include "ground_2/notation/relations/rcoafter_3.ma".
include "ground_2/relocation/rtmap_sor.ma".
-include "ground_2/relocation/rtmap_istot.ma".
+include "ground_2/relocation/rtmap_after.ma".
(* RELOCATION MAP ***********************************************************)
#f2 #f1 @eq_repl_sym /2 width=3 by coafter_eq_repl_back0/
qed-.
-(* Main properties **********************************************************)
-(*
-corec theorem coafter_trans1: ∀f0,f3,f4. f0 ~⊚ f3 ≡ f4 →
- ∀f1,f2. f1 ~⊚ f2 ≡ f0 →
- ∀f. f2 ~⊚ f3 ≡ f → f1 ~⊚ f ≡ f4.
-#f0 #f3 #f4 * -f0 -f3 -f4 #f0 #f3 #f4 #g0 [1,2: #g3 ] #g4
-[ #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
- cases (coafter_inv_xxp … Hg0 … H0) -g0
- #f1 #f2 #Hf0 #H1 #H2
- cases (coafter_inv_ppx … Hg … H2 H3) -g2 -g3
- #f #Hf #H /3 width=7 by coafter_refl/
-| #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
- cases (coafter_inv_xxp … Hg0 … H0) -g0
- #f1 #f2 #Hf0 #H1 #H2
- cases (coafter_inv_pnx … Hg … H2 H3) -g2 -g3
- #f #Hf #H /3 width=7 by coafter_push/
-| #Hf4 #H0 #H4 #g1 #g2 #Hg0 #g #Hg
- cases (coafter_inv_xxn … Hg0 … H0) -g0 *
- [ #f1 #f2 #Hf0 #H1 #H2
- cases (coafter_inv_nxx … Hg … H2) -g2
- #f #Hf #H /3 width=7 by coafter_push/
- | #f1 #Hf0 #H1 /3 width=6 by coafter_next/
- ]
-]
-qed-.
-
-corec theorem coafter_trans2: ∀f1,f0,f4. f1 ~⊚ f0 ≡ f4 →
- ∀f2, f3. f2 ~⊚ f3 ≡ f0 →
- ∀f. f1 ~⊚ f2 ≡ f → f ~⊚ f3 ≡ f4.
-#f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #g1 [1,2: #g0 ] #g4
-[ #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
- cases (coafter_inv_xxp … Hg0 … H0) -g0
- #f2 #f3 #Hf0 #H2 #H3
- cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
- #f #Hf #H /3 width=7 by coafter_refl/
-| #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
- cases (coafter_inv_xxn … Hg0 … H0) -g0 *
- [ #f2 #f3 #Hf0 #H2 #H3
- cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
- #f #Hf #H /3 width=7 by coafter_push/
- | #f2 #Hf0 #H2
- cases (coafter_inv_pnx … Hg … H1 H2) -g1 -g2
- #f #Hf #H /3 width=6 by coafter_next/
- ]
-| #Hf4 #H1 #H4 #f2 #f3 #Hf0 #g #Hg
- cases (coafter_inv_nxx … Hg … H1) -g1
- #f #Hg #H /3 width=6 by coafter_next/
-]
-qed-.
-*)
(* Main inversion lemmas ****************************************************)
corec theorem coafter_mono: ∀f1,f2,x,y. f1 ~⊚ f2 ≡ x → f1 ~⊚ f2 ≡ y → x ≗ y.
f1 ≗ g1 → f2 ≗ g2 → f ≗ g.
/4 width=4 by coafter_mono, coafter_eq_repl_back1, coafter_eq_repl_back2/ qed-.
+(* Inversion lemmas with pushs **********************************************)
+
+lemma coafter_fwd_pushs: ∀n,g2,g1,g. g2 ~⊚ g1 ≡ g → @⦃0, g2⦄ ≡ n →
+ ∃f. ↑*[n]f = g.
+#n elim n -n /2 width=2 by ex_intro/
+#n #IH #g2 #g1 #g #Hg #Hg2
+cases (at_inv_pxn … Hg2) -Hg2 [ |*: // ] #f2 #Hf2 #H2
+cases (coafter_inv_nxx … Hg … H2) -Hg -H2 #f #Hf #H0 destruct
+elim (IH … Hf Hf2) -g1 -g2 -f2 /2 width=2 by ex_intro/
+qed-.
+
(* Inversion lemmas with tail ***********************************************)
lemma coafter_inv_tl1: ∀g2,g1,g. g2 ~⊚ ⫱g1 ≡ g →
]
qed-.
-(* Properties on tls ********************************************************)
+(* Properties with iterated tail ********************************************)
lemma coafter_tls: ∀n,f1,f2,f. @⦃0, f1⦄ ≡ n →
f1 ~⊚ f2 ≡ f → ⫱*[n]f1 ~⊚ f2 ≡ ⫱*[n]f.
