Hashtbl.iter iter graph
let rec read ich =
- let _ = Scanf.fscanf ich "./%s@:include \"%s@\". " init in
+ let _ = Scanf.fscanf ich "%s@:include \"%s@\". " init in
read ich
let _ =
(**************************************************************************)
include "basic_2/reducibility/fpr_cpr.ma".
-include "basic_2/computation/cprs.ma".
-include "basic_2/computation/fprs.ma".
+include "basic_2/computation/cprs_lfprs.ma".
+include "basic_2/computation/lfprs_ltprs.ma".
+include "basic_2/computation/lfprs_fprs.ma".
(* CONTEXT-FREE PARALLEL COMPUTATION ON CLOSURES ****************************)
+(* Advanced inversion lemmas ************************************************)
+
+lemma fprs_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡* ⦃L2, T2⦄ →
+ ∃∃K2,V2. ⦃K1, V1⦄ ➡* ⦃K2, V2⦄ &
+ ⦃K1, -ⓑ{I}V1.T1⦄ ➡* ⦃K2, -ⓑ{I}V2.T2⦄ &
+ L2 = K2.ⓑ{I}V2.
+#I #K1 #L2 #V1 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 /2 width=5/
+#L #L2 #T #T2 #_ #HT2 * #K #V #HV1 #HT1 #H destruct
+elim (fpr_inv_pair1 … HT2) -HT2 #K2 #V2 #HV2 #HT2 #H destruct /3 width=5/
+qed-.
+
+lemma fprs_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡* ⦃K2.ⓑ{I}V2, T2⦄ →
+ ∃∃K1,V1. ⦃K1, V1⦄ ➡* ⦃K2, V2⦄ &
+ ⦃K1, -ⓑ{I}V1.T1⦄ ➡* ⦃K2, -ⓑ{I}V2.T2⦄ &
+ L1 = K1.ⓑ{I}V1.
+#I2 #L1 #K2 #V2 #T1 #T2 #H @(fprs_ind_dx … H) -L1 -T1 /2 width=5/
+#L1 #L #T1 #T #HT1 #_ * #K #V #HV2 #HT2 #H destruct
+elim (fpr_inv_pair3 … HT1) -HT1 #K1 #V1 #HV1 #HT1 #H destruct /3 width=5/
+qed-.
+
+(* Advanced forward lemmas **************************************************)
+
+lemma fprs_fwd_bind2_minus: ∀I,L1,L,V1,T1,T. ⦃L1, -ⓑ{I}V1.T1⦄ ➡* ⦃L, T⦄ → ∀b.
+ ∃∃V2,T2. ⦃L1, ⓑ{b,I}V1.T1⦄ ➡* ⦃L, ⓑ{b,I}V2.T2⦄ &
+ T = -ⓑ{I}V2.T2.
+#I #L1 #L #V1 #T1 #T #H1 #b @(fprs_ind … H1) -L -T /2 width=4/
+#L0 #L #T0 #T #_ #H0 * #W1 #U1 #HTU1 #H destruct
+elim (fpr_fwd_bind2_minus … H0 b) -H0 /3 width=4/
+qed-.
+
+lemma fprs_fwd_pair1_full: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡* ⦃L2, T2⦄ →
+ ∀b. ∃∃K2,V2. ⦃K1, V1⦄ ➡* ⦃K2, V2⦄ &
+ ⦃K1, ⓑ{b,I}V1.T1⦄ ➡* ⦃K2, ⓑ{b,I}V2.T2⦄ &
+ L2 = K2.ⓑ{I}V2.
+#I #K1 #L2 #V1 #T1 #T2 #H #b
+elim (fprs_inv_pair1 … H) -H #K2 #V2 #HV12 #HT12 #H destruct
+elim (fprs_fwd_bind2_minus … HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /2 width=5/
+qed-.
+
(* Properties on context-sensitive parallel computation for terms ***********)
lemma cprs_fprs: ∀L,T1,T2. L ⊢ T1 ➡* T2 → ⦃L, T1⦄ ➡* ⦃L, T2⦄.
#L #T1 #T2 #H @(cprs_ind … H) -T2 // /3 width=4/
qed.
+
+(* Forward lemmas on context-sensitive parallel computation for terms *******)
+
+lemma fprs_fwd_cprs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → L1 ⊢ T1 ➡* T2.
