- lfpxs based on tc_lfxs

[helm.git] / matita / matita / contribs / lambdadelta / basic_2 / static / lfdeq_lfdeq.ma

diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_lfdeq.ma b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_lfdeq.ma
index 9f92ecd272e3c8aa1c5e13eedeb557b1a5c79a19..f3fbeda428510b67e04c70acfdfe3f67359a4dd8 100644 (file)
--- a/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_lfdeq.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/lfdeq_lfdeq.ma
@@ -20,20 +20,24 @@ include "basic_2/static/lfdeq.ma".
 
 (* Advanced properties ******************************************************)
 
+(* Basic_2A1: uses: lleq_dec *)
 lemma lfdeq_dec: ∀h,o,L1,L2. ∀T:term. Decidable (L1 ≡[h, o, T] L2).
 /3 width=1 by lfxs_dec, tdeq_dec/ qed-.
 
 (* Main properties **********************************************************)
 
+(* Basic_2A1: uses: lleq_bind lleq_bind_O *) 
 theorem lfdeq_bind: ∀h,o,p,I,L1,L2,V1,V2,T.
                     L1 ≡[h, o, V1] L2 → L1.ⓑ{I}V1 ≡[h, o, T] L2.ⓑ{I}V2 →
                     L1 ≡[h, o, ⓑ{p,I}V1.T] L2.
 /2 width=2 by lfxs_bind/ qed.
 
+(* Basic_2A1: uses: lleq_flat *)
 theorem lfdeq_flat: ∀h,o,I,L1,L2,V,T. L1 ≡[h, o, V] L2 → L1 ≡[h, o, T] L2 →
                     L1 ≡[h, o, ⓕ{I}V.T] L2.
 /2 width=1 by lfxs_flat/ qed.
 
+(* Basic_2A1: uses: lleq_trans *)
 theorem lfdeq_trans: ∀h,o,T. Transitive … (lfdeq h o T).
 #h #o #T #L1 #L * #f1 #Hf1 #HL1 #L2 * #f2 #Hf2 #HL2
 lapply (frees_tdeq_conf_lexs … Hf1 T … HL1) // #H0
@@ -41,9 +45,11 @@ lapply (frees_mono … Hf2 … H0) -Hf2 -H0
 /4 width=7 by lexs_trans, lexs_eq_repl_back, tdeq_trans, ex2_intro/
 qed-.
 
+(* Basic_2A1: uses: lleq_canc_sn *)
 theorem lfdeq_canc_sn: ∀h,o,T. left_cancellable … (lfdeq h o T).
 /3 width=3 by lfdeq_trans, lfdeq_sym/ qed-.
 
+(* Basic_2A1: uses: lleq_canc_dx *)
 theorem lfdeq_canc_dx: ∀h,o,T. right_cancellable … (lfdeq h o T).
 /3 width=3 by lfdeq_trans, lfdeq_sym/ qed-.
 
@@ -51,13 +57,31 @@ theorem lfdeq_repl: ∀h,o,L1,L2. ∀T:term. L1 ≡[h, o, T] L2 →
                     ∀K1. L1 ≡[h, o, T] K1 → ∀K2. L2 ≡[h, o, T] K2 → K1 ≡[h, o, T] K2.
 /3 width=3 by lfdeq_canc_sn, lfdeq_trans/ qed-.
 
-(* Advanced properies on negated lazy equivalence *****************************)
+(* Negated properties *******************************************************)
 
-(* Note: auto works with /4 width=8/ so lfdeq_canc_sn is preferred ************) 
-lemma lfdeq_nlfdeq_trans: ∀h,o.∀T:term.∀L1,L. L1 ≡[h, o, T] L →
+(* Note: auto works with /4 width=8/ so lfdeq_canc_sn is preferred **********) 
+(* Basic_2A1: uses: lleq_nlleq_trans *)
+lemma lfdeq_lfdneq_trans: ∀h,o.∀T:term.∀L1,L. L1 ≡[h, o, T] L →
                           ∀L2. (L ≡[h, o, T] L2 → ⊥) → (L1 ≡[h, o, T] L2 → ⊥).
 /3 width=3 by lfdeq_canc_sn/ qed-.
 
-lemma nlfdeq_lfdeq_div: ∀h,o.∀T:term.∀L2,L. L2 ≡[h, o, T] L →
+(* Basic_2A1: uses: nlleq_lleq_div *)
+lemma lfdneq_lfdeq_div: ∀h,o.∀T:term.∀L2,L. L2 ≡[h, o, T] L →
                         ∀L1. (L1 ≡[h, o, T] L → ⊥) → (L1 ≡[h, o, T] L2 → ⊥).
 /3 width=3 by lfdeq_trans/ qed-.
+
+theorem lfdneq_lfdeq_canc_dx: ∀h,o,L1,L. ∀T:term. (L1 ≡[h, o, T] L → ⊥) →
+                              ∀L2. L2 ≡[h, o, T] L → L1 ≡[h, o, T] L2 → ⊥.
+/3 width=3 by lfdeq_trans/ qed-.
+
+(* Negated inversion lemmas *************************************************)
+
+(* Basic_2A1: uses: nlleq_inv_bind nlleq_inv_bind_O *)
+lemma lfdneq_inv_bind: ∀h,o,p,I,L1,L2,V,T. (L1 ≡[h, o, ⓑ{p,I}V.T] L2 → ⊥) →
+                       (L1 ≡[h, o, V] L2 → ⊥) ∨ (L1.ⓑ{I}V ≡[h, o, T] L2.ⓑ{I}V → ⊥).
+/3 width=2 by lfnxs_inv_bind, tdeq_dec/ qed-.
+
+(* Basic_2A1: uses: nlleq_inv_flat *)
+lemma lfdneq_inv_flat: ∀h,o,I,L1,L2,V,T. (L1 ≡[h, o, ⓕ{I}V.T] L2 → ⊥) →
+                       (L1 ≡[h, o, V] L2 → ⊥) ∨ (L1 ≡[h, o, T] L2 → ⊥).
+/3 width=2 by lfnxs_inv_flat, tdeq_dec/ qed-.