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 ***********************************************************)
definition H_coafter_fwd_isid2: predicate rtmap ≝
λf1. ∀f2,f. f1 ~⊚ f2 ≡ f → 𝐓⦃f1⦄ → 𝐈⦃f⦄ → 𝐈⦃f2⦄.
+definition H_coafter_isfin2_fwd: predicate rtmap ≝
+ λf1. ∀f2. 𝐅⦃f2⦄ → 𝐓⦃f1⦄ → ∀f. f1 ~⊚ f2 ≡ f → 𝐅⦃f⦄.
+
(* Basic inversion lemmas ***************************************************)
lemma coafter_inv_ppx: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f1,f2. ↑f1 = g1 → ↑f2 = g2 →
#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 →
+ ∃∃f. ↑g2 ~⊚ g1 ≡ f & ⫱f = g.
+#g2 #g1 #g elim (pn_split g1) * #f1 #H1 #H destruct
+[ /3 width=7 by coafter_refl, ex2_intro/
+| @(ex2_intro … (⫯g)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
+]
+qed-.
+
lemma coafter_inv_tl0: ∀g2,g1,g. g2 ~⊚ g1 ≡ ⫱g →
∃∃f1. ↑g2 ~⊚ f1 ≡ g & ⫱f1 = g1.
-#g1 #g2 #g elim (pn_split g) * #f #H0 #H destruct
+#g2 #g1 #g elim (pn_split g) * #f #H0 #H destruct
[ /3 width=7 by coafter_refl, ex2_intro/
-| @(ex2_intro … (⫯g2)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
+| @(ex2_intro … (⫯g1)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
]
qed-.
#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.
+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 on isid *******************************************************)
corec lemma coafter_isid_sn: ∀f1. 𝐈⦃f1⦄ → ∀f2. f1 ~⊚ f2 ≡ f2.
/3 width=7 by coafter_fwd_isid2_aux, coafter_fwd_isid2_O_aux/
qed-.
+fact coafter_isfin2_fwd_O_aux: ∀f1. @⦃0, f1⦄ ≡ 0 →
+ H_coafter_isfin2_fwd f1.
+#f1 #Hf1 #f2 #H
+generalize in match Hf1; generalize in match f1; -f1
+@(isfin_ind … H) -f2
+[ /3 width=4 by coafter_isid_inv_dx, isfin_isid/ ]
+#f2 #_ #IH #f1 #H #Hf1 #f #Hf
+elim (at_inv_pxp … H) -H [ |*: // ] #g1 #H1
+lapply (istot_inv_push … Hf1 … H1) -Hf1 #Hg1
+elim (Hg1 0) #n #Hn
+[ elim (coafter_inv_ppx … Hf) | elim (coafter_inv_pnx … Hf)
+] -Hf [1,6: |*: // ] #g #Hg #H0 destruct
+/5 width=6 by isfin_next, isfin_push, isfin_inv_tls, istot_tls, at_pxx_tls, coafter_tls/
+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.
+#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
+elim (coafter_inv_nxx … Hf … H1) -Hf #g #Hg #H0
+lapply (IH … Hg1 … Hg) -i2 -Hg
+/2 width=4 by istot_inv_next, isfin_push/ (**) (* full auto fails *)
+qed-.
+
+lemma coafter_isfin2_fwd: ∀f1. H_coafter_isfin2_fwd f1.
+#f1 #f2 #Hf2 #Hf1 cases (Hf1 0)
+/3 width=7 by coafter_isfin2_fwd_aux, coafter_isfin2_fwd_O_aux/
+qed-.
+
lemma coafter_inv_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀f1. f2 ~⊚ f1 ≡ f → ∀fa,fb. fa ⋓ fb ≡ f →
∃∃f1a,f1b. f2 ~⊚ f1a ≡ fa & f2 ~⊚ f1b ≡ fb & f1a ⋓ f1b ≡ f1.
@isfin_ind
elim (IH … Hg2 … H1f … H2f) -f -Hg2
/3 width=11 by sor_np, sor_pn, sor_nn, ex3_2_intro, coafter_refl, coafter_push/
]
-qed-.
+qed-.
+
+(* Properties with istot ****************************************************)
+
+lemma coafter_sor: ∀f. 𝐅⦃f⦄ → ∀f2. 𝐓⦃f2⦄ → ∀f1. f2 ~⊚ f1 ≡ f → ∀f1a,f1b. f1a ⋓ f1b ≡ f1 →
+ ∃∃fa,fb. f2 ~⊚ f1a ≡ fa & f2 ~⊚ f1b ≡ fb & fa ⋓ fb ≡ f.
+@isfin_ind
+[ #f #Hf #f2 #Hf2 #f1 #Hf #f1a #f1b #Hf1
+ lapply (coafter_fwd_isid2 … Hf ??) -Hf // #H2f1
+ elim (sor_inv_isid3 … Hf1) -Hf1 //
+ /3 width=5 by coafter_isid_dx, sor_refl, ex3_2_intro/
+| #f #_ #IH #f2 #Hf2 #f1 #H1 #f1a #f1b #H2
+ elim (coafter_inv_xxp … H1) -H1 [1,3: * |*: // ]
+ [ #g2 #g1 #Hf #Hgf2 #Hgf1
+ elim (sor_inv_xxp … H2) -H2 [ |*: // ] #ga #gb #Hg1
+ lapply (istot_inv_push … Hf2 … Hgf2) -Hf2 #Hg2
+ elim (IH … Hf … Hg1) // -f1 -g1 -IH -Hg2
+ /3 width=11 by coafter_refl, sor_pp, ex3_2_intro/
+ | #g2 #Hf #Hgf2
+ lapply (istot_inv_next … Hf2 … Hgf2) -Hf2 #Hg2
+ elim (IH … Hf … H2) // -f1 -IH -Hg2
+ /3 width=11 by coafter_next, sor_pp, ex3_2_intro/
+ ]
+| #f #_ #IH #f2 #Hf2 #f1 #H1 #f1a #f1b #H2
+ elim (coafter_inv_xxn … H1) -H1 [ |*: // ] #g2 #g1 #Hf #Hgf2 #Hgf1
+ lapply (istot_inv_push … Hf2 … Hgf2) -Hf2 #Hg2
+ elim (sor_inv_xxn … H2) -H2 [1,3,4: * |*: // ] #ga #gb #Hg1
+ elim (IH … Hf … Hg1) // -f1 -g1 -IH -Hg2
+ /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-.
+*)
projects / helm.git / blobdiff