+#L1 #L2 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 //
+#L #L2 #T #T2 #H1 #H2 #IH1
+elim (fpr_inv_all … H2) -H2 #L0 #HL0 #HT2 #_ -L2
+lapply (lfprs_cpr_trans L1 … HT2) -HT2 /3 width=3/
+qed-.
(*
-(* Advanced propertis *******************************************************)
+(* Advanced properties ******************************************************)
lamma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 →
∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄.
#V #T #HV1 #_ #H destruct /3 width=4/
]
qed-.
-
-(* Advanced inversion lemmas ************************************************)
-
-lamma fpr_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
- ∃∃K2,V2. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
- ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
- L2 = K2.ⓑ{I}V2.
-#I1 #K1 #X #V1 #T1 #T2 #H
-elim (fpr_fwd_pair1 … H) -H #I2 #K2 #V2 #HT12 #H destruct
-elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
-qed-.
-
-lamma fpr_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I}V2, T2⦄ →
- ∃∃K1,V1. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
- ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
- L1 = K1.ⓑ{I}V1.
-#I2 #X #K2 #V2 #T1 #T2 #H
-elim (fpr_fwd_pair3 … H) -H #I1 #K1 #V1 #HT12 #H destruct
-elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
-qed-.
*)
#I #L1 #L2 #HL12 #V1 #V2 #H @(cprs_ind … H) -V2
/3 width=1/ /3 width=5/
qed.
+(*
+lamma lfprs_cprs_conf: ∀L1,L,L2,T1,T2. ⦃L1⦄ ➡* ⦃L2⦄ → L1 ⊢ T1 ➡* T2 → ⦃L1, T1⦄ ➡* ⦃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/reducibility/lfpr_fpr.ma".
+include "basic_2/computation/fprs_fprs.ma".
+include "basic_2/computation/lfprs.ma".
+
+(* FOCALIZED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *********************)
+
+(* Inversion lemmas on context-free parallel reduction for closures *********)
+
+lemma lfprs_inv_fprs: ∀L1,L2. ⦃L1⦄ ➡* ⦃L2⦄ → ∀T. ⦃L1, T⦄ ➡* ⦃L2, T⦄.
+#L1 #L2 #H @(lfprs_ind … H) -L2 //
+#L #L2 #_ #HL2 #IHL1 #T
+lapply (lfpr_inv_fpr … HL2 T) -HL2 /3 width=4/
+qed-.
+
+(* Properties on context-free parallel computation for closures *************)
+
+lemma fprs_lfprs: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡* ⦃L2, T2⦄ → ⦃L1⦄ ➡* ⦃L2⦄.
+#L1 #L2 #T1 #T2 #H @(fprs_ind … H) -L2 -T2 // /3 width=5/
+qed.
+
+lemma lfprs_fprs_trans: ∀L1,L,L2,T1,T2. ⦃L1⦄ ➡* ⦃L⦄ → ⦃L, T1⦄ ➡* ⦃L2, T2⦄ → ⦃L1, T1⦄ ➡* ⦃L2, T2⦄.
+#L1 #L #L2 #T1 #T2 #HL1 #HL2
+lapply (lfprs_inv_fprs … HL1 T1) -HL1 /2 width=4/
+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/ltprs.ma".
+include "basic_2/computation/lfprs.ma".
+
+(* FOCALIZED PARALLEL COMPUTATION ON LOCAL ENVIRONMENTS *********************)
+
+(* Properties on context-free parallel computation for local environments ***)
+
+lemma ltprs_lfprs: ∀L1,L2. L1 ➡* L2 → ⦃L1⦄ ➡* ⦃L2⦄.
+/3 width=3/ qed.
lemma ltprs_refl: reflexive … ltprs.
/2 width=1/ qed.
+lemma ltprs_strap1: ∀L1,L,L2. L1 ➡* L → L ➡ L2 → L1 ➡* L2.
+/2 width=3/ qed.
+
+lemma ltprs_strap2: ∀L1,L,L2. L1 ➡ L → L ➡* L2 → L1 ➡* L2.
+/2 width=3/ qed.
+
(* Basic inversion lemmas ***************************************************)
lemma ltprs_inv_atom1: ∀L2. ⋆ ➡* L2 → L2 = ⋆.
