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\98 f â\86\92 f1 ~â\8a\9a f22 â\89\98 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\98 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\98 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\98 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\98 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\98 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\98 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\98 g â\86\92 â\88\80f1. ⫯f1 = g1 →
- â\88\83â\88\83f. f1 ~â\8a\9a f2 â\89\98 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: ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
- â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g → f1 ~⊚ f2 ≘ f.
+ â\88\80f1,f2,f. ⫯f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g → f1 ~⊚ f2 ≘ 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: ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
- â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 ⫯f = 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: ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
- â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 ⫯f = g → f1 ~⊚ f2 ≘ f.
+ â\88\80f1,f2,f. ⫯f1 = g1 â\86\92 â\86\91f2 = g2 â\86\92 â\86\91f = g → f1 ~⊚ f2 ≘ 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: ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
- â\88\80f1,f2,f. â\86\91f1 = g1 â\86\92 ⫯f2 = g2 â\86\92 â\86\91f = 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: ∀g1,f2,g. g1 ~⊚ f2 ≘ g →
- â\88\80f1,f. ⫯f1 = g1 â\86\92 â\86\91f = g → f1 ~⊚ f2 ≘ f.
+ â\88\80f1,f. â\86\91f1 = g1 â\86\92 ⫯f = g → f1 ~⊚ f2 ≘ 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: ∀g1,f2,g. g1 ~⊚ f2 ≘ g →
- â\88\80f1,f. ⫯f1 = g1 â\86\92 ⫯f = 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: ∀g1,g2,g. g1 ~⊚ g2 ≘ 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\98 f & â\86\91f2 = g2.
+ 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: ∀g1,g2,g. g1 ~⊚ g2 ≘ 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\98 f & ⫯f2 = g2.
+ 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\98 g â\86\92 â\88\80f. ⫯f = g →
- â\88\83â\88\83f1,f2. f1 ~â\8a\9a f2 â\89\98 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_xnn: ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
- â\88\80f2,f. ⫯f2 = g2 â\86\92 ⫯f = g →
- â\88\83â\88\83f1. f1 ~â\8a\9a f2 â\89\98 f & â\86\91f1 = g1.
+ â\88\80f2,f. â\86\91f2 = g2 â\86\92 â\86\91f = g →
+ â\88\83â\88\83f1. f1 ~â\8a\9a f2 â\89\98 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: 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 & â\86\91f1 = g1 & â\86\91f2 = g2) ∨
- â\88\83â\88\83f1. f1 ~â\8a\9a g2 â\89\98 f & ⫯f1 = g1.
+lemma coafter_inv_xxp: â\88\80g1,g2,g. g1 ~â\8a\9a g2 â\89\98 g â\86\92 â\88\80f. ⫯f = g →
+ (â\88\83â\88\83f1,f2. f1 ~â\8a\9a f2 â\89\98 f & ⫯f1 = g1 & ⫯f2 = g2) ∨
+ â\88\83â\88\83f1. f1 ~â\8a\9a g2 â\89\98 f & â\86\91f1 = 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\98 g â\86\92 â\88\80f1. â\86\91f1 = g1 →
- (â\88\83â\88\83f2,f. f1 ~â\8a\9a f2 â\89\98 f & â\86\91f2 = g2 & â\86\91f = g) ∨
- (â\88\83â\88\83f2,f. f1 ~â\8a\9a f2 â\89\98 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/
(* Main inversion lemmas ****************************************************)
-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\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: ∀f1,f2,f. f1 ~⊚ f2 ≘ f → ∀g1,g2,g. g1 ~⊚ g2 ≘ g →
- f1 â\89\97 g1 â\86\92 f2 â\89\97 g2 â\86\92 f â\89\97 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-.
(* Forward lemmas with pushs ************************************************)
-lemma coafter_fwd_pushs: â\88\80j,i,g2,f1,g. g2 ~â\8a\9a â\86\91*[i]f1 â\89\98 g â\86\92 @â¦\83i, g2â¦\84 ≘ j →
- â\88\83f. â\86\91*[j] f = g.
