(* Basic properties *********************************************************)
lemma cpxs_refl: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ T ➡*[h, g] T.
-/2 width=1/ qed.
+/2 width=1 by inj/ qed.
lemma cpx_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
-/2 width=1/ qed.
+/2 width=1 by inj/ qed.
lemma cpxs_strap1: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T →
∀T2. ⦃G, L⦄ ⊢ T ➡[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
-normalize /2 width=3/ qed.
+normalize /2 width=3 by step/ qed.
lemma cpxs_strap2: ∀h,g,G,L,T1,T. ⦃G, L⦄ ⊢ T1 ➡[h, g] T →
∀T2. ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
-normalize /2 width=3/ qed.
+normalize /2 width=3 by TC_strap/ qed.
lemma lsubr_cpxs_trans: ∀h,g,G. lsub_trans … (cpxs h g G) lsubr.
/3 width=5 by lsubr_cpx_trans, TC_lsub_trans/
qed-.
lemma cprs_cpxs: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2.
-#h #g #G #L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=3/
+#h #g #G #L #T1 #T2 #H @(cprs_ind … H) -T2 /3 width=3 by cpxs_strap1, cpr_cpx/
qed.
lemma cpxs_bind_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
∀I,T1,T2. ⦃G, L. ⓑ{I}V1⦄ ⊢ T1 ➡*[h, g] T2 →
∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
#h #g #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12 #a @(cpxs_ind_dx … HT12) -T1
-/3 width=1/ /3 width=3/
+/3 width=3 by cpxs_strap2, cpx_cpxs, cpx_pair_sn, cpx_bind/
qed.
lemma cpxs_flat_dx: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡[h, g] V2 →
∀T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2.
-#h #g #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2 /3 width=1/ /3 width=5/
+#h #g #G #L #V1 #V2 #HV12 #T1 #T2 #HT12 @(cpxs_ind … HT12) -T2
+/3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
qed.
lemma cpxs_flat_sn: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡[h, g] T2 →
∀V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
∀I. ⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2.
-#h #g #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2 /3 width=1/ /3 width=5/
+#h #g #G #L #T1 #T2 #HT12 #V1 #V2 #H @(cpxs_ind … H) -V2
+/3 width=5 by cpxs_strap1, cpx_cpxs, cpx_pair_sn, cpx_flat/
+qed.
+
+lemma cpxs_pair_sn: ∀h,g,I,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
+ ∀T. ⦃G, L⦄ ⊢ ②{I}V1.T ➡*[h, g] ②{I}V2.T.
+#h #g #I #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
+/3 width=3 by cpxs_strap1, cpx_pair_sn/
qed.
lemma cpxs_zeta: ∀h,g,G,L,V,T1,T,T2. ⇧[0, 1] T2 ≡ T →
⦃G, L.ⓓV⦄ ⊢ T1 ➡*[h, g] T → ⦃G, L⦄ ⊢ +ⓓV.T1 ➡*[h, g] T2.
-#h #g #G #L #V #T1 #T #T2 #HT2 #H @(TC_ind_dx … T1 H) -T1 /3 width=3/
+#h #g #G #L #V #T1 #T #T2 #HT2 #H @(cpxs_ind_dx … H) -T1
+/3 width=3 by cpxs_strap2, cpx_cpxs, cpx_bind, cpx_zeta/
qed.
lemma cpxs_tau: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
∀V. ⦃G, L⦄ ⊢ ⓝV.T1 ➡*[h, g] T2.
-#h #g #G #L #T1 #T2 #H elim H -T2 /2 width=3/ /3 width=1/
+#h #g #G #L #T1 #T2 #H @(cpxs_ind … H) -T2
+/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_tau/
qed.
lemma cpxs_ti: ∀h,g,G,L,V1,V2. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
∀T. ⦃G, L⦄ ⊢ ⓝV1.T ➡*[h, g] V2.
-#h #g #G #L #V1 #V2 #H elim H -V2 /2 width=3/ /3 width=1/
+#h #g #G #L #V1 #V2 #H @(cpxs_ind … H) -V2
+/3 width=3 by cpxs_strap1, cpx_cpxs, cpx_ti/
qed.
lemma cpxs_beta_dx: ∀h,g,a,G,L,V1,V2,W1,W2,T1,T2.
⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡[h, g] W2 →
⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, g] ⓓ{a}ⓝW2.V2.T2.
-#h #g #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2 /3 width=1/
-/4 width=7 by cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_beta/ (**) (* auto too slow without trace *)
+#h #g #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 * -T2
+/4 width=7 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_beta/
qed.
lemma cpxs_theta_dx: ∀h,g,a,G,L,V1,V,V2,W1,W2,T1,T2.
⦃G, L⦄ ⊢ V1 ➡[h, g] V → ⇧[0, 1] V ≡ V2 → ⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 →
⦃G, L⦄ ⊢ W1 ➡[h, g] W2 → ⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, g] ⓓ{a}W2.ⓐV2.T2.
-#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2 [ /3 width=3/ ]
-/4 width=9 by cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_theta/ (**) (* auto too slow without trace *)
+#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 * -T2
+/4 width=9 by cpx_cpxs, cpxs_strap1, cpxs_bind_dx, cpxs_flat_dx, cpx_theta/
qed.
(* Basic inversion lemmas ***************************************************)
@(ex2_2_intro … 0 … Hkl) -Hkl //
| #U #U2 #_ #HU2 * #n #l #Hknl #H destruct
elim (cpx_inv_sort1 … HU2) -HU2
- [ #H destruct /2 width=4/
+ [ #H destruct /2 width=4 by ex2_2_intro/
| * #l0 #Hkl0 #H destruct -l
@(ex2_2_intro … (n+1) l0) /2 width=1 by deg_inv_prec/ >iter_SO //
]
∨∨ ∃∃W2,T2. ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 & ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 & U2 = ⓝW2.T2
| ⦃G, L⦄ ⊢ T1 ➡*[h, g] U2
| ⦃G, L⦄ ⊢ W1 ➡*[h, g] U2.
