+++ /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 "labelled_sequential_computation.ma".
-include "labelled_hap_reduction.ma".
-
-(* KASHIMA'S "HAP" COMPUTATION (LABELLED MULTISTEP) *************************)
-
-(* Note: this is the "head in application" computation of:
- R. Kashima: "A proof of the Standization Theorem in λ-Calculus". Typescript note, (2000).
-*)
-definition lhap: pseq → relation term ≝ lstar … lhap1.
-
-interpretation "labelled 'hap' computation"
- 'HApStar M p N = (lhap p M N).
-
-notation "hvbox( M break ⓗ↦* [ term 46 p ] break term 46 N )"
- non associative with precedence 45
- for @{ 'HApStar $M $p $N }.
-
-lemma lhap_step_rc: ∀p,M1,M2. M1 ⓗ↦[p] M2 → M1 ⓗ↦*[p::◊] M2.
-/2 width=1/
-qed.
-
-lemma lhap_inv_nil: ∀s,M1,M2. M1 ⓗ↦*[s] M2 → ◊ = s → M1 = M2.
-/2 width=5 by lstar_inv_nil/
-qed-.
-
-lemma lhap_inv_cons: ∀s,M1,M2. M1 ⓗ↦*[s] M2 → ∀q,r. q::r = s →
- ∃∃M. M1 ⓗ↦[q] M & M ⓗ↦*[r] M2.
-/2 width=3 by lstar_inv_cons/
-qed-.
-
-lemma lhap_inv_step_rc: ∀p,M1,M2. M1 ⓗ↦*[p::◊] M2 → M1 ⓗ↦[p] M2.
-/2 width=1 by lstar_inv_step/
-qed-.
-
-lemma lhap_inv_pos: ∀s,M1,M2. M1 ⓗ↦*[s] M2 → 0 < |s| →
- ∃∃p,r,M. p::r = s & M1 ⓗ↦[p] M & M ⓗ↦*[r] M2.
-/2 width=1 by lstar_inv_pos/
-qed-.
-
-lemma lhap_compatible_dx: ho_compatible_dx lhap.
-/3 width=1/
-qed.
-
-lemma lhap_lift: ∀s. liftable (lhap s).
-/2 width=1/
-qed.
-
-lemma lhap_inv_lift: ∀s. deliftable_sn (lhap s).
-/3 width=3 by lstar_deliftable_sn, lhap1_inv_lift/
-qed-.
-
-lemma lhap_dsubst: ∀s. dsubstable_dx (lhap s).
-/2 width=1/
-qed.
-
-theorem lhap_mono: ∀s. singlevalued … (lhap s).
-/3 width=7 by lstar_singlevalued, lhap1_mono/
-qed-.
-
-theorem lhap_trans: ltransitive … lhap.
-/2 width=3 by lstar_ltransitive/
-qed-.
-
-lemma lhap_step_dx: ∀s,M1,M. M1 ⓗ↦*[s] M →
- ∀p,M2. M ⓗ↦[p] M2 → M1 ⓗ↦*[s@p::◊] M2.
-#s #M1 #M #HM1 #p #M2 #HM2
-@(lhap_trans … HM1) /2 width=1/
-qed-.
-
-lemma head_lsreds_lhap: ∀s,M1,M2. M1 ↦*[s] M2 → is_head s → M1 ⓗ↦*[s] M2.
-#s #M1 #M2 #H @(lstar_ind_l ????????? H) -s -M1 //
-#a #s #M1 #M #HM1 #_ #IHM2 * /3 width=3/
-qed.
-
-lemma lhap_inv_head: ∀s,M1,M2. M1 ⓗ↦*[s] M2 → is_head s.
-#s #M1 #M2 #H @(lstar_ind_l ????????? H) -s -M1 //
-#p #s #M1 #M #HM1 #_ #IHM2
-lapply (lhap1_inv_head … HM1) -HM1 /2 width=1/
-qed-.
-
-lemma lhap_inv_lsreds: ∀s,M1,M2. M1 ⓗ↦*[s] M2 → M1 ↦*[s] M2.
-#s #M1 #M2 #H @(lstar_ind_l ????????? H) -s -M1 //
-#p #s #M1 #M #HM1 #_ #IHM2
-lapply (lhap1_inv_lsred … HM1) -HM1 /2 width=3/
-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 "labelled_sequential_reduction.ma".
