+ reversing paths completed updating component "reduction"
include "delayed_updating/syntax/prototerm_eq.ma".
include "delayed_updating/syntax/path_head.ma".
include "delayed_updating/syntax/path_depth.ma".
include "delayed_updating/syntax/prototerm_eq.ma".
include "delayed_updating/syntax/path_head.ma".
include "delayed_updating/syntax/path_depth.ma".
-include "delayed_updating/syntax/path_reverse.ma".
include "delayed_updating/notation/relations/black_rightarrow_df_4.ma".
(* DELAYED FOCUSED REDUCTION ************************************************)
definition dfr (p) (q): relation2 prototerm prototerm ≝
include "delayed_updating/notation/relations/black_rightarrow_df_4.ma".
(* DELAYED FOCUSED REDUCTION ************************************************)
definition dfr (p) (q): relation2 prototerm prototerm ≝
- ∧∧ (𝗟◗q)ᴿ = ↳[n](𝗟◗q)ᴿ & r◖𝗱n ϵ t1 &
- t1[⋔r←𝛕n.(t1⋔(p◖𝗦))] ⇔ t2
+ ∧∧ 𝗟◗q = ↳[k](𝗟◗q) & r◖𝗱k ϵ t1 &
+ t1[⋔r←𝛕k.(t1⋔(p◖𝗦))] ⇔ t2
include "delayed_updating/syntax/prototerm_proper_constructors.ma".
include "delayed_updating/syntax/path_head_structure.ma".
include "delayed_updating/syntax/path_structure_depth.ma".
include "delayed_updating/syntax/prototerm_proper_constructors.ma".
include "delayed_updating/syntax/path_head_structure.ma".
include "delayed_updating/syntax/path_structure_depth.ma".
-include "delayed_updating/syntax/path_structure_reverse.ma".
-include "delayed_updating/syntax/path_depth_reverse.ma".
(* DELAYED FOCUSED REDUCTION ************************************************)
(* DELAYED FOCUSED REDUCTION ************************************************)
theorem dfr_des_ifr (f) (p) (q) (t1) (t2): t1 ϵ 𝐓 →
t1 ➡𝐝𝐟[p,q] t2 → ▼[f]t1 ➡𝐢𝐟[⊗p,⊗q] ▼[f]t2.
#f #p #q #t1 #t2 #H0t1
theorem dfr_des_ifr (f) (p) (q) (t1) (t2): t1 ϵ 𝐓 →
t1 ➡𝐝𝐟[p,q] t2 → ▼[f]t1 ➡𝐢𝐟[⊗p,⊗q] ▼[f]t2.
#f #p #q #t1 #t2 #H0t1
@(ex_intro … (↑♭q)) @and3_intro
[ -H0t1 -Ht1 -Ht2
@(ex_intro … (↑♭q)) @and3_intro
[ -H0t1 -Ht1 -Ht2
- >structure_L_sn >structure_reverse
- >H1n in ⊢ (??%?); >path_head_structure_depth <H1n -H1n //
+ >structure_L_sn
+ >H1k in ⊢ (??%?); >path_head_structure_depth <H1k -H1k //
| lapply (in_comp_unwind2_path_term f … Ht1) -Ht2 -Ht1 -H0t1
| lapply (in_comp_unwind2_path_term f … Ht1) -Ht2 -Ht1 -H0t1
- <unwind2_path_d_dx >(list_append_rcons_sn … p) <reverse_append
- lapply (unwind2_rmap_append_pap_closed f … (p◖𝗔)ᴿ … H1n) -H1n
- <reverse_lcons <depth_L_dx #H2n
- lapply (eq_inv_ninj_bi … H2n) -H2n #H2n <H2n -H2n #Ht1 //
+ <unwind2_path_d_dx <list_append_rcons_sn
+ lapply (unwind2_rmap_append_pap_closed f … (p◖𝗔) … H1k) -H1k
+ <depth_L_sn #H2k
+ lapply (eq_inv_ninj_bi … H2k) -H2k #H2k <H2k -H2k #Ht1 //
| lapply (unwind2_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
@(subset_eq_trans … Ht2) -t2
