+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||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/relocation/lifts_lifts_bind.ma".
-include "basic_2/relocation/drops_weight.ma".
-
-(* GENERIC SLICING FOR LOCAL ENVIRONMENTS ***********************************)
-
-(* Main properties **********************************************************)
-
-(* Basic_2A1: includes: drop_conf_ge drop_conf_be drop_conf_le *)
-theorem drops_conf: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L →
- ∀b2,f,L2. ⬇*[b2, f] L1 ≘ L2 →
- ∀f2. f1 ⊚ f2 ≘ f → ⬇*[b2, f2] L ≘ L2.
-#b1 #f1 #L1 #L #H elim H -f1 -L1 -L
-[ #f1 #_ #b2 #f #L2 #HL2 #f2 #Hf12 elim (drops_inv_atom1 … HL2) -b1 -HL2
- #H #Hf destruct @drops_atom
- #H elim (after_inv_isid3 … Hf12) -Hf12 /2 width=1 by/
-| #f1 #I1 #K1 #K #_ #IH #b2 #f #L2 #HL2 #f2 #Hf elim (after_inv_nxx … Hf) -Hf [2,3: // ]
- #g #Hg #H destruct /3 width=3 by drops_inv_drop1/
-| #f1 #I1 #I #K1 #K #_ #HI1 #IH #b2 #f #L2 #HL2 #f2 #Hf elim (after_inv_pxx … Hf) -Hf [1,3: * |*:// ]
- #g2 #g #Hf #H1 #H2 destruct
- [ elim (drops_inv_skip1 … HL2) -HL2 /3 width=6 by drops_skip, liftsb_div3/
- | /4 width=3 by drops_inv_drop1, drops_drop/
- ]
-]
-qed-.
-
-(* Basic_1: was: drop1_trans *)
-(* Basic_2A1: includes: drop_trans_ge drop_trans_le drop_trans_ge_comm
- drops_drop_trans
-*)
-theorem drops_trans: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L →
- ∀b2,f2,L2. ⬇*[b2, f2] L ≘ L2 →
- ∀f. f1 ⊚ f2 ≘ f → ⬇*[b1∧b2, f] L1 ≘ L2.
-#b1 #f1 #L1 #L #H elim H -f1 -L1 -L
-[ #f1 #Hf1 #b2 #f2 #L2 #HL2 #f #Hf elim (drops_inv_atom1 … HL2) -HL2
- #H #Hf2 destruct @drops_atom #H elim (andb_inv_true_dx … H) -H
- #H1 #H2 lapply (after_isid_inv_sn … Hf ?) -Hf
- /3 width=3 by isid_eq_repl_back/
-| #f1 #I1 #K1 #K #_ #IH #b2 #f2 #L2 #HL2 #f #Hf elim (after_inv_nxx … Hf) -Hf
- /3 width=3 by drops_drop/
-| #f1 #I1 #I #K1 #K #_ #HI1 #IH #b2 #f2 #L2 #HL2 #f #Hf elim (after_inv_pxx … Hf) -Hf [1,3: * |*: // ]
- #g2 #g #Hg #H1 #H2 destruct
- [ elim (drops_inv_skip1 … HL2) -HL2 /3 width=6 by drops_skip, liftsb_trans/
- | /4 width=3 by drops_inv_drop1, drops_drop/
- ]
-]
-qed-.
-
-theorem drops_conf_div: ∀f1,L,K. ⬇*[Ⓣ,f1] L ≘ K → ∀f2. ⬇*[Ⓣ,f2] L ≘ K →
- 𝐔⦃f1⦄ → 𝐔⦃f2⦄ → f1 ≡ f2.
-#f1 #L #K #H elim H -f1 -L -K
-[ #f1 #Hf1 #f2 #Hf2 elim (drops_inv_atom1 … Hf2) -Hf2
- /3 width=1 by isid_inv_eq_repl/
-| #f1 #I #L #K #Hf1 #IH #f2 elim (pn_split f2) *
- #g2 #H2 #Hf2 #HU1 #HU2 destruct
- [ elim (drops_inv_skip1 … Hf2) -IH -HU1 -Hf2 #Y2 #X2 #HY2 #_ #H destruct
- lapply (drops_fwd_isid … HY2 ?) -HY2 /2 width=3 by isuni_inv_push/ -HU2
- #H destruct elim (drops_inv_x_bind_xy … Hf1)
- | /4 width=5 by drops_inv_drop1, isuni_inv_next, eq_next/
- ]
-| #f1 #I1 #I2 #L #K #Hf1 #_ #IH #f2 elim (pn_split f2) *
- #g2 #H2 #Hf2 #HU1 #HU2 destruct
- [ elim (drops_inv_skip1 … Hf2) -Hf2 #Y2 #X2 #HY2 #_ #H destruct -Hf1
- /4 width=5 by isuni_fwd_push, eq_push/
- | lapply (drops_inv_drop1 … Hf2) -Hf2 -IH -HU2 #Hg2
- lapply (drops_fwd_isid … Hf1 ?) -Hf1 /2 width=3 by isuni_inv_push/ -HU1
- #H destruct elim (drops_inv_x_bind_xy … Hg2)
- ]
-]
-qed-.