-
-(* KASHIMA'S "HAP" COMPUTATION (LABELLED SINGLE STEP) ***********************)
-
-(* Note: this is one step of the "head in application" computation of:
- R. Kashima: "A proof of the Standization Theorem in λ-Calculus". Typescript note, (2000).
-*)
-inductive lhap1: ptr → relation term ≝
-| hap1_beta: ∀B,A. lhap1 (◊) (@B.𝛌.A) ([↙B]A)
-| hap1_appl: ∀p,B,A1,A2. lhap1 p A1 A2 → lhap1 (dx::p) (@B.A1) (@B.A2)
-.
-
-interpretation "labelled 'hap' reduction"
- 'HAp M p N = (lhap1 p M N).
-
-notation "hvbox( M break ⓗ↦ [ term 46 p ] break term 46 N )"
- non associative with precedence 45
- for @{ 'HAp $M $p $N }.
-
-lemma lhap1_inv_nil: ∀p,M,N. M ⓗ↦[p] N → ◊ = p →
- ∃∃B,A. @B.𝛌.A = M & [↙B]A = N.
-#p #M #N * -p -M -N
-[ #B #A #_ /2 width=4/
-| #p #B #A1 #A2 #_ #H destruct
-]
-qed-.
-
-lemma lhap1_inv_cons: ∀p,M,N. M ⓗ↦[p] N → ∀c,q. c::q = p →
- ∃∃B,A1,A2. dx = c & A1 ⓗ↦[q] A2 & @B.A1 = M & @B.A2 = N.
-#p #M #N * -p -M -N
-[ #B #A #c #q #H destruct
-| #p #B #A1 #A2 #HA12 #c #q #H destruct /2 width=6/
-]
-qed-.
-
-lemma lhap1_lift: ∀p. liftable (lhap1 p).
-#p #h #M1 #M2 #H elim H -p -M1 -M2 normalize /2 width=1/
-#B #A #d <dsubst_lift_le //
-qed.
-
-lemma lhap1_inv_lift: ∀p. deliftable_sn (lhap1 p).
-#p #h #N1 #N2 #H elim H -p -N1 -N2
-[ #D #C #d #M1 #H
- elim (lift_inv_appl … H) -H #B #M #H0 #HM #H destruct
- elim (lift_inv_abst … HM) -HM #A #H0 #H destruct /3 width=3/
-| #p #D1 #C1 #C2 #_ #IHC12 #d #M1 #H
- elim (lift_inv_appl … H) -H #B #A1 #H1 #H2 #H destruct
- elim (IHC12 ???) -IHC12 [4: // |2,3: skip ] #A2 #HA12 #H destruct (**) (* simplify line *)
- @(ex2_intro … (@B.A2)) // /2 width=1/
-]
-qed-.
-
-lemma lhap1_dsubst: ∀p. dsubstable_dx (lhap1 p).
-#p #D1 #M1 #M2 #H elim H -p -M1 -M2 normalize /2 width=1/
-#D2 #A #d >dsubst_dsubst_ge //
-qed.
-
-lemma head_lsred_lhap1: ∀p. in_head p → ∀M,N. M ↦[p] N → M ⓗ↦[p] N.
-#p #H @(in_head_ind … H) -p
-[ #M #N #H elim (lsred_inv_nil … H ?) -H //
-| #p #_ #IHp #M #N #H
- elim (lsred_inv_dx … H ??) -H [3: // |2: skip ] /3 width=1/ (**) (* simplify line *)
-]
-qed.
-
-lemma lhap1_inv_head: ∀p,M,N. M ⓗ↦[p] N → in_head p.
-#p #M #N #H elim H -p -M -N // /2 width=1/
-qed-.
-
-lemma lhap1_inv_lsred: ∀p,M,N. M ⓗ↦[p] N → M ↦[p] N.
-#p #M #N #H elim H -p -M -N // /2 width=1/
-qed-.
-
-theorem lhap1_mono: ∀p. singlevalued … (lhap1 p).
-#p #M #N1 #H elim H -p -M -N1
-[ #B #A #N2 #H
- elim (lhap1_inv_nil … H ?) -H // #D #C #H #HN2 destruct //
-| #p #B #A1 #A2 #_ #IHA12 #N2 #H
- elim (lhap1_inv_cons … H ???) -H [4: // |2,3: skip ] (**) (* simplify line *)
- #D #C1 #C2 #_ #HC12 #H #HN2 destruct /3 width=1/
-]
-qed-.