@(subset_eq_trans … (unwind2_term_fsubst …))
| lapply (unwind2_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
@(subset_eq_trans … Ht2) -t2
@(subset_eq_trans … (unwind2_term_fsubst …))
@(subset_eq_trans … (lift_unwind2_term_after …))
@unwind2_term_eq_repl_sn
(* Note: crux of the proof begins *)
@(subset_eq_trans … (lift_unwind2_term_after …))
@unwind2_term_eq_repl_sn
(* Note: crux of the proof begins *)
- >list_append_rcons_sn <reverse_append
@(stream_eq_trans … (tr_compose_uni_dx …))
@tr_compose_eq_repl
[ <unwind2_rmap_append_pap_closed //
@(stream_eq_trans … (tr_compose_uni_dx …))
@tr_compose_eq_repl
[ <unwind2_rmap_append_pap_closed //
- | >unwind2_rmap_A_sn <reverse_rcons
/2 width=1 by tls_unwind2_rmap_closed/
]
(* Note: crux of the proof ends *)
/2 width=1 by tls_unwind2_rmap_closed/
]
(* Note: crux of the proof ends *)
theorem dfr_lift_bi (f) (p) (q) (t1) (t2):
t1 ➡𝐝𝐟[p,q] t2 → ↑[f]t1 ➡𝐝𝐟[↑[f]p,↑[↑[p◖𝗔◖𝗟]f]q] ↑[f]t2.
#f #p #q #t1 #t2
theorem dfr_lift_bi (f) (p) (q) (t1) (t2):
t1 ➡𝐝𝐟[p,q] t2 → ↑[f]t1 ➡𝐝𝐟[↑[f]p,↑[↑[p◖𝗔◖𝗟]f]q] ↑[f]t2.
#f #p #q #t1 #t2
-* #n * #H1n #Ht1 #Ht2
-@(ex_intro … ((↑[p●𝗔◗𝗟◗q]f)@⧣❨n❩)) @and3_intro
+* #k * #H1k #Ht1 #Ht2
+@(ex_intro … ((↑[p●𝗔◗𝗟◗q]f)@⧣❨k❩)) @and3_intro
[ -Ht1 -Ht2
<lift_rmap_L_dx >lift_path_L_sn
[ -Ht1 -Ht2
<lift_rmap_L_dx >lift_path_L_sn
- <(lift_path_head … H1n) in ⊢ (??%?); -H1n //
-| lapply (in_comp_lift_path_term f … Ht1) -Ht2 -Ht1 -H1n
+ <(lift_path_head_closed … H1k) in ⊢ (??%?); -H1k //
+| lapply (in_comp_lift_path_term f … Ht1) -Ht2 -Ht1 -H1k
<lift_path_d_dx #Ht1 //
| lapply (lift_term_eq_repl_dx f … Ht2) -Ht2 #Ht2 -Ht1
@(subset_eq_trans … Ht2) -t2
<lift_path_d_dx #Ht1 //
| lapply (lift_term_eq_repl_dx f … Ht2) -Ht2 #Ht2 -Ht1
@(subset_eq_trans … Ht2) -t2
@(subset_eq_canc_sn … (lift_term_grafted_S …))
@lift_term_eq_repl_sn
(* Note: crux of the proof begins *)
@(subset_eq_canc_sn … (lift_term_grafted_S …))
@lift_term_eq_repl_sn
(* Note: crux of the proof begins *)
- >list_append_rcons_sn in H1n; #H1n >lift_rmap_A_dx
+ >list_append_rcons_sn in H1k; #H1k >lift_rmap_A_dx
/2 width=1 by tls_lift_rmap_closed/
(* Note: crux of the proof ends *)
]
/2 width=1 by tls_lift_rmap_closed/
(* Note: crux of the proof ends *)
]
include "delayed_updating/substitution/lift_prototerm.ma".
include "delayed_updating/syntax/prototerm_eq.ma".
include "delayed_updating/syntax/path_head.ma".
include "delayed_updating/substitution/lift_prototerm.ma".
include "delayed_updating/syntax/prototerm_eq.ma".
include "delayed_updating/syntax/path_head.ma".
-include "delayed_updating/syntax/path_reverse.ma".
include "delayed_updating/notation/relations/black_rightarrow_if_4.ma".
include "delayed_updating/notation/relations/black_rightarrow_if_4.ma".
+include "ground/relocation/tr_uni.ma".