-#h #g #G #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5/
-#U2 #U #_ #HU2 * /3 width=3/ *
+#h #g #G #L #W1 #T1 #U2 #H @(cpxs_ind … H) -U2 /3 width=5 by or3_intro0, ex3_2_intro/
+#U2 #U #_ #HU2 * /3 width=3 by cpxs_strap1, or3_intro1, or3_intro2/ *
#W #T #HW1 #HT1 #H destruct
-elim (cpx_inv_cast1 … HU2) -HU2 /3 width=3/ *
+elim (cpx_inv_cast1 … HU2) -HU2 /3 width=3 by cpxs_strap1, or3_intro1, or3_intro2/ *
#W2 #T2 #HW2 #HT2 #H destruct
lapply (cpxs_strap1 … HW1 … HW2) -W
-lapply (cpxs_strap1 … HT1 … HT2) -T /3 width=5/
+lapply (cpxs_strap1 … HT1 … HT2) -T /3 width=5 by or3_intro0, ex3_2_intro/
qed-.
lemma cpxs_inv_cnx1: ∀h,g,G,L,T,U. ⦃G, L⦄ ⊢ T ➡*[h, g] U → ⦃G, L⦄ ⊢ 𝐍[h, g]⦃T⦄ → T = U.
#h #g #G #L #T #U #H @(cpxs_ind_dx … H) -T //
#T0 #T #H1T0 #_ #IHT #H2T0
-lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1/
+lapply (H2T0 … H1T0) -H1T0 #H destruct /2 width=1 by/
qed-.
lemma cpxs_neq_inv_step_sn: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) →
(* Main properties **********************************************************)
theorem cpxs_trans: ∀h,g,G,L. Transitive … (cpxs h g G L).
-#h #g #G #L #T1 #T #HT1 #T2 @trans_TC @HT1 qed-. (**) (* auto /3 width=3/ does not work because a δ-expansion gets in the way *)
+#h #g #G #L #T1 #T #HT1 #T2
+@trans_TC @HT1 qed-. (**) (* auto /3 width=3/ does not work because a δ-expansion gets in the way *)
theorem cpxs_bind: ∀h,g,a,I,G,L,V1,V2,T1,T2. ⦃G, L.ⓑ{I}V1⦄ ⊢ T1 ➡*[h, g] T2 →
⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
-#h #g #a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2 /2 width=1/
-#V #V2 #_ #HV2 #IHV1
-@(cpxs_trans … IHV1) -V1 /2 width=1/
+#h #g #a #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
+/3 width=5 by cpxs_trans, cpxs_bind_dx/
qed.
theorem cpxs_flat: ∀h,g,I,G,L,V1,V2,T1,T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 →
⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
⦃G, L⦄ ⊢ ⓕ{I}V1.T1 ➡*[h, g] ⓕ{I}V2.T2.
-#h #g #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2 /2 width=1/
-#V #V2 #_ #HV2 #IHV1
-@(cpxs_trans … IHV1) -IHV1 /2 width=1/
+#h #g #I #G #L #V1 #V2 #T1 #T2 #HT12 #H @(cpxs_ind … H) -V2
+/3 width=5 by cpxs_trans, cpxs_flat_dx/
qed.
theorem cpxs_beta_rc: ∀h,g,a,G,L,V1,V2,W1,W2,T1,T2.
⦃G, L⦄ ⊢ V1 ➡[h, g] V2 → ⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 →
⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, g] ⓓ{a}ⓝW2.V2.T2.
-#h #g #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cpxs_ind … H) -W2 /2 width=1/
-#W #W2 #_ #HW2 #IHW1
-@(cpxs_trans … IHW1) -IHW1 /3 width=1/
+#h #g #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HV12 #HT12 #H @(cpxs_ind … H) -W2
+/4 width=5 by cpxs_trans, cpxs_beta_dx, cpxs_bind_dx, cpx_pair_sn/
qed.
theorem cpxs_beta: ∀h,g,a,G,L,V1,V2,W1,W2,T1,T2.
⦃G, L.ⓛW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 → ⦃G, L⦄ ⊢ V1 ➡*[h, g] V2 →
⦃G, L⦄ ⊢ ⓐV1.ⓛ{a}W1.T1 ➡*[h, g] ⓓ{a}ⓝW2.V2.T2.
-#h #g #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cpxs_ind … H) -V2 /2 width=1/
-#V #V2 #_ #HV2 #IHV1
-@(cpxs_trans … IHV1) -IHV1 /3 width=1/
+#h #g #a #G #L #V1 #V2 #W1 #W2 #T1 #T2 #HT12 #HW12 #H @(cpxs_ind … H) -V2
+/4 width=5 by cpxs_trans, cpxs_beta_rc, cpxs_bind_dx, cpx_flat/
qed.
theorem cpxs_theta_rc: ∀h,g,a,G,L,V1,V,V2,W1,W2,T1,T2.
⦃G, L⦄ ⊢ V1 ➡[h, g] V → ⇧[0, 1] V ≡ V2 →
⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 →
⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, g] ⓓ{a}W2.ⓐV2.T2.
-#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H elim H -W2 /2 width=3/
-#W #W2 #_ #HW2 #IHW1
-@(cpxs_trans … IHW1) -IHW1 /2 width=1/
+#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV1 #HV2 #HT12 #H @(cpxs_ind … H) -W2
+/3 width=5 by cpxs_trans, cpxs_theta_dx, cpxs_bind_dx/
qed.
theorem cpxs_theta: ∀h,g,a,G,L,V1,V,V2,W1,W2,T1,T2.
⇧[0, 1] V ≡ V2 → ⦃G, L⦄ ⊢ W1 ➡*[h, g] W2 →
⦃G, L.ⓓW1⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ V1 ➡*[h, g] V →
⦃G, L⦄ ⊢ ⓐV1.ⓓ{a}W1.T1 ➡*[h, g] ⓓ{a}W2.ⓐV2.T2.
-#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1 /2 width=3/
-#V1 #V0 #HV10 #_ #IHV0
-@(cpxs_trans … IHV0) -IHV0 /2 width=1/
+#h #g #a #G #L #V1 #V #V2 #W1 #W2 #T1 #T2 #HV2 #HW12 #HT12 #H @(TC_ind_dx … V1 H) -V1
+/3 width=5 by cpxs_trans, cpxs_theta_rc, cpxs_flat_dx/
qed.
(* Advanced inversion lemmas ************************************************)
| ∃∃a,W,T. ⦃G, L⦄ ⊢ T1 ➡*[h, g] ⓛ{a}W.T & ⦃G, L⦄ ⊢ ⓓ{a}ⓝW.V1.T ➡*[h, g] U2
| ∃∃a,V0,V2,V,T. ⦃G, L⦄ ⊢ V1 ➡*[h, g] V0 & ⇧[0,1] V0 ≡ V2 &
⦃G, L⦄ ⊢ T1 ➡*[h, g] ⓓ{a}V.T & ⦃G, L⦄ ⊢ ⓓ{a}V.ⓐV2.T ➡*[h, g] U2.
