- some renaming

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / dynamic / snv_cpcs.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma
index 643bf6569fde0a82ea166f1278fccf5117e9fc90..f00ae1832ae6ad453f776a795a6ae18e9d58e01d 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/dynamic/snv_cpcs.ma
@@ -29,11 +29,11 @@ definition IH_ssta_cpr_lpr: ∀h:sh. sd h → relation2 lenv term ≝
                             λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[g] →
                             ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
                             ∀T2. L1 ⊢ T1 ➡ T2 → ∀L2. L1 ⊢ ➡ L2 →
-                            ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L1 ⊢ U1 ⬌* U2.
+                            ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L2 ⊢ U1 ⬌* U2.
 
 definition IH_snv_ssta: ∀h:sh. sd h → relation2 lenv term ≝
-                        λh,g,L1,T1. ⦃h, L1⦄ ⊢ T1 ¡[g] →
-                        ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l+1, U1⦄ → ⦃h, L1⦄ ⊢ U1 ¡[g].
+                        λh,g,L,T. ⦃h, L⦄ ⊢ T ¡[g] →
+                        ∀U,l. ⦃h, L⦄ ⊢ T •[g] ⦃l+1, U⦄ → ⦃h, L⦄ ⊢ U ¡[g].
 
 definition IH_snv_lsubsv: ∀h:sh. sd h → relation2 lenv term ≝
                           λh,g,L2,T. ⦃h, L2⦄ ⊢ T ¡[g] →
@@ -55,41 +55,51 @@ fact ssta_cprs_lpr_aux: ∀h,g,L0,T0.
                         (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
                         ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
                         ∀U1,l. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l, U1⦄ →
-                        ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 → 
-                        ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L1 ⊢ U1 ⬌* U2.
+                        ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
+                        ∃∃U2. ⦃h, L2⦄ ⊢ T2 •[g] ⦃l, U2⦄ & L2 ⊢ U1 ⬌* U2.
 #h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #l #HTU1 #T2 #H
-elim H -T2 [ /2 width=5/ ]
+elim H -T2 [ /2 width=7/ ]
 #T #T2 #HT1T #HTT2 #IHT1 #L2 #HL12
 elim (IHT1 L1) // -IHT1 #U #HTU #HU1
-elim (IH1 … HTU … HTT2 … HL12) -IH1 -HTU -HTT2 -HL12
+elim (IH1 … HTU … HTT2 … HL12) -IH1 -HTU -HTT2
 [2: /3 width=9 by snv_cprs_lpr_aux/
 |3: /5 width=6 by ygt_yprs_trans, cprs_yprs/
 ] -L0 -T0 -T1 -T #U2 #HTU2 #HU2
+lapply (cpcs_lpr_conf … HL12 … HU1) -L1 #HU1
 lapply (cpcs_trans … HU1 … HU2) -U /2 width=3/
 qed-.
 
-fact ssta_cpcs_aux: ∀h,g,L0,T0.
-                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
-                    (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
-                    ∀L,T1,T2. h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T1⦄ → h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T2⦄ →
-                        ⦃h, L⦄ ⊢ T1 ¡[g] → ⦃h, L⦄ ⊢ T2 ¡[g] →
-                    ∀U1,l1. ⦃h, L⦄ ⊢ T1 •[g] ⦃l1, U1⦄ →
-                    ∀U2,l2. ⦃h, L⦄ ⊢ T2 •[g] ⦃l2, U2⦄ →
-                    L ⊢ T1 ⬌* T2 → l1 = l2 ∧ L ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH2 #IH1 #L #T1 #T2 #HLT01 #HLT02 #HT1 #HT2 #U1 #l1 #HTU1 #U2 #l2 #HTU2 #H
+fact ssta_cpcs_lpr_aux: ∀h,g,L0,T0.
+                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+                        ∀L1,T1,T2. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T2⦄ →
+                        ⦃h, L1⦄ ⊢ T1 ¡[g] → ⦃h, L1⦄ ⊢ T2 ¡[g] →
+                        ∀U1,l1. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l1, U1⦄ →
+                        ∀U2,l2. ⦃h, L1⦄ ⊢ T2 •[g] ⦃l2, U2⦄ →
+                        L1 ⊢ T1 ⬌* T2 → ∀L2. L1 ⊢ ➡ L2 →
+                        l1 = l2 ∧ L2 ⊢ U1 ⬌* U2.
+#h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #T2 #H01 #H02 #HT1 #HT2 #U1 #l1 #HTU1 #U2 #l2 #HTU2 #H #L2 #HL12
 elim (cpcs_inv_cprs … H) -H #T #H1 #H2
-elim (ssta_cprs_lpr_aux … HLT01 HT1 … HTU1 … H1 L) -T1 // /2 width=1/ #W1 #H1 #HUW1
-elim (ssta_cprs_lpr_aux … HLT02 HT2 … HTU2 … H2 L) -T2 // /2 width=1/ #W2 #H2 #HUW2 -L0 -T0
+elim (ssta_cprs_lpr_aux … H01 HT1 … HTU1 … H1 … HL12) -T1 /2 width=1/ #W1 #H1 #HUW1
+elim (ssta_cprs_lpr_aux … H02 HT2 … HTU2 … H2 … HL12) -T2 /2 width=1/ #W2 #H2 #HUW2 -L0 -T0
 elim (ssta_mono … H1 … H2) -h -T #H1 #H2 destruct
 lapply (cpcs_canc_dx … HUW1 … HUW2) -W2 /2 width=1/
 qed-.
 