+lemma coafter_fwd_pushs: â\88\80j,i,g2,f1,g. g2 ~â\8a\9a ⫯*[i]f1 â\89\98 g â\86\92 @â\9dªi, g2â\9d« ≘ j →
+ â\88\83f. ⫯*[j] f = g.
#j elim j -j
[ #i #g2 #f1 #g #Hg #H
elim (at_inv_xxp … H) -H [|*: // ] #f2 #H1 #H2 destruct
(* Inversion lemmas with tail ***********************************************)
lemma coafter_inv_tl1: ∀g2,g1,g. g2 ~⊚ ⫱g1 ≘ g →
- â\88\83â\88\83f. â\86\91g2 ~⊚ g1 ≘ f & ⫱f = g.
+ â\88\83â\88\83f. ⫯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 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: ∀g2,g1,g. g2 ~⊚ g1 ≘ ⫱g →
- â\88\83â\88\83f1. â\86\91g2 ~⊚ f1 ≘ g & ⫱f1 = g1.
+ â\88\83â\88\83f1. ⫯g2 ~⊚ f1 ≘ 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 with iterated tail ********************************************)
-lemma coafter_tls: â\88\80j,i,f1,f2,f. @â¦\83i, f1â¦\84 ≘ j →
+lemma coafter_tls: â\88\80j,i,f1,f2,f. @â\9dªi, f1â\9d« ≘ 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 //
]
qed.
-lemma coafter_tls_O: â\88\80n,f1,f2,f. @â¦\830, f1â¦\84 ≘ n →
+lemma coafter_tls_O: â\88\80n,f1,f2,f. @â\9dª0, f1â\9d« ≘ 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 →
- â\88\80n. @â¦\830, g2â¦\84 â\89\98 n â\86\92 ⫱*[⫯n]g2 ~â\8a\9a ⫱g1 â\89\98 ⫱*[⫯n]g.
+ â\88\80n. @â\9dª0, g2â\9d« â\89\98 n â\86\92 ⫱*[â\86\91n]g2 ~â\8a\9a ⫱g1 â\89\98 ⫱*[â\86\91n]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: â\88\80g2,f1,g,i,j. @â¦\83i, g2â¦\84 â\89\98 j â\86\92 g2 ~â\8a\9a â\86\91*[⫯i]f1 ≘ g →
- â\88\83â\88\83f. ⫱*[⫯j]g2 ~â\8a\9a f1 â\89\98 f & â\86\91*[⫯j]f = g.
+lemma coafter_fwd_xpx_pushs: â\88\80g2,f1,g,i,j. @â\9dªi, g2â\9d« â\89\98 j â\86\92 g2 ~â\8a\9a ⫯*[â\86\91i]f1 ≘ g →
+ â\88\83â\88\83f. ⫱*[â\86\91j]g2 ~â\8a\9a f1 â\89\98 f & ⫯*[â\86\91j]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 <tls_pushs #Hf
/2 width=3 by ex2_intro/
qed-.
-lemma coafter_fwd_xnx_pushs: â\88\80g2,f1,g,i,j. @â¦\83i, g2â¦\84 â\89\98 j â\86\92 g2 ~â\8a\9a â\86\91*[i]⫯f1 ≘ g →
- â\88\83â\88\83f. ⫱*[⫯j]g2 ~â\8a\9a f1 â\89\98 f & â\86\91*[j] ⫯f = g.
+lemma coafter_fwd_xnx_pushs: â\88\80g2,f1,g,i,j. @â\9dªi, g2â\9d« â\89\98 j â\86\92 g2 ~â\8a\9a ⫯*[i]â\86\91f1 ≘ g →
+ â\88\83â\88\83f. ⫱*[â\86\91j]g2 ~â\8a\9a f1 â\89\98 f & ⫯*[j] â\86\91f = 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 <tls_pushs #Hf
(* Properties with test for identity ****************************************)
-corec lemma coafter_isid_sn: â\88\80f1. ð\9d\90\88â¦\83f1â¦\84 → ∀f2. f1 ~⊚ f2 ≘ f2.
