coinductive coafter: relation3 rtmap rtmap rtmap ≝
| coafter_refl: ∀f1,f2,f,g1,g2,g. coafter f1 f2 f →
- â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g → coafter g1 g2 g
+ ⫯f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g → coafter g1 g2 g
| coafter_push: ∀f1,f2,f,g1,g2,g. coafter f1 f2 f →
- â\86\91f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g → coafter g1 g2 g
+ ⫯f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g → coafter g1 g2 g
| coafter_next: ∀f1,f2,f,g1,g. coafter f1 f2 f →
- ⫯f1 = g1 â\86\92 â\86\91f = g → coafter g1 f2 g
+ â\86\91f1 = g1 â\86\92 ⫯f = g → coafter g1 f2 g
.
interpretation "relational co-composition (rtmap)"
'RCoAfter f1 f2 f = (coafter f1 f2 f).
definition H_coafter_inj: predicate rtmap ≝
- λf1. ð\9d\90\93â¦\83f1â¦\84 →
- â\88\80f,f21,f22. f1 ~â\8a\9a f21 â\89¡ f â\86\92 f1 ~â\8a\9a f22 â\89¡ f â\86\92 f21 â\89\97 f22.
+ λf1. ð\9d\90\93â\9dªf1â\9d« →
+ â\88\80f,f21,f22. f1 ~â\8a\9a f21 â\89\98 f â\86\92 f1 ~â\8a\9a f22 â\89\98 f â\86\92 f21 â\89¡ f22.
definition H_coafter_fwd_isid2: predicate rtmap ≝
- λf1. â\88\80f2,f. f1 ~â\8a\9a f2 â\89¡ f â\86\92 ð\9d\90\93â¦\83f1â¦\84 â\86\92 ð\9d\90\88â¦\83fâ¦\84 â\86\92 ð\9d\90\88â¦\83f2â¦\84.
+ λf1. â\88\80f2,f. f1 ~â\8a\9a f2 â\89\98 f â\86\92 ð\9d\90\93â\9dªf1â\9d« â\86\92 ð\9d\90\88â\9dªfâ\9d« â\86\92 ð\9d\90\88â\9dªf2â\9d«.
definition H_coafter_isfin2_fwd: predicate rtmap ≝
- λf1. â\88\80f2. ð\9d\90\85â¦\83f2â¦\84 â\86\92 ð\9d\90\93â¦\83f1â¦\84 â\86\92 â\88\80f. f1 ~â\8a\9a f2 â\89¡ f â\86\92 ð\9d\90\85â¦\83fâ¦\84.
+ λf1. â\88\80f2. ð\9d\90\85â\9dªf2â\9d« â\86\92 ð\9d\90\93â\9dªf1â\9d« â\86\92 â\88\80f. f1 ~â\8a\9a f2 â\89\98 f â\86\92 ð\9d\90\85â\9dªfâ\9d«.
(* Basic inversion lemmas ***************************************************)
-lemma coafter_inv_ppx: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89¡ g â\86\92 â\88\80f1,f2. â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 →
- â\88\83â\88\83f. f1 ~â\8a\9a f2 â\89¡ f & â\86\91f = g.
+lemma coafter_inv_ppx: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89\98 g â\86\92 â\88\80f1,f2. ⫯f1 = g1 â\86\92 ⫯f2 = g2 →
+ â\88\83â\88\83f. f1 ~â\8a\9a f2 â\89\98 f & ⫯f = g.
#g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
[ #g2 #g #Hf #H1 #H2 #H #x1 #x2 #Hx1 #Hx2 destruct
>(injective_push … Hx1) >(injective_push … Hx2) -x2 -x1
]
qed-.
-lemma coafter_inv_pnx: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89¡ g â\86\92 â\88\80f1,f2. â\86\91f1 = g1 â\86\92 ⫯f2 = g2 →
- â\88\83â\88\83f. f1 ~â\8a\9a f2 â\89¡ f & ⫯f = g.
+lemma coafter_inv_pnx: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89\98 g â\86\92 â\88\80f1,f2. ⫯f1 = g1 â\86\92 â\86\91f2 = g2 →
+ â\88\83â\88\83f. f1 ~â\8a\9a f2 â\89\98 f & â\86\91f = g.
#g1 #g2 #g * -g1 -g2 -g #f1 #f2 #f #g1
[ #g2 #g #_ #_ #H2 #_ #x1 #x2 #_ #Hx2 destruct
elim (discr_next_push … Hx2)
]
qed-.
-lemma coafter_inv_nxx: â\88\80g1,f2,g. g1 ~â\8a\9a f2 â\89¡ g â\86\92 â\88\80f1. ⫯f1 = g1 →
- â\88\83â\88\83f. f1 ~â\8a\9a f2 â\89¡ f & â\86\91f = g.
+lemma coafter_inv_nxx: â\88\80g1,f2,g. g1 ~â\8a\9a f2 â\89\98 g â\86\92 â\88\80f1. â\86\91f1 = g1 →
+ â\88\83â\88\83f. f1 ~â\8a\9a f2 â\89\98 f & ⫯f = g.
#g1 #f2 #g * -g1 -f2 -g #f1 #f2 #f #g1
[ #g2 #g #_ #H1 #_ #_ #x1 #Hx1 destruct
elim (discr_next_push … Hx1)
(* Advanced inversion lemmas ************************************************)
-lemma coafter_inv_ppp: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89¡ g →
- â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g â\86\92 f1 ~â\8a\9a f2 â\89¡ f.
+lemma coafter_inv_ppp: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89\98 g →
+ â\88\80f1,f2,f. ⫯f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g â\86\92 f1 ~â\8a\9a f2 â\89\98 f.
#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H
elim (coafter_inv_ppx … Hg … H1 H2) -g1 -g2 #x #Hf #Hx destruct
<(injective_push … Hx) -f //
qed-.
-lemma coafter_inv_ppn: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89¡ g →
- â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 ⫯f = g → ⊥.
+lemma coafter_inv_ppn: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89\98 g →
+ â\88\80f1,f2,f. ⫯f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 â\86\91f = g → ⊥.
#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H
elim (coafter_inv_ppx … Hg … H1 H2) -g1 -g2 #x #Hf #Hx destruct
elim (discr_push_next … Hx)
qed-.
-lemma coafter_inv_pnn: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89¡ g →
- â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g â\86\92 f1 ~â\8a\9a f2 â\89¡ f.