 fact snv_sstas_aux: ∀h,g,L0,T0.
                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                    ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
-                    ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → ⦃h, L1⦄ ⊢ U1 ¡[g].
-#h #g #L0 #T0 #IH #L1 #T1 #HLT0 #HT1 #U1 #H
-@(sstas_ind … H) -U1 // -HT1 /4 width=5 by ygt_yprs_trans, sstas_yprs/
+                    ∀L,T. h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T⦄ → ⦃h, L⦄ ⊢ T ¡[g] →
+                    ∀U. ⦃h, L⦄ ⊢ T •*[g] U → ⦃h, L⦄ ⊢ U ¡[g].
+#h #g #L0 #T0 #IH #L #T #H01 #HT #U #H
+@(sstas_ind … H) -U // -HT /4 width=5 by ygt_yprs_trans, sstas_yprs/
+qed-.
+
+fact snv_sstas_lpr_aux: ∀h,g,L0,T0.
+                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                        (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+                        ∀L1,T. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T⦄ → ⦃h, L1⦄ ⊢ T ¡[g] →
+                        ∀U. ⦃h, L1⦄ ⊢ T •*[g] U → ∀L2. L1 ⊢ ➡ L2 → ⦃h, L2⦄ ⊢ U ¡[g].
+/4 width=7 by snv_sstas_aux, ygt_yprs_trans, sstas_yprs/
 qed-.
 
 fact sstas_cprs_lpr_aux: ∀h,g,L0,T0.
@@ -97,45 +107,50 @@ fact sstas_cprs_lpr_aux: ∀h,g,L0,T0.
                          (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
                          (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
                          ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
-                         ∀T2. L1 ⊢ T1 ➡* T2 → ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → ∀L2. L1 ⊢ ➡ L2 →
-                         ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L1 ⊢ U1 ⬌* U2.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #L2 #HL12 #T2 #HT12 #U1 #H
-@(sstas_ind … H) -U1 [ /3 width=3/ ]
-#U1 #W1 #l1 #HTU1 #HUW1 * #U2 #HTU2 #HU12
-lapply (snv_ltpr_cprs_aux … IH2 … HT1 … HT12) // #HT2
+                         ∀U1. ⦃h, L1⦄ ⊢ T1 •*[g] U1 → ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
+                         ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*[g] U2 & L2 ⊢ U1 ⬌* U2.
+#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 #H
+@(sstas_ind … H) -U1 [ /3 width=5 by cprs_lpr_conf, ex2_intro/ ]
+#U1 #W1 #l1 #HTU1 #HUW1 #IHTU1 #T2 #HT12 #L2 #HL12
+elim (IHTU1 … HT12 … HL12) -IHTU1 #U2 #HTU2 #HU12
+lapply (snv_cprs_lpr_aux … IH2 … HT1 … HT12 … HL12) // #HT2
 elim (snv_sstas_fwd_correct … HTU2) // #W2 #l2 #HUW2
-elim (ssta_ltpr_cpcs_aux … IH2 IH1 … HUW1 … HUW2 … HU12) -IH2 -IH1 -HUW1 -HU12 //
-[2: /4 width=8 by snv_sstas_aux, ygt_yprs_trans, ltpr_cprs_yprs/
-|3: /3 width=7 by snv_sstas_aux, ygt_yprs_trans, cprs_yprs/
-|4: /4 width=5 by ygt_yprs_trans, ltpr_cprs_yprs, sstas_yprs/
-|5: /3 width=4 by ygt_yprs_trans, cprs_yprs, sstas_yprs/
-] -L0 -T0 -T1 -HT2 #H #HW12 destruct /3 width=4/
+elim (IH1 … HUW1 U1 … HL12) -HUW1 //
+[2: /3 width=7 by snv_sstas_aux/
+|3: /3 width=4 by ygt_yprs_trans, sstas_yprs/
+] #W #HU1W #HW1
+elim (ssta_cpcs_lpr_aux … IH2 IH1 … HU1W … HUW2 … HU12 L2) // -IH1 -HU1W -HU12
+[2: /4 width=8 by snv_sstas_aux, ygt_yprs_trans, cprs_lpr_yprs/
+|3: /3 width=10 by snv_sstas_lpr_aux/
+|4,5: /4 width=5 by ygt_yprs_trans, cprs_lpr_yprs, sstas_yprs/
+] -L0 -T0 -L1 -T1 -HT2 #H #HW12 destruct
+lapply (cpcs_trans … HW1 … HW12) -W /3 width=4/
 qed-.
 