(* *)
(**************************************************************************)
-include "labelled_hap_computation.ma".
+include "labelled_sequential_computation.ma".
(* KASHIMA'S "ST" COMPUTATION ***********************************************)
R. Kashima: "A proof of the Standization Theorem in λ-Calculus". Typescript note, (2000).
*)
inductive st: relation term ≝
-| st_vref: ∀s,M,i. M ⓗ↦*[s] #i → st M (#i)
-| st_abst: ∀s,M,A1,A2. M ⓗ↦*[s] 𝛌.A1 → st A1 A2 → st M (𝛌.A2)
-| st_appl: ∀s,M,B1,B2,A1,A2. M ⓗ↦*[s] @B1.A1 → st B1 B2 → st A1 A2 → st M (@B2.A2)
+| st_vref: ∀s,M,i. is_head s → M ↦*[s] #i → st M (#i)
+| st_abst: ∀s,M,A1,A2. is_head s → M ↦*[s] 𝛌.A1 → st A1 A2 → st M (𝛌.A2)
+| st_appl: ∀s,M,B1,B2,A1,A2. is_head s → M ↦*[s] @B1.A1 → st B1 B2 → st A1 A2 → st M (@B2.A2)
.
interpretation "'st' computation"
for @{ 'Std $M $N }.
lemma st_inv_lref: ∀M,N. M ⓢ⤇* N → ∀j. #j = N →
- ∃s. M ⓗ↦*[s] #j.
+ ∃∃s. is_head 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
+[ /2 width=3/
+| #s #M #A1 #A2 #_ #_ #_ #j #H destruct
+| #s #M #B1 #B2 #A1 #A2 #_ #_ #_ #_ #j #H destruct
]
qed-.
lemma st_inv_abst: ∀M,N. M ⓢ⤇* N → ∀C2. 𝛌.C2 = N →
- ∃∃s,C1. M ⓗ↦*[s] 𝛌.C1 & C1 ⓢ⤇* C2.
+ ∃∃s,C1. is_head s & 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
+[ #s #M #i #_ #_ #C2 #H destruct
+| #s #M #A1 #A2 #Hs #HM #A12 #C2 #H destruct /2 width=5/
+| #s #M #B1 #B2 #A1 #A2 #_ #_ #_ #_ #C2 #H destruct
]
qed-.
lemma st_inv_appl: ∀M,N. M ⓢ⤇* N → ∀D2,C2. @D2.C2 = N →
- ∃∃s,D1,C1. M ⓗ↦*[s] @D1.C1 & D1 ⓢ⤇* D2 & C1 ⓢ⤇* C2.
+ ∃∃s,D1,C1. is_head s & 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/
+[ #s #M #i #_ #_ #D2 #C2 #H destruct
+| #s #M #A1 #A2 #_ #_ #_ #D2 #C2 #H destruct
+| #s #M #B1 #B2 #A1 #A2 #Hs #HM #HB12 #HA12 #D2 #C2 #H destruct /2 width=7/
]
qed-.
lemma st_refl: reflexive … st.
-#M elim M -M /2 width=2/ /2 width=4/ /2 width=6/
+#M elim M -M /2 width=3/ /2 width=5/ /2 width=7/
qed.
-lemma st_step_sn: ∀N1,N2. N1 ⓢ⤇* N2 → ∀s,M. M ⓗ↦*[s] N1 → M ⓢ⤇* N2.
+lemma st_step_sn: ∀N1,N2. N1 ⓢ⤇* N2 → ∀s,M. is_head s → 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/
+[ #r #N #i #Hr #HN #s #M #Hs #HMN
+ lapply (lsreds_trans … HMN … HN) -N /3 width=3/
+| #r #N #C1 #C2 #Hr #HN #_ #IHC12 #s #M #Hs #HMN
+ lapply (lsreds_trans … HMN … HN) -N /3 width=7/
+| #r #N #D1 #D2 #C1 #C2 #Hr #HN #_ #_ #IHD12 #IHC12 #s #M #Hs #HMN
+ lapply (lsreds_trans … HMN … HN) -N /3 width=9/
]
qed-.
-lemma st_step_rc: ∀s,M1,M2. M1 ⓗ↦*[s] M2 → M1 ⓢ⤇* M2.
-/3 width=4 by st_step_sn/
+lemma st_step_rc: ∀s,M1,M2. is_head s → M1 ↦*[s] M2 → M1 ⓢ⤇* M2.