-#h #g #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 [ /3 width=5/ ]
+#h #g #G #L #V1 #T1 #U2 #H @(cpxs_ind … H) -U2 [ /3 width=5 by or3_intro0, ex3_2_intro/ ]
#U #U2 #_ #HU2 * *
[ #V0 #T0 #HV10 #HT10 #H destruct
elim (cpx_inv_appl1 … HU2) -HU2 *
- [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/
+ [ #V2 #T2 #HV02 #HT02 #H destruct /4 width=5 by cpxs_strap1, or3_intro0, ex3_2_intro/
| #a #V2 #W #W2 #T #T2 #HV02 #HW2 #HT2 #H1 #H2 destruct
lapply (cpxs_strap1 … HV10 … HV02) -V0 #HV12
- lapply (lsubr_cpx_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2 /2 width=1/ #HT2
- @or3_intro1 @(ex2_3_intro … HT10) -HT10 /3 width=1/ (**) (* explicit constructor. /5 width=8/ is too slow because TC_transitive gets in the way *)
+ lapply (lsubr_cpx_trans … HT2 (L.ⓓⓝW.V1) ?) -HT2
+ /5 width=5 by cpxs_bind, cpxs_flat_dx, cpx_cpxs, lsubr_abst, ex2_3_intro, or3_intro1/
| #a #V #V2 #W0 #W2 #T #T2 #HV0 #HV2 #HW02 #HT2 #H1 #H2 destruct
- @or3_intro2 @(ex4_5_intro … HV2 HT10) /2 width=3/ /3 width=1/ (**) (* explicit constructor. /5 width=8/ is too slow because TC_transitive gets in the way *)
+ /5 width=10 by cpxs_flat_sn, cpxs_bind_dx, cpxs_strap1, ex4_5_intro, or3_intro2/
]
-| /4 width=9/
-| /4 width=11/
+| /4 width=9 by cpxs_strap1, or3_intro1, ex2_3_intro/
+| /4 width=11 by cpxs_strap1, or3_intro2, ex4_5_intro/
]
qed-.
lemma lpx_cpx_trans: ∀h,g,G. s_r_trans … (cpx h g G) (lpx h g G).
#h #g #G #L2 #T1 #T2 #HT12 elim HT12 -G -L2 -T1 -T2
-[ /2 width=3/
-| /3 width=2/
+[ /2 width=3 by /
+| /3 width=2 by cpx_cpxs, cpx_sort/
| #I #G #L2 #K2 #V0 #V2 #W2 #i #HLK2 #_ #HVW2 #IHV02 #L1 #HL12
elim (lpx_ldrop_trans_O1 … HL12 … HLK2) -L2 #X #HLK1 #H
elim (lpx_inv_pair2 … H) -H #K1 #V1 #HK12 #HV10 #H destruct
lapply (IHV02 … HK12) -K2 #HV02
- lapply (cpxs_strap2 … HV10 … HV02) -V0 /2 width=7/
+ lapply (cpxs_strap2 … HV10 … HV02) -V0 /2 width=7 by cpxs_delta/
| #a #I #G #L2 #V1 #V2 #T1 #T2 #_ #_ #IHV12 #IHT12 #L1 #HL12
- lapply (IHT12 (L1.ⓑ{I}V1) ?) -IHT12 /2 width=1/ /3 width=1/
-|5,7,8: /3 width=1/
+ lapply (IHT12 (L1.ⓑ{I}V1) ?) -IHT12 /3 width=1 by cpxs_bind, lpx_pair/
+|5,7,8: /3 width=1 by cpxs_flat, cpxs_ti, cpxs_tau/
| #G #L2 #V2 #T1 #T #T2 #_ #HT2 #IHT1 #L1 #HL12
- lapply (IHT1 (L1.ⓓV2) ?) -IHT1 /2 width=1/ /2 width=3/
+ lapply (IHT1 (L1.ⓓV2) ?) -IHT1 /2 width=3 by cpxs_zeta, lpx_pair/
| #a #G #L2 #V1 #V2 #W1 #W2 #T1 #T2 #_ #_ #_ #IHV12 #IHW12 #IHT12 #L1 #HL12
- lapply (IHT12 (L1.ⓛW1) ?) -IHT12 /2 width=1/ /3 width=1/
+ lapply (IHT12 (L1.ⓛW1) ?) -IHT12 /3 width=1 by cpxs_beta, lpx_pair/
| #a #G #L2 #V1 #V #V2 #W1 #W2 #T1 #T2 #_ #HV2 #_ #_ #IHV1 #IHW12 #IHT12 #L1 #HL12
- lapply (IHT12 (L1.ⓓW1) ?) -IHT12 /2 width=1/ /3 width=3/
+ lapply (IHT12 (L1.ⓓW1) ?) -IHT12 /3 width=3 by cpxs_theta, cpxs_strap1, lpx_pair/
]
qed-.
∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡[h, g] T2 →
∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
#h #g #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
-lapply (lpx_cpx_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
+lapply (lpx_cpx_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1 by cpxs_bind_dx, lpx_pair/
qed.
(* Advanced properties ******************************************************)
∀I,T1,T2. ⦃G, L.ⓑ{I}V2⦄ ⊢ T1 ➡*[h, g] T2 →
∀a. ⦃G, L⦄ ⊢ ⓑ{a,I}V1.T1 ➡*[h, g] ⓑ{a,I}V2.T2.
#h #g #G #L #V1 #V2 #HV12 #I #T1 #T2 #HT12
-lapply (lpx_cpxs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1/
+lapply (lpx_cpxs_trans … HT12 (L.ⓑ{I}V1) ?) /2 width=1 by cpxs_bind_dx, lpx_pair/
qed.
