for @{ 'StdStar $F $p $G }.
lemma pl_sts_fwd_pl_sreds: ∀s,F1,F2. F1 Ⓡ↦*[s] F2 → ⇓F1 ↦*[s] ⇓F2.
-#s #F1 #F2 #H @(lstar_ind_r ????????? H) -s -F2 //
+#s #F1 #F2 #H @(lstar_ind_r … s F2 H) -s -F2 //
#p #s #F #F2 #_ #HF2 #IHF1
lapply (pl_st_fwd_pl_sred … HF2) -HF2 /2 width=3/
qed-.
+lemma pl_sts_inv_pl_sreds: ∀s,M1,F2. {⊤}⇑M1 Ⓡ↦*[s] F2 → is_whd s →
+ ∃∃M2. M1 ↦*[s] M2 & {⊤}⇑M2 = F2.
+#s #M1 #F2 #H @(lstar_ind_r … s F2 H) -s -F2 /2 width=3/
+#p #s #F #F2 #_ #HF2 #IHF #H
+elim (is_whd_inv_append … H) -H #Hs * #Hp #_
+elim (IHF Hs) -IHF -Hs #M #HM #H destruct
+elim (pl_st_inv_pl_sred … HF2) -HF2 // -Hp #M2 #HM2 #H
+lapply (pl_sreds_step_dx … HM … HM2) -M /2 width=3/
+qed-.
+
+lemma pl_sts_inv_empty: ∀s,M1,F2. {⊥}⇑M1 Ⓡ↦*[s] F2 → ◊ = s ∧ {⊥}⇑M1 = F2.
+#s #M1 #F2 #H @(lstar_ind_r … s F2 H) -s -F2 /2 width=1/ #p #s #F #F2 #_ #HF2 * #_ #H
+elim (pl_st_inv_empty … HF2 … H)
+qed-.
+
lemma pl_sts_refl: reflexive … (pl_sts (◊)).
//
qed.
/2 width=1 by lstar_inv_pos/
qed-.
+lemma pl_sts_inv_rc_abst_dx: ∀b2,s,F1,T2. F1 Ⓡ↦*[s] {b2}𝛌.T2 → ∀r. rc:::r = s →
+ ∃∃b1,T1. T1 Ⓡ↦*[r] T2 & {b1}𝛌.T1 = F1.
+#b2 #s #F1 #T2 #H @(lstar_ind_l … s F1 H) -s -F1
+[ #r #H lapply (map_cons_inv_nil … r H) -H #H destruct /2 width=4/
+| #p #s #F1 #F #HF1 #_ #IHF2 #r #H -b2
+ elim (map_cons_inv_cons … r H) -H #q #r0 #Hp #Hs #Hr
+ elim (pl_st_inv_rc … HF1 … Hp) -HF1 -p #b1 #T1 #T #HT1 #HF1 #HF destruct
+ elim (IHF2 …) -IHF2 [3: // |2: skip ] (**) (* simplify line *)
+ #b0 #T0 #HT02 #H destruct
+ lapply (pl_sts_step_sn … HT1 … HT02) -T /2 width=4/
+]
+qed-.
+
+lemma pl_sts_inv_sn_appl_dx: ∀b2,s,F1,V2,T2. F1 Ⓡ↦*[s] {b2}@V2.T2 → ∀r. sn:::r = s →
+ ∃∃b1,V1,T1. V1 Ⓡ↦*[r] V2 & {b1}@V1.T1 = F1.
+#b2 #s #F1 #V2 #T2 #H @(lstar_ind_l … s F1 H) -s -F1
+[ #r #H lapply (map_cons_inv_nil … r H) -H #H destruct /2 width=5/
+| #p #s #F1 #F #HF1 #_ #IHF2 #r #H -b2
+ elim (map_cons_inv_cons … r H) -H #q #r0 #Hp #Hs #Hr
+ elim (pl_st_inv_sn … HF1 … Hp) -HF1 -p #b1 #V1 #V #T1 #HV1 #HF1 #HF destruct
+ elim (IHF2 …) -IHF2 [3: // |2: skip ] (**) (* simplify line *)
+ #b0 #V0 #T0 #HV02 #H destruct
+ lapply (pl_sts_step_sn … HV1 … HV02) -V /2 width=5/
+]
+qed-.
