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15 include "ground/relocation/gr_pat_lt.ma".
16 include "ground/relocation/gr_ist.ma".
17 include "ground/relocation/gr_after_pat.ma".
19 (* RELATIONAL COMPOSITION FOR GENERIC RELOCATION MAPS ***********************************************************)
21 (* Forward lemmas on istot **************************************************)
23 (*** after_istot_fwd *)
24 lemma gr_after_ist_des:
25 ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓❪f2❫ → 𝐓❪f1❫ → 𝐓❪f❫.
26 #f2 #f1 #f #Hf #Hf2 #Hf1 #i1 elim (Hf1 i1) -Hf1
27 #i2 #Hf1 elim (Hf2 i2) -Hf2
28 /3 width=7 by gr_after_des_pat, ex_intro/
31 (*** after_fwd_istot_dx *)
32 lemma gr_after_des_ist_dx:
33 ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓❪f❫ → 𝐓❪f1❫.
34 #f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
35 #i2 #Hf elim (gr_after_pat_des … Hf … H) -f /2 width=2 by ex_intro/
38 (*** after_fwd_istot_sn *)
39 lemma gr_after_des_ist_sn:
40 ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓❪f❫ → 𝐓❪f2❫.
41 #f2 #f1 #f #H #Hf #i1 elim (Hf i1) -Hf
42 #i #Hf elim (gr_after_pat_des … Hf … H) -f
43 #i2 #Hf1 #Hf2 lapply (gr_pat_increasing … Hf1) -f1
44 #Hi12 elim (gr_pat_le_ex … Hf2 … Hi12) -i2 /2 width=2 by ex_intro/
48 lemma gr_after_des_ist_pat:
49 ∀f1,i1,i2. @❪i1, f1❫ ≘ i2 → ∀f2. 𝐓❪f2❫ → ∀f. f2 ⊚ f1 ≘ f →
50 ∃∃i. @❪i2, f2❫ ≘ i & @❪i1, f❫ ≘ i.
51 #f1 #i1 #i2 #Hf1 #f2 #Hf2 #f #Hf elim (Hf2 i2) -Hf2
52 /3 width=8 by gr_after_des_pat, ex2_intro/
55 (* Inversions with gr_ist *)
57 (*** after_inv_istot *)
58 lemma gr_after_inv_ist:
59 ∀f2,f1,f. f2 ⊚ f1 ≘ f → 𝐓❪f❫ → ∧∧ 𝐓❪f2❫ & 𝐓❪f1❫.
60 /3 width=4 by gr_after_des_ist_sn, gr_after_des_ist_dx, conj/ qed-.