-
-(* Advanced properties ******************************************************)
-
-(* Basic_2A1: includes: drop_mono *)
-lemma drops_mono: ∀b1,f,L,L1. ⬇*[b1, f] L ≘ L1 →
- ∀b2,L2. ⬇*[b2, f] L ≘ L2 → L1 = L2.
-#b1 #f #L #L1 lapply (after_isid_dx 𝐈𝐝 … f)
-/3 width=8 by drops_conf, drops_fwd_isid/
-qed-.
-
-(* Basic_2A1: includes: drop_conf_lt *)
-lemma drops_conf_skip1: ∀b2,f,L,L2. ⬇*[b2, f] L ≘ L2 →
- ∀b1,f1,I1,K1. ⬇*[b1, f1] L ≘ K1.ⓘ{I1} →
- ∀f2. f1 ⊚ ⫯f2 ≘ f →
- ∃∃I2,K2. L2 = K2.ⓘ{I2} &
- ⬇*[b2, f2] K1 ≘ K2 & ⬆*[f2] I2 ≘ I1.
-#b2 #f #L #L2 #H2 #b1 #f1 #I1 #K1 #H1 #f2 #Hf lapply (drops_conf … H1 … H2 … Hf) -L -Hf
-#H elim (drops_inv_skip1 … H) -H /2 width=5 by ex3_2_intro/
-qed-.
-
-(* Basic_2A1: includes: drop_trans_lt *)
-lemma drops_trans_skip2: ∀b1,f1,L1,L. ⬇*[b1, f1] L1 ≘ L →
- ∀b2,f2,I2,K2. ⬇*[b2, f2] L ≘ K2.ⓘ{I2} →
- ∀f. f1 ⊚ f2 ≘ ⫯f →
- ∃∃I1,K1. L1 = K1.ⓘ{I1} &
- ⬇*[b1∧b2, f] K1 ≘ K2 & ⬆*[f] I2 ≘ I1.
-#b1 #f1 #L1 #L #H1 #b2 #f2 #I2 #K2 #H2 #f #Hf
-lapply (drops_trans … H1 … H2 … Hf) -L -Hf
-#H elim (drops_inv_skip2 … H) -H /2 width=5 by ex3_2_intro/
-qed-.
-
-(* Basic_2A1: includes: drops_conf_div *)
-lemma drops_conf_div_bind: ∀f1,f2,I1,I2,L,K.
- ⬇*[Ⓣ, f1] L ≘ K.ⓘ{I1} → ⬇*[Ⓣ, f2] L ≘ K.ⓘ{I2} →
- 𝐔⦃f1⦄ → 𝐔⦃f2⦄ → f1 ≡ f2 ∧ I1 = I2.
-#f1 #f2 #I1 #I2 #L #K #Hf1 #Hf2 #HU1 #HU2
-lapply (drops_isuni_fwd_drop2 … Hf1) // #H1
-lapply (drops_isuni_fwd_drop2 … Hf2) // #H2
-lapply (drops_conf_div … H1 … H2 ??) /2 width=3 by isuni_next/ -H1 -H2 -HU1 -HU2 #H
-lapply (eq_inv_nn … H ????) -H [5: |*: // ] #H12
-lapply (drops_eq_repl_back … Hf1 … H12) -Hf1 #H0
-lapply (drops_mono … H0 … Hf2) -L #H
-destruct /2 width=1 by conj/
-qed-.
-
-lemma drops_inv_uni: ∀L,i. ⬇*[Ⓕ, 𝐔❴i❵] L ≘ ⋆ → ∀I,K. ⬇*[i] L ≘ K.ⓘ{I} → ⊥.
-#L #i #H1 #I #K #H2
-lapply (drops_F … H2) -H2 #H2
-lapply (drops_mono … H2 … H1) -L -i #H destruct
-qed-.
projects / helm.git / blobdiff