+
+lemma pl_sts_inv_dx_appl_dx: ∀b,s,F1,V,T2. F1 Ⓡ↦*[s] {b}@V.T2 → ∀r. dx:::r = s →
+ ∃∃T1. T1 Ⓡ↦*[r] T2 & {b}@V.T1 = F1.
+#b #s #F1 #V #T2 #H @(lstar_ind_l … s F1 H) -s -F1
+[ #r #H lapply (map_cons_inv_nil … r H) -H #H destruct /2 width=3/
+| #p #s #F1 #F #HF1 #_ #IHF2 #r #H
+ elim (map_cons_inv_cons … r H) -H #q #r0 #Hp #Hs #Hr
+ elim (pl_st_inv_dx … HF1 … Hp) -HF1 -p #b0 #V0 #T1 #T #HT1 #HF1 #HF destruct
+ elim (IHF2 …) -IHF2 [3: // |2: skip ] (**) (* simplify line *)
+ #T0 #HT02 #H destruct
+ lapply (pl_sts_step_sn … HT1 … HT02) -T /2 width=3/
+]
+qed-.
+
lemma pl_sts_lift: ∀s. sliftable (pl_sts s).
/2 width=1/
qed.
/2 width=3 by lstar_ltransitive/
qed-.
+lemma pl_sts_inv_trans: inv_ltransitive … pl_sts.
+/2 width=3 by lstar_inv_ltransitive/
+qed-.
+
+lemma pl_sts_fwd_dx_sn_appl_dx: ∀b2,s,r,F1,V2,T2. F1 Ⓡ↦*[(dx:::s)@(sn:::r)] {b2}@V2.T2 →
+ ∃∃b1,V1,T1,T0. V1 Ⓡ↦*[r] V2 & T1 Ⓡ↦*[s] T0 & {b1}@V1.T1 = F1.
+#b2 #s #r #F1 #V2 #T2 #H
+elim (pl_sts_inv_trans … H) -H #F #HF1 #H
+elim (pl_sts_inv_sn_appl_dx … H …) -H [3: // |2: skip ] (**) (* simplify line *)
+#b #V #T #HV2 #H destruct
+elim (pl_sts_inv_dx_appl_dx … HF1 …) -HF1 [3: // |2: skip ] (**) (* simplify line *)
+#T1 #HT1 #H destruct /2 width=7/
+qed-.
+
theorem pl_sts_fwd_is_standard: ∀s,F1,F2. F1 Ⓡ↦*[s] F2 → is_standard s.
#s elim s -s // #p1 * //
#p2 #s #IHs #F1 #F2 #H
-elim (pl_sts_inv_cons … H ???) -H [4: // |2,3: skip ] #F3 #HF13 #H (**) (* simplify line *)
-elim (pl_sts_inv_cons … H ???) [2: // |3,4: skip ] #F4 #HF34 #_ (**) (* simplify line *)
+elim (pl_sts_inv_cons … H …) -H [4: // |2,3: skip ] #F3 #HF13 #H (**) (* simplify line *)
+elim (pl_sts_inv_cons … H …) [2: // |3,4: skip ] #F4 #HF34 #_ (**) (* simplify line *)
lapply (pl_st_fwd_sle … HF13 … HF34) -F1 -F4 /3 width=3/
qed-.
-axiom pl_sred_is_standard_pl_st: ∀p,M,M2. M ↦[p] M2 → ∀F. ⇓F = M →
+lemma pl_sts_fwd_abst_dx: ∀b2,s,F1,T2. F1 Ⓡ↦*[s] {b2}𝛌.T2 →
+ ∃∃r1,r2. is_whd r1 & r1@rc:::r2 = s.
