non associative with precedence 45
for @{ 'Std $M $N }.
-axiom st_refl: reflexive … st.
+lemma st_inv_lref: ∀M,N. M ⓢ⥤* N → ∀j. #j = N →
+ ∃s. M ⓗ⇀*[s] #j.
+#M #N * -M -N
+[ /2 width=2/
+| #s #M #A1 #A2 #_ #_ #j #H destruct
+| #s #M #B1 #B2 #A1 #A2 #_ #_ #_ #j #H destruct
+]
+qed-.
-axiom st_step_sn: ∀N1,N2. N1 ⓢ⥤* N2 → ∀s,M. M ⓗ⇀*[s] N1 → M ⓢ⥤* N2.
+lemma st_inv_abst: ∀M,N. M ⓢ⥤* N → ∀C2. 𝛌.C2 = N →
+ ∃∃s,C1. M ⓗ⇀*[s] 𝛌.C1 & C1 ⓢ⥤* C2.
+#M #N * -M -N
+[ #s #M #i #_ #C2 #H destruct
+| #s #M #A1 #A2 #HM #A12 #C2 #H destruct /2 width=4/
+| #s #M #B1 #B2 #A1 #A2 #_ #_ #_ #C2 #H destruct
+]
+qed-.
-axiom st_lift: liftable st.
+lemma st_inv_appl: ∀M,N. M ⓢ⥤* N → ∀D2,C2. @D2.C2 = N →
+ ∃∃s,D1,C1. M ⓗ⇀*[s] @D1.C1 & D1 ⓢ⥤* D2 & C1 ⓢ⥤* C2.
+#M #N * -M -N
+[ #s #M #i #_ #D2 #C2 #H destruct
+| #s #M #A1 #A2 #_ #_ #D2 #C2 #H destruct
+| #s #M #B1 #B2 #A1 #A2 #HM #HB12 #HA12 #D2 #C2 #H destruct /2 width=6/
+]
+qed-.
-axiom st_inv_lift: deliftable_sn st.
+lemma st_refl: reflexive … st.
+#M elim M -M /2 width=2/ /2 width=4/ /2 width=6/
+qed.
-axiom st_dsubst: dsubstable st.
+lemma st_step_sn: ∀N1,N2. N1 ⓢ⥤* N2 → ∀s,M. M ⓗ⇀*[s] N1 → M ⓢ⥤* N2.
+#N1 #N2 #H elim H -N1 -N2
+[ #r #N #i #HN #s #M #HMN
+ lapply (lhap_trans … HMN … HN) -N /2 width=2/
+| #r #N #C1 #C2 #HN #_ #IHC12 #s #M #HMN
+ lapply (lhap_trans … HMN … HN) -N /3 width=5/
+| #r #N #D1 #D2 #C1 #C2 #HN #_ #_ #IHD12 #IHC12 #s #M #HMN
+ lapply (lhap_trans … HMN … HN) -N /3 width=7/
+]
+qed-.
+
+lemma st_step_rc: ∀s,M1,M2. M1 ⓗ⇀*[s] M2 → M1 ⓢ⥤* M2.
+/3 width=4 by st_step_sn/
+qed.
+
+lemma st_lift: liftable st.
+#h #M1 #M2 #H elim H -M1 -M2
+[ /3 width=2/
+| #s #M #A1 #A2 #HM #_ #IHA12 #d
+ @st_abst [3: @(lhap_lift … HM) |1,2: skip ] -M // (**) (* auto fails here *)
+| #s #M #B1 #B2 #A1 #A2 #HM #_ #_ #IHB12 #IHA12 #d
+ @st_appl [4: @(lhap_lift … HM) |1,2,3: skip ] -M // (**) (* auto fails here *)
+]
+qed.
+
+lemma st_inv_lift: deliftable_sn st.