+lemma coafter_inv_pnn: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89\98 g →
+ â\88\80f1,f2,f. ⫯f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g â\86\92 f1 ~â\8a\9a f2 â\89\98 f.
#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H
elim (coafter_inv_pnx … Hg … H1 H2) -g1 -g2 #x #Hf #Hx destruct
<(injective_next … Hx) -f //
qed-.
-lemma coafter_inv_pnp: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89¡ g →
- â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 â\86\91f = g → ⊥.
+lemma coafter_inv_pnp: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89\98 g →
+ â\88\80f1,f2,f. ⫯f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 ⫯f = g → ⊥.
#g1 #g2 #g #Hg #f1 #f2 #f #H1 #H2 #H
elim (coafter_inv_pnx … Hg … H1 H2) -g1 -g2 #x #Hf #Hx destruct
elim (discr_next_push … Hx)
qed-.
-lemma coafter_inv_nxp: â\88\80g1,f2,g. g1 ~â\8a\9a f2 â\89¡ g →
- â\88\80f1,f. ⫯f1 = g1 â\86\92 â\86\91f = g â\86\92 f1 ~â\8a\9a f2 â\89¡ f.
+lemma coafter_inv_nxp: â\88\80g1,f2,g. g1 ~â\8a\9a f2 â\89\98 g →
+ â\88\80f1,f. â\86\91f1 = g1 â\86\92 ⫯f = g â\86\92 f1 ~â\8a\9a f2 â\89\98 f.
#g1 #f2 #g #Hg #f1 #f #H1 #H
elim (coafter_inv_nxx … Hg … H1) -g1 #x #Hf #Hx destruct
<(injective_push … Hx) -f //
qed-.
-lemma coafter_inv_nxn: â\88\80g1,f2,g. g1 ~â\8a\9a f2 â\89¡ g →
- â\88\80f1,f. ⫯f1 = g1 â\86\92 ⫯f = g → ⊥.
+lemma coafter_inv_nxn: â\88\80g1,f2,g. g1 ~â\8a\9a f2 â\89\98 g →
+ â\88\80f1,f. â\86\91f1 = g1 â\86\92 â\86\91f = g → ⊥.
#g1 #f2 #g #Hg #f1 #f #H1 #H
elim (coafter_inv_nxx … Hg … H1) -g1 #x #Hf #Hx destruct
elim (discr_push_next … Hx)
qed-.
-lemma coafter_inv_pxp: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89¡ g →
- â\88\80f1,f. â\86\91f1 = g1 â\86\92 â\86\91f = g →
- â\88\83â\88\83f2. f1 ~â\8a\9a f2 â\89¡ f & â\86\91f2 = g2.
+lemma coafter_inv_pxp: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89\98 g →
+ â\88\80f1,f. ⫯f1 = g1 â\86\92 ⫯f = g →
+ â\88\83â\88\83f2. f1 ~â\8a\9a f2 â\89\98 f & ⫯f2 = g2.
#g1 #g2 #g #Hg #f1 #f #H1 #H elim (pn_split g2) * #f2 #H2
[ lapply (coafter_inv_ppp … Hg … H1 H2 H) -g1 -g /2 width=3 by ex2_intro/
| elim (coafter_inv_pnp … Hg … H1 H2 H)
]
qed-.
-lemma coafter_inv_pxn: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89¡ g →
- â\88\80f1,f. â\86\91f1 = g1 â\86\92 ⫯f = g →
- â\88\83â\88\83f2. f1 ~â\8a\9a f2 â\89¡ f & ⫯f2 = g2.
+lemma coafter_inv_pxn: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89\98 g →
+ â\88\80f1,f. ⫯f1 = g1 â\86\92 â\86\91f = g →
+ â\88\83â\88\83f2. f1 ~â\8a\9a f2 â\89\98 f & â\86\91f2 = g2.
#g1 #g2 #g #Hg #f1 #f #H1 #H elim (pn_split g2) * #f2 #H2
[ elim (coafter_inv_ppn … Hg … H1 H2 H)
| lapply (coafter_inv_pnn … Hg … H1 … H) -g1 -g /2 width=3 by ex2_intro/
]
qed-.
-lemma coafter_inv_xxn: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89¡ g â\86\92 â\88\80f. ⫯f = g →
- â\88\83â\88\83f1,f2. f1 ~â\8a\9a f2 â\89¡ f & â\86\91f1 = g1 & ⫯f2 = g2.
+lemma coafter_inv_xxn: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89\98 g â\86\92 â\88\80f. â\86\91f = g →
+ â\88\83â\88\83f1,f2. f1 ~â\8a\9a f2 â\89\98 f & ⫯f1 = g1 & â\86\91f2 = g2.
#g1 #g2 #g #Hg #f #H elim (pn_split g1) * #f1 #H1
[ elim (coafter_inv_pxn … Hg … H1 H) -g /2 width=5 by ex3_2_intro/
| elim (coafter_inv_nxn … Hg … H1 H)
]
qed-.
-lemma coafter_inv_xxp: ∀g1,g2,g. g1 ~⊚ g2 ≡ g → ∀f. ↑f = g →
- (∃∃f1,f2. f1 ~⊚ f2 ≡ f & ↑f1 = g1 & ↑f2 = g2) ∨
- ∃∃f1. f1 ~⊚ g2 ≡ f & ⫯f1 = g1.
+lemma coafter_inv_xnn: ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
+ ∀f2,f. ↑f2 = g2 → ↑f = g →
+ ∃∃f1. f1 ~⊚ f2 ≘ f & ⫯f1 = g1.
+#g1 #g2 #g #Hg #f2 #f #H2 destruct #H
+elim (coafter_inv_xxn … Hg … H) -g
+#z1 #z2 #Hf #H1 #H2 destruct /2 width=3 by ex2_intro/
+qed-.
+
+lemma coafter_inv_xxp: ∀g1,g2,g. g1 ~⊚ g2 ≘ g → ∀f. ⫯f = g →
+ (∃∃f1,f2. f1 ~⊚ f2 ≘ f & ⫯f1 = g1 & ⫯f2 = g2) ∨
+ ∃∃f1. f1 ~⊚ g2 ≘ f & ↑f1 = g1.
#g1 #g2 #g #Hg #f #H elim (pn_split g1) * #f1 #H1
[ elim (coafter_inv_pxp … Hg … H1 H) -g
/3 width=5 by or_introl, ex3_2_intro/
]
qed-.