+#b2 #s #F1 #T2 #H
+lapply (pl_sts_fwd_is_standard … H)
+@(lstar_ind_l … s F1 H) -s -F1
+[ #_ @(ex2_2_intro … ◊ ◊) // (**) (* auto needs some help here *)
+| #p #s #F1 #F #HF1 #HF2 #IHF1 #Hs
+ lapply (is_standard_fwd_cons … Hs) #H
+ elim (IHF1 …) // -IHF1 -H #r1 #r2 #Hr1 #H destruct
+ elim (in_whd_or_in_inner p) #Hp
+ [ -Hs -F1 -F -T2 -b2
+ @(ex2_2_intro … (p::r1) r2) // /2 width=1/ (**) (* auto needs some help here *)
+ | lapply (is_standard_fwd_append_sn (p::r1) ? Hs) -Hs #H
+ lapply (is_standard_fwd_in_inner … H ?) -H // #H
+ lapply (is_whd_is_inner_inv … Hr1 ?) -Hr1 // -H #H destruct
+ elim (in_inner_inv … Hp) -Hp * #q [3: #IHq ] #Hp
+(* case 1: dx *)
+ [ -IHq
+ elim (pl_sts_inv_rc_abst_dx … HF2 …) -b2 [3: // |2: skip ] (**) (* simplify line *)
+ #b #T #_ #HT -T2
+ elim (pl_st_inv_dx … HF1 …) -HF1 [3: // |2: skip ] (**) (* simplify line *)
+ #c #V #T1 #T0 #_ #_ #HT0 -q -T1 -F1 destruct
+(* case 2: rc *)
+ | destruct -F1 -F -T2 -b2
+ @(ex2_2_intro … ◊ (q::r2)) // (**) (* auto needs some help here *)
+(* case 3: sn *)
+ | elim (pl_sts_inv_rc_abst_dx … HF2 …) -b2 [3: // |2: skip ] (**) (* simplify line *)
+ #b #T #_ #HT -T2
+ elim (pl_st_inv_sn … HF1 …) -HF1 [3: // |2: skip ] (**) (* simplify line *)
+ #c #V1 #V #T0 #_ #_ #HT0 -c -q -V1 -F1 destruct
+ ]
+ ]
+]
+qed-.
+
+lemma pl_sts_fwd_appl_dx: ∀b2,s,F1,V2,T2. F1 Ⓡ↦*[s] {b2}@V2.T2 →
+ ∃∃r1,r2,r3. is_whd r1 & is_inner r2 &
+ r1@(dx:::r2)@sn:::r3 = s.
+#b2 #s #F1 #V2 #T2 #H
+lapply (pl_sts_fwd_is_standard … H)
+@(lstar_ind_l … s F1 H) -s -F1
+[ #_ @(ex3_3_intro … ◊ ◊ ◊) // (**) (* auto needs some help here *)
+| #p #s #F1 #F #HF1 #HF2 #IHF1 #Hs
+ lapply (is_standard_fwd_cons … Hs) #H
+ elim (IHF1 …) // -IHF1 -H #r1 #r2 #r3 #Hr1 #Hr2 #H destruct
+ elim (in_whd_or_in_inner p) #Hp
+ [ -Hs -F1 -F -V2 -T2 -b2
+ @(ex3_3_intro … (p::r1) r2 r3) // /2 width=1/ (**) (* auto needs some help here *)
+ | lapply (is_standard_fwd_append_sn (p::r1) ? Hs) -Hs #H
+ lapply (is_standard_fwd_in_inner … H ?) -H // #H
+ lapply (is_whd_is_inner_inv … Hr1 ?) -Hr1 // -H #H destruct
+ elim (in_inner_inv … Hp) -Hp * #q [3: #IHq ] #Hp
+(* case 1: dx *)
+ [ destruct -F1 -F -V2 -T2 -b2
+ @(ex3_3_intro … ◊ (q::r2) r3) // /2 width=1/ (**) (* auto needs some help here *)
+(* case 2: rc *)
+ | -Hr2
+ elim (pl_sts_fwd_dx_sn_appl_dx … HF2) -b2 #b #V #T #T0 #_ #_ #HT -V2 -T2 -T0
+ elim (pl_st_inv_rc … HF1 … Hp) -HF1 #c #V0 #T0 #_ #_ #HT0 -c -V0 -q -F1 destruct
+(* case 3: sn *)
+ | -Hr2
+ elim (pl_sts_fwd_dx_sn_appl_dx … HF2) -b2 #b #V #T #T0 #_ #HT0 #HT -V2 -T2
+ elim (pl_st_inv_sn … HF1 … Hp) -HF1 #c #V1 #V0 #T1 #_ #_ #H -c -V1 -F1 destruct -V
+ elim (pl_sts_inv_empty … HT0) -HT0 #H #_ -T0 -T1 destruct
+ @(ex3_3_intro … ◊ ◊ (q::r3)) // (**) (* auto needs some help here *)
+ ]
+ ]
+]
+qed-.