+corec lemma coafter_isid_sn: â\88\80f1. ð\9d\90\88â\9dªf1â\9d« → ∀f2. f1 ~⊚ f2 ≘ 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 → ∀f1. f1 ~⊚ f2 ≘ 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« → ∀f1. f1 ~⊚ f2 ≘ 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/
(* Inversion lemmas with test for identity **********************************)
-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¦\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\98 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 with test for uniform relocations *****************************)
-lemma coafter_isuni_isid: â\88\80f2. ð\9d\90\88â¦\83f2â¦\84 â\86\92 â\88\80f1. ð\9d\90\94â¦\83f1â¦\84 → f1 ~⊚ f2 ≘ f2.
+lemma coafter_isuni_isid: â\88\80f2. ð\9d\90\88â\9dªf2â\9d« â\86\92 â\88\80f1. ð\9d\90\94â\9dªf1â\9d« → 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: â\88\80f1,f2. ð\9d\90\88â¦\83f2â¦\84 â\86\92 ð\9d\90\94â¦\83f1â¦\84 â\86\92 f1 ~â\8a\9a ⫯f2 â\89\98 ⫯f1.
+lemma coafter_isid_isuni: â\88\80f1,f2. ð\9d\90\88â\9dªf2â\9d« â\86\92 ð\9d\90\94â\9dªf1â\9d« â\86\92 f1 ~â\8a\9a â\86\91f2 â\89\98 â\86\91f1.
#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\98 f â\86\92 f2 ~â\8a\9a ⫯f1 ≘ 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 ≘ f.
#f2 #H elim H -f2
[ #f2 #Hf2 #f1 #f #Hf
elim (coafter_inv_nxx … Hf) -Hf [2,3: // ] #g #Hg #H0 destruct
(* Properties with uniform relocations **************************************)
-lemma coafter_uni_sn: â\88\80i,f. ð\9d\90\94â\9d´iâ\9dµ ~â\8a\9a f â\89\98 â\86\91*[i] f.
+lemma coafter_uni_sn: â\88\80i,f. ð\9d\90\94â\9d¨iâ\9d© ~â\8a\9a f â\89\98 ⫯*[i] f.
#i elim i -i /2 width=5 by coafter_isid_sn, coafter_next/
qed.
(*
-lemma coafter_uni: â\88\80n1,n2. ð\9d\90\94â\9d´n1â\9dµ ~â\8a\9a ð\9d\90\94â\9d´n2â\9dµ â\89\98 ð\9d\90\94â\9d´n1+n2â\9dµ.
+lemma coafter_uni: â\88\80n1,n2. ð\9d\90\94â\9d¨n1â\9d© ~â\8a\9a ð\9d\90\94â\9d¨n2â\9d© â\89\98 ð\9d\90\94â\9d¨n1+n2â\9d©.
@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 ≘ i → ∀f2,f1. f2 ~⊚ f1 ≘ f →
- â\88\83â\88\83i2. @â¦\83i1, f1â¦\84 â\89\98 i2 & @â¦\83i2, f2â¦\84 ≘ i.