+/3 width=5 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 *)
+[ /3 width=3/
+| #s #M #A1 #A2 #Hs #HM #_ #IHA12 #d
+ @(st_abst … Hs) [2: @(lsreds_lift … HM) | skip ] -M // (**) (* auto fails here *)
+| #s #M #B1 #B2 #A1 #A2 #Hs #HM #_ #_ #IHB12 #IHA12 #d
+ @(st_appl … Hs) [3: @(lsreds_lift … HM) |1,2: 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
+[ #s #N1 #i #Hs #HN1 #d #M1 #HMN1
+ elim (lsreds_inv_lift … HN1 … HMN1) -N1 /3 width=3/
+| #s #N1 #C1 #C2 #Hs #HN1 #_ #IHC12 #d #M1 #HMN1
+ elim (lsreds_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
+ @(ex2_intro … (𝛌.A2)) // /2 width=5/
+| #s #N1 #D1 #D2 #C1 #C2 #Hs #HN1 #_ #_ #IHD12 #IHC12 #d #M1 #HMN1
+ elim (lsreds_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/
+ @(ex2_intro … (@B2.A2)) // /2 width=7/
]
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/
+[ #s #M #i #Hs #HM #d elim (lt_or_eq_or_gt i d) #Hid
+ [ lapply (lsreds_dsubst … N1 … HM d) -HM
+ >(dsubst_vref_lt … Hid) >(dsubst_vref_lt … Hid) /2 width=3/
| 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/
+ | lapply (lsreds_dsubst … N1 … HM d) -HM
+ >(dsubst_vref_gt … Hid) >(dsubst_vref_gt … Hid) /2 width=3/
]
-| #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 *)
+| #s #M #A1 #A2 #Hs #HM #_ #IHA12 #d
+ lapply (lsreds_dsubst … N1 … HM d) -HM /2 width=5/ (**) (* auto needs some help here *)
+| #s #M #B1 #B2 #A1 #A2 #Hs #HM #_ #_ #IHB12 #IHA12 #d
+ lapply (lsreds_dsubst … N1 … HM d) -HM /2 width=7/ (**) (* auto needs some help here *)
]
qed.
lemma st_inv_lsreds_is_le: ∀M,N. M ⓢ⤇* N →
∃∃r. M ↦*[r] N & is_le r.
#M #N #H elim H -M -N
-[ #s #M #i #H
- lapply (lhap_inv_lsreds … H)
- lapply (lhap_inv_head … H) -H #H
- lapply (is_head_is_le … H) -H /2 width=3/
-| #s #M #A1 #A2 #H #_ * #r #HA12 #Hr
- lapply (lhap_inv_lsreds … H) #HM
- lapply (lhap_inv_head … H) -H #Hs
+[ #s #M #i #Hs #HM
+ lapply (is_head_is_le … Hs) -Hs /2 width=3/
+| #s #M #A1 #A2 #Hs #HM #_ * #r #HA12 #Hr
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
- lapply (lhap_inv_head … H) -H #Hs
+| #s #M #B1 #B2 #A1 #A2 #Hs #HM #_ #_ * #rb #HB12 #Hrb * #ra #HA12 #Hra
lapply (lsreds_trans … HM (dx:::ra) (@B1.A2) ?) /2 width=1/ -A1 #HM
lapply (lsreds_trans … HM (sn:::rb) (@B2.A2) ?) /2 width=1/ -B1 #HM
@(ex2_intro … HM) -M -B2 -A2 >associative_append /3 width=1/
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/
+ elim (st_inv_appl … H ???) -H [4: // |2,3: skip ] #s #B1 #M #Hs #HM1 #HB1 #H (**) (* simplify line *)
+ elim (st_inv_abst … H ??) -H [3: // |2: skip ] #r #A1 #Hr #HM #HA1 (**) (* simplify line *)
+ lapply (lsreds_trans … HM1 … (dx:::r) (@B1.𝛌.A1) ?) /2 width=1/ -M #HM1
+ lapply (lsreds_step_dx … HM1 (◊) ([↙B1]A1) ?) -HM1 // #HM1
+ @(st_step_sn … HM1) /2 width=1/ /4 width=1/
| #p #A #A2 #_ #IHA2 #M1 #H
- elim (st_inv_abst … H ??) -H [3: // |2: skip ] /3 width=4/ (**) (* simplify line *)
+ elim (st_inv_abst … H ??) -H [3: // |2: skip ] /3 width=5/ (**) (* simplify line *)
| #p #B #B2 #A #_ #IHB2 #M1 #H
- elim (st_inv_appl … H ???) -H [4: // |2,3: skip ] /3 width=6/ (**) (* simplify line *)
+ elim (st_inv_appl … H ???) -H [4: // |2,3: skip ] /3 width=7/ (**) (* simplify line *)
| #p #B #A #A2 #_ #IHA2 #M1 #H
- elim (st_inv_appl … H ???) -H [4: // |2,3: skip ] /3 width=6/ (**) (* simplify line *)
+ elim (st_inv_appl … H ???) -H [4: // |2,3: skip ] /3 width=7/ (**) (* 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
+(* Note: we use "lapply (rewrite_r ?? is_head … Hq)" (procedural)
+ in place of "cut (is_head (q::r)) [ >Hq ]" (declarative)
+*)
+lemma st_lsred_swap: ∀p. in_head p → ∀N1,N2. N1 ↦[p] N2 → ∀M1. M1 ⓢ⤇* N1 →
+ ∃∃q,M2. in_head q & M1 ↦[q] M2 & M2 ⓢ⤇* N2.
