+ lprs started ...
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "ground_2/lib/star.ma".
+include "basic_2/rt_transition/lpr.ma".
+include "basic_2/rt_computation/cpms.ma".
+
+(* T-BOUND CONTEXT-SENSITIVE PARALLEL RT-COMPUTATION FOR TERMS **************)
+
+(* Properties concerning sn parallel reduction on local environments ********)
+
+(* Basic_1: uses: pr3_pr2_pr2_t *)
+(* Basic_1: includes: pr3_pr0_pr2_t *)
+(* Basic_2A1: uses: lpr_cpr_trans *)
+lemma lpr_cpm_trans (n) (h) (G): s_r_transitive … (λL. cpm h G L n) (λ_. lpr h G).
+#n #h #G #L2 #T1 #T2 #H @(cpm_ind … H) -G -L2 -T1 -T2
+[ /2 width=3 by/
+| /3 width=2 by cpx_cpxs, cpx_ess/
+| #I #G #K2 #V2 #V4 #W4 #_ #IH #HVW4 #L1 #H
+ elim (lpx_inv_pair_dx … H) -H #K1 #V1 #HK12 #HV12 #H destruct
+ /4 width=3 by cpxs_delta, cpxs_strap2/
+| #I2 #G #K2 #T #U #i #_ #IH #HTU #L1 #H
+ elim (lpx_inv_bind_dx … H) -H #I1 #K1 #HK12 #HI12 #H destruct
+ /4 width=3 by cpxs_lref, cpxs_strap2/
+|5,10: /4 width=1 by cpxs_beta, cpxs_bind, lpx_bind_refl_dx/
+|6,8,9: /3 width=1 by cpxs_flat, cpxs_ee, cpxs_eps/
+| /4 width=3 by cpxs_zeta, lpx_bind_refl_dx/
+| /4 width=3 by cpxs_theta, cpxs_strap1, lpx_bind_refl_dx/
+]
+qed-.
+
+
+
+
+
+
+
+#G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
+[ /2 width=3 by/
+| #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
+ elim (lpr_drop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
+ elim (lpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV10 #H destruct
+ /4 width=6 by cprs_strap2, cprs_delta/
+|3,7: /4 width=1 by lpr_pair, cprs_bind, cprs_beta/
+|4,6: /3 width=1 by cprs_flat, cprs_eps/
+|5,8: /4 width=3 by lpr_pair, cprs_zeta, cprs_theta, cprs_strap1/
+]
+qed-.
+
+lemma cpr_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡ T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
+/4 width=5 by lpr_cpr_trans, cprs_bind_dx, lpr_pair/ qed.
+
+(* Advanced properties ******************************************************)
+
+(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
+lemma lpr_cprs_trans: ∀G. b_rs_transitive … (cpr G) (λ_. lpr G).
+#G @b_c_trans_CTC1 /2 width=3 by lpr_cpr_trans/ (**) (* full auto fails *)
+qed-.
+
+lemma cprs_bind2_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
+ ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
+ ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
+/4 width=5 by lpr_cprs_trans, cprs_bind_dx, lpr_pair/ qed.
]
qed-.
-(* Properties concerning sn parallel reduction on local environments ********)
-
-(* Basic_1: was just: pr3_pr2_pr2_t *)
-(* Basic_1: includes: pr3_pr0_pr2_t *)
-lemma lpr_cpr_trans: ∀G. b_c_transitive … (cpr G) (λ_. lpr G).
-#G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
-[ /2 width=3 by/
-| #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
- elim (lpr_drop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
- elim (lpr_inv_pair2 … H) -H #K1 #V1 #HK12 #HV10 #H destruct
- /4 width=6 by cprs_strap2, cprs_delta/
-|3,7: /4 width=1 by lpr_pair, cprs_bind, cprs_beta/
-|4,6: /3 width=1 by cprs_flat, cprs_eps/
-|5,8: /4 width=3 by lpr_pair, cprs_zeta, cprs_theta, cprs_strap1/
-]
-qed-.
-
-lemma cpr_bind2: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 → ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡ T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-/4 width=5 by lpr_cpr_trans, cprs_bind_dx, lpr_pair/ qed.