-lemma coafter_inv_pxx: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89¡ g â\86\92 â\88\80f1. â\86\91f1 = g1 →
- (â\88\83â\88\83f2,f. f1 ~â\8a\9a f2 â\89¡ f & â\86\91f2 = g2 & â\86\91f = g) ∨
- (â\88\83â\88\83f2,f. f1 ~â\8a\9a f2 â\89¡ f & ⫯f2 = g2 & ⫯f = g).
+lemma coafter_inv_pxx: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89\98 g â\86\92 â\88\80f1. ⫯f1 = g1 →
+ (â\88\83â\88\83f2,f. f1 ~â\8a\9a f2 â\89\98 f & ⫯f2 = g2 & ⫯f = g) ∨
+ (â\88\83â\88\83f2,f. f1 ~â\8a\9a f2 â\89\98 f & â\86\91f2 = g2 & â\86\91f = g).
#g1 #g2 #g #Hg #f1 #H1 elim (pn_split g2) * #f2 #H2
[ elim (coafter_inv_ppx … Hg … H1 H2) -g1
/3 width=5 by or_introl, ex3_2_intro/
(* Basic properties *********************************************************)
-corec lemma coafter_eq_repl_back2: â\88\80f1,f. eq_repl_back (λf2. f2 ~â\8a\9a f1 â\89¡ f).
+corec lemma coafter_eq_repl_back2: â\88\80f1,f. eq_repl_back (λf2. f2 ~â\8a\9a f1 â\89\98 f).
#f1 #f #f2 * -f2 -f1 -f
#f21 #f1 #f #g21 [1,2: #g1 ] #g #Hf #H21 [1,2: #H1 ] #H #g22 #H0
[ cases (eq_inv_px … H0 … H21) -g21 /3 width=7 by coafter_refl/
]
qed-.
-lemma coafter_eq_repl_fwd2: â\88\80f1,f. eq_repl_fwd (λf2. f2 ~â\8a\9a f1 â\89¡ f).
+lemma coafter_eq_repl_fwd2: â\88\80f1,f. eq_repl_fwd (λf2. f2 ~â\8a\9a f1 â\89\98 f).
#f1 #f @eq_repl_sym /2 width=3 by coafter_eq_repl_back2/
qed-.
-corec lemma coafter_eq_repl_back1: â\88\80f2,f. eq_repl_back (λf1. f2 ~â\8a\9a f1 â\89¡ f).
+corec lemma coafter_eq_repl_back1: â\88\80f2,f. eq_repl_back (λf1. f2 ~â\8a\9a f1 â\89\98 f).
#f2 #f #f1 * -f2 -f1 -f
#f2 #f11 #f #g2 [1,2: #g11 ] #g #Hf #H2 [1,2: #H11 ] #H #g2 #H0
[ cases (eq_inv_px … H0 … H11) -g11 /3 width=7 by coafter_refl/
]
qed-.
-lemma coafter_eq_repl_fwd1: â\88\80f2,f. eq_repl_fwd (λf1. f2 ~â\8a\9a f1 â\89¡ f).
+lemma coafter_eq_repl_fwd1: â\88\80f2,f. eq_repl_fwd (λf1. f2 ~â\8a\9a f1 â\89\98 f).
#f2 #f @eq_repl_sym /2 width=3 by coafter_eq_repl_back1/
qed-.
-corec lemma coafter_eq_repl_back0: â\88\80f1,f2. eq_repl_back (λf. f2 ~â\8a\9a f1 â\89¡ f).
+corec lemma coafter_eq_repl_back0: â\88\80f1,f2. eq_repl_back (λf. f2 ~â\8a\9a f1 â\89\98 f).
#f2 #f1 #f * -f2 -f1 -f
#f2 #f1 #f01 #g2 [1,2: #g1 ] #g01 #Hf01 #H2 [1,2: #H1 ] #H01 #g02 #H0
[ cases (eq_inv_px … H0 … H01) -g01 /3 width=7 by coafter_refl/
]
qed-.
-lemma coafter_eq_repl_fwd0: â\88\80f2,f1. eq_repl_fwd (λf. f2 ~â\8a\9a f1 â\89¡ f).
+lemma coafter_eq_repl_fwd0: â\88\80f2,f1. eq_repl_fwd (λf. f2 ~â\8a\9a f1 â\89\98 f).
#f2 #f1 @eq_repl_sym /2 width=3 by coafter_eq_repl_back0/
qed-.
(* Main inversion lemmas ****************************************************)
-corec theorem coafter_mono: â\88\80f1,f2,x,y. f1 ~â\8a\9a f2 â\89¡ x â\86\92 f1 ~â\8a\9a f2 â\89¡ y â\86\92 x â\89\97 y.
+corec theorem coafter_mono: â\88\80f1,f2,x,y. f1 ~â\8a\9a f2 â\89\98 x â\86\92 f1 ~â\8a\9a f2 â\89\98 y â\86\92 x â\89¡ y.
#f1 #f2 #x #y * -f1 -f2 -x
#f1 #f2 #x #g1 [1,2: #g2 ] #g #Hx #H1 [1,2: #H2 ] #H0x #Hy
[ cases (coafter_inv_ppx … Hy … H1 H2) -g1 -g2 /3 width=8 by eq_push/
]
qed-.
-lemma coafter_mono_eq: â\88\80f1,f2,f. f1 ~â\8a\9a f2 â\89¡ f â\86\92 â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89¡ g →
- f1 â\89\97 g1 â\86\92 f2 â\89\97 g2 â\86\92 f â\89\97 g.
+lemma coafter_mono_eq: â\88\80f1,f2,f. f1 ~â\8a\9a f2 â\89\98 f â\86\92 â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89\98 g →
+ f1 â\89¡ g1 â\86\92 f2 â\89¡ g2 â\86\92 f â\89¡ 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/
+(* Forward lemmas with pushs ************************************************)
+
+lemma coafter_fwd_pushs: ∀j,i,g2,f1,g. g2 ~⊚ ⫯*[i]f1 ≘ g → @❪i, g2❫ ≘ j →
+ ∃f. ⫯*[j] f = g.