#n elim n -n //
#n #IH #f1 #f2 #f #Hf1 #Hf
cases (at_inv_pxn … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1
-cases (coafter_inv_nxx … Hf … H1) -Hf /2 width=1 by/
+cases (coafter_inv_nxx … Hf … H1) -Hf #g #Hg #H0 destruct
+<tls_xn <tls_xn /2 width=1 by/
+qed.
+
+lemma coafter_tls_succ: ∀g2,g1,g. g2 ~⊚ g1 ≡ g →
+ ∀n. @⦃0, g2⦄ ≡ n → ⫱*[⫯n]g2 ~⊚ ⫱g1 ≡ ⫱*[⫯n]g.
+#g2 #g1 #g #Hg #n #Hg2
+lapply (coafter_tls … Hg2 … Hg) -Hg #Hg
+lapply (at_pxx_tls … Hg2) -Hg2 #H
+elim (at_inv_pxp … H) -H [ |*: // ] #f2 #H2
+elim (coafter_inv_pxx … Hg … H2) -Hg * #f1 #f #Hf #H1 #H0 destruct
+<tls_S <tls_S <H2 <H0 -g2 -g -n //
qed.
-(* Properties on isid *******************************************************)
+lemma coafter_fwd_xpx_pushs: ∀g2,f1,g,n. g2 ~⊚ ↑f1 ≡ g → @⦃0, g2⦄ ≡ n →
+ ∃f. ↑*[⫯n]f = g.
+#g2 #g1 #g #n #Hg #Hg2
+elim (coafter_fwd_pushs … Hg Hg2) #f #H0 destruct
+lapply (coafter_tls … Hg2 Hg) -Hg <tls_pushs #Hf
+lapply (at_pxx_tls … Hg2) -Hg2 #H
+elim (at_inv_pxp … H) -H [ |*: // ] #f2 #H2
+elim (coafter_inv_pxx … Hf … H2) -Hf -H2 * #f1 #g #_ #H1 #H0 destruct
+[ /2 width=2 by ex_intro/
+| elim (discr_next_push … H1)
+]
+qed-.
+
+lemma coafter_fwd_xnx_pushs: ∀g2,f1,g,n. g2 ~⊚ ⫯f1 ≡ g → @⦃0, g2⦄ ≡ n →
+ ∃f. ↑*[n] ⫯f = g.
+#g2 #g1 #g #n #Hg #Hg2
+elim (coafter_fwd_pushs … Hg Hg2) #f #H0 destruct
+lapply (coafter_tls … Hg2 Hg) -Hg <tls_pushs #Hf
+lapply (at_pxx_tls … Hg2) -Hg2 #H
+elim (at_inv_pxp … H) -H [ |*: // ] #f2 #H2
+elim (coafter_inv_pxx … Hf … H2) -Hf -H2 * #f1 #g #_ #H1 #H0 destruct
+[ elim (discr_push_next … H1)
+| /2 width=2 by ex_intro/
+]
+qed-.
+
+(* Properties with test for identity ****************************************)
corec lemma coafter_isid_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ~⊚ f2 ≡ f2.
#f1 * -f1 #f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) * #g2 #H2
]
qed.
-(* Inversion lemmas on isid *************************************************)
+(* Inversion lemmas with test for identity **********************************)
lemma coafter_isid_inv_sn: ∀f1,f2,f. f1 ~⊚ f2 ≡ f → 𝐈⦃f1⦄ → f2 ≗ f.
/3 width=6 by coafter_isid_sn, coafter_mono/ qed-.
lemma coafter_isid_inv_dx: ∀f1,f2,f. f1 ~⊚ f2 ≡ f → 𝐈⦃f2⦄ → 𝐈⦃f⦄.
/4 width=4 by eq_id_isid, coafter_isid_dx, coafter_mono/ qed-.
-(*
-(* Properties on isuni ******************************************************)
+(* Properties with test for uniform relocations *****************************)
+
+lemma coafter_isuni_isid: ∀f2. 𝐈⦃f2⦄ → ∀f1. 𝐔⦃f1⦄ → f1 ~⊚ f2 ≡ f2.
+#f #Hf #g #H elim H -g
+/3 width=5 by coafter_isid_sn, coafter_eq_repl_back0, coafter_next, eq_push_inv_isid/
+qed.
+
+
+(*
lemma coafter_isid_isuni: ∀f1,f2. 𝐈⦃f2⦄ → 𝐔⦃f1⦄ → f1 ~⊚ ⫯f2 ≡ ⫯f1.
#f1 #f2 #Hf2 #H elim H -H
/5 width=7 by coafter_isid_dx, coafter_eq_repl_back2, coafter_next, coafter_push, eq_push_inv_isid/
/3 width=5 by coafter_next/
]
qed.