(* *)
(**************************************************************************)
+
include "basic_2/reducibility/ltpr.ma".
include "basic_2/computation/xprs_lift.ma".
include "basic_2/equivalence/cpcs_cpcs.ma".
include "basic_2/equivalence/lsubse_ssta.ma".
-include "basic_2/equivalence/fpcs.ma".
+include "basic_2/equivalence/fpcs_cpcs.ma".
+include "basic_2/equivalence/fpcs_fpcs.ma".
include "basic_2/dynamic/snv_ssta.ma".
(*
include "basic_2/static/ssta_ltpss_dx.ma".
-include "basic_2/equivalence/fpcs_fpcs.ma".
include "basic_2/dynamic/snv_lift.ma".
*)
(* STRATIFIED NATIVE VALIDITY FOR TERMS *************************************)
elim (IH1 … HVW1 … HL12 … HV12) -HVW1 // -HV1 #W2 #HVW2 #HW12
elim (IH1 … HWV … HL12 W) -HWV // -HW #V0 #HWV0 #_
elim (IH1 … HTU2 (L2.ⓛW) … HT20 HT2) -IH1 -HTU2 -HT20 -HT2 // [2: /2 width=1/ ] #U20 #HTU20 #HU20
+ lapply (lfpr_inv_fpr L1 L2 … W) [ /2 width=1/ ] -HL12 #HL12
elim (lsubse_ssta_trans … HTU20 (L2.ⓓV2) ?) -HTU20
- [ #U #HTU20 #HU20 -HWV0 -HL12 -W1 -W2
- @(ex2_intro … (ⓓ{b}V2.U)) [ /2 width=1/ ] -T20 -l
- | @(lsubse_abbr … HVW2) // -g -h -b -l -l1 -V -V0 -V1 -V2 -T2 -T20 -U0
+ [ #U #HTU20 #HU20 -HWV0 -HL12 -W1 -W2
+ @(ex2_intro … (ⓓ{b}V2.U)) [ /2 width=1/ ] -h -l -l1 -V -V0 -T2 -T20 -U0
+ @(fpcs_fprs_strap2 ? L1 … (ⓓ{b}V2.U2)) [ /4 width=1/ ] -V1
+ @fpcs_shift_full -b
+ @(fpcs_canc_dx ?? (L2.ⓓV2) … U20) [2: /2 width=1/ ] -U
+ | -b -l -V -V1 -T2 -T20 -U0 -U2 -U20
+ /6 width=6 by lsubse_abbr, fpcs_inv_cpcs, fpcs_canc_sn, fpcs_fprs_strap1, cpcs_fpcs, bi_inj/
]
]
|
include "basic_2/computation/fprs_cprs.ma".
include "basic_2/equivalence/cpcs_cpcs.ma".
-include "basic_2/equivalence/fpcs_fprs.ma".
+include "basic_2/equivalence/fpcs_fpcs.ma".
(* CONTEXT-FREE PARALLEL EQUIVALENCE ON CLOSURES ****************************)
+(* Advanced properties ******************************************************)
+
+lemma fpcs_shift: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄ →
+ ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄.
+#I #L1 #L2 #V1 #V2 #T1 #T2 #H12
+elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
+elim (fprs_inv_pair1 … H1) -H1 #K1 #U1 #_ #HTU1 #H destruct
+elim (fprs_inv_pair1 … H2) -H2 #K2 #U2 #_ #HTU2 #H destruct /2 width=4/
+qed.
+
+lemma fpcs_bind_minus: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1, -ⓑ{I}V1.T1⦄ ⬌* ⦃L2, -ⓑ{I}V2.T2⦄ →
+ ∀b. ⦃L1, ⓑ{b,I}V1.T1⦄ ⬌* ⦃L2, ⓑ{b,I}V2.T2⦄.
+#I #L1 #L2 #V1 #V2 #T1 #T2 #H12 #b
+elim (fpcs_inv_fprs … H12) -H12 #L #T #H1 #H2
+elim (fprs_fwd_bind2_minus … H1 b) -H1 #W1 #U1 #HTU1 #H destruct
+elim (fprs_fwd_bind2_minus … H2 b) -H2 #W2 #U2 #HTU2 #H destruct /2 width=4/
+qed.