+#j elim j -j
+[ #i #g2 #f1 #g #Hg #H
+ elim (at_inv_xxp … H) -H [|*: // ] #f2 #H1 #H2 destruct
+ /2 width=2 by ex_intro/
+| #j #IH * [| #i ] #g2 #f1 #g #Hg #H
+ [ elim (at_inv_pxn … H) -H [|*: // ] #f2 #Hij #H destruct
+ elim (coafter_inv_nxx … Hg) -Hg [|*: // ] #f #Hf #H destruct
+ elim (IH … Hf Hij) -f1 -f2 -IH /2 width=2 by ex_intro/
+ | elim (at_inv_nxn … H) -H [1,4: * |*: // ] #f2 #Hij #H destruct
+ [ elim (coafter_inv_ppx … Hg) -Hg [|*: // ] #f #Hf #H destruct
+ elim (IH … Hf Hij) -f1 -f2 -i /2 width=2 by ex_intro/
+ | elim (coafter_inv_nxx … Hg) -Hg [|*: // ] #f #Hf #H destruct
+ elim (IH … Hf Hij) -f1 -f2 -i /2 width=2 by ex_intro/
+ ]
+ ]
+]
qed-.
(* Inversion lemmas with tail ***********************************************)
-lemma coafter_inv_tl1: â\88\80g2,g1,g. g2 ~â\8a\9a ⫱g1 â\89¡ g →
- â\88\83â\88\83f. â\86\91g2 ~â\8a\9a g1 â\89¡ f & ⫱f = g.
+lemma coafter_inv_tl1: â\88\80g2,g1,g. g2 ~â\8a\9a ⫱g1 â\89\98 g →
+ â\88\83â\88\83f. ⫯g2 ~â\8a\9a g1 â\89\98 f & ⫱f = g.
#g2 #g1 #g elim (pn_split g1) * #f1 #H1 #H destruct
[ /3 width=7 by coafter_refl, ex2_intro/
-| @(ex2_intro â\80¦ (⫯g)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
+| @(ex2_intro â\80¦ (â\86\91g)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
]
qed-.
-lemma coafter_inv_tl0: â\88\80g2,g1,g. g2 ~â\8a\9a g1 â\89¡ ⫱g →
- â\88\83â\88\83f1. â\86\91g2 ~â\8a\9a f1 â\89¡ g & ⫱f1 = g1.
+lemma coafter_inv_tl0: â\88\80g2,g1,g. g2 ~â\8a\9a g1 â\89\98 ⫱g →
+ â\88\83â\88\83f1. ⫯g2 ~â\8a\9a f1 â\89\98 g & ⫱f1 = g1.
#g2 #g1 #g elim (pn_split g) * #f #H0 #H destruct
[ /3 width=7 by coafter_refl, ex2_intro/
-| @(ex2_intro â\80¦ (⫯g1)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
+| @(ex2_intro â\80¦ (â\86\91g1)) /2 width=7 by coafter_push/ (**) (* full auto fails *)
]
qed-.
-(* Properties on tls ********************************************************)
-
-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 #g #Hg #H0 destruct
-<tls_xn <tls_xn /2 width=1 by/
+(* Properties with iterated tail ********************************************)
+
+lemma coafter_tls: ∀j,i,f1,f2,f. @❪i, f1❫ ≘ j →
+ f1 ~⊚ f2 ≘ f → ⫱*[j]f1 ~⊚ ⫱*[i]f2 ≘ ⫱*[j]f.
+#j elim j -j [ #i | #j #IH * [| #i ] ] #f1 #f2 #f #Hf1 #Hf
+[ elim (at_inv_xxp … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1 destruct //
+| elim (at_inv_pxn … Hf1) -Hf1 [ |*: // ] #g1 #Hg1 #H1
+ elim (coafter_inv_nxx … Hf … H1) -Hf #g #Hg #H0 destruct
+ lapply (IH … Hg1 Hg) -IH -Hg1 -Hg //
+| elim (at_inv_nxn … Hf1) -Hf1 [1,4: * |*: // ] #g1 #Hg1 #H1
+ [ elim (coafter_inv_pxx … Hf … H1) -Hf * #g2 #g #Hg #H2 #H0 destruct
+ lapply (IH … Hg1 Hg) -IH -Hg1 -Hg #H //
+ | elim (coafter_inv_nxx … Hf … H1) -Hf #g #Hg #H0 destruct
+ lapply (IH … Hg1 Hg) -IH -Hg1 -Hg #H //
+ ]
+]
qed.
-lemma coafter_tls_succ: ∀g2,g1,g. g2 ~⊚ g1 ≡ g →
- ∀n. @⦃0, g2⦄ ≡ n → ⫱*[⫯n]g2 ~⊚ ⫱g1 ≡ ⫱*[⫯n]g.
+lemma coafter_tls_O: ∀n,f1,f2,f. @❪0, f1❫ ≘ n →
+ f1 ~⊚ f2 ≘ f → ⫱*[n]f1 ~⊚ f2 ≘ ⫱*[n]f.
+/2 width=1 by coafter_tls/ 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
<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
+lemma coafter_fwd_xpx_pushs: ∀g2,f1,g,i,j. @❪i, g2❫ ≘ j → g2 ~⊚ ⫯*[↑i]f1 ≘ g →
+ ∃∃f. ⫱*[↑j]g2 ~⊚ f1 ≘ f & ⫯*[↑j]f = g.
+#g2 #g1 #g #i #j #Hg2 <pushs_xn #Hg
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)
-]
+lapply (coafter_tls … Hg2 Hg) -Hg <tls_pushs <tls_pushs #Hf
+lapply (at_inv_tls … Hg2) -Hg2 #H
+lapply (coafter_eq_repl_fwd2 … Hf … H) -H -Hf #Hf
+elim (coafter_inv_ppx … Hf) [|*: // ] -Hf #g #Hg #H destruct
+/2 width=3 by ex2_intro/
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
+lemma coafter_fwd_xnx_pushs: ∀g2,f1,g,i,j. @❪i, g2❫ ≘ j → g2 ~⊚ ⫯*[i]↑f1 ≘ g →
+ ∃∃f. ⫱*[↑j]g2 ~⊚ f1 ≘ f & ⫯*[j] ↑f = g.
+#g2 #g1 #g #i #j #Hg2 #Hg
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/
-]
+lapply (coafter_tls … Hg2 Hg) -Hg <tls_pushs <tls_pushs #Hf
+lapply (at_inv_tls … Hg2) -Hg2 #H
+lapply (coafter_eq_repl_fwd2 … Hf … H) -H -Hf #Hf
+elim (coafter_inv_pnx … Hf) [|*: // ] -Hf #g #Hg #H destruct
+/2 width=3 by ex2_intro/
qed-.
