(**************************************************************************)
include "ground_2/notation/relations/rat_3.ma".
-include "ground_2/relocation/rtmap_id.ma".
+include "ground_2/relocation/rtmap_uni.ma".
(* RELOCATION MAP ***********************************************************)
interpretation "relational application (rtmap)"
'RAt i1 f i2 = (at f i1 i2).
+definition H_at_div: relation4 rtmap rtmap rtmap rtmap ≝ λf2,g2,f1,g1.
+ ∀jf,jg,j. @⦃jf, f2⦄ ≡ j → @⦃jg, g2⦄ ≡ j →
+ ∃∃j0. @⦃j0, f1⦄ ≡ jf & @⦃j0, g1⦄ ≡ jg.
+
(* Basic inversion lemmas ***************************************************)
lemma at_inv_ppx: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g. 0 = i1 → ↑g = f → 0 = i2.
#x2 #Hg * -i2 #H destruct //
qed-.
-(* --------------------------------------------------------------------------*)
-
lemma at_inv_pxp: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → 0 = i1 → 0 = i2 → ∃g. ↑g = f.
#f elim (pn_split … f) * /2 width=2 by ex_intro/
#g #H #i1 #i2 #Hf #H1 #H2 cases (at_inv_xnp … Hf … H H2)
/4 width=7 by at_inv_xnn, at_inv_npn, ex2_intro, or_intror, or_introl/
qed-.
-(* --------------------------------------------------------------------------*)
-
+(* Note: the following inversion lemmas must be checked *)
lemma at_inv_xpx: ∀f,i1,i2. @⦃i1, f⦄ ≡ i2 → ∀g. ↑g = f →
(0 = i1 ∧ 0 = i2) ∨
∃∃j1,j2. @⦃j1, g⦄ ≡ j2 & ⫯j1 = i1 & ⫯j2 = i2.
(* Basic properties *********************************************************)
-let corec at_eq_repl_back: ∀i1,i2. eq_repl_back (λf. @⦃i1, f⦄ ≡ i2) ≝ ?.
+corec lemma at_eq_repl_back: ∀i1,i2. eq_repl_back (λf. @⦃i1, f⦄ ≡ i2).
#i1 #i2 #f1 #H1 cases H1 -f1 -i1 -i2
[ #f1 #g1 #j1 #j2 #H #H1 #H2 #f2 #H12 cases (eq_inv_px … H12 … H) -g1 /2 width=2 by at_refl/
| #f1 #i1 #i2 #Hf1 #g1 #j1 #j2 #H #H1 #H2 #f2 #H12 cases (eq_inv_px … H12 … H) -g1 /3 width=7 by at_push/
#Hi elim (lt_le_false i i) /3 width=6 by at_monotonic, eq_sym/
qed-.
-(* Properties on minus ******************************************************)
+theorem at_div_comm: ∀f2,g2,f1,g1.
+ H_at_div f2 g2 f1 g1 → H_at_div g2 f2 g1 f1.
+#f2 #g2 #f1 #g1 #IH #jg #jf #j #Hg #Hf
+elim (IH … Hf Hg) -IH -j /2 width=3 by ex2_intro/
+qed-.
+
+theorem at_div_pp: ∀f2,g2,f1,g1.
+ H_at_div f2 g2 f1 g1 → H_at_div (↑f2) (↑g2) (↑f1) (↑g1).
+#f2 #g2 #f1 #g1 #IH #jf #jg #j #Hf #Hg
+elim (at_inv_xpx … Hf) -Hf [1,2: * |*: // ]
+[ #H1 #H2 destruct -IH
+ lapply (at_inv_xpp … Hg ???) -Hg [4: |*: // ] #H destruct
+ /3 width=3 by at_refl, ex2_intro/
+| #xf #i #Hf2 #H1 #H2 destruct
+ lapply (at_inv_xpn … Hg ????) -Hg [5: * |*: // ] #xg #Hg2 #H destruct
+ elim (IH … Hf2 Hg2) -IH -i /3 width=9 by at_push, ex2_intro/
+]
+qed-.
+
+theorem at_div_nn: ∀f2,g2,f1,g1.
+ H_at_div f2 g2 f1 g1 → H_at_div (⫯f2) (⫯g2) (f1) (g1).
+#f2 #g2 #f1 #g1 #IH #jf #jg #j #Hf #Hg
+elim (at_inv_xnx … Hf) -Hf [ |*: // ] #i #Hf2 #H destruct
+lapply (at_inv_xnn … Hg ????) -Hg [5: |*: // ] #Hg2
+elim (IH … Hf2 Hg2) -IH -i /2 width=3 by ex2_intro/
+qed-.
+
+theorem at_div_np: ∀f2,g2,f1,g1.
+ H_at_div f2 g2 f1 g1 → H_at_div (⫯f2) (↑g2) (f1) (⫯g1).
+#f2 #g2 #f1 #g1 #IH #jf #jg #j #Hf #Hg
+elim (at_inv_xnx … Hf) -Hf [ |*: // ] #i #Hf2 #H destruct
+lapply (at_inv_xpn … Hg ????) -Hg [5: * |*: // ] #xg #Hg2 #H destruct
+elim (IH … Hf2 Hg2) -IH -i /3 width=7 by at_next, ex2_intro/
+qed-.
+
+theorem at_div_pn: ∀f2,g2,f1,g1.
+ H_at_div f2 g2 f1 g1 → H_at_div (↑f2) (⫯g2) (⫯f1) (g1).
+/4 width=6 by at_div_np, at_div_comm/ qed-.