-
-(* Advanced properties ******************************************************)
-
-(* Basic_1: was only: pr3_pr2_pr3_t pr3_wcpr0_t *)
-lemma lpr_cprs_trans: ∀G. b_rs_transitive … (cpr G) (λ_. lpr G).
-#G @b_c_trans_CTC1 /2 width=3 by lpr_cpr_trans/ (**) (* full auto fails *)
-qed-.
-
(* Basic_1: was: pr3_strip *)
(* Basic_1: includes: pr1_strip *)
lemma cprs_strip: ∀G,L. confluent2 … (cprs G L) (cpr G L).
#G #L0 #T0 #T1 #HT01 #L1 #HL01 elim (cprs_lpr_conf_dx … HT01 … HL01) -HT01
/3 width=3 by lpr_cprs_trans, ex2_intro/
qed-.
-
-lemma cprs_bind2_dx: ∀G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡ V2 →
- ∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡* T2 →
- ∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡* ⓑ{a,I}V2.T2.
-/4 width=5 by lpr_cprs_trans, cprs_bind_dx, lpr_pair/ qed.
(* *)
(**************************************************************************)
-include "basic_2/notation/relations/predsnstar_3.ma".
-include "basic_2/substitution/lpx_sn_tc.ma".
-include "basic_2/reduction/lpr.ma".
+include "basic_2/notation/relations/predsnstar_4.ma".
+include "basic_2/relocation/lex.ma".
+include "basic_2/rt_computation/cpms.ma".
-(* SN PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS ****************************)
+(* PARALLEL R-COMPUTATION FOR FULL LOCAL ENVIRONMENTS ***********************)
-definition lprs: relation3 genv lenv lenv ≝
- λG. TC … (lpr G).
+definition lprs (h) (G): relation lenv ≝
+ lex (λL.cpms h G L 0).
-interpretation "parallel computation (local environment, sn variant)"
- 'PRedSnStar G L1 L2 = (lprs G L1 L2).
-
-(* Basic eliminators ********************************************************)
-
-lemma lprs_ind: ∀G,L1. ∀R:predicate lenv. R L1 →
- (∀L,L2. ⦃G, L1⦄ ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → R L → R L2) →
- ∀L2. ⦃G, L1⦄ ⊢ ➡* L2 → R L2.
-#G #L1 #R #HL1 #IHL1 #L2 #HL12
-@(TC_star_ind … HL1 IHL1 … HL12) //
-qed-.
-
-lemma lprs_ind_dx: ∀G,L2. ∀R:predicate lenv. R L2 →
- (∀L1,L. ⦃G, L1⦄ ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → R L → R L1) →
- ∀L1. ⦃G, L1⦄ ⊢ ➡* L2 → R L1.
-#G #L2 #R #HL2 #IHL2 #L1 #HL12
-@(TC_star_ind_dx … HL2 IHL2 … HL12) //
-qed-.
+interpretation
+ "parallel r-computation on all entries (local environment)"
+ 'PRedSnStar h G L1 L2 = (lprs h G L1 L2).
(* Basic properties *********************************************************)
-lemma lpr_lprs: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡* L2.
-/2 width=1 by inj/ qed.
-
-lemma lprs_refl: ∀G,L. ⦃G, L⦄ ⊢ ➡* L.
-/2 width=1 by lpr_lprs/ qed.
+lemma lprs_refl (h) (G): ∀L. ⦃G, L⦄ ⊢ ➡*[h] L.
+/2 width=1 by lex_refl/ qed.
-lemma lprs_strap1: ∀G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡* L → ⦃G, L⦄ ⊢ ➡ L2 → ⦃G, L1⦄ ⊢ ➡* L2.
-/2 width=3 by step/ qed-.
-
-lemma lprs_strap2: ∀G,L1,L,L2. ⦃G, L1⦄ ⊢ ➡ L → ⦃G, L⦄ ⊢ ➡* L2 → ⦃G, L1⦄ ⊢ ➡* L2.
-/2 width=3 by TC_strap/ qed-.
-
-lemma lprs_pair_refl: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → ∀I,V. ⦃G, L1.ⓑ{I}V⦄ ⊢ ➡* L2.ⓑ{I}V.
