From 93db728b3f690d2c683f64833353da685a6f1b65 Mon Sep 17 00:00:00 2001
From: Ferruccio Guidi <fguidi@maelstrom.helm.cs.unibo.it>
Date: Fri, 16 Feb 2018 17:33:02 +0100
Subject: [PATCH] lref case closed in cpx_lfxs_conf_fle

+ premise with h.o equality added to the theorem
---
 .../basic_2/rt_transition/cpx_etc.ma          | 44 ++++++++++++++-----
 .../lambdadelta/basic_2/static/fle_drops.ma   | 10 +++++
 2 files changed, 42 insertions(+), 12 deletions(-)

diff --git a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_etc