+
+(* Properties on supclosure *************************************************)
+
+lemma fqu_cpxs_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊃ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G1 -G2 -L1 -L2 -T1 -T2
+[ #I #G #L #V1 #V2 #HV12 #_ elim (lift_total V2 0 1)
+ #U2 #HVU2 @(ex3_intro … U2)
+ [1,3: /3 width=7 by fqu_drop, cpxs_delta, ldrop_pair, ldrop_ldrop/
+ | #H destruct /2 width=7 by lift_inv_lref2_be/
+ ]
+| #I #G #L #V1 #T #V2 #HV12 #H @(ex3_intro … (②{I}V2.T))
+ [1,3: /2 width=4 by fqu_pair_sn, cpxs_pair_sn/
+ | #H0 destruct /2 width=1 by/
+ ]
+| #a #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓑ{a,I}V.T2))
+ [1,3: /2 width=4 by fqu_bind_dx, cpxs_bind/
+ | #H0 destruct /2 width=1 by/
+ ]
+| #I #G #L #V #T1 #T2 #HT12 #H @(ex3_intro … (ⓕ{I}V.T2))
+ [1,3: /2 width=4 by fqu_flat_dx, cpxs_flat/
+ | #H0 destruct /2 width=1 by/
+ ]
+| #G #L #K #T1 #U1 #e #HLK #HTU1 #T2 #HT12 #H elim (lift_total T2 0 (e+1))
+ #U2 #HTU2 @(ex3_intro … U2)
+ [1,3: /2 width=9 by cpxs_lift, fqu_drop/
+ | #H0 destruct /3 width=5 by lift_inj/
+]
+qed-.
+
+lemma fquq_cpxs_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃⸮ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊃⸮ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fquq_inv_gen … H12) -H12
+[ #H12 elim (fqu_cpxs_trans_neq … H12 … HTU2 H) -T2
+ /3 width=4 by fqu_fquq, ex3_intro/
+| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
+]
+qed-.
+
+lemma fqup_cpxs_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊃+ ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H @(fqup_ind_dx … H) -G1 -L1 -T1
+[ #G1 #L1 #T1 #H12 #U2 #HTU2 #H elim (fqu_cpxs_trans_neq … H12 … HTU2 H) -T2
+ /3 width=4 by fqu_fqup, ex3_intro/
+| #G #G1 #L #L1 #T #T1 #H1 #_ #IH12 #U2 #HTU2 #H elim (IH12 … HTU2 H) -T2
+ #U1 #HTU1 #H #H12 elim (fqu_cpxs_trans_neq … H1 … HTU1 H) -T1
+ /3 width=8 by fqup_strap2, ex3_intro/
+]
+qed-.
+
+lemma fqus_cpxs_trans_neq: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ →
+ ∀U2. ⦃G2, L2⦄ ⊢ T2 ➡*[h, g] U2 → (T2 = U2 → ⊥) →
+ ∃∃U1. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] U1 & T1 = U1 → ⊥ & ⦃G1, L1, U1⦄ ⊃* ⦃G2, L2, U2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H12 #U2 #HTU2 #H elim (fqus_inv_gen … H12) -H12
+[ #H12 elim (fqup_cpxs_trans_neq … H12 … HTU2 H) -T2
+ /3 width=4 by fqup_fqus, ex3_intro/
+| * #HG #HL #HT destruct /3 width=4 by ex3_intro/
+]
+qed-.
(∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
) →
∀T. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T → R T.
-#h #g #G #L #R #H0 #T1 #H elim H -T1 #T1 #HT1 #IHT1
-@H0 -H0 /3 width=1/ -IHT1 /4 width=1/
+#h #g #G #L #R #H0 #T1 #H elim H -T1 /5 width=1 by SN_intro/
qed-.
(* Basic properties *********************************************************)
+lemma csx_intro_cprs: ∀h,g,G,L,T1.
+ (∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊*[h, g] T2) →
+ ⦃G, L⦄ ⊢ ⬊*[h, g] T1.
+/4 width=1 by cpx_cpxs, csx_intro/ qed.
+
(* Basic_1: was just: sn3_intro *)
lemma csxa_intro: ∀h,g,G,L,T1.
(∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2) →
⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
-/4 width=1/ qed.
+/4 width=1 by SN_intro/ qed.
fact csxa_intro_aux: ∀h,g,G,L,T1. (
∀T,T2. ⦃G, L⦄ ⊢ T ➡*[h, g] T2 → T1 = T → (T1 = T2 → ⊥) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T2
) → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1.
-/4 width=3/ qed-.
+/4 width=3 by csxa_intro/ qed-.
(* Basic_1: was just: sn3_pr3_trans (old version) *)
lemma csxa_cpxs_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T1 →
#h #g #G #L #T1 #H elim H -T1 #T1 #HT1 #IHT1 #T2 #HLT12
@csxa_intro #T #HLT2 #HT2
elim (eq_term_dec T1 T2) #HT12
-[ -IHT1 -HLT12 destruct /3 width=1/
-| -HT1 -HT2 /3 width=4/
+[ -IHT1 -HLT12 destruct /3 width=1 by/
+| -HT1 -HT2 /3 width=4 by/
qed.
(* Basic_1: was just: sn3_pr2_intro (old version) *)
elim H //
| #T0 #T #HLT1 #HLT2 #IHT #HT10 #HT12 destruct
elim (eq_term_dec T0 T) #HT0
- [ -HLT1 -HLT2 -H /3 width=1/
- | -IHT -HT12 /4 width=3/
+ [ -HLT1 -HLT2 -H /3 width=1 by/
+ | -IHT -HT12 /4 width=3 by csxa_cpxs_trans/
]
]
qed.
(* Main properties **********************************************************)
theorem csx_csxa: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T.
-#h #g #G #L #T #H @(csx_ind … H) -T /4 width=1/
+#h #g #G #L #T #H @(csx_ind … H) -T /4 width=1 by csxa_intro_cpx/
qed.
theorem csxa_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⬊⬊*[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #T #H @(csxa_ind … H) -T /4 width=1/
+#h #g #G #L #T #H @(csxa_ind … H) -T /4 width=1 by cpx_cpxs, csx_intro/
qed.
(* Basic_1: was just: sn3_pr3_trans *)
lemma csx_cpxs_trans: ∀h,g,G,L,T1. ⦃G, L⦄ ⊢ ⬊*[h, g] T1 →
∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → ⦃G, L⦄ ⊢ ⬊*[h, g] T2.
-#h #g #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2 // /2 width=3 by csx_cpx_trans/
+#h #g #G #L #T1 #HT1 #T2 #H @(cpxs_ind … H) -T2 /2 width=3 by csx_cpx_trans/
qed-.
(* Main eliminators *********************************************************)
(∀T2. ⦃G, L⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → R T2) → R T1
) →
∀T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R T.
