partial commit: "computation" component ...

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / computation / lprs_cprs.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/computation/lprs_cprs.ma b/matita/matita/contribs/lambdadelta/basic_2/computation/lprs_cprs.ma
index c3ba54ef23b5d66880e95fd16b74c9c04cc062d3..a1c4086603883fa52155bd5291d4a0656765cb51 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/computation/lprs_cprs.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/computation/lprs_cprs.ma
@@ -19,32 +19,35 @@ include "basic_2/computation/lprs.ma".
 
 (* Advanced properties ******************************************************)
 
-lemma lprs_pair: ∀I,L1,L2. L1 ⊢ ➡* L2 → ∀V1,V2. L1 ⊢ V1 ➡* V2 →
-                 L1. ⓑ{I} V1 ⊢ ➡* L2.ⓑ{I} V2.
+lemma lprs_pair: ∀I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
+                 ∀V1,V2. ⦃G, L1⦄ ⊢ V1 ➡* V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡* L2.ⓑ{I}V2.
 /2 width=1 by TC_lpx_sn_pair/ qed.
 
 (* Advanced inversion lemmas ************************************************)
 
-lemma lprs_inv_pair1: ∀I,K1,L2,V1. K1. ⓑ{I} V1 ⊢ ➡* L2 →
-                      ∃∃K2,V2. K1 ⊢ ➡* K2 & K1 ⊢ V1 ➡* V2 & L2 = K2. ⓑ{I} V2.
+lemma lprs_inv_pair1: ∀I,G,K1,L2,V1. ⦃G, K1.ⓑ{I}V1⦄ ⊢ ➡* L2 →
+                      ∃∃K2,V2. ⦃G, K1⦄ ⊢ ➡* K2 & ⦃G, K1⦄ ⊢ V1 ➡* V2 &
+                               L2 = K2.ⓑ{I}V2.
 /3 width=3 by TC_lpx_sn_inv_pair1, lpr_cprs_trans/ qed-.
 
-lemma lprs_inv_pair2: ∀I,L1,K2,V2. L1 ⊢ ➡* K2. ⓑ{I} V2 →
-                      ∃∃K1,V1. K1 ⊢ ➡* K2 & K1 ⊢ V1 ➡* V2 & L1 = K1. ⓑ{I} V1.
+lemma lprs_inv_pair2: ∀I,G,L1,K2,V2. ⦃G, L1⦄ ⊢ ➡* K2.ⓑ{I}V2 →
+                      ∃∃K1,V1. ⦃G, K1⦄ ⊢ ➡* K2 & ⦃G, K1⦄ ⊢ V1 ➡* V2 &
+                               L1 = K1.ⓑ{I}V1.
 /3 width=3 by TC_lpx_sn_inv_pair2, lpr_cprs_trans/ qed-.
 
 (* Properties on context-sensitive parallel computation for terms ***********)
 
-lemma lprs_cpr_trans: s_r_trans … cpr lprs.
+lemma lprs_cpr_trans: ∀G. s_r_trans … (cpr G) (lprs G).
 /3 width=5 by s_r_trans_TC2, lpr_cprs_trans/ qed-.
 
 (* Basic_1: was just: pr3_pr3_pr3_t *)
-lemma lprs_cprs_trans: s_rs_trans … cpr lprs.
+lemma lprs_cprs_trans: ∀G. s_rs_trans … (cpr G) (lprs G).
 /3 width=5 by s_r_trans_TC1, lprs_cpr_trans/ qed-.
 
-lemma lprs_cprs_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡* T1 → ∀L1. L0 ⊢ ➡* L1 →
-                         ∃∃T. L1 ⊢ T1 ➡* T & L1 ⊢ T0 ➡* T.
-#L0 #T0 #T1 #HT01 #L1 #H elim H -L1
+lemma lprs_cprs_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
+                         ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
+                         ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+#G #L0 #T0 #T1 #HT01 #L1 #H elim H -L1
 [ #L1 #HL01
   elim (cprs_lpr_conf_dx … HT01 … HL01) -L0 /2 width=3/
 | #L #L1 #_ #HL1 * #T #HT1 #HT0 -L0
@@ -56,34 +59,38 @@ lemma lprs_cprs_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡* T1 → ∀L1. L0 ⊢ ➡* L
 ]
 qed-.
 
-lemma lprs_cpr_conf_dx: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ➡* L1 →
-                        ∃∃T. L1 ⊢ T1 ➡* T & L1 ⊢ T0 ➡* T.
+lemma lprs_cpr_conf_dx: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
+                        ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
+                        ∃∃T. ⦃G, L1⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
 /3 width=3 by lprs_cprs_conf_dx, cpr_cprs/ qed-.
 
-lemma lprs_cprs_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡* T1 → ∀L1. L0 ⊢ ➡* L1 →
-                         ∃∃T. L0 ⊢ T1 ➡* T & L1 ⊢ T0 ➡* T.
-#L0 #T0 #T1 #HT01 #L1 #HL01
+lemma lprs_cprs_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡* T1 →
+                         ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
+                         ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
+#G #L0 #T0 #T1 #HT01 #L1 #HL01
 elim (lprs_cprs_conf_dx … HT01 … HL01) -HT01 #T #HT1
 lapply (lprs_cprs_trans … HT1 … HL01) -HT1 /2 width=3/
 qed-.
 