+lemma coafter_at_fwd: â\88\80i,i1,f. @â\9dªi1, fâ\9d« ≘ i → ∀f2,f1. f2 ~⊚ f1 ≘ f →
+ â\88\83â\88\83i2. @â\9dªi1, f1â\9d« â\89\98 i2 & @â\9dªi2, f2â\9d« ≘ 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\98 i2 â\86\92 @â¦\83i2, f2â¦\84 ≘ i →
- â\88\80f. f2 ~â\8a\9a f1 â\89\98 f â\86\92 @â¦\83i1, fâ¦\84 ≘ 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« ≘ i →
+ â\88\80f. f2 ~â\8a\9a f1 â\89\98 f â\86\92 @â\9dªi1, fâ\9d« ≘ 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\98 i â\86\92 â\88\80f1,i2. @â¦\83i1, f1â¦\84 ≘ i2 →
- â\88\80f2. f2 ~â\8a\9a f1 â\89\98 f â\86\92 @â¦\83i2, f2â¦\84 ≘ 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« ≘ i2 →
+ â\88\80f2. f2 ~â\8a\9a f1 â\89\98 f â\86\92 @â\9dªi2, f2â\9d« ≘ 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\98 i â\86\92 @â¦\83i2, f2â¦\84 ≘ i →
- â\88\80f1. f2 ~â\8a\9a f1 â\89\98 f â\86\92 @â¦\83i1, f1â¦\84 ≘ 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« ≘ i →
+ â\88\80f1. f2 ~â\8a\9a f1 â\89\98 f â\86\92 @â\9dªi1, f1â\9d« ≘ 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 ≘ i2 →
- â\88\80f. f2 ~â\8a\9a ð\9d\90\94â\9d´i1â\9dµ â\89\98 f â\86\92 ð\9d\90\94â\9d´i2â\9dµ ~⊚ ⫱*[i2] f2 ≘ f.
+lemma coafter_uni_dx: â\88\80i2,i1,f2. @â\9dªi1, f2â\9d« ≘ i2 →
+ â\88\80f. f2 ~â\8a\9a ð\9d\90\94â\9d¨i1â\9d© â\89\98 f â\86\92 ð\9d\90\94â\9d¨i2â\9d© ~⊚ ⫱*[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 ≘ i2 →
- â\88\80f. ð\9d\90\94â\9d´i2â\9dµ ~â\8a\9a ⫱*[i2] f2 â\89\98 f â\86\92 f2 ~â\8a\9a ð\9d\90\94â\9d´i1â\9dµ ≘ f.
+lemma coafter_uni_sn: â\88\80i2,i1,f2. @â\9dªi1, f2â\9d« ≘ i2 →
+ â\88\80f. ð\9d\90\94â\9d¨i2â\9d© ~â\8a\9a ⫱*[i2] f2 â\89\98 f â\86\92 f2 ~â\8a\9a ð\9d\90\94â\9d¨i1â\9d© ≘ 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 ≘ i2 →
- â\88\80f. f2 ~â\8a\9a ð\9d\90\94â\9d´â«¯i1â\9dµ â\89\98 f â\86\92 ð\9d\90\94â\9d´â«¯i2â\9dµ ~â\8a\9a ⫱*[⫯i2] f2 ≘ f.
+lemma coafter_uni_succ_dx: â\88\80i2,i1,f2. @â\9dªi1, f2â\9d« ≘ i2 →
+ â\88\80f. f2 ~â\8a\9a ð\9d\90\94â\9d¨â\86\91i1â\9d© â\89\98 f â\86\92 ð\9d\90\94â\9d¨â\86\91i2â\9d© ~â\8a\9a ⫱*[â\86\91i2] 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 ≘ i2 →
- â\88\80f. ð\9d\90\94â\9d´â«¯i2â\9dµ ~â\8a\9a ⫱*[⫯i2] f2 â\89\98 f â\86\92 f2 ~â\8a\9a ð\9d\90\94â\9d´â«¯i1â\9dµ ≘ f.