-(* Properties on isid *******************************************************)
+(* Properties with test for identity ****************************************)
-corec lemma coafter_isid_sn: â\88\80f1. ð\9d\90\88â¦\83f1â¦\84 â\86\92 â\88\80f2. f1 ~â\8a\9a f2 â\89¡ f2.
+corec lemma coafter_isid_sn: â\88\80f1. ð\9d\90\88â\9dªf1â\9d« â\86\92 â\88\80f2. f1 ~â\8a\9a f2 â\89\98 f2.
#f1 * -f1 #f1 #g1 #Hf1 #H1 #f2 cases (pn_split f2) * #g2 #H2
/3 width=7 by coafter_push, coafter_refl/
qed.
-corec lemma coafter_isid_dx: â\88\80f2,f. ð\9d\90\88â¦\83f2â¦\84 â\86\92 ð\9d\90\88â¦\83fâ¦\84 â\86\92 â\88\80f1. f1 ~â\8a\9a f2 â\89¡ f.
+corec lemma coafter_isid_dx: â\88\80f2,f. ð\9d\90\88â\9dªf2â\9d« â\86\92 ð\9d\90\88â\9dªfâ\9d« â\86\92 â\88\80f1. f1 ~â\8a\9a f2 â\89\98 f.
#f2 #f * -f2 #f2 #g2 #Hf2 #H2 * -f #f #g #Hf #H #f1 cases (pn_split f1) * #g1 #H1
[ /3 width=7 by coafter_refl/
| @(coafter_next … H1 … H) /3 width=3 by isid_push/
]
qed.
-(* Inversion lemmas on isid *************************************************)
+(* Inversion lemmas with test for identity **********************************)
-lemma coafter_isid_inv_sn: â\88\80f1,f2,f. f1 ~â\8a\9a f2 â\89¡ f â\86\92 ð\9d\90\88â¦\83f1â¦\84 â\86\92 f2 â\89\97 f.
+lemma coafter_isid_inv_sn: â\88\80f1,f2,f. f1 ~â\8a\9a f2 â\89\98 f â\86\92 ð\9d\90\88â\9dªf1â\9d« â\86\92 f2 â\89¡ f.
/3 width=6 by coafter_isid_sn, coafter_mono/ qed-.
-lemma coafter_isid_inv_dx: â\88\80f1,f2,f. f1 ~â\8a\9a f2 â\89¡ f â\86\92 ð\9d\90\88â¦\83f2â¦\84 â\86\92 ð\9d\90\88â¦\83fâ¦\84.
+lemma coafter_isid_inv_dx: â\88\80f1,f2,f. f1 ~â\8a\9a f2 â\89\98 f â\86\92 ð\9d\90\88â\9dªf2â\9d« â\86\92 ð\9d\90\88â\9dªfâ\9d«.
/4 width=4 by eq_id_isid, coafter_isid_dx, coafter_mono/ qed-.
-(*
-(* Properties on isuni ******************************************************)
-lemma coafter_isid_isuni: ∀f1,f2. 𝐈⦃f2⦄ → 𝐔⦃f1⦄ → f1 ~⊚ ⫯f2 ≡ ⫯f1.
+(* 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/
qed.
-lemma coafter_uni_next2: â\88\80f2. ð\9d\90\94â¦\83f2â¦\84 â\86\92 â\88\80f1,f. ⫯f2 ~â\8a\9a f1 â\89¡ f â\86\92 f2 ~â\8a\9a ⫯f1 â\89¡ f.
+lemma coafter_uni_next2: â\88\80f2. ð\9d\90\94â\9dªf2â\9d« â\86\92 â\88\80f1,f. â\86\91f2 ~â\8a\9a f1 â\89\98 f â\86\92 f2 ~â\8a\9a â\86\91f1 â\89\98 f.
#f2 #H elim H -f2
[ #f2 #Hf2 #f1 #f #Hf
elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
/3 width=5 by coafter_next/
]
qed.
+*)
-(* Properties on uni ********************************************************)
+(* Properties with uniform relocations **************************************)
-lemma coafter_uni: ∀n1,n2. 𝐔❴n1❵ ~⊚ 𝐔❴n2❵ ≡ 𝐔❴n1+n2❵.
+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/
qed.
(* Forward lemmas on at *****************************************************)
-lemma coafter_at_fwd: â\88\80i,i1,f. @â¦\83i1, fâ¦\84 â\89¡ i â\86\92 â\88\80f2,f1. f2 ~â\8a\9a f1 â\89¡ f →
- ∃∃i2. @⦃i1, f1⦄ ≡ i2 & @⦃i2, f2⦄ ≡ i.
+lemma coafter_at_fwd: â\88\80i,i1,f. @â\9dªi1, fâ\9d« â\89\98 i â\86\92 â\88\80f2,f1. f2 ~â\8a\9a f1 â\89\98 f →
+ ∃∃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 ]
/3 width=9 by at_refl, at_push, at_next, ex2_intro/
qed-.
-lemma coafter_fwd_at: â\88\80i,i2,i1,f1,f2. @â¦\83i1, f1â¦\84 â\89¡ i2 â\86\92 @â¦\83i2, f2â¦\84 â\89¡ i →
- ∀f. f2 ~⊚ f1 ≡ f → @⦃i1, f⦄ ≡ i.
+lemma coafter_fwd_at: â\88\80i,i2,i1,f1,f2. @â\9dªi1, f1â\9d« â\89\98 i2 â\86\92 @â\9dªi2, f2â\9d« â\89\98 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: â\88\80f,i1,i. @â¦\83i1, fâ¦\84 â\89¡ i â\86\92 â\88\80f1,i2. @â¦\83i1, f1â¦\84 â\89¡ i2 →
- ∀f2. f2 ~⊚ f1 ≡ f → @⦃i2, f2⦄ ≡ i.
+lemma coafter_fwd_at2: â\88\80f,i1,i. @â\9dªi1, fâ\9d« â\89\98 i â\86\92 â\88\80f1,i2. @â\9dªi1, f1â\9d« â\89\98 i2 →
+ ∀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: â\88\80i,i2,i1,f,f2. @â¦\83i1, fâ¦\84 â\89¡ i â\86\92 @â¦\83i2, f2â¦\84 â\89¡ i →
- ∀f1. f2 ~⊚ f1 ≡ f → @⦃i1, f1⦄ ≡ i2.