-/2 width=1 by TC_lpx_sn_pair_refl/ qed.
+(* Basic_2A1: uses: lprs_pair_refl *)
+lemma lprs_bind_refl_dx (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 →
+ ∀I. ⦃G, L1.ⓘ{I}⦄ ⊢ ➡*[h] L2.ⓘ{I}.
+/2 width=1 by lex_bind_refl_dx/ qed.
(* Basic inversion lemmas ***************************************************)
-lemma lprs_inv_atom1: ∀G,L2. ⦃G, ⋆⦄ ⊢ ➡* L2 → L2 = ⋆.
-/2 width=2 by TC_lpx_sn_inv_atom1/ qed-.
-
-lemma lprs_inv_atom2: ∀G,L1. ⦃G, L1⦄ ⊢ ➡* ⋆ → L1 = ⋆.
-/2 width=2 by TC_lpx_sn_inv_atom2/ qed-.
-
-(* Basic forward lemmas *****************************************************)
+(* Basic_2A1: uses: lprs_inv_atom1 *)
+lemma lprs_inv_atom_sn (h) (G): ∀L2. ⦃G, ⋆⦄ ⊢ ➡*[h] L2 → L2 = ⋆.
+/2 width=2 by lex_inv_atom_sn/ qed-.
-lemma lprs_fwd_length: ∀G,L1,L2. ⦃G, L1⦄ ⊢ ➡* L2 → |L1| = |L2|.
-/2 width=2 by TC_lpx_sn_fwd_length/ qed-.
+(* Basic_2A1: uses: lprs_inv_atom2 *)
+lemma lprs_inv_atom_dx (h) (G): ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] ⋆ → L1 = ⋆.
+/2 width=2 by lex_inv_atom_dx/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lex_length.ma".
+include "basic_2/rt_computation/lprs.ma".
+
+(* PARALLEL R-COMPUTATION FOR FULL LOCAL ENVIRONMENTS ***********************)
+
+(* Forward lemmas with length for local environments ************************)
+
+lemma lprs_fwd_length (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → |L1| = |L2|.
+/2 width=2 by lex_fwd_length/ qed-.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lex_tc.ma".
+include "basic_2/rt_computation/cprs_lpr.ma".
+include "basic_2/rt_computation/lprs.ma".
+
+(* PARALLEL R-COMPUTATION FOR FULL LOCAL ENVIRONMENTS ***********************)
+
+(* Eliminators with r-transition for full local environments ****************)
+
+(* Basic_2A1: was: lprs_ind_dx *)
+lemma lprs_ind_sn (h) (G) (L2): ∀R:predicate lenv. R L2 →
+ (∀L1,L. ⦃G, L1⦄ ⊢ ➡[h] L → ⦃G, L⦄ ⊢ ➡*[h] L2 → R L → R L1) →
+ ∀L1. ⦃G, L1⦄ ⊢ ➡*[h] L2 → R L1.
+/3 width=7 by lpr_cprs_trans, cpr_refl, lex_CTC_ind_sn/ qed-.
+
+(* Basic_2A1: was: lprs_ind *)
+lemma lprs_ind_dx (h) (G) (L1): ∀R:predicate lenv. R L1 →
+ (∀L,L2. ⦃G, L1⦄ ⊢ ➡*[h] L → ⦃G, L⦄ ⊢ ➡[h] L2 → R L → R L2) →
+ ∀L2. ⦃G, L1⦄ ⊢ ➡*[h] L2 → R L2.
+/3 width=7 by lpr_cprs_trans, cpr_refl, lex_CTC_ind_dx/ qed-.
+
+(* Properties with unbound rt-transition for full local environments ********)
+
+lemma lpr_lprs (h) (G): ∀L1,L2. ⦃G, L1⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2.
+/3 width=3 by lpr_cprs_trans, lex_CTC_inj/ qed.
+
+(* Basic_2A1: was: lprs_strap2 *)
+lemma lprs_step_sn (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ➡[h] L →
+ ∀L2.⦃G, L⦄ ⊢ ➡*[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2.
+/3 width=3 by lpr_cprs_trans, lex_CTC_step_sn/ qed-.