-#h #g #G #L #R #H0 #T1 #H @(csxa_ind … (csx_csxa … H)) -T1 #T1 #HT1 #IHT1
-@H0 -H0 /2 width=1/ -HT1 /3 width=1/
+#h #g #G #L #R #H0 #T1 #H @(csxa_ind … (csx_csxa … H)) -T1 /4 width=1 by csxa_csx/
qed-.
--- /dev/null
+include "basic_2/computation/fpbc.ma".
+
+(* Advanced eliminators *****************************************************)
+
+fact csx_ind_fpbc_aux: ∀h,g. ∀R:relation3 genv lenv term.
+ (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
+ (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ R G1 L1 T1
+ ) →
+ ∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
+ ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ → R G2 L2 T2.
+#h #g #R #IH1 #G1 #L1 #T1 #H @(csx_ind … H) -T1
+#T1 @(fqup_wf_ind … G1 L1 T1) -G1 -L1 -T1
+#G1 #L1 #T1 #IH2 #H1 #IH3 #G2 #L2 #T2 #H12 @IH1 -IH1 /2 width=5 by csx_fqus_conf/
+#G #L #T *
+[ #G0 #L0 #T0 #H20 lapply (fqus_fqup_trans … H12 H20) -G2 -L2 -T2
+ #H10 @(IH2 … H10) -IH2 /2 width=5 by csx_fqup_conf/
+ #T2 #HT02 #H #G3 #L3 #T3 #HT23 elim (fqup_cpx_trans_neq … H10 … HT02 H) -T0
+ /4 width=8 by fqup_fqus_trans, fqup_fqus/
+| #T0 #HT20 #H elim (fqus_cpxs_trans_neq … H12 … HT20 H) -T2 /3 width=4 by/
+]
+qed-.
+
+lemma csx_ind_fpbc: ∀h,g. ∀R:relation3 genv lenv term.
+ (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
+ (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
+ R G1 L1 T1
+ ) →
+ ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R G L T.
+/4 width=8 by csx_ind_fpbc_fqus/ qed-.
(**************************************************************************)
include "basic_2/reduction/cnx_lift.ma".
-include "basic_2/reduction/fpbc.ma".
include "basic_2/computation/acp.ma".
include "basic_2/computation/csx.ma".
]
qed-.
-(* Advanced eliminators *****************************************************)
-
-lemma csx_ind_fpbc_fqus: ∀h,g. ∀R:relation3 genv lenv term.
- (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
- R G1 L1 T1
- ) →
- ∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
- ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃* ⦃G2, L2, T2⦄ → R G2 L2 T2.
-#h #g #R #IH1 #G1 #L1 #T1 #H @(csx_ind … H) -T1
-#T1 @(fqup_wf_ind … G1 L1 T1) -G1 -L1 -T1
-#G1 #L1 #T1 #IH2 #H1 #IH3 #G2 #L2 #T2 #H12 @IH1 -IH1 /2 width=5 by csx_fqus_conf/
-#G #L #T *
-[ #G0 #L0 #T0 #H20 lapply (fqus_strap1_fqu … H12 H20) -G2 -L2 -T2
- #H10 @(IH2 … H10) -IH2 /2 width=5 by csx_fqup_conf/
- #T2 #HT02 #H #G3 #L3 #T3 #HT23 elim (fqup_cpx_trans_neq … H10 … HT02 H) -T0
- /4 width=8 by fqup_fqus_trans, fqup_fqus/
-| #T0 #HT20 #H elim (fqus_cpx_trans_neq … H12 … HT20 H) -T2 /3 width=4 by/
-]
-qed-.
-
-lemma csx_ind_fpbc: ∀h,g. ∀R:relation3 genv lenv term.
- (∀G1,L1,T1. ⦃G1, L1⦄ ⊢ ⬊*[h, g] T1 →
- (∀G2,L2,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ → R G2 L2 T2) →
- R G1 L1 T1
- ) →
- ∀G,L,T. ⦃G, L⦄ ⊢ ⬊*[h, g] T → R G L T.
-/4 width=8 by csx_ind_fpbc_fqus/ qed-.
-
(* Main properties **********************************************************)
theorem csx_acp: ∀h,g. acp (cpx h g) (eq …) (csx h g).
+++ /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/notation/relations/lazyeq_8.ma".
-include "basic_2/computation/lpxs.ma".
-
-(* EQUIVALENT "BIG TREE" NORMAL FORMS ***************************************)
-
-(* Note: this definition works but is not symmetric nor decidable *)
-definition fleq: ∀h. sd h → tri_relation genv lenv term ≝
- λh,g,G1,L1,T1,G2,L2,T2.
- ∧∧ G1 = G2 & ⦃G1, L1⦄ ⊢ ➡*[h, g] L2 & T1 = T2.
-
-interpretation
- "equivalent 'big tree' normal forms (closure)"
- 'LazyEq h g G1 L1 T1 G2 L2 T2 = (fleq h g G1 L1 T1 G2 L2 T2).
-
-(* Basic_properties *********************************************************)
-
-lemma fleq_refl: ∀h,g. tri_reflexive … (fleq h g).
-/2 width=1 by and3_intro/ 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/notation/relations/btpredproper_8.ma".
+include "basic_2/relocation/lleq.ma".
+include "basic_2/computation/fpbs.ma".
+
+(* ATOMIC "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************)
+
+inductive fpbc (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
+| fpbc_fqup: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃+ ⦃G2, L2, T2⦄ → fpbc h g G1 L1 T1 G2 L2 T2
+| fpbc_cpxs: ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T2 → (T1 = T2 → ⊥) → fpbc h g G1 L1 T1 G1 L1 T2
+| fpbc_lpxs: ∀L2. ⦃G1, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ⋕[0, T1] L2 → ⊥) → fpbc h g G1 L1 T1 G1 L2 T1
+.
+
+interpretation
+ "'big tree' proper parallel reduction (closure)"
+ 'BTPRedProper h g G1 L1 T1 G2 L2 T2 = (fpbc h g G1 L1 T1 G2 L2 T2).
+
+(* Basic properties *********************************************************)
+
+lemma cprs_fpbc: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡* T2 → (T1 = T2 → ⊥) →
+ ⦃G, L, T1⦄ ≻[h, g] ⦃G, L, T2⦄.
+/3 width=1 by fpbc_cpxs, cprs_cpxs/ qed.