+lemma coafter_fwd_at1: â\88\80i,i2,i1,f,f2. @â\9dªi1, fâ\9d« â\89\98 i â\86\92 @â\9dªi2, f2â\9d« â\89\98 i →
+ ∀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: â\88\80i2,i1,f2. @â¦\83i1, f2â¦\84 â\89¡ i2 →
- ∀f. f2 ~⊚ 𝐔❴i1❵ ≡ f → 𝐔❴i2❵ ~⊚ ⫱*[i2] f2 ≡ f.
+lemma coafter_uni_dx: â\88\80i2,i1,f2. @â\9dªi1, f2â\9d« â\89\98 i2 →
+ ∀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: â\88\80i2,i1,f2. @â¦\83i1, f2â¦\84 â\89¡ i2 →
- ∀f. 𝐔❴i2❵ ~⊚ ⫱*[i2] f2 ≡ f → f2 ~⊚ 𝐔❴i1❵ ≡ f.
+lemma coafter_uni_sn: â\88\80i2,i1,f2. @â\9dªi1, f2â\9d« â\89\98 i2 →
+ ∀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: â\88\80i2,i1,f2. @â¦\83i1, f2â¦\84 â\89¡ i2 →
- ∀f. f2 ~⊚ 𝐔❴⫯i1❵ ≡ f → 𝐔❴⫯i2❵ ~⊚ ⫱*[⫯i2] f2 ≡ f.
+lemma coafter_uni_succ_dx: â\88\80i2,i1,f2. @â\9dªi1, f2â\9d« â\89\98 i2 →
+ ∀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: â\88\80i2,i1,f2. @â¦\83i1, f2â¦\84 â\89¡ i2 →
- ∀f. 𝐔❴⫯i2❵ ~⊚ ⫱*[⫯i2] f2 ≡ f → f2 ~⊚ 𝐔❴⫯i1❵ ≡ f.
+lemma coafter_uni_succ_sn: â\88\80i2,i1,f2. @â\9dªi1, f2â\9d« â\89\98 i2 →
+ ∀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_one_dx: â\88\80f2,f. â\86\91f2 ~â\8a\9a ð\9d\90\94â\9d´â«¯Oâ\9dµ â\89¡ f â\86\92 ð\9d\90\94â\9d´â«¯Oâ\9dµ ~â\8a\9a f2 â\89¡ f.
-#f2 #f #H @(coafter_uni_succ_dx â\80¦ (â\86\91f2)) /2 width=3 by at_refl/
+lemma coafter_uni_one_dx: â\88\80f2,f. ⫯f2 ~â\8a\9a ð\9d\90\94â\9d¨â\86\91Oâ\9d© â\89\98 f â\86\92 ð\9d\90\94â\9d¨â\86\91Oâ\9d© ~â\8a\9a f2 â\89\98 f.
+#f2 #f #H @(coafter_uni_succ_dx â\80¦ (⫯f2)) /2 width=3 by at_refl/
qed.
-lemma coafter_uni_one_sn: â\88\80f1,f. ð\9d\90\94â\9d´â«¯Oâ\9dµ ~â\8a\9a f1 â\89¡ f â\86\92 â\86\91f1 ~â\8a\9a ð\9d\90\94â\9d´â«¯Oâ\9dµ â\89¡ f.
+lemma coafter_uni_one_sn: â\88\80f1,f. ð\9d\90\94â\9d¨â\86\91Oâ\9d© ~â\8a\9a f1 â\89\98 f â\86\92 ⫯f1 ~â\8a\9a ð\9d\90\94â\9d¨â\86\91Oâ\9d© â\89\98 f.
/3 width=3 by coafter_uni_succ_sn, at_refl/ qed-.
*)
(* Forward lemmas with istot ************************************************)
(*
-lemma coafter_istot_fwd: â\88\80f2,f1,f. f2 ~â\8a\9a f1 â\89¡ f â\86\92 ð\9d\90\93â¦\83f2â¦\84 â\86\92 ð\9d\90\93â¦\83f1â¦\84 â\86\92 ð\9d\90\93â¦\83fâ¦\84.
+lemma coafter_istot_fwd: â\88\80f2,f1,f. f2 ~â\8a\9a f1 â\89\98 f â\86\92 ð\9d\90\93â\9dªf2â\9d« â\86\92 ð\9d\90\93â\9dªf1â\9d« â\86\92 ð\9d\90\93â\9dªfâ\9d«.
#f2 #f1 #f #Hf #Hf2 #Hf1 #i1 elim (Hf1 i1) -Hf1
#i2 #Hf1 elim (Hf2 i2) -Hf2
/3 width=7 by coafter_fwd_at, ex_intro/
qed-.
-lemma coafter_fwd_istot_dx: â\88\80f2,f1,f. f2 ~â\8a\9a f1 â\89¡ f â\86\92 ð\9d\90\93â¦\83fâ¦\84 â\86\92 ð\9d\90\93â¦\83f1â¦\84.
+lemma coafter_fwd_istot_dx: â\88\80f2,f1,f. f2 ~â\8a\9a f1 â\89\98 f â\86\92 ð\9d\90\93â\9dªfâ\9d« â\86\92 ð\9d\90\93â\9dªf1â\9d«.
#f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
#i2 #Hf elim (coafter_at_fwd … Hf … H) -f /2 width=2 by ex_intro/
qed-.
-lemma coafter_fwd_istot_sn: â\88\80f2,f1,f. f2 ~â\8a\9a f1 â\89¡ f â\86\92 ð\9d\90\93â¦\83fâ¦\84 â\86\92 ð\9d\90\93â¦\83f2â¦\84.
+lemma coafter_fwd_istot_sn: â\88\80f2,f1,f. f2 ~â\8a\9a f1 â\89\98 f â\86\92 ð\9d\90\93â\9dªfâ\9d« â\86\92 ð\9d\90\93â\9dªf2â\9d«.
#f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
#i #Hf elim (coafter_at_fwd … Hf … H) -f
#i2 #Hf1 #Hf2 lapply (at_increasing … Hf1) -f1
#Hi12 elim (at_le_ex … Hf2 … Hi12) -i2 /2 width=2 by ex_intro/
qed-.