+*)
-(* Properties on uni ********************************************************)
+(* Properties with uniform relocations **************************************)
+lemma coafter_uni_sn: ∀i,f. 𝐔❴i❵ ~⊚ f ≡ ↑*[i] f.
+#i elim i -i /2 width=5 by coafter_isid_sn, coafter_next/
+qed.
+
+(*
lemma coafter_uni: ∀n1,n2. 𝐔❴n1❵ ~⊚ 𝐔❴n2❵ ≡ 𝐔❴n1+n2❵.
@nat_elim2
/4 width=5 by coafter_uni_next2, coafter_isid_sn, coafter_isid_dx, coafter_next/
(* Forward lemmas on at *****************************************************)
lemma coafter_at_fwd: ∀i,i1,f. @⦃i1, f⦄ ≡ i → ∀f2,f1. f2 ~⊚ f1 ≡ f →
- ∃∃i2. @⦃i1, f1⦄ ≡ i2 & @⦃i2, f2⦄ ≡ i.
+ ∃∃i2. @⦃i1, f1⦄ ≡ i2 & @⦃i2, f2⦄ ≡ i.
#i elim i -i [2: #i #IH ] #i1 #f #Hf #f2 #f1 #Hf21
[ elim (at_inv_xxn … Hf) -Hf [1,3:* |*: // ]
[1: #g #j1 #Hg #H0 #H |2,4: #g #Hg #H ]
qed-.
lemma coafter_fwd_at: ∀i,i2,i1,f1,f2. @⦃i1, f1⦄ ≡ i2 → @⦃i2, f2⦄ ≡ i →
- ∀f. f2 ~⊚ f1 ≡ f → @⦃i1, f⦄ ≡ i.
+ ∀f. f2 ~⊚ f1 ≡ f → @⦃i1, f⦄ ≡ i.
#i elim i -i [2: #i #IH ] #i2 #i1 #f1 #f2 #Hf1 #Hf2 #f #Hf
[ elim (at_inv_xxn … Hf2) -Hf2 [1,3: * |*: // ]
#g2 [ #j2 ] #Hg2 [ #H22 ] #H20
qed-.
lemma coafter_fwd_at2: ∀f,i1,i. @⦃i1, f⦄ ≡ i → ∀f1,i2. @⦃i1, f1⦄ ≡ i2 →
- ∀f2. f2 ~⊚ f1 ≡ f → @⦃i2, f2⦄ ≡ i.
+ ∀f2. f2 ~⊚ f1 ≡ f → @⦃i2, f2⦄ ≡ i.
#f #i1 #i #Hf #f1 #i2 #Hf1 #f2 #H elim (coafter_at_fwd … Hf … H) -f
#j1 #H #Hf2 <(at_mono … Hf1 … H) -i1 -i2 //
qed-.
lemma coafter_fwd_at1: ∀i,i2,i1,f,f2. @⦃i1, f⦄ ≡ i → @⦃i2, f2⦄ ≡ i →
- ∀f1. f2 ~⊚ f1 ≡ f → @⦃i1, f1⦄ ≡ i2.
+ ∀f1. f2 ~⊚ f1 ≡ f → @⦃i1, f1⦄ ≡ i2.
#i elim i -i [2: #i #IH ] #i2 #i1 #f #f2 #Hf #Hf2 #f1 #Hf1
[ elim (at_inv_xxn … Hf) -Hf [1,3: * |*: // ]
#g [ #j1 ] #Hg [ #H01 ] #H00
(* Properties with at *******************************************************)
lemma coafter_uni_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
- ∀f. f2 ~⊚ 𝐔❴i1❵ ≡ f → 𝐔❴i2❵ ~⊚ ⫱*[i2] f2 ≡ f.
+ ∀f. f2 ~⊚ 𝐔❴i1❵ ≡ f → 𝐔❴i2❵ ~⊚ ⫱*[i2] f2 ≡ f.
#i2 elim i2 -i2
[ #i1 #f2 #Hf2 #f #Hf
elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
qed.
lemma coafter_uni_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
- ∀f. 𝐔❴i2❵ ~⊚ ⫱*[i2] f2 ≡ f → f2 ~⊚ 𝐔❴i1❵ ≡ f.
+ ∀f. 𝐔❴i2❵ ~⊚ ⫱*[i2] f2 ≡ f → f2 ~⊚ 𝐔❴i1❵ ≡ f.
#i2 elim i2 -i2
[ #i1 #f2 #Hf2 #f #Hf
elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
qed-.
lemma coafter_uni_succ_dx: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
- ∀f. f2 ~⊚ 𝐔❴⫯i1❵ ≡ f → 𝐔❴⫯i2❵ ~⊚ ⫱*[⫯i2] f2 ≡ f.
+ ∀f. f2 ~⊚ 𝐔❴⫯i1❵ ≡ f → 𝐔❴⫯i2❵ ~⊚ ⫱*[⫯i2] f2 ≡ f.