+
+lemma lprs_fpbc: ∀h,g,G,L1,L2,T. ⦃G, L1⦄ ⊢ ➡* L2 → (L1 ⋕[0, T] L2 → ⊥) →
+ ⦃G, L1, T⦄ ≻[h, g] ⦃G, L2, T⦄.
+/3 width=1 by fpbc_lpxs, lprs_lpxs/ qed.
+
+(* Basic forward lemmas *****************************************************)
+
+lemma fpbc_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
+/3 width=1 by lpxs_fpbs, cpxs_fpbs, fqup_fpbs/
+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/computation/lpxs_lleq.ma".
+include "basic_2/computation/fpns.ma".
+include "basic_2/computation/fpbs_alt.ma".
+include "basic_2/computation/fpbc.ma".
+
+(* ATOMIC "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************)
+
+(* Forward lemmas on parallel computation for "big tree" normal forms *******)
+
+lemma fpbs_fwd_fpbc_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
+ ⦃G1, L1, T1⦄ ⊢ ⋕➡*[h, g] ⦃G2, L2, T2⦄ ∨
+ ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim(fpbs_fpbsa … H) -H
+#L #T #HT1 #HT2 #HL2 elim (eq_term_dec T1 T) #H destruct
+[ -HT1 elim (fqus_inv_gen … HT2) -HT2
+ [ #HT2 elim (fqup_inv_step_sn … HT2) -HT2
+ #H6 #H7 #H8 #H9 #H10 @or_intror @(ex2_3_intro … H6 H7 H8)
+ /5 width=9 by fpbsa_inv_fpbs, fpbc_fqup, fqu_fqup, ex3_2_intro, ex2_3_intro, or_intror/
+ | * #HG #HL #HT destruct elim (lleq_dec T2 L L2 0) #H
+ [ /3 width=1 by fpns_intro, or_introl/
+ | elim (lpxs_nlleq_inv_step_sn … HL2 H) -HL2 -H
+ /5 width=5 by fpbc_lpxs, lpxs_fpbs, ex2_3_intro, or_intror/
+ ]
+ ]
+| elim (cpxs_neq_inv_step_sn … HT1 H) -HT1 -H
+ /5 width=9 by fpbsa_inv_fpbs, fpbc_cpxs, cpx_cpxs, ex3_2_intro, ex2_3_intro, or_intror/
+]
+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/static/ssta_ssta.ma".
+include "basic_2/computation/cpxs_lift.ma".
+include "basic_2/computation/fpbc.ma".
+
+(* ATOMIC "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES ***************)
+
+(* Advanced properties ******************************************************)
+
+lemma lsstas_fpbc: ∀h,g,G,L,T1,T2,l2. ⦃G, L⦄ ⊢ T1 •*[h, g, l2] T2 → (T1 = T2 → ⊥) →
+ ∀l1. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ⦃G, L, T1⦄ ≻[h, g] ⦃G, L, T2⦄.
+/4 width=5 by fpbc_cpxs, lsstas_cpxs/ qed.
+
+lemma ssta_fpbc: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 →
+ ⦃G, L⦄ ⊢ T1 •[h, g] T2 → ⦃G, L, T1⦄ ≻[h, g] ⦃G, L, T2⦄.
+#h #g #G #L #T1 #T2 #l #HT1 #HT12 elim (eq_term_dec T1 T2)
+/3 width=5 by ssta_lsstas, lsstas_fpbc/ #H destruct
+elim (ssta_inv_refl_pos … HT1 … HT12)
+qed.
(**************************************************************************)
include "basic_2/notation/relations/btpredstarproper_8.ma".
-include "basic_2/reduction/fpbc.ma".
-include "basic_2/computation/fpbs.ma".
+include "basic_2/computation/fpbc.ma".
(* GENEARAL "BIG TREE" PROPER PARALLEL COMPUTATION FOR CLOSURES *************)
fpbg h g G1 L1 T1 G1 L2 T2
| fpbg_fqup: ∀G2,L,L2,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T → ⦃G1, L1, T⦄ ⊃+ ⦃G2, L, T2⦄ → ⦃G2, L⦄ ⊢ ➡*[h, g] L2 →
fpbg h g G1 L1 T1 G2 L2 T2
+| fpbg_lpxs: ∀G2,L,L0,L2,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T → ⦃G1, L1, T⦄ ⊃* ⦃G2, L, T2⦄ → ⦃G2, L⦄ ⊢ ➡*[h, g] L0 →
+ (L ⋕[0, T2] L0 → ⊥) → ⦃G2, L0⦄ ⊢ ➡*[h, g] L2 → L0 ⋕[0, T2] L2 →
+ fpbg h g G1 L1 T1 G2 L2 T2
.
interpretation "'big tree' proper parallel computation (closure)"
lemma fpbc_fpbg: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-/3 width=5 by fpbg_cpxs, fpbg_fqup, fqu_fqup, cpx_cpxs/
+/3 width=9 by fpbg_fqup, fpbg_cpxs, fpbg_lpxs/
qed.
(* *)
(**************************************************************************)
+include "basic_2/computation/lpxs_lpxs.ma".
include "basic_2/computation/fpbs_alt.ma".
include "basic_2/computation/fpbg.ma".
lemma fpbg_fwd_fpbs: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄ →
⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G2 -L2 -T2
-/3 width=5 by cpxs_fqup_fpbs, fpbs_trans, lpxs_fpbs, cpxs_fpbs/
+/3 width=5 by cpxs_fqus_lpxs_fpbs, cpxs_fqup_fpbs, fpbs_trans, lpxs_fpbs, cpxs_fpbs/
qed-.