+
+(* Basic_2A1: was: lpxs_strap1 *)
+lemma lprs_step_dx (h) (G): ∀L1,L. ⦃G, L1⦄ ⊢ ➡*[h] L →
+ ∀L2. ⦃G, L⦄ ⊢ ➡[h] L2 → ⦃G, L1⦄ ⊢ ➡*[h] L2.
+/3 width=3 by lpr_cprs_trans, lex_CTC_step_dx/ qed-.
fsb.ma fsb_ffdeq.ma fsb_aaa.ma fsb_csx.ma fsb_fpbg.ma
cpms.ma cpms_drops.ma cpms_lsubr.ma cpms_aaa.ma cpms_cpxs.ma
cprs.ma cprs_drops.ma
+lprs.ma lprs_length.ma
#n #h #I #G #L #V1 #T1 #T * #c #Hc #H #p elim (cpg_fwd_bind1_minus … H p) -H
/3 width=4 by ex2_2_intro, ex2_intro/
qed-.
+
+(* Basic eliminators ********************************************************)
+
+lemma isrt_inv_max_shift_sn: ∀n,c1,c2. 𝐑𝐓⦃n, ↕*c1 ∨ c2⦄ →
+ ∧∧ 𝐑𝐓⦃0, c1⦄ & 𝐑𝐓⦃n, c2⦄.
+#n #c1 #c2 #H
+elim (isrt_inv_max … H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
+elim (isrt_inv_shift … Hc1) -Hc1 #Hc1 * -n1
+/2 width=1 by conj/
+qed-.
+
+lemma isrt_inv_max_eq_t: ∀n,c1,c2. 𝐑𝐓⦃n, c1 ∨ c2⦄ → eq_t c1 c2 →
+ ∧∧ 𝐑𝐓⦃n, c1⦄ & 𝐑𝐓⦃n, c2⦄.
+#n #c1 #c2 #H #Hc12
+elim (isrt_inv_max … H) -H #n1 #n2 #Hc1 #Hc2 #H destruct
+lapply (isrt_eq_t_trans … Hc1 … Hc12) -Hc12 #H
+<(isrt_inj … H … Hc2) -Hc2
+<idempotent_max /2 width=1 by conj/
+qed-.
+
+lemma cpm_ind (h): ∀R:relation5 nat genv lenv term term.
+ (∀I,G,L. R 0 G L (⓪{I}) (⓪{I})) →
+ (∀G,L,s. R 1 G L (⋆s) (⋆(next h s))) →
+ (∀n,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 → R n G K V1 V2 →
+ ⬆*[1] V2 ≘ W2 → R n G (K.ⓓV1) (#0) W2
+ ) → (∀n,G,K,V1,V2,W2. ⦃G, K⦄ ⊢ V1 ➡[n, h] V2 → R n G K V1 V2 →
+ ⬆*[1] V2 ≘ W2 → R (↑n) G (K.ⓛV1) (#0) W2
+ ) → (∀n,I,G,K,T,U,i. ⦃G, K⦄ ⊢ #i ➡[n, h] T → R n G K (#i) T →
+ ⬆*[1] T ≘ U → R n G (K.ⓘ{I}) (#↑i) (U)
+ ) → (∀n,p,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡[n, h] T2 →
+ R 0 G L V1 V2 → R n G (L.ⓑ{I}V1) T1 T2 → R n G L (ⓑ{p,I}V1.T1) (ⓑ{p,I}V2.T2)
+ ) → (∀n,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
+ R 0 G L V1 V2 → R n G L T1 T2 → R n G L (ⓐV1.T1) (ⓐV2.T2)
+ ) → (∀n,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 → ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
+ R n G L V1 V2 → R n G L T1 T2 → R n G L (ⓝV1.T1) (ⓝV2.T2)
+ ) → (∀n,G,L,V,T1,T,T2. ⦃G, L.ⓓV⦄ ⊢ T1 ➡[n, h] T → R n G (L.ⓓV) T1 T →
+ ⬆*[1] T2 ≘ T → R n G L (+ⓓV.T1) T2
+ ) → (∀n,G,L,V,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 →
+ R n G L T1 T2 → R n G L (ⓝV.T1) T2
+ ) → (∀n,G,L,V1,V2,T. ⦃G, L⦄ ⊢ V1 ➡[n, h] V2 →
+ R n G L V1 V2 → R (↑n) G L (ⓝV1.T) V2
+ ) → (∀n,p,G,L,V1,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V2 → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡[n, h] T2 →
+ R 0 G L V1 V2 → R 0 G L W1 W2 → R n G (L.ⓛW1) T1 T2 →
+ R n G L (ⓐV1.ⓛ{p}W1.T1) (ⓓ{p}ⓝW2.V2.T2)
+ ) → (∀n,p,G,L,V1,V,V2,W1,W2,T1,T2. ⦃G, L⦄ ⊢ V1 ➡[h] V → ⦃G, L⦄ ⊢ W1 ➡[h] W2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡[n, h] T2 →
+ R 0 G L V1 V → R 0 G L W1 W2 → R n G (L.ⓓW1) T1 T2 →
+ ⬆*[1] V ≘ V2 → R n G L (ⓐV1.ⓓ{p}W1.T1) (ⓓ{p}W2.ⓐV2.T2)
+ ) →
+ ∀n,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[n, h] T2 → R n G L T1 T2.