-lemma coafter_inv_istot: â\88\80f2,f1,f. f2 ~â\8a\9a f1 â\89¡ f â\86\92 ð\9d\90\93â¦\83fâ¦\84 â\86\92 ð\9d\90\93â¦\83f2â¦\84 â\88§ ð\9d\90\93â¦\83f1â¦\84.
+lemma coafter_inv_istot: â\88\80f2,f1,f. f2 ~â\8a\9a f1 â\89\98 f â\86\92 ð\9d\90\93â\9dªfâ\9d« â\86\92 ð\9d\90\93â\9dªf2â\9d« â\88§ ð\9d\90\93â\9dªf1â\9d«.
/3 width=4 by coafter_fwd_istot_sn, coafter_fwd_istot_dx, conj/ qed-.
-lemma coafter_at1_fwd: â\88\80f1,i1,i2. @â¦\83i1, f1â¦\84 â\89¡ i2 â\86\92 â\88\80f2. ð\9d\90\93â¦\83f2â¦\84 â\86\92 â\88\80f. f2 ~â\8a\9a f1 â\89¡ f →
- â\88\83â\88\83i. @â¦\83i2, f2â¦\84 â\89¡ i & @â¦\83i1, fâ¦\84 â\89¡ i.
+lemma coafter_at1_fwd: â\88\80f1,i1,i2. @â\9dªi1, f1â\9d« â\89\98 i2 â\86\92 â\88\80f2. ð\9d\90\93â\9dªf2â\9d« â\86\92 â\88\80f. f2 ~â\8a\9a f1 â\89\98 f →
+ â\88\83â\88\83i. @â\9dªi2, f2â\9d« â\89\98 i & @â\9dªi1, fâ\9d« â\89\98 i.
#f1 #i1 #i2 #Hf1 #f2 #Hf2 #f #Hf elim (Hf2 i2) -Hf2
/3 width=8 by coafter_fwd_at, ex2_intro/
qed-.
-lemma coafter_fwd_isid_sn: â\88\80f2,f1,f. ð\9d\90\93â¦\83fâ¦\84 â\86\92 f2 ~â\8a\9a f1 â\89¡ f â\86\92 f1 â\89\97 f â\86\92 ð\9d\90\88â¦\83f2â¦\84.
+lemma coafter_fwd_isid_sn: â\88\80f2,f1,f. ð\9d\90\93â\9dªfâ\9d« â\86\92 f2 ~â\8a\9a f1 â\89\98 f â\86\92 f1 â\89¡ f â\86\92 ð\9d\90\88â\9dªf2â\9d«.
#f2 #f1 #f #H #Hf elim (coafter_inv_istot … Hf H) -H
#Hf2 #Hf1 #H @isid_at_total // -Hf2
#i2 #i #Hf2 elim (Hf1 i2) -Hf1
/3 width=7 by at_eq_repl_back, at_mono, at_id_le/
qed-.
-lemma coafter_fwd_isid_dx: â\88\80f2,f1,f. ð\9d\90\93â¦\83fâ¦\84 â\86\92 f2 ~â\8a\9a f1 â\89¡ f â\86\92 f2 â\89\97 f â\86\92 ð\9d\90\88â¦\83f1â¦\84.
+lemma coafter_fwd_isid_dx: â\88\80f2,f1,f. ð\9d\90\93â\9dªfâ\9d« â\86\92 f2 ~â\8a\9a f1 â\89\98 f â\86\92 f2 â\89¡ f â\86\92 ð\9d\90\88â\9dªf1â\9d«.
#f2 #f1 #f #H #Hf elim (coafter_inv_istot … Hf H) -H
#Hf2 #Hf1 #H2 @isid_at_total // -Hf1
#i1 #i2 #Hi12 elim (coafter_at1_fwd … Hi12 … Hf) -f1
/3 width=8 by at_inj, at_eq_repl_back/
qed-.
*)
-corec fact coafter_inj_O_aux: â\88\80f1. @â¦\830, f1â¦\84 â\89¡ 0 → H_coafter_inj f1.
+corec fact coafter_inj_O_aux: â\88\80f1. @â\9dª0, f1â\9d« â\89\98 0 → H_coafter_inj f1.
#f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
cases (at_inv_pxp … H1f1) -H1f1 [ |*: // ] #g1 #H1
lapply (istot_inv_push … H2f1 … H1) -H2f1 #H2g1
/2 width=1 by coafter_tls, istot_tls, at_pxx_tls/
qed-.
-fact coafter_inj_aux: (â\88\80f1. @â¦\830, f1â¦\84 â\89¡ 0 → H_coafter_inj f1) →
- â\88\80i2,f1. @â¦\830, f1â¦\84 â\89¡ i2 → H_coafter_inj f1.
+fact coafter_inj_aux: (â\88\80f1. @â\9dª0, f1â\9d« â\89\98 0 → H_coafter_inj f1) →
+ â\88\80i2,f1. @â\9dª0, f1â\9d« â\89\98 i2 → H_coafter_inj f1.
#H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
#i2 #IH #f1 #H1f1 #H2f1 #f #f21 #f22 #H1f #H2f
elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #H1g1 #H1
#f1 #H cases (H 0) /3 width=7 by coafter_inj_aux, coafter_inj_O_aux/
qed-.
-corec fact coafter_fwd_isid2_O_aux: â\88\80f1. @â¦\830, f1â¦\84 â\89¡ 0 →
+corec fact coafter_fwd_isid2_O_aux: â\88\80f1. @â\9dª0, f1â\9d« â\89\98 0 →
H_coafter_fwd_isid2 f1.
#f1 #H1f1 #f2 #f #H #H2f1 #Hf
cases (at_inv_pxp … H1f1) -H1f1 [ |*: // ] #g1 #H1
cases (H2g1 0) #n #Hn
cases (coafter_inv_pxx … H … H1) -H * #g2 #g #H #H2 #H0
[ lapply (isid_inv_push … Hf … H0) -Hf #Hg
- @(isid_push … H2)
- /3 width=7 by coafter_tls, istot_tls, at_pxx_tls, isid_tls/
+ @(isid_push … H2) -H2
+ /3 width=7 by coafter_tls_O, at_pxx_tls, istot_tls, isid_tls/
| cases (isid_inv_next … Hf … H0)
]
qed-.
-fact coafter_fwd_isid2_aux: (â\88\80f1. @â¦\830, f1â¦\84 â\89¡ 0 → H_coafter_fwd_isid2 f1) →
- â\88\80i2,f1. @â¦\830, f1â¦\84 â\89¡ i2 → H_coafter_fwd_isid2 f1.