-lemma fpbg_fwd_fpb_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄ →
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+lemma fpbg_fwd_fpbc_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ >[h, g] ⦃G2, L2, T2⦄ →
+ ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim H -G2 -L2 -T2
-[ #L2 #T2 #HT12 #H #HL12
- elim (cpxs_neq_inv_step_sn … HT12 H) -HT12 -H
- /4 width=9 by fpbsa_inv_fpbs, fpbc_cpx, ex3_2_intro, ex2_3_intro/
+[ /4 width=5 by fpbc_cpxs, lpxs_fpbs, ex2_3_intro/
| #G2 #L #L2 #T #T2 #HT1 #HT2 #HL2 elim (eq_term_dec T1 T) #H destruct
- [ -HT1 elim (fqup_inv_step_sn … HT2) -HT2
- /4 width=9 by fpbsa_inv_fpbs, fpbc_fqu, ex3_2_intro, ex2_3_intro/
- | elim (cpxs_neq_inv_step_sn … HT1 H) -HT1 -H
- /5 width=9 by fpbsa_inv_fpbs, fpbc_cpx, fqup_fqus, ex3_2_intro, ex2_3_intro/
+ [ -HT1 /3 width=5 by fpbc_fqup, lpxs_fpbs, ex2_3_intro/
+ | /5 width=9 by fpbc_cpxs, fpbsa_inv_fpbs, fqup_fqus, ex3_2_intro, ex2_3_intro/
+ ]
+| #G2 #L #L0 #L2 #T #T2 #HT1 #HT2 #HL0 #H0 #HL02 #H02
+ lapply (lpxs_trans … HL0 … HL02) #HL2
+ elim (eq_term_dec T1 T) #H destruct
+ [ -HT1 elim (fqus_inv_gen … HT2) -HT2
+ [ /3 width=5 by fpbc_fqup, lpxs_fpbs, ex2_3_intro/
+ | * #H1 #H2 #H3 destruct
+ /4 width=5 by fpbc_lpxs, lpxs_fpbs, ex2_3_intro/
+ ]
+ | /4 width=9 by fpbc_cpxs, fpbsa_inv_fpbs, ex3_2_intro, ex2_3_intro/
]
]
qed-.
(* *)
(**************************************************************************)
-include "basic_2/computation/cpxs_lift.ma".
+include "basic_2/computation/fpbc_lift.ma".
include "basic_2/computation/fpbg.ma".
(* GENERAL "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *********************)
lemma lsstas_fpbg: ∀h,g,G,L,T1,T2,l2. ⦃G, L⦄ ⊢ T1 •*[h, g, l2] T2 → (T1 = T2 → ⊥) →
∀l1. l2 ≤ l1 → ⦃G, L⦄ ⊢ T1 ▪[h, g] l1 → ⦃G, L, T1⦄ >[h, g] ⦃G, L, T2⦄.
-/4 width=5 by fpbg_cpxs, lsstas_cpxs/ qed.
+/4 width=5 by fpbc_fpbg, lsstas_fpbc/ qed.
+
+lemma ssta_fpbg: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 →
+ ⦃G, L⦄ ⊢ T1 •[h, g] T2 → ⦃G, L, T1⦄ >[h, g] ⦃G, L, T2⦄.
+/3 width=2 by fpbc_fpbg, ssta_fpbc/ 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/reduction/fpbc.ma".
-include "basic_2/computation/fleq.ma".
-include "basic_2/computation/fpbs_alt.ma".
-
-(* "BIG TREE" PARALLEL COMPUTATION FOR CLOSURES *****************************)
-
-(* Forward lemmas on equivalent "big tree" normal forms *********************)
-
-lemma fpbs_fwd_fpb_sn: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ⋕[h, g] ⦃G2, L2, T2⦄ ∨
- ∃∃G,L,T. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G, L, T⦄ & ⦃G, L, T⦄ ≥[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 #H elim(fpbs_fpbsa … H) -H
-#L #T #HT1 #HT2 #HL2 elim (eq_term_dec T1 T) #H destruct
-[ -HT1 elim (fqus_inv_gen … HT2) -HT2
- [ #HT2 elim (fqup_inv_step_sn … HT2) -HT2
- /5 width=9 by fpbsa_inv_fpbs, fpbc_fqu, ex3_2_intro, ex2_3_intro, or_intror/
- | * #HG #HL #HT destruct /3 width=1 by and3_intro, or_introl/
- ]
-| elim (cpxs_neq_inv_step_sn … HT1 H) -HT1 -H
- /5 width=9 by fpbsa_inv_fpbs, fpbc_cpx, ex3_2_intro, ex2_3_intro, or_intror/
-]
-qed-.
lemma cpxs_fqup_fpbs: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
⦃G1, L1, T⦄ ⊃+ ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
/3 width=5 by fpbs_trans, cpxs_fpbs, fqup_fpbs/ qed.
+
+lemma cpxs_fqus_fpbs: ∀h,g,G1,G2,L1,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
+ ⦃G1, L1, T⦄ ⊃* ⦃G2, L2, T2⦄ → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_trans, cpxs_fpbs, fqus_fpbs/ qed.
+
+lemma cpxs_fqus_lpxs_fpbs: ∀h,g,G1,G2,L1,L,L2,T1,T,T2. ⦃G1, L1⦄ ⊢ T1 ➡*[h, g] T →
+ ⦃G1, L1, T⦄ ⊃* ⦃G2, L, T2⦄ → ⦃G2, L⦄ ⊢ ➡*[h, g] L2 → ⦃G1, L1, T1⦄ ≥[h, g] ⦃G2, L2, T2⦄.
+/3 width=5 by fpbs_trans, cpxs_fqus_fpbs, lpxs_fpbs/ qed.
(**************************************************************************)
include "basic_2/notation/relations/btsn_5.ma".
-include "basic_2/reduction/fpbc.ma".
-include "basic_2/computation/csx.ma".
+include "basic_2/computation/fpbc.ma".
+include "basic_2/computation/csx_alt.ma".
(* "BIG TREE" STRONGLY NORMALIZING TERMS ************************************)
(* Basic inversion lemmas ***************************************************)
lemma fsb_inv_csx: ∀h,g,G,L,T. ⦃G, L⦄ ⊢ ⦥[h, g] T → ⦃G, L⦄ ⊢ ⬊*[h, g] T.
-#h #g #G #L #T #H elim H -G -L -T /5 width=1 by csx_intro, fpbc_cpx/
+#h #g #G #L #T #H elim H -G -L -T /5 width=1 by csx_intro_cprs, fpbc_cpxs/
qed-.
(* SN EXTENDED PARALLEL COMPUTATION FOR LOCAL ENVIRONMENTS ******************)
+(* Inversion lemmas on lazy equivalence for local environments **************)
+
+lemma lpxs_nlleq_inv_step_sn: ∀h,g,G,L1,L2,T,d. ⦃G, L1⦄ ⊢ ➡*[h, g] L2 → (L1 ⋕[d, T] L2 → ⊥) →
+ ∃∃L. ⦃G, L1⦄ ⊢ ➡*[h, g] L & L1 ⋕[d, T] L → ⊥ & ⦃G, L⦄ ⊢ ➡*[h, g] L2.