#i2 elim i2 -i2
[ #i1 #f2 #Hf2 #f #Hf
elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
qed.
lemma coafter_uni_succ_sn: ∀i2,i1,f2. @⦃i1, f2⦄ ≡ i2 →
- ∀f. 𝐔❴⫯i2❵ ~⊚ ⫱*[⫯i2] f2 ≡ f → f2 ~⊚ 𝐔❴⫯i1❵ ≡ f.
+ ∀f. 𝐔❴⫯i2❵ ~⊚ ⫱*[⫯i2] f2 ≡ f → f2 ~⊚ 𝐔❴⫯i1❵ ≡ f.
#i2 elim i2 -i2
[ #i1 #f2 #Hf2 #f #Hf
elim (at_inv_xxp … Hf2) -Hf2 // #g2 #H1 #H2 destruct
qed-.
fact coafter_isfin2_fwd_aux: (∀f1. @⦃0, f1⦄ ≡ 0 → H_coafter_isfin2_fwd f1) →
- ∀i2,f1. @⦃0, f1⦄ ≡ i2 → H_coafter_isfin2_fwd f1.
+ ∀i2,f1. @⦃0, f1⦄ ≡ i2 → H_coafter_isfin2_fwd f1.
#H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
#i2 #IH #f1 #H1f1 #f2 #Hf2 #H2f1 #f #Hf
elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #Hg1 #H1
/3 width=11 by coafter_refl, coafter_push, sor_np, sor_pn, sor_nn, ex3_2_intro/
]
qed-.
+
+(* Properties with after ****************************************************)
+(*
+corec theorem coafter_trans1: ∀f0,f3,f4. f0 ~⊚ f3 ≡ f4 →
+ ∀f1,f2. f1 ~⊚ f2 ≡ f0 →
+ ∀f. f2 ~⊚ f3 ≡ f → f1 ~⊚ f ≡ f4.
+#f0 #f3 #f4 * -f0 -f3 -f4 #f0 #f3 #f4 #g0 [1,2: #g3 ] #g4
+[ #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
+ cases (coafter_inv_xxp … Hg0 … H0) -g0
+ #f1 #f2 #Hf0 #H1 #H2
+ cases (coafter_inv_ppx … Hg … H2 H3) -g2 -g3
+ #f #Hf #H /3 width=7 by coafter_refl/
+| #Hf4 #H0 #H3 #H4 #g1 #g2 #Hg0 #g #Hg
+ cases (coafter_inv_xxp … Hg0 … H0) -g0
+ #f1 #f2 #Hf0 #H1 #H2
+ cases (coafter_inv_pnx … Hg … H2 H3) -g2 -g3
+ #f #Hf #H /3 width=7 by coafter_push/
+| #Hf4 #H0 #H4 #g1 #g2 #Hg0 #g #Hg
+ cases (coafter_inv_xxn … Hg0 … H0) -g0 *
+ [ #f1 #f2 #Hf0 #H1 #H2
+ cases (coafter_inv_nxx … Hg … H2) -g2
+ #f #Hf #H /3 width=7 by coafter_push/
+ | #f1 #Hf0 #H1 /3 width=6 by coafter_next/
+ ]
+]
+qed-.
+
+corec theorem coafter_trans2: ∀f1,f0,f4. f1 ~⊚ f0 ≡ f4 →
+ ∀f2, f3. f2 ~⊚ f3 ≡ f0 →
+ ∀f. f1 ~⊚ f2 ≡ f → f ~⊚ f3 ≡ f4.
+#f1 #f0 #f4 * -f1 -f0 -f4 #f1 #f0 #f4 #g1 [1,2: #g0 ] #g4
+[ #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
+ cases (coafter_inv_xxp … Hg0 … H0) -g0
+ #f2 #f3 #Hf0 #H2 #H3
+ cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
+ #f #Hf #H /3 width=7 by coafter_refl/
+| #Hf4 #H1 #H0 #H4 #g2 #g3 #Hg0 #g #Hg
+ cases (coafter_inv_xxn … Hg0 … H0) -g0 *
+ [ #f2 #f3 #Hf0 #H2 #H3
+ cases (coafter_inv_ppx … Hg … H1 H2) -g1 -g2
+ #f #Hf #H /3 width=7 by coafter_push/
+ | #f2 #Hf0 #H2
+ cases (coafter_inv_pnx … Hg … H1 H2) -g1 -g2
+ #f #Hf #H /3 width=6 by coafter_next/
+ ]
+| #Hf4 #H1 #H4 #f2 #f3 #Hf0 #g #Hg
+ cases (coafter_inv_nxx … Hg … H1) -g1
+ #f #Hg #H /3 width=6 by coafter_next/
+]
+qed-.
+*)
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