(* *)
(**************************************************************************)
-include "basic_2/unwind/sstas.ma".
include "basic_2/reducibility/ysc.ma".
-include "basic_2/computation/cprs.ma".
+include "basic_2/computation/yprs.ma".
-(* "BIG TREE" ORDER FOR CLOSURES ********************************************)
+(* "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES **********************)
-definition ygt: ∀h. sd h → bi_relation lenv term ≝
- λh,g. bi_TC … (ysc h g).
+inductive ygt (h) (g) (L1) (T1): relation2 lenv term ≝
+| ygt_inj : ∀L,L2,T,T2. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≻[g] ⦃L2, T2⦄ →
+ ygt h g L1 T1 L2 T2
+| ygt_step: ∀L,L2,T. ygt h g L1 T1 L T → L ➡ L2 → ygt h g L1 T1 L2 T
+.
-interpretation "'big tree' order (closure)"
- 'BTGreaterThan h g L1 T1 L2 T2 = (ygt h g L1 T1 L2 T2).
+interpretation "'big tree' proper parallel computation (closure)"
+ 'BTPRedStarProper h g L1 T1 L2 T2 = (ygt h g L1 T1 L2 T2).
-(* Basic eliminators ********************************************************)
+(* Basic forvard lemmas *****************************************************)
-lemma ygt_ind: ∀h,g,L1,T1. ∀R:relation2 lenv term.
- (∀L2,T2. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L2, T2⦄ → R L2 T2) →
- (∀L,T,L2,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ ≻[g] ⦃L2, T2⦄ → R L T → R L2 T2) →
- ∀L2,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄ → R L2 T2.
-#h #g #L1 #T1 #R #IH1 #IH2 #L2 #T2 #H
-@(bi_TC_ind … IH1 IH2 L2 T2 H)
-qed-. (**) (* /3 width=6 by bi_TC_ind/ fails *)
-
-lemma ygt_ind_dx: ∀h,g,L2,T2. ∀R:relation2 lenv term.
- (∀L1,T1. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L2, T2⦄ → R L1 T1) →
- (∀L1,L,T1,T. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L, T⦄ → h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → R L T → R L1 T1) →
- ∀L1,T1. h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄ → R L1 T1.
-/3 width=6 by bi_TC_ind_dx/ qed-.
+lemma ygt_fwd_yprs: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄ →
+ h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L2, T2⦄.
+#h #g #L1 #L2 #T1 #T2 #H elim H -L2 -T2
+/3 width=4 by yprs_strap1, ysc_ypr, ypr_ltpr/
+qed-.
(* Basic properties *********************************************************)
-lemma ygt_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
- h ⊢ ⦃L, T⦄ ≻[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
-/2 width=4/ qed-.
-
-lemma ygt_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L, T⦄ →
- h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
-/2 width=4/ qed-.
+lemma ysc_ygt: ∀h,g,L1,L2,T1,T2. h ⊢ ⦃L1, T1⦄ ≻[g] ⦃L2, T2⦄ →
+ h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+/3 width=4/ qed.
-lemma ygt_cprs_trans: ∀h,g,L1,L,T1,T. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
- ∀T2. L ⊢ T ➡* T2 → h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T2⦄.
-#h #g #L1 #L #T1 #T #HLT1 #T2 #H @(cprs_ind … H) -T2 //
-#T0 #T2 #_ #HT02 #IHT0 -HLT1
-elim (term_eq_dec T0 T2) #HT02 destruct //
-@(ygt_strap1 … IHT0) /3 width=1/
+lemma ygt_strap1: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
+ h ⊢ ⦃L, T⦄ ≽[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+#h #g #L1 #L #L2 #T1 #T #T2 #H1 #H2
+lapply (ygt_fwd_yprs … H1) #H0
+elim (ypr_inv_ysc … H2) -H2 [| * #HL2 #H destruct ] /2 width=4/
qed-.
-lemma ygt_sstas_trans: ∀h,g,L1,L,T1,T. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
- ∀T2. ⦃h, L⦄ ⊢ T •*[g] T2 → h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T2⦄.
-#h #g #L1 #L #T1 #T #HLT1 #T2 #H @(sstas_ind … H) -T2 //
-#T0 #T2 #l #_ #HT02 #IHT0 -HLT1
-@(ygt_strap1 … IHT0) -IHT0 /2 width=2/
+lemma ygt_strap2: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ ≽[g] ⦃L, T⦄ →
+ h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+#h #g #L1 #L #L2 #T1 #T #T2 #H1 #H2 elim H2 -L2 -T2
+[ /3 width=4 by ygt_inj, yprs_strap2/ | /2 width=3/ ]
qed-.