+
+lemma pl_sred_is_standard_pl_st: ∀p,M,M2. M ↦[p] M2 → ∀F. ⇓F = M →
∀s,M1.{⊤}⇑ M1 Ⓡ↦*[s] F →
is_standard (s@(p::◊)) →
∃∃F2. F Ⓡ↦[p] F2 & ⇓F2 = M2.
+#p #M #M2 #H elim H -p -M -M2
+[ #B #A #F #HF #s #M1 #HM1 #Hs
+ lapply (is_standard_fwd_is_whd … Hs) -Hs // #Hs
+ elim (pl_sts_inv_pl_sreds … HM1 Hs) -HM1 -Hs #M #_ #H -s -M1 destruct
+ >carrier_boolean in HF; #H destruct normalize /2 width=3/
+| #p #A1 #A2 #_ #IHA12 #F #HF #s #M1 #HM1 #Hs
+ elim (carrier_inv_abst … HF) -HF #b #T #HT #HF destruct
+ elim (pl_sts_fwd_abst_dx … HM1) #r1 #r2 #Hr1 #H destruct
+ elim (pl_sts_inv_trans … HM1) -HM1 #F0 #HM1 #HT
+ elim (pl_sts_inv_pl_sreds … HM1 …) // #M0 #_ #H -M1 -Hr1 destruct
+ >associative_append in Hs; #Hs
+ lapply (is_standard_fwd_append_dx … Hs) -r1
+ <(map_cons_append … r2 (p::◊)) #H
+ lapply (is_standard_inv_compatible_rc … H) -H #Hp
+ elim (pl_sts_inv_rc_abst_dx … HT …) -HT [3: // |2: skip ] #b0 #T0 #HT02 #H (**) (* simplify line *)
+ elim (boolean_inv_abst … (sym_eq … H)) -H #A0 #_ #H #_ -b0 -M0 destruct
+ elim (IHA12 … HT02 …) // -r2 -A0 -IHA12 #F2 #HF2 #H
+ @(ex2_intro … ({⊥}𝛌.F2)) normalize // /2 width=1/ (**) (* auto needs some help here *)
+| #p #B1 #B2 #A #_ #IHB12 #F #HF #s #M1 #HM1 #Hs
+ elim (carrier_inv_appl … HF) -HF #b #V #T #HV #HT #HF destruct
+ elim (pl_sts_fwd_appl_dx … HM1) #r1 #r2 #r3 #Hr1 #_ #H destruct
+ elim (pl_sts_inv_trans … HM1) -HM1 #F0 #HM1 #HT
+ elim (pl_sts_inv_pl_sreds … HM1 …) // #M0 #_ #H -M1 -Hr1 destruct
+ >associative_append in Hs; #Hs
+ lapply (is_standard_fwd_append_dx … Hs) -r1
+ >associative_append #Hs
+ lapply (is_standard_fwd_append_dx … Hs) -Hs
+ <(map_cons_append … r3 (p::◊)) #H
+ lapply (is_standard_inv_compatible_sn … H) -H #Hp
+ elim (pl_sts_fwd_dx_sn_appl_dx … HT) -HT #b0 #V0 #T0 #T1 #HV0 #_ #H -T1 -r2
+ elim (boolean_inv_appl … (sym_eq … H)) -H #B0 #A0 #_ #H #_ #_ -b0 -M0 -T0 destruct
+ elim (IHB12 … HV0 …) // -r3 -B0 -IHB12 #G2 #HG2 #H
+ @(ex2_intro … ({⊥}@G2.{⊥}⇕T)) normalize // /2 width=1/ (**) (* auto needs some help here *)
+| #p #B #A1 #A2 #_ #IHA12 #F #HF #s #M1 #HM1 #Hs
+ elim (carrier_inv_appl … HF) -HF #b #V #T #HV #HT #HF destruct