-lemma lprs_cpr_conf_sn: ∀L0,T0,T1. L0 ⊢ T0 ➡ T1 → ∀L1. L0 ⊢ ➡* L1 →
-                        ∃∃T. L0 ⊢ T1 ➡* T & L1 ⊢ T0 ➡* T.
+lemma lprs_cpr_conf_sn: ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ➡ T1 →
+                        ∀L1. ⦃G, L0⦄ ⊢ ➡* L1 →
+                        ∃∃T. ⦃G, L0⦄ ⊢ T1 ➡* T & ⦃G, L1⦄ ⊢ T0 ➡* T.
 /3 width=3 by lprs_cprs_conf_sn, cpr_cprs/ qed-.
 
-lemma cprs_bind2: ∀L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 → ∀I,T1,T2. L. ⓑ{I}V2 ⊢ T1 ➡* T2 →
-                  ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1. T1 ➡* ⓑ{a,I}V2. T2.
-#L #V1 #V2 #HV12 #I #T1 #T2 #HT12
+lemma cprs_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡* V2 →
+                  ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
+                  ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
+#G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
 lapply (lprs_cprs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
 qed.
 
 (* Inversion lemmas on context-sensitive parallel computation for terms *****)
 
 (* Basic_1: was: pr3_gen_abst *)
-lemma cprs_inv_abst1: ∀a,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* U2 →
-                      ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & L.ⓛW1 ⊢ T1 ➡* T2 &
+lemma cprs_inv_abst1: ∀a,G,L,W1,T1,U2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* U2 →
+                      ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡* W2 & ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2 &
                                U2 = ⓛ{a}W2.T2.
-#a #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5/
+#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /2 width=5/
 #U0 #U2 #_ #HU02 * #V0 #T0 #HV10 #HT10 #H destruct
 elim (cpr_inv_abst1 … HU02) -HU02 #V2 #T2 #HV02 #HT02 #H destruct
 lapply (lprs_cpr_trans … HT02 (L.ⓛV1) ?) /2 width=1/ -HT02 #HT02
@@ -91,19 +98,19 @@ lapply (cprs_strap1 … HV10 … HV02) -V0
 lapply (cprs_trans … HT10 … HT02) -T0 /2 width=5/
 qed-.
 
-lemma cprs_inv_abst: ∀a,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2 →
-                     ⦃G, L⦄ ⊢ W1 ➡* W2 ∧ L.ⓛW1 ⊢ T1 ➡* T2.
-#a #L #W1 #W2 #T1 #T2 #H
+lemma cprs_inv_abst: ∀a,G,L,W1,W2,T1,T2. ⦃G, L⦄ ⊢ ⓛ{a}W1.T1 ➡* ⓛ{a}W2.T2 →
+                     ⦃G, L⦄ ⊢ W1 ➡* W2 ∧ ⦃G, L.ⓛW1⦄ ⊢ T1 ➡* T2.
+#a #G #L #W1 #W2 #T1 #T2 #H
 elim (cprs_inv_abst1 … H) -H #W #T #HW1 #HT1 #H destruct /2 width=1/
 qed-.
 
 (* Basic_1: was pr3_gen_abbr *)
-lemma cprs_inv_abbr1: ∀a,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡* U2 → (
-                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & L. ⓓV1 ⊢ T1 ➡* T2 &
+lemma cprs_inv_abbr1: ∀a,G,L,V1,T1,U2. ⦃G, L⦄ ⊢ ⓓ{a}V1.T1 ➡* U2 → (
+                      ∃∃V2,T2. ⦃G, L⦄ ⊢ V1 ➡* V2 & ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 &
                                U2 = ⓓ{a}V2.T2
                       ) ∨
-                      ∃∃T2. L. ⓓV1 ⊢ T1 ➡* T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
-#a #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
+                      ∃∃T2. ⦃G, L.ⓓV1⦄ ⊢ T1 ➡* T2 & ⇧[0, 1] U2 ≡ T2 & a = true.
+#a #G #L #V1 #T1 #U2 #H @(cprs_ind … H) -U2 /3 width=5/
 #U0 #U2 #_ #HU02 * *
 [ #V0 #T0 #HV10 #HT10 #H destruct
   elim (cpr_inv_abbr1 … HU02) -HU02 *
@@ -123,6 +130,6 @@ qed-.
 
 (* More advanced properties *************************************************)
 
-lemma lprs_pair2: ∀I,L1,L2. L1 ⊢ ➡* L2 → ∀V1,V2. L2 ⊢ V1 ➡* V2 →
-                  L1. ⓑ{I} V1 ⊢ ➡* L2. ⓑ{I} V2.
+lemma lprs_pair2: ∀I,G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 →
+                  ∀V1,V2. ⦃G, L2⦄ ⊢ V1 ➡* V2 → ⦃G, L1.ⓑ{I}V1⦄ ⊢ ➡* L2.ⓑ{I}V2.
 /3 width=3 by lprs_pair, lprs_cprs_trans/ qed.