #T #HA #HV #L0 #V10 #X #cs #HL0 #H #HB + elim (lifts_inv_applv1 … H) -H #V10c #Y #HV10c #HY #H destruct + elim (lifts_inv_bind1 … HY) -HY #V0 #T0 #HV0 #HT0 #H destruct + elim (lift_total V10 0 1) #V20 #HV120 + elim (liftv_total 0 1 V10c) #V20c #HV120c + @(c6 … IHA … (V10 @ V10c) (V20 @ V20c)) /3 width=7 by gcp2_lifts, liftv_cons/ + @(HA … (cs + 1)) /2 width=2 by drops_skip/ + [ @lifts_applv // + elim (liftsv_liftv_trans_le … HV10c … HV120c) -V10c #V10c #HV10c #HV120c + >(liftv_mono … HV12c … HV10c) -V1c // + | @(gcr_lift … H1RP … HB … HV120) /2 width=2 by drop_drop/ + ] +| #G #L #Vs #T #W #HA #HW #L0 #V0 #X #cs #HL0 #H #HB + elim (lifts_inv_applv1 … H) -H #V0c #Y #HV0c #HY #H destruct + elim (lifts_inv_flat1 … HY) -HY #W0 #T0 #HW0 #HT0 #H destruct + @(c7 … IHA … (V0 @ V0c)) /3 width=5 by lifts_applv/ +] +qed. + +lemma acr_abst: ∀RR,RS,RP. gcp RR RS RP → gcr RR RS RP RP → + ∀a,G,L,W,T,A,B. ⦃G, L, W⦄ ϵ[RP] 〚B〛 → ( + ∀L0,V0,W0,T0,cs. ⬇*[Ⓕ, cs] L0 ≡ L → ⬆*[cs] W ≡ W0 → ⬆*[cs + 1] T ≡ T0 → + ⦃G, L0, V0⦄ ϵ[RP] 〚B〛 → ⦃G, L0, W0⦄ ϵ[RP] 〚B〛 → ⦃G, L0.ⓓⓝW0.V0, T0⦄ ϵ[RP] 〚A〛 + ) → + ⦃G, L, ⓛ{a}W.T⦄ ϵ[RP] 〚②B.A〛. +#RR #RS #RP #H1RP #H2RP #a #G #L #W #T #A #B #HW #HA #L0 #V0 #X #cs #HL0 #H #HB +lapply (acr_gcr … H1RP H2RP A) #HCA +lapply (acr_gcr … H1RP H2RP B) #HCB +elim (lifts_inv_bind1 … H) -H #W0 #T0 #HW0 #HT0 #H destruct +lapply (gcr_lifts … H1RP … HL0 … HW0 HW) -HW #HW0 +lapply (c3 … HCA … a G L0 (◊)) #H @H -H +lapply (c6 … HCA G L0 (◊) (◊) ?) // #H @H -H +[ @(HA … HL0) // +| lapply (c1 … HCB) -HCB #HCB + lapply (c7 … H2RP G L0 (◊)) /3 width=1 by/ +] +qed. + +(* Basic_1: removed theorems 2: sc3_arity_gen sc3_repl *) +(* Basic_1: removed local theorems 1: sc3_sn3_abst *)