+
+lemma fpcs_shift_full: ∀I,L1,L2,V1,V2,T1,T2. ⦃L1.ⓑ{I}V1, T1⦄ ⬌* ⦃L2.ⓑ{I}V2, T2⦄ →
+ ∀b. ⦃L1, ⓑ{b,I}V1.T1⦄ ⬌* ⦃L2, ⓑ{b,I}V2.T2⦄.
+/3 width=1/ qed.
+
(* Properties on context-sensitive parallel equivalence for terms ***********)
lemma cpcs_fpcs: ∀L,T1,T2. L ⊢ T1 ⬌* T2 → ⦃L, T1⦄ ⬌* ⦃L, T2⦄.
#L #T1 #T2 #H
elim (cpcs_inv_cprs … H) -H /3 width=4 by fprs_div, cprs_fprs/ (**) (* too slow without trace *)
qed.
+
+(* Inversion lemmas on context-sensitive parallel equivalence for terms *****)
+
+lemma fpcs_inv_cpcs: ∀L,T1,T2. ⦃L, T1⦄ ⬌* ⦃L, T2⦄ → L ⊢ T1 ⬌* T2.
+#L #T1 #T2 #H
+elim (fpcs_inv_fprs … H) -H /3 width=4 by cprs_div, fprs_fwd_cprs/
+qed-.
(* Basic forward lemmas *****************************************************)
+lemma cpr_fwd_bind1_minus: ∀I,L,V1,T1,T. L ⊢ -ⓑ{I}V1.T1 ➡ T → ∀b.
+ ∃∃V2,T2. L ⊢ ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
+ T = -ⓑ{I}V2.T2.
+#I #L #V1 #T1 #T * #X #H1 #H2 #b
+elim (tpr_fwd_bind1_minus … H1 b) -H1 #V0 #T0 #HT10 #H destruct
+elim (tpss_inv_bind1 … H2) -H2 #V2 #T2 #HV02 #HT02 #H destruct /4 width=5/
+qed-.
+
lemma cpr_fwd_shift1: ∀L,L1,T1,T. L ⊢ L1 @@ T1 ➡ T →
∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
#L #L1 #T1 #T * #X #H1 #H2
lemma cpr_fpr: ∀L,T1,T2. L ⊢ T1 ➡ T2 → ⦃L, T1⦄ ➡ ⦃L, T2⦄.
/2 width=4/ qed.
-(* Advanced propertis *******************************************************)
+(* Advanced properties ******************************************************)
lemma fpr_bind_sn: ∀L1,L2,V1,V2. ⦃L1, V1⦄ ➡ ⦃L2, V2⦄ → ∀T1,T2. T1 ➡ T2 →
∀a,I. ⦃L1, ⓑ{a,I}V1.T1⦄ ➡ ⦃L2, ⓑ{a,I}V2.T2⦄.
(* Advanced forward lemmas **************************************************)
+lemma fpr_fwd_bind2_minus: ∀I,L1,L,V1,T1,T. ⦃L1, -ⓑ{I}V1.T1⦄ ➡ ⦃L, T⦄ → ∀b.
+ ∃∃V2,T2. ⦃L1, ⓑ{b,I}V1.T1⦄ ➡ ⦃L, ⓑ{b,I}V2.T2⦄ &
+ T = -ⓑ{I}V2.T2.
+#I #L1 #L #V1 #T1 #T #H1 #b
+elim (fpr_inv_all … H1) -H1 #L0 #HL10 #HT1 #HL0
+elim (cpr_fwd_bind1_minus … HT1 b) -HT1 /3 width=4/
+qed-.
+
lemma fpr_fwd_shift_bind_minus: ∀I1,I2,L1,L2,V1,V2,T1,T2.
⦃L1, -ⓑ{I1}V1.T1⦄ ➡ ⦃L2, -ⓑ{I2}V2.T2⦄ →
⦃L1, V1⦄ ➡ ⦃L2, V2⦄ ∧ I1 = I2.
lemma fpr_inv_pair1: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
∃∃K2,V2. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
- ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
+ ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
L2 = K2.ⓑ{I}V2.