+fact coafter_fwd_isid2_aux: (â\88\80f1. @â\9dª0, f1â\9d« â\89\98 0 → H_coafter_fwd_isid2 f1) →
+ â\88\80i2,f1. @â\9dª0, f1â\9d« â\89\98 i2 → H_coafter_fwd_isid2 f1.
#H0 #i2 elim i2 -i2 /2 width=1 by/ -H0
#i2 #IH #f1 #H1f1 #f2 #f #H #H2f1 #Hf
elim (at_inv_pxn … H1f1) -H1f1 [ |*: // ] #g1 #Hg1 #H1
/3 width=7 by coafter_fwd_isid2_aux, coafter_fwd_isid2_O_aux/
qed-.
-fact coafter_isfin2_fwd_O_aux: â\88\80f1. @â¦\830, f1â¦\84 â\89¡ 0 →
+fact coafter_isfin2_fwd_O_aux: â\88\80f1. @â\9dª0, f1â\9d« â\89\98 0 →
H_coafter_isfin2_fwd f1.
#f1 #Hf1 #f2 #H
generalize in match Hf1; generalize in match f1; -f1
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/
+/5 width=6 by isfin_next, isfin_push, isfin_inv_tls, istot_tls, at_pxx_tls, coafter_tls_O/
qed-.
-fact coafter_isfin2_fwd_aux: (â\88\80f1. @â¦\830, f1â¦\84 â\89¡ 0 → H_coafter_isfin2_fwd f1) →
- â\88\80i2,f1. @â¦\830, f1â¦\84 â\89¡ i2 → H_coafter_isfin2_fwd f1.
+fact coafter_isfin2_fwd_aux: (â\88\80f1. @â\9dª0, f1â\9d« â\89\98 0 → H_coafter_isfin2_fwd f1) →
+ â\88\80i2,f1. @â\9dª0, f1â\9d« â\89\98 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=7 by coafter_isfin2_fwd_aux, coafter_isfin2_fwd_O_aux/
qed-.
-lemma coafter_inv_sor: â\88\80f. ð\9d\90\85â¦\83fâ¦\84 â\86\92 â\88\80f2. ð\9d\90\93â¦\83f2â¦\84 â\86\92 â\88\80f1. f2 ~â\8a\9a f1 â\89¡ f â\86\92 â\88\80fa,fb. fa â\8b\93 fb â\89¡ f →
- â\88\83â\88\83f1a,f1b. f2 ~â\8a\9a f1a â\89¡ fa & f2 ~â\8a\9a f1b â\89¡ fb & f1a â\8b\93 f1b â\89¡ f1.
+lemma coafter_inv_sor: â\88\80f. ð\9d\90\85â\9dªfâ\9d« â\86\92 â\88\80f2. ð\9d\90\93â\9dªf2â\9d« â\86\92 â\88\80f1. f2 ~â\8a\9a f1 â\89\98 f â\86\92 â\88\80fa,fb. fa â\8b\93 fb â\89\98 f →
+ â\88\83â\88\83f1a,f1b. f2 ~â\8a\9a f1a â\89\98 fa & f2 ~â\8a\9a f1b â\89\98 fb & f1a â\8b\93 f1b â\89\98 f1.
@isfin_ind
[ #f #Hf #f2 #Hf2 #f1 #H1f #fa #fb #H2f
elim (sor_inv_isid3 … H2f) -H2f //
(* Properties with istot ****************************************************)
-lemma coafter_sor: â\88\80f. ð\9d\90\85â¦\83fâ¦\84 â\86\92 â\88\80f2. ð\9d\90\93â¦\83f2â¦\84 â\86\92 â\88\80f1. f2 ~â\8a\9a f1 â\89¡ f â\86\92 â\88\80f1a,f1b. f1a â\8b\93 f1b â\89¡ f1 →
- â\88\83â\88\83fa,fb. f2 ~â\8a\9a f1a â\89¡ fa & f2 ~â\8a\9a f1b â\89¡ fb & fa â\8b\93 fb â\89¡ f.
+lemma coafter_sor: â\88\80f. ð\9d\90\85â\9dªfâ\9d« â\86\92 â\88\80f2. ð\9d\90\93â\9dªf2â\9d« â\86\92 â\88\80f1. f2 ~â\8a\9a f1 â\89\98 f â\86\92 â\88\80f1a,f1b. f1a â\8b\93 f1b â\89\98 f1 →
+ â\88\83â\88\83fa,fb. f2 ~â\8a\9a f1a â\89\98 fa & f2 ~â\8a\9a f1b â\89\98 fb & fa â\8b\93 fb â\89\98 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/
+ /3 width=5 by coafter_isid_dx, sor_idem, 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
(* Properties with after ****************************************************)
(*
-corec theorem coafter_trans1: â\88\80f0,f3,f4. f0 ~â\8a\9a f3 â\89¡ f4 →
- â\88\80f1,f2. f1 ~â\8a\9a f2 â\89¡ f0 →
- â\88\80f. f2 ~â\8a\9a f3 â\89¡ f â\86\92 f1 ~â\8a\9a f â\89¡ f4.
+corec theorem coafter_trans1: â\88\80f0,f3,f4. f0 ~â\8a\9a f3 â\89\98 f4 →
+ â\88\80f1,f2. f1 ~â\8a\9a f2 â\89\98 f0 →
+ â\88\80f. f2 ~â\8a\9a f3 â\89\98 f â\86\92 f1 ~â\8a\9a f â\89\98 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
]
qed-.
-corec theorem coafter_trans2: â\88\80f1,f0,f4. f1 ~â\8a\9a f0 â\89¡ f4 →
- â\88\80f2, f3. f2 ~â\8a\9a f3 â\89¡ f0 →
- â\88\80f. f1 ~â\8a\9a f2 â\89¡ f â\86\92 f ~â\8a\9a f3 â\89¡ f4.
+corec theorem coafter_trans2: â\88\80f1,f0,f4. f1 ~â\8a\9a f0 â\89\98 f4 →
+ â\88\80f2, f3. f2 ~â\8a\9a f3 â\89\98 f0 →
+ â\88\80f. f1 ~â\8a\9a f2 â\89\98 f â\86\92 f ~â\8a\9a f3 â\89\98 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
projects / helm.git / blobdiff