-fact dxprs_ltpr_cprs_aux: ∀h,g,L0,T0.
-                          (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
-                          (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
-                          (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
-                          ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
-                          ∀U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
-                          ∀L2. L1 ➡ L2 → ∀T2. L2 ⊢ T1 ➡* T2 →
-                          ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[g] U2 & L2 ⊢ U1 ➡* U2.
-#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 * #W1 #HTW1 #HWU1 #L2 #HL12 #T2 #HT12
-elim (sstas_ltpr_cprs_aux … IH3 IH2 IH1 … H01 … HT12 … HTW1) // -L0 -T0 -T1 #W2 #HTW2 #HW12
-lapply (ltpr_cprs_conf … HL12 … HWU1) -L1 #HWU1
+fact dxprs_cprs_lpr_aux: ∀h,g,L0,T0.
+                         (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ssta h g L1 T1) →
+                         (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                         (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+                         ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
+                         ∀U1. ⦃h, L1⦄ ⊢ T1 •*➡*[g] U1 →
+                         ∀T2. L1 ⊢ T1 ➡* T2 → ∀L2. L1 ⊢ ➡ L2 →
+                         ∃∃U2. ⦃h, L2⦄ ⊢ T2 •*➡*[g] U2 & L2 ⊢ U1 ➡* U2.
+#h #g #L0 #T0 #IH3 #IH2 #IH1 #L1 #T1 #H01 #HT1 #U1 * #W1 #HTW1 #HWU1 #T2 #HT12 #L2 #HL12
+elim (sstas_cprs_lpr_aux … IH3 IH2 IH1 … H01 … HTW1 … HT12 … HL12) // -L0 -T0 -T1 #W2 #HTW2 #HW12
+lapply (cprs_lpr_conf … HL12 … HWU1) -L1 #HWU1
 lapply (cpcs_canc_sn … HW12 HWU1) -W1 #H
 elim (cpcs_inv_cprs … H) -H /3 width=3/
 qed-.
 
 fact ssta_dxprs_aux: ∀h,g,L0,T0.
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_ltpr_tpr h g L1 T1) →
-                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_ltpr_tpr h g L1 T1) →
-                     ∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → ⦃h, L1⦄ ⊢ T1 ¡[g] →
-                     ∀l,U1. ⦃h, L1⦄ ⊢ T1 •[g] ⦃l+1, U1⦄ → ∀T2. ⦃h, L1⦄ ⊢ T1 •*➡*[g] T2 →
-                     ∃∃U,U2. ⦃h, L1⦄ ⊢ U1 •*[g] U & ⦃h, L1⦄ ⊢ T2 •*[g] U2 & L1 ⊢ U ⬌* U2.
-#h #g #L0 #T0 #IH2 #IH1 #L1 #T1 #H01 #HT1 #l #U1 #HTU1 #T2 * #T #HT1T #HTT2
+                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_snv_cpr_lpr h g L1 T1) →
+                     (∀L1,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L1, T1⦄ → IH_ssta_cpr_lpr h g L1 T1) →
+                     ∀L,T1. h ⊢ ⦃L0, T0⦄ >[g] ⦃L, T1⦄ → ⦃h, L⦄ ⊢ T1 ¡[g] →
+                     ∀l,U1. ⦃h, L⦄ ⊢ T1 •[g] ⦃l+1, U1⦄ → ∀T2. ⦃h, L⦄ ⊢ T1 •*➡*[g] T2 →
+                     ∃∃U,U2. ⦃h, L⦄ ⊢ U1 •*[g] U & ⦃h, L⦄ ⊢ T2 •*[g] U2 & L ⊢ U ⬌* U2.
+#h #g #L0 #T0 #IH2 #IH1 #L #T1 #H01 #HT1 #l #U1 #HTU1 #T2 * #T #HT1T #HTT2
 elim (sstas_strip … HT1T … HTU1) #HU1T destruct [ -HT1T | -L0 -T0 -T1 ]
-[ elim (ssta_ltpr_cprs_aux … IH2 IH1 … HTU1 L1 … HTT2) // -L0 -T0 -T /3 width=5/
+[ elim (ssta_cprs_lpr_aux … IH2 IH1 … HTU1 … HTT2 L) // -L0 -T0 -T /3 width=5/
 | @(ex3_2_intro …T2 HU1T) // /2 width=1/
 ]
 qed-.