+#h #N1 #N2 #H elim H -N1 -N2
+[ #s #N1 #i #HN1 #d #M1 #HMN1
+ elim (lhap_inv_lift … HN1 … HMN1) -N1 /3 width=3/
+| #s #N1 #C1 #C2 #HN1 #_ #IHC12 #d #M1 #HMN1
+ elim (lhap_inv_lift … HN1 … HMN1) -N1 #M2 #HM12 #HM2
+ elim (lift_inv_abst … HM2) -HM2 #A1 #HAC1 #HM2 destruct
+ elim (IHC12 ???) -IHC12 [4: // |2,3: skip ] #A2 #HA12 #HAC2 destruct (**) (* simplify line *)
+ @(ex2_intro … (𝛌.A2)) // /2 width=4/
+| #s #N1 #D1 #D2 #C1 #C2 #HN1 #_ #_ #IHD12 #IHC12 #d #M1 #HMN1
+ elim (lhap_inv_lift … HN1 … HMN1) -N1 #M2 #HM12 #HM2
+ elim (lift_inv_appl … HM2) -HM2 #B1 #A1 #HBD1 #HAC1 #HM2 destruct
+ elim (IHD12 ???) -IHD12 [4: // |2,3: skip ] #B2 #HB12 #HBD2 destruct (**) (* simplify line *)
+ elim (IHC12 ???) -IHC12 [4: // |2,3: skip ] #A2 #HA12 #HAC2 destruct (**) (* simplify line *)
+ @(ex2_intro … (@B2.A2)) // /2 width=6/
+]
+qed-.
+
+lemma st_dsubst: dsubstable st.
+#N1 #N2 #HN12 #M1 #M2 #H elim H -M1 -M2
+[ #s #M #i #HM #d elim (lt_or_eq_or_gt i d) #Hid
+ [ lapply (lhap_dsubst … N1 … HM d) -HM
+ >(dsubst_vref_lt … Hid) >(dsubst_vref_lt … Hid) /2 width=2/
+ | destruct >dsubst_vref_eq
+ @(st_step_sn (↑[0,i]N1) … s) /2 width=1/
+ | lapply (lhap_dsubst … N1 … HM d) -HM
+ >(dsubst_vref_gt … Hid) >(dsubst_vref_gt … Hid) /2 width=2/
+ ]
+| #s #M #A1 #A2 #HM #_ #IHA12 #d
+ lapply (lhap_dsubst … N1 … HM d) -HM /2 width=4/ (**) (* auto needs some help here *)
+| #s #M #B1 #B2 #A1 #A2 #HM #_ #_ #IHB12 #IHA12 #d
+ lapply (lhap_dsubst … N1 … HM d) -HM /2 width=6/ (**) (* auto needs some help here *)
+]
+qed.
lemma st_inv_lsreds_is_le: ∀M,N. M ⓢ⥤* N →
∃∃r. M ⇀*[r] N & is_le r.
| #s #M #A1 #A2 #H #_ * #r #HA12 #Hr
lapply (lhap_inv_lsreds … H) #HM
lapply (lhap_inv_head … H) -H #Hs
- lapply (lsreds_trans … HM (rc:::r) (𝛌.A2) ?) /2 width=1/ -A1 #HM
+ lapply (lsreds_trans … HM (sn:::r) (𝛌.A2) ?) /2 width=1/ -A1 #HM
@(ex2_intro … HM) -M -A2 /3 width=1/
| #s #M #B1 #B2 #A1 #A2 #H #_ #_ * #rb #HB12 #Hrb * #ra #HA12 #Hra
lapply (lhap_inv_lsreds … H) #HM
@(ex2_intro … HM) -M -B2 -A2 >associative_append /3 width=1/
]
qed-.
+
+lemma st_step_dx: ∀p,M,M2. M ⇀[p] M2 → ∀M1. M1 ⓢ⥤* M → M1 ⓢ⥤* M2.
+#p #M #M2 #H elim H -p -M -M2
+[ #B #A #M1 #H
+ elim (st_inv_appl … H ???) -H [4: // |2,3: skip ] #s #B1 #M #HM1 #HB1 #H (**) (* simplify line *)
+ elim (st_inv_abst … H ??) -H [3: // |2: skip ] #r #A1 #HM #HA1 (**) (* simplify line *)
+ lapply (lhap_trans … HM1 … (dx:::r) (@B1.𝛌.A1) ?) /2 width=1/ -M #HM1
+ lapply (lhap_step_dx … HM1 (◊) ([⬐B1]A1) ?) -HM1 // #HM1
+ @(st_step_sn … HM1) /2 width=1/
+| #p #A #A2 #_ #IHA2 #M1 #H
+ elim (st_inv_abst … H ??) -H [3: // |2: skip ] /3 width=4/ (**) (* simplify line *)
+| #p #B #B2 #A #_ #IHB2 #M1 #H
+ elim (st_inv_appl … H ???) -H [4: // |2,3: skip ] /3 width=6/ (**) (* simplify line *)
+| #p #B #A #A2 #_ #IHA2 #M1 #H
+ elim (st_inv_appl … H ???) -H [4: // |2,3: skip ] /3 width=6/ (**) (* simplify line *)
+]
+qed-.