+#h #g #G #L1 #L2 #T #d #H @(lpxs_ind_dx … H) -L1
+[ #H elim H -H //
+| #L1 #L #H1 #H2 #IH2 #H12 elim (lleq_dec T L1 L d) #H
+ [ -H2 elim IH2 -IH2
+ /4 width=4 by lpxs_strap2, lleq_canc_sn, lleq_trans, ex3_intro/
+ | -IH2 -H12 /3 width=4 by lpx_lpxs, ex3_intro/ (**) (* auto fails without clear *)
+ ]
+]
+qed-.
+
(* Properties on lazy equivalence for local environments ********************)
lemma lpxs_lleq_fqu_trans: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ →
+++ /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 *)
-(* *)
-(**************************************************************************)
-
-(* NOTATION FOR THE FORMAL SYSTEM λδ ****************************************)
-
-notation "hvbox( ⦃ term 46 G1, break term 46 L1, break term 46 T1 ⦄ ⋕ break [ term 46 h , break term 46 g ] break ⦃ term 46 G2, break term 46 L2 , break term 46 T2 ⦄ )"
- non associative with precedence 45
- for @{ 'LazyEq $h $g $G1 $L1 $T1 $G2 $L2 $T2 }.
+++ /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/notation/relations/btpredproper_8.ma".
-include "basic_2/reduction/fpb.ma".
-
-(* "BIG TREE" PROPER PARALLEL REDUCTION FOR CLOSURES ************************)
-
-inductive fpbc (h) (g) (G1) (L1) (T1): relation3 genv lenv term ≝
-| fpbc_fqu: ∀G2,L2,T2. ⦃G1, L1, T1⦄ ⊃ ⦃G2, L2, T2⦄ → fpbc h g G1 L1 T1 G2 L2 T2
-| fpbc_cpx: ∀T2. ⦃G1, L1⦄ ⊢ T1 ➡[h, g] T2 → (T1 = T2 → ⊥) → fpbc h g G1 L1 T1 G1 L1 T2
-.
-
-interpretation
- "'big tree' proper parallel reduction (closure)"
- 'BTPRedProper h g G1 L1 T1 G2 L2 T2 = (fpbc h g G1 L1 T1 G2 L2 T2).
-
-(* Basic properties *********************************************************)
-
-lemma cpr_fpbc: ∀h,g,G,L,T1,T2. ⦃G, L⦄ ⊢ T1 ➡ T2 → (T1 = T2 → ⊥) →
- ⦃G, L, T1⦄ ≻[h, g] ⦃G, L, T2⦄.
-/3 width=1 by fpbc_cpx, cpr_cpx/ qed.
-
-(* Basic forward lemmas *****************************************************)
-
-lemma fpbc_fwd_fpb: ∀h,g,G1,G2,L1,L2,T1,T2. ⦃G1, L1, T1⦄ ≻[h, g] ⦃G2, L2, T2⦄ →
- ⦃G1, L1, T1⦄ ≽[h, g] ⦃G2, L2, T2⦄.
-#h #g #G1 #G2 #L1 #L2 #T1 #T2 * -G2 -L2 -T2
-/3 width=1 by fpb_fquq, fpb_cpx, fqu_fquq/
-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/static/ssta_ssta.ma".
-include "basic_2/reduction/cpx_lift.ma".
-include "basic_2/reduction/fpbc.ma".
-
-(* "BIG TREE" PROPER PARALLEL REDUCTION FOR CLOSURES ************************)
-
-(* Advanced properties ******************************************************)
-
-lemma ssta_fpbc: ∀h,g,G,L,T1,T2,l. ⦃G, L⦄ ⊢ T1 ▪[h, g] l+1 →
- ⦃G, L⦄ ⊢ T1 •[h, g] T2 → ⦃G, L, T1⦄ ≻[h, g] ⦃G, L, T2⦄.
-#h #g #G #L #T1 #T2 #l #HT1 #HT12 elim (eq_term_dec T1 T2)
-/3 width=5 by fpbc_cpx, ssta_cpx/ #H destruct
-elim (ssta_inv_refl_pos … HT1 … HT12)
-qed.
[ "csx ( ⦃?,?⦄ ⊢ ⬊*[?,?] ? )" "csx_alt ( ⦃?,?⦄ ⊢ ⬊⬊*[?,?] ? )" "csx_lift" + "csx_lpx" * ]
}
]
- [ { "parallel computation for \"big tree\" normal forms" * } {
- [ "fpns ( ⦃?,?,?⦄ ⊢ ⋕➡*[?,?] ⦃?,?,?⦄ )" "fpns_fpns" * ]
- }
- ]
[ { "\"big tree\" parallel computation" * } {
[ "fpbr ( ⦃?,?,?⦄ ⊃≥[?,?] ⦃?,?,?⦄ )" "fpbr_fpbr" * ]
[ "fpbg ( ⦃?,?,?⦄ >[?,?] ⦃?,?,?⦄ )" "fpbg_lift" + "fpbg_fpbg" * ]
- [ "fpbs ( ⦃?,?,?⦄ ≥[?,?] ⦃?,?,?⦄ )" "fpbs_alt ( ⦃?,?,?⦄ ≥≥[?,?] ⦃?,?,?⦄ )" "fpbs_lift" + "fpbs_fleq" + "fpbs_fpbs" * ]
+ [ "fpbc ( ⦃?,?,?⦄ ≻[?,?] ⦃?,?,?⦄ )" "fpbc_lift" + "fpbc_fpns" * ]
+ [ "fpbs ( ⦃?,?,?⦄ ≥[?,?] ⦃?,?,?⦄ )" "fpbs_alt ( ⦃?,?,?⦄ ≥≥[?,?] ⦃?,?,?⦄ )" "fpbs_lift" + "fpbs_fpbs" * ]
+ }
+ ]
+ [ { "parallel computation for \"big tree\" normal forms" * } {
+ [ "fpns ( ⦃?,?,?⦄ ⊢ ⋕➡*[?,?] ⦃?,?,?⦄ )" "fpns_fpns" * ]
}
]
[ { "decomposed extended computation" * } {
class "water"
[ { "reduction" * } {
[ { "\"big tree\" parallel reduction" * } {
- [ "fpbc ( ⦃?,?,?⦄ ≻[?,?] ⦃?,?,?⦄ )" "fpbc_lift" * ]
[ "fpb ( ⦃?,?,?⦄ ≽[?,?] ⦃?,?,?⦄ )" "fpb_lift" * ]
}
]
projects / helm.git / commitdiff