+
+(* Properties on tls ********************************************************)
-lemma at_pxx_minus: ∀n,f. @⦃0, f⦄ ≡ n → @⦃0, f-n⦄ ≡ 0.
+lemma at_pxx_tls: ∀n,f. @⦃0, f⦄ ≡ n → @⦃0, ⫱*[n]f⦄ ≡ 0.
#n elim n -n //
-#n #IH #f #Hf cases (at_inv_pxn … Hf) -Hf /2 width=3 by/
+#n #IH #f #Hf
+cases (at_inv_pxn … Hf) -Hf [ |*: // ] #g #Hg #H0 destruct
+<tls_xn /2 width=1 by/
qed.
+lemma at_tls: ∀i2,f. ↑⫱*[⫯i2]f ≗ ⫱*[i2]f → ∃i1. @⦃i1, f⦄ ≡ i2.
+#i2 elim i2 -i2
+[ /4 width=4 by at_eq_repl_back, at_refl, ex_intro/
+| #i2 #IH #f <tls_xn <tls_xn in ⊢ (??%→?); #H
+ elim (IH … H) -IH -H #i1 #Hf
+ elim (pn_split f) * #g #Hg destruct /3 width=8 by at_push, at_next, ex_intro/
+]
+qed-.
+
+(* Inversion lemmas with tls ************************************************)
+
+lemma at_inv_nxx: ∀n,g,i1,j2. @⦃⫯i1, g⦄ ≡ j2 → @⦃0, g⦄ ≡ n →
+ ∃∃i2. @⦃i1, ⫱*[⫯n]g⦄ ≡ i2 & ⫯(n+i2) = j2.
+#n elim n -n
+[ #g #i1 #j2 #Hg #H
+ elim (at_inv_pxp … H) -H [ |*: // ] #f #H0
+ elim (at_inv_npx … Hg … H0) -Hg [ |*: // ] #x2 #Hf #H2 destruct
+ /2 width=3 by ex2_intro/
+| #n #IH #g #i1 #j2 #Hg #H
+ elim (at_inv_pxn … H) -H [ |*: // ] #f #Hf2 #H0
+ elim (at_inv_xnx … Hg … H0) -Hg #x2 #Hf1 #H2 destruct
+ elim (IH … Hf1 Hf2) -IH -Hf1 -Hf2 #i2 #Hf #H2 destruct
+ /2 width=3 by ex2_intro/
+]
+qed-.
+
+lemma at_inv_tls: ∀i2,i1,f. @⦃i1, f⦄ ≡ i2 → ↑⫱*[⫯i2]f ≗ ⫱*[i2]f.
+#i2 elim i2 -i2
+[ #i1 #f #Hf elim (at_inv_xxp … Hf) -Hf // #g #H1 #H destruct
+ /2 width=1 by eq_refl/
+| #i2 #IH #i1 #f #Hf
+ elim (at_inv_xxn … Hf) -Hf [1,3: * |*: // ]
+ [ #g #j1 #Hg #H1 #H2 | #g #Hg #Ho ] destruct
+ <tls_xn /2 width=2 by/
+]
+qed-.
+
(* Advanced inversion lemmas on isid ****************************************)
lemma isid_inv_at: ∀i,f. 𝐈⦃f⦄ → @⦃i, f⦄ ≡ i.
lemma isid_inv_at_mono: ∀f,i1,i2. 𝐈⦃f⦄ → @⦃i1, f⦄ ≡ i2 → i1 = i2.
/3 width=6 by isid_inv_at, at_mono/ qed-.
-(* Advancedd properties on isid *********************************************)
+(* Advanced properties on isid **********************************************)
-let corec isid_at: ∀f. (∀i. @⦃i, f⦄ ≡ i) → 𝐈⦃f⦄ ≝ ?.
+corec lemma isid_at: ∀f. (∀i. @⦃i, f⦄ ≡ i) → 𝐈⦃f⦄.
#f #Hf lapply (Hf 0)
#H cases (at_fwd_id_ex … H) -H
#g #H @(isid_push … H) /3 width=7 by at_inv_npn/
lemma id_at: ∀i. @⦃i, 𝐈𝐝⦄ ≡ i.
/2 width=1 by isid_inv_at/ qed.
+
+(* Advanced forward lemmas on id ********************************************)
+
+lemma at_id_fwd: ∀i1,i2. @⦃i1, 𝐈𝐝⦄ ≡ i2 → i1 = i2.
+/2 width=4 by at_mono/ qed.
+
+(* Main properties on id ****************************************************)
+
+theorem at_div_id_dx: ∀f. H_at_div f 𝐈𝐝 𝐈𝐝 f.
+#f #jf #j0 #j #Hf #H0
+lapply (at_id_fwd … H0) -H0 #H destruct
+/2 width=3 by ex2_intro/
+qed-.
+
+theorem at_div_id_sn: ∀f. H_at_div 𝐈𝐝 f f 𝐈𝐝.
+/3 width=6 by at_div_id_dx, at_div_comm/ qed-.
+
+(* Properties with uniform relocations **************************************)
+
+lemma at_uni: ∀n,i. @⦃i,𝐔❴n❵⦄ ≡ n+i.
+#n elim n -n /2 width=5 by at_next/
+qed.
+
+(* Inversion lemmas with uniform relocations ********************************)
+
+lemma at_inv_uni: ∀n,i,j. @⦃i,𝐔❴n❵⦄ ≡ j → j = n+i.
+/2 width=4 by at_mono/ qed-.
projects / helm.git / blobdiff