+
+lemma st_lhap1_swap: ∀p,N1,N2. N1 ⓗ⇀[p] N2 → ∀M1. M1 ⓢ⥤* N1 →
+ ∃∃q,M2. M1 ⓗ⇀[q] M2 & M2 ⓢ⥤* N2.
+#p #N1 #N2 #H elim H -p -N1 -N2
+[ #D #C #M1 #H
+ elim (st_inv_appl … H ???) -H [4: // |2,3: skip ] #s #D1 #N #HM1 #HD1 #H (**) (* simplify line *)
+ elim (st_inv_abst … H ??) -H [3: // |2: skip ] #r #C1 #HN #HC1 (**) (* simplify line *)
+ lapply (lhap_trans … HM1 … (dx:::r) (@D1.𝛌.C1) ?) /2 width=1/ -N #HM1
+ lapply (lhap_step_dx … HM1 (◊) ([⬐D1]C1) ?) -HM1 // #HM1
+ elim (lhap_inv_pos … HM1 ?) -HM1
+ [2: >length_append normalize in ⊢ (??(??%)); // ]
+ #q #r #M #_ #HM1 #HM -s
+ @(ex2_2_intro … HM1) -M1
+ @(st_step_sn … HM) /2 width=1/
+| #p #D #C1 #C2 #_ #IHC12 #M1 #H -p
+ elim (st_inv_appl … H ???) -H [4: // |2,3: skip ] #s #B #A1 #HM1 #HBD #HAC1 (**) (* simplify line *)
+ elim (IHC12 … HAC1) -C1 #p #C1 #HAC1 #HC12
+ lapply (lhap_step_dx … HM1 (dx::p) (@B.C1) ?) -HM1 /2 width=1/ -A1 #HM1
+ elim (lhap_inv_pos … HM1 ?) -HM1
+ [2: >length_append normalize in ⊢ (??(??%)); // ]
+ #q #r #M #_ #HM1 #HM -p -s
+ @(ex2_2_intro … HM1) -M1 /2 width=6/
+]
+qed-.
+
+lemma st_lsreds: ∀s,M1,M2. M1 ⇀*[s] M2 → M1 ⓢ⥤* M2.
+#s #M1 #M2 #H @(lstar_ind_r ????????? H) -s -M2 // /2 width=4 by st_step_dx/
+qed.
+
+theorem st_trans: transitive … st.
+#M1 #M #M2 #HM1 #HM2
+elim (st_inv_lsreds_is_le … HM1) -HM1 #s1 #HM1 #_
+elim (st_inv_lsreds_is_le … HM2) -HM2 #s2 #HM2 #_
+lapply (lsreds_trans … HM1 … HM2) -M /2 width=2/
+qed-.
+
+theorem lsreds_standard: ∀s,M,N. M ⇀*[s] N →
+ ∃∃r. M ⇀*[r] N & is_le r.
+#s #M #N #H
+@st_inv_lsreds_is_le /2 width=2/
+qed-.
+
+theorem lsreds_lhap1_swap: ∀s,M1,N1. M1 ⇀*[s] N1 → ∀p,N2. N1 ⓗ⇀[p] N2 →
+ ∃∃q,r,M2. M1 ⓗ⇀[q] M2 & M2 ⇀*[r] N2 & is_le (q::r).
+#s #M1 #N1 #HMN1 #p #N2 #HN12
+lapply (st_lsreds … HMN1) -s #HMN1
+elim (st_lhap1_swap … HN12 … HMN1) -p -N1 #q #M2 #HM12 #HMN2
+elim (st_inv_lsreds_is_le … HMN2) -HMN2 #r #HMN2 #Hr
+lapply (lhap1_inv_head … HM12) /3 width=7/
+qed-.
projects / helm.git / blobdiff