+#p #H @(in_head_ind … H) -p
+[ #N1 #N2 #H1 #M1 #H2
+ elim (lsred_inv_nil … H1 ?) -H1 // #D #C #HN1 #HN2
+ elim (st_inv_appl … H2 … HN1) -N1 #s1 #D1 #N #Hs1 #HM1 #HD1 #H
+ elim (st_inv_abst … H ??) -H [3: // |2: skip ] #s2 #C1 #Hs2 #HN #HC1 (**) (* simplify line *)
+ lapply (lsreds_trans … HM1 … (dx:::s2) (@D1.𝛌.C1) ?) /2 width=1/ -N #HM1
+ lapply (lsreds_step_dx … HM1 (◊) ([↙D1]C1) ?) -HM1 // #HM1
+ elim (lsreds_inv_pos … HM1 ?) -HM1
[2: >length_append normalize in ⊢ (??(??%)); // ]
- #q #r #M #_ #HM1 #HM -s
- @(ex2_2_intro … HM1) -M1
+ #q #r #M #Hq #HM1 #HM
+ lapply (rewrite_r ?? is_head … Hq) -Hq /4 width=1/ -s1 -s2 * #Hq #Hr
+ @(ex3_2_intro … HM1) -M1 // -q
@(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
+| #p #_ #IHp #N1 #N2 #H1 #M1 #H2
+ elim (lsred_inv_dx … H1 ??) -H1 [3: // |2: skip ] #D #C1 #C2 #HC12 #HN1 #HN2 (**) (* simplify line *)
+ elim (st_inv_appl … H2 … HN1) -N1 #s #B #A1 #Hs #HM1 #HBD #HAC1
+ elim (IHp … HC12 … HAC1) -p -C1 #p #C1 #Hp #HAC1 #HC12
+ lapply (lsreds_step_dx … HM1 (dx::p) (@B.C1) ?) -HM1 /2 width=1/ -A1 #HM1
+ elim (lsreds_inv_pos … HM1 ?) -HM1
[2: >length_append normalize in ⊢ (??(??%)); // ]
- #q #r #M #_ #HM1 #HM -p -s
- @(ex2_2_intro … HM1) -M1 /2 width=6/
+ #q #r #M #Hq #HM1 #HM
+ lapply (rewrite_r ?? is_head … Hq) -Hq /4 width=1/ -p -s * #Hq #Hr
+ @(ex3_2_intro … HM1) -M1 // -q /2 width=7/
]
qed-.
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.
+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
+theorem lsreds_lhap1_swap: ∀s,M1,N1. M1 ↦*[s] N1 →
+ ∀p,N2. in_head p → N1 ↦[p] N2 →
+ ∃∃q,r,M2. in_head q & M1 ↦[q] M2 & M2 ↦*[r] N2 &
+ is_le (q::r).
+#s #M1 #N1 #HMN1 #p #N2 #Hp #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/
+elim (st_lsred_swap … Hp … HN12 … HMN1) -p -N1 #q #M2 #Hq #HM12 #HMN2
+elim (st_inv_lsreds_is_le … HMN2) -HMN2 /3 width=8/
qed-.
projects / helm.git / commitdiff