(* IMMEDIATE FOCUSED REDUCTION ************************************************)
definition ifr (p) (q): relation2 prototerm prototerm ≝
(* IMMEDIATE FOCUSED REDUCTION ************************************************)
definition ifr (p) (q): relation2 prototerm prototerm ≝
- ∧∧ (𝗟◗q)ᴿ = ↳[n](𝗟◗q)ᴿ & r◖𝗱n ϵ t1 &
- t1[⋔r←↑[𝐮❨n❩](t1⋔(p◖𝗦))] ⇔ t2
+ ∧∧ 𝗟◗q = ↳[k](𝗟◗q) & r◖𝗱k ϵ t1 &
+ t1[⋔r←↑[𝐮❨k❩](t1⋔(p◖𝗦))] ⇔ t2
include "delayed_updating/substitution/fsubst_lift.ma".
include "delayed_updating/substitution/fsubst_eq.ma".
include "delayed_updating/substitution/fsubst_lift.ma".
include "delayed_updating/substitution/fsubst_eq.ma".
+include "delayed_updating/substitution/lift_prototerm_after.ma".
include "delayed_updating/substitution/lift_path_head.ma".
include "delayed_updating/substitution/lift_rmap_head.ma".
include "delayed_updating/substitution/lift_path_head.ma".
include "delayed_updating/substitution/lift_rmap_head.ma".
theorem ifr_lift_bi (f) (p) (q) (t1) (t2):
t1 ➡𝐢𝐟[p,q] t2 → ↑[f]t1 ➡𝐢𝐟[↑[f]p,↑[↑[p◖𝗔◖𝗟]f]q] ↑[f]t2.
#f #p #q #t1 #t2
theorem ifr_lift_bi (f) (p) (q) (t1) (t2):
t1 ➡𝐢𝐟[p,q] t2 → ↑[f]t1 ➡𝐢𝐟[↑[f]p,↑[↑[p◖𝗔◖𝗟]f]q] ↑[f]t2.
#f #p #q #t1 #t2
-* #n * #H1n #Ht1 #Ht2
-@(ex_intro … ((↑[p●𝗔◗𝗟◗q]f)@⧣❨n❩)) @and3_intro
+* #k * #H1k #Ht1 #Ht2
+@(ex_intro … ((↑[p●𝗔◗𝗟◗q]f)@⧣❨k❩)) @and3_intro
[ -Ht1 -Ht2
<lift_rmap_L_dx >lift_path_L_sn
[ -Ht1 -Ht2
<lift_rmap_L_dx >lift_path_L_sn
- <(lift_path_head … H1n) in ⊢ (??%?); -H1n //
-| lapply (in_comp_lift_path_term f … Ht1) -Ht2 -Ht1 -H1n
+ <(lift_path_head_closed … H1k) in ⊢ (??%?); -H1k //
+| lapply (in_comp_lift_path_term f … Ht1) -Ht2 -Ht1 -H1k
<lift_path_d_dx #Ht1 //
| lapply (lift_term_eq_repl_dx f … Ht2) -Ht2 #Ht2 -Ht1
@(subset_eq_trans … Ht2) -t2
<lift_path_d_dx #Ht1 //
| lapply (lift_term_eq_repl_dx f … Ht2) -Ht2 #Ht2 -Ht1
@(subset_eq_trans … Ht2) -t2
(* Note: crux of the proof begins *)
@(stream_eq_trans … (tr_compose_uni_dx …))
@tr_compose_eq_repl //
(* Note: crux of the proof begins *)
@(stream_eq_trans … (tr_compose_uni_dx …))
@tr_compose_eq_repl //
- >list_append_rcons_sn in H1n; #H1n >lift_rmap_A_dx
+ >list_append_rcons_sn in H1k; #H1k >lift_rmap_A_dx
/2 width=1 by tls_lift_rmap_closed/
(* Note: crux of the proof ends *)
]
/2 width=1 by tls_lift_rmap_closed/
(* Note: crux of the proof ends *)
]
include "delayed_updating/syntax/path_head_structure.ma".
include "delayed_updating/syntax/path_structure_depth.ma".
include "delayed_updating/syntax/path_head_structure.ma".
include "delayed_updating/syntax/path_structure_depth.ma".