-lemma cprs_ygt_trans: ∀h,g,L,T1,T. L ⊢ T1 ➡* T →
- ∀L2,T2. h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L, T1⦄ >[g] ⦃L2, T2⦄.
-#h #g #L #T1 #T #H @(cprs_ind … H) -T //
-#T0 #T #_ #HT0 #IHT10 #L2 #T2 #HLT2
-elim (term_eq_dec T0 T) #HT0 destruct /2 width=1/
-@IHT10 -IHT10 @(ygt_strap2 … HLT2) /3 width=1/
+lemma ygt_yprs_trans: ∀h,g,L1,L,L2,T1,T,T2. h ⊢ ⦃L1, T1⦄ >[g] ⦃L, T⦄ →
+ h ⊢ ⦃L, T⦄ ≥[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+#h #g #L1 #L #L2 #T1 #T #T2 #HT1 #HT2 @(yprs_ind … HT2) -L2 -T2 //
+/2 width=4 by ygt_strap1/
qed-.
-lemma sstas_ygt_trans: ∀h,g,L,T1,T. ⦃h, L⦄ ⊢ T1 •*[g] T →
- ∀L2,T2. h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L, T1⦄ >[g] ⦃L2, T2⦄.
-#h #g #L #T1 #T #H @(sstas_ind … H) -T //
-#T0 #T #l #_ #HT0 #IHT10 #L2 #T2 #HLT2
-@IHT10 -IHT10 @(ygt_strap2 … HLT2) /2 width=2/
+lemma yprs_ygt_trans: ∀h,g,L1,L,T1,T. h ⊢ ⦃L1, T1⦄ ≥[g] ⦃L, T⦄ →
+ ∀L2,T2. h ⊢ ⦃L, T⦄ >[g] ⦃L2, T2⦄ → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
+#h #g #L1 #L #T1 #T #HT1 @(yprs_ind … HT1) -L -T //
+/3 width=4 by ygt_strap2/
qed-.
lemma fw_ygt: ∀h,g,L1,L2,T1,T2. ♯{L2, T2} < ♯{L1, T1} → h ⊢ ⦃L1, T1⦄ >[g] ⦃L2, T2⦄.
/3 width=1/ qed.
-lemma cprs_ygt: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) → h ⊢ ⦃L, T1⦄ >[g] ⦃L, T2⦄.
+lemma cprs_ygt: ∀h,g,L,T1,T2. L ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) →
+ h ⊢ ⦃L, T1⦄ >[g] ⦃L, T2⦄.
#h #g #L #T1 #T2 #H @(cprs_ind … H) -T2
-[ #H elim H -H //
-| #T #T2 #_ #HT2 #IHT1 #H
- elim (term_eq_dec T1 T) #HT1 destruct
- [ -IHT1 /4 width=1 by ysc_cpr, bi_inj/ (**) (* auto too slow without trace *)
- | -H /4 width=3 by inj, ygt_cprs_trans/
+[ #H elim H //
+| #T #T2 #_ #HT2 #IHT1 #HT12
+ elim (term_eq_dec T1 T) #H destruct
+ [ -IHT1 /4 width=1/
+ | lapply (IHT1 … H) -IHT1 -H -HT12 #HT1
+ @(ygt_strap1 … HT1) -HT1 /2 width=1/
+ ]
+]
+qed.
+
+lemma sstas_ygt: ∀h,g,L,T1,T2. ⦃h, L⦄ ⊢ T1 •*[g] T2 → (T1 = T2 → ⊥) →
+ h ⊢ ⦃L, T1⦄ >[g] ⦃L, T2⦄.
+#h #g #L #T1 #T2 #H @(sstas_ind … H) -T2
+[ #H elim H //
+| #T #T2 #l #_ #HT2 #IHT1 #HT12 -HT12
+ elim (term_eq_dec T1 T) #H destruct
+ [ -IHT1 /3 width=2/
+ | lapply (IHT1 … H) -IHT1 -H #HT1
+ @(ygt_strap1 … HT1) -HT1 /2 width=2/
]
]
qed.
projects / helm.git / blobdiff