+lemma coafter_uni_succ_sn: â\88\80i2,i1,f2. @â\9dªi1, f2â\9d« ≘ i2 →
+ â\88\80f. ð\9d\90\94â\9d¨â\86\91i2â\9d© ~â\8a\9a ⫱*[â\86\91i2] f2 â\89\98 f â\86\92 f2 ~â\8a\9a ð\9d\90\94â\9d¨â\86\91i1â\9d© ≘ 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\98 f â\86\92 ð\9d\90\94â\9d´â«¯Oâ\9dµ ~⊚ f2 ≘ 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© ~⊚ f2 ≘ 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\98 f â\86\92 â\86\91f1 ~â\8a\9a ð\9d\90\94â\9d´â«¯Oâ\9dµ ≘ 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© ≘ 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\98 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\98 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\98 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\98 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\98 i2 â\86\92 â\88\80f2. ð\9d\90\93â¦\83f2â¦\84 → ∀f. f2 ~⊚ f1 ≘ f →
- â\88\83â\88\83i. @â¦\83i2, f2â¦\84 â\89\98 i & @â¦\83i1, fâ¦\84 ≘ 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« → ∀f. f2 ~⊚ f1 ≘ f →
+ â\88\83â\88\83i. @â\9dªi2, f2â\9d« â\89\98 i & @â\9dªi1, fâ\9d« ≘ 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\98 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\98 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 ≘ 0 → H_coafter_inj f1.
+corec fact coafter_inj_O_aux: â\88\80f1. @â\9dª0, f1â\9d« ≘ 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 ≘ 0 → H_coafter_inj f1) →
- â\88\80i2,f1. @â¦\830, f1â¦\84 ≘ i2 → H_coafter_inj f1.
+fact coafter_inj_aux: (â\88\80f1. @â\9dª0, f1â\9d« ≘ 0 → H_coafter_inj f1) →
+ â\88\80i2,f1. @â\9dª0, f1â\9d« ≘ 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 ≘ 0 →
+corec fact coafter_fwd_isid2_O_aux: â\88\80f1. @â\9dª0, f1â\9d« ≘ 0 →
H_coafter_fwd_isid2 f1.
#f1 #H1f1 #f2 #f #H #H2f1 #Hf
cases (at_inv_pxp … H1f1) -H1f1 [ |*: // ] #g1 #H1
]
qed-.
-fact coafter_fwd_isid2_aux: (â\88\80f1. @â¦\830, f1â¦\84 ≘ 0 → H_coafter_fwd_isid2 f1) →
- â\88\80i2,f1. @â¦\830, f1â¦\84 ≘ i2 → H_coafter_fwd_isid2 f1.
+fact coafter_fwd_isid2_aux: (â\88\80f1. @â\9dª0, f1â\9d« ≘ 0 → H_coafter_fwd_isid2 f1) →
+ â\88\80i2,f1. @â\9dª0, f1â\9d« ≘ 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 ≘ 0 →
+fact coafter_isfin2_fwd_O_aux: â\88\80f1. @â\9dª0, f1â\9d« ≘ 0 →
H_coafter_isfin2_fwd f1.
#f1 #Hf1 #f2 #H
generalize in match Hf1; generalize in match f1; -f1
/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 ≘ 0 → H_coafter_isfin2_fwd f1) →
- â\88\80i2,f1. @â¦\830, f1â¦\84 ≘ i2 → H_coafter_isfin2_fwd f1.
+fact coafter_isfin2_fwd_aux: (â\88\80f1. @â\9dª0, f1â\9d« ≘ 0 → H_coafter_isfin2_fwd f1) →
+ â\88\80i2,f1. @â\9dª0, f1â\9d« ≘ 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 → ∀f1. f2 ~⊚ f1 ≘ f → ∀fa,fb. fa ⋓ fb ≘ f →
+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« → ∀f1. f2 ~⊚ f1 ≘ f → ∀fa,fb. fa ⋓ fb ≘ f →
∃∃f1a,f1b. f2 ~⊚ f1a ≘ fa & f2 ~⊚ f1b ≘ fb & f1a ⋓ f1b ≘ f1.
@isfin_ind
[ #f #Hf #f2 #Hf2 #f1 #H1f #fa #fb #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 → ∀f1. f2 ~⊚ f1 ≘ f → ∀f1a,f1b. f1a ⋓ f1b ≘ f1 →
+lemma coafter_sor: â\88\80f. ð\9d\90\85â\9dªfâ\9d« â\86\92 â\88\80f2. ð\9d\90\93â\9dªf2â\9d« → ∀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
projects / helm.git / blobdiff