-include "delayed_updating/syntax/path_structure_reverse.ma".
-include "delayed_updating/syntax/path_depth_reverse.ma".
(* IMMEDIATE FOCUSED REDUCTION **********************************************)
(* IMMEDIATE FOCUSED REDUCTION **********************************************)
-(* Constructions with unwind ************************************************)
+(* Constructions with unwind2 ***********************************************)
lemma ifr_unwind_bi (f) (p) (q) (t1) (t2):
t1 ϵ 𝐓 → t1⋔(p◖𝗦) ϵ 𝐏 →
t1 ➡𝐢𝐟[p,q] t2 → ▼[f]t1 ➡𝐢𝐟[⊗p,⊗q] ▼[f]t2.
#f #p #q #t1 #t2 #H1t1 #H2t1
lemma ifr_unwind_bi (f) (p) (q) (t1) (t2):
t1 ϵ 𝐓 → t1⋔(p◖𝗦) ϵ 𝐏 →
t1 ➡𝐢𝐟[p,q] t2 → ▼[f]t1 ➡𝐢𝐟[⊗p,⊗q] ▼[f]t2.
#f #p #q #t1 #t2 #H1t1 #H2t1
@(ex_intro … (↑♭q)) @and3_intro
[ -H1t1 -H2t1 -Ht1 -Ht2
@(ex_intro … (↑♭q)) @and3_intro
[ -H1t1 -H2t1 -Ht1 -Ht2
- >structure_L_sn >structure_reverse
- >H1n in ⊢ (??%?); >path_head_structure_depth <H1n -H1n //
+ >structure_L_sn
+ >H1k in ⊢ (??%?); >path_head_structure_depth <H1k -H1k //
| lapply (in_comp_unwind2_path_term f … Ht1) -Ht2 -Ht1 -H1t1 -H2t1
| lapply (in_comp_unwind2_path_term f … Ht1) -Ht2 -Ht1 -H1t1 -H2t1
- <unwind2_path_d_dx >(list_append_rcons_sn … p) <reverse_append
- lapply (unwind2_rmap_append_pap_closed f … (p◖𝗔)ᴿ … H1n) -H1n
- <reverse_lcons <depth_L_dx #H2n
- lapply (eq_inv_ninj_bi … H2n) -H2n #H2n <H2n -H2n #Ht1 //
+ <unwind2_path_d_dx <list_append_rcons_sn
+ lapply (unwind2_rmap_append_pap_closed f … (p◖𝗔) … H1k) -H1k
+ <depth_L_sn #H2k
+ lapply (eq_inv_ninj_bi … H2k) -H2k #H2k <H2k -H2k #Ht1 //
| lapply (unwind2_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
@(subset_eq_trans … Ht2) -t2
@(subset_eq_trans … (unwind2_term_fsubst …))
| lapply (unwind2_term_eq_repl_dx f … Ht2) -Ht2 #Ht2
@(subset_eq_trans … Ht2) -t2
@(subset_eq_trans … (unwind2_term_fsubst …))
@(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
[ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
@(subset_eq_trans … (lift_unwind2_term_after …))
@(subset_eq_canc_sn … (lift_term_eq_repl_dx …))
[ @unwind2_term_grafted_S /2 width=2 by ex_intro/ | skip ] -Ht1
@(subset_eq_trans … (lift_unwind2_term_after …))
- @(subset_eq_canc_dx … (unwind2_term_after_lift …))
+ @(subset_eq_canc_dx … (unwind2_lift_term_after …))
@unwind2_term_eq_repl_sn
(* Note: crux of the proof begins *)
@unwind2_term_eq_repl_sn
(* Note: crux of the proof begins *)
- >list_append_rcons_sn <reverse_append
@(stream_eq_trans … (tr_compose_uni_dx …))
@tr_compose_eq_repl
[ <unwind2_rmap_append_pap_closed //
@(stream_eq_trans … (tr_compose_uni_dx …))
@tr_compose_eq_repl
[ <unwind2_rmap_append_pap_closed //
- | >unwind2_rmap_A_sn <reverse_rcons
/2 width=1 by tls_unwind2_rmap_closed/
]
(* Note: crux of the proof ends *)
/2 width=1 by tls_unwind2_rmap_closed/
]
(* Note: crux of the proof ends *)