#I1 #K1 #X #V1 #T1 #T2 #H
elim (fpr_fwd_pair1 … H) -H #I2 #K2 #V2 #HT12 #H destruct
lemma fpr_inv_pair3: ∀I,L1,K2,V2,T1,T2. ⦃L1, T1⦄ ➡ ⦃K2.ⓑ{I}V2, T2⦄ →
∃∃K1,V1. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
- ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
+ ⦃K1, -ⓑ{I}V1.T1⦄ ➡ ⦃K2, -ⓑ{I}V2.T2⦄ &
L1 = K1.ⓑ{I}V1.
#I2 #X #K2 #V2 #T1 #T2 #H
elim (fpr_fwd_pair3 … H) -H #I1 #K1 #V1 #HT12 #H destruct
elim (fpr_fwd_shift_bind_minus … HT12) #HV12 #H destruct /2 width=5/
qed-.
+
+(* More advanced forward lemmas *********************************************)
+
+lemma fpr_fwd_pair1_full: ∀I,K1,L2,V1,T1,T2. ⦃K1.ⓑ{I}V1, T1⦄ ➡ ⦃L2, T2⦄ →
+ ∀b. ∃∃K2,V2. ⦃K1, V1⦄ ➡ ⦃K2, V2⦄ &
+ ⦃K1, ⓑ{b,I}V1.T1⦄ ➡ ⦃K2, ⓑ{b,I}V2.T2⦄ &
+ L2 = K2.ⓑ{I}V2.
+#I #K1 #L2 #V1 #T1 #T2 #H #b
+elim (fpr_inv_pair1 … H) -H #K2 #V2 #HV12 #HT12 #H destruct
+elim (fpr_fwd_bind2_minus … HT12 b) -HT12 #W1 #U1 #HTU1 #H destruct /2 width=5/
+qed-.
lemma ltpss_sn_lfpr: ∀L1,L2,d,e. L1 ⊢ ▶* [d, e] L2 → ⦃L1⦄ ➡ ⦃L2⦄.
/3 width=5/ qed.
+lemma ltpr_lfpr: ∀L1,L2. L1 ➡ L2 → ⦃L1⦄ ➡ ⦃L2⦄.
+/3 width=3/ qed.
+
(* Basic inversion lemmas ***************************************************)
lemma lfpr_inv_atom1: ∀L2. ⦃⋆⦄ ➡ ⦃L2⦄ → L2 = ⋆.
(* FOCALIZED PARALLEL REDUCTION FOR LOCAL ENVIRONMENTS **********************)
-(* Advanced properties ****************************************************)
+(* Advanced properties ******************************************************)
lemma lfpr_pair_cpr: ∀L1,L2. ⦃L1⦄ ➡ ⦃L2⦄ → ∀V1,V2. L2 ⊢ V1 ➡ V2 →
∀I. ⦃L1. ⓑ{I} V1⦄ ➡ ⦃L2. ⓑ{I} V2⦄.
(* FOCALIZED PARALLEL REDUCTION ON LOCAL ENVIRONMENTS ***********************)
-(* Inversion lemmas on context-free parallel reduction for closures *********)
+(* Properties on context-free parallel reduction for closures ***************)
lemma fpr_lfpr: ∀L1,L2,T1,T2. ⦃L1, T1⦄ ➡ ⦃L2, T2⦄ → ⦃L1⦄ ➡ ⦃L2⦄.
#L1 #L2 #T1 #T2 #H
(* Basic forward lemmas *****************************************************)
+lemma tpr_fwd_bind1_minus: ∀I,V1,T1,T. -ⓑ{I}V1.T1 ➡ T → ∀b.
+ ∃∃V2,T2. ⓑ{b,I}V1.T1 ➡ ⓑ{b,I}V2.T2 &
+ T = -ⓑ{I}V2.T2.
+#I #V1 #T1 #T #H #b elim (tpr_inv_bind1 … H) -H *
+[ #V2 #T0 #T2 #HV12 #HT10 #HT02 #H destruct /3 width=4/
+| #T2 #_ #_ #H destruct
+]
+qed-.
+
lemma tpr_fwd_shift1: ∀L1,T1,T. L1 @@ T1 ➡ T →
∃∃L2,T2. |L1| = |L2| & T = L2 @@ T2.
#L1 @(lenv_ind_dx … L1) -L1 normalize
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