+#h #R #IH1 #IH2 #IH3 #IH4 #IH5 #IH6 #IH7 #IH8 #IH9 #IH10 #IH11 #IH12 #IH13 #n #G #L #T1 #T2
+* #c #HC #H generalize in match HC; -HC generalize in match n; -n
+elim H -c -G -L -T1 -T2
+[ #I #G #L #n #H <(isrt_inv_00 … H) -H //
+| #G #L #s #n #H <(isrt_inv_01 … H) -H //
+| /3 width=4 by ex2_intro/
+| #c #G #L #V1 #V2 #W2 #HV12 #HVW2 #IH #x #H
+ elim (isrt_inv_plus_SO_dx … H) -H // #n #Hc #H destruct
+ /3 width=4 by ex2_intro/
+| /3 width=4 by ex2_intro/
+| #cV #cT #p #I #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #IHV #IHT #n #H
+ elim (isrt_inv_max_shift_sn … H) -H #HcV #HcT
+ /3 width=3 by ex2_intro/
+| #cV #cT #G #L #V1 #V2 #T1 #T2 #HV12 #HT12 #IHV #IHT #n #H
+ elim (isrt_inv_max_shift_sn … H) -H #HcV #HcT
+ /3 width=3 by ex2_intro/
+| #cU #cT #G #L #U1 #U2 #T1 #T2 #HUT #HU12 #HT12 #IHU #IHT #n #H
+ elim (isrt_inv_max_eq_t … H) -H // #HcV #HcT
+ /3 width=3 by ex2_intro/
+| #c #G #L #V #T1 #T2 #T #HT12 #HT2 #IH #n #H
+ lapply (isrt_inv_plus_O_dx … H ?) -H // #Hc
+ /3 width=4 by ex2_intro/
+| #c #G #L #U #T1 #T2 #HT12 #IH #n #H
+ lapply (isrt_inv_plus_O_dx … H ?) -H // #Hc
+ /3 width=3 by ex2_intro/
+| #c #G #L #U1 #U2 #T #HU12 #IH #x #H
+ elim (isrt_inv_plus_SO_dx … H) -H // #n #Hc #H destruct
+ /3 width=3 by ex2_intro/
+| #cV #cW #cT #p #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HW12 #HT12 #IHV #IHW #IHT #n #H
+ lapply (isrt_inv_plus_O_dx … H ?) -H // >max_shift #H
+ elim (isrt_inv_max_shift_sn … H) -H #H #HcT
+ elim (isrt_O_inv_max … H) -H #HcV #HcW
+ /3 width=3 by ex2_intro/
+| #cV #cW #cT #p #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HW12 #HT12 #IHV #IHW #IHT #n #H
+ lapply (isrt_inv_plus_O_dx … H ?) -H // >max_shift #H
+ elim (isrt_inv_max_shift_sn … H) -H #H #HcT
+ elim (isrt_O_inv_max … H) -H #HcV #HcW
+ /3 width=4 by ex2_intro/
+]
+qed-.
projects / helm.git / commitdiff