+ elim (pl_sts_fwd_appl_dx … HM1) #r1 #r2 #r3 #Hr1 #Hr2 #H destruct
+ elim (pl_sts_inv_trans … HM1) -HM1 #F0 #HM1 #HT
+ elim (pl_sts_inv_pl_sreds … HM1 …) // #M0 #_ #H -M1 -Hr1 destruct
+ >associative_append in Hs; #Hs
+ lapply (is_standard_fwd_append_dx … Hs) -r1
+ >associative_append #Hs
+ elim (list_inv … r3)
+ [ #H destruct
+ elim (in_whd_or_in_inner p) #Hp
+ [ lapply (is_standard_fwd_is_whd … Hs) -Hs /2 width=1/ -Hp #Hs
+ lapply (is_whd_inv_dx … Hs) -Hs #H
+ lapply (is_whd_is_inner_inv … Hr2) -Hr2 // -H #H destruct
+ lapply (pl_sts_inv_nil … HT ?) -HT // #H
+ elim (boolean_inv_appl … H) -H #B0 #A0 #_ #_ #H #_ -M0 -B0 destruct
+ elim (IHA12 … A0 …) -IHA12 [3,5,6: // |2,4: skip ] (* simplify line *)
+ #F2 #HF2 #H
+ @(ex2_intro … ({b}@V.F2)) normalize // /2 width=1/ (**) (* auto needs some help here *)
+ | <(map_cons_append … r2 (p::◊)) in Hs; #H
+ lapply (is_standard_inv_compatible_dx … H ?) -H /3 width=1/ -Hp #Hp
+ >append_nil in HT; #HT
+ elim (pl_sts_inv_dx_appl_dx … HT …) -HT [3: // |2: skip ] (* simplify line *)
+ #T0 #HT0 #H
+ elim (boolean_inv_appl … (sym_eq … H)) -H #B0 #A0 #_ #_ #H #_ -M0 -B0 destruct
+ elim (IHA12 … HT0 …) // -r2 -A0 -IHA12 #F2 #HF2 #H
+ @(ex2_intro … ({b}@V.F2)) normalize // /2 width=1/ (**) (* auto needs some help here *)
+ ]
+ | -IHA12 -Hr2 -M0 * #q #r #H destruct
+ lapply (is_standard_fwd_append_dx … Hs) -r2 #Hs
+ lapply (is_standard_fwd_sle … Hs) -r #H
+ elim (sle_inv_sn … H …) -H [3: // |2: skip ] (**) (* simplify line *)
+ #q0 #_ #H destruct
+ ]
+]
+qed-.
theorem pl_sreds_is_standard_pl_sts: ∀s,M1,M2. M1 ↦*[s] M2 → is_standard s →
∃∃F2. {⊤}⇑ M1 Ⓡ↦*[s] F2 & ⇓F2 = M2.
-#s #M1 #M2 #H @(lstar_ind_r ????????? H) -s -M2 /2 width=3/
+#s #M1 #M2 #H @(lstar_ind_r … s M2 H) -s -M2 /2 width=3/
#p #s #M #M2 #_ #HM2 #IHM1 #Hsp
lapply (is_standard_fwd_append_sn … Hsp) #Hs
elim (IHM1 Hs) -IHM1 -Hs #F #HM1 #H
-elim (pl_sred_is_standard_pl_st … HM2 … HM1 ?) -HM2 // -M -Hsp #F2 #HF2 #HFM2
+elim (pl_sred_is_standard_pl_st … HM2 … HM1 …) -HM2 // -M -Hsp #F2 #HF2 #HFM2
lapply (pl_sts_step_dx … HM1 … HF2) -F
#H @(ex2_intro … F2) // (**) (* auto needs some help here *)
qed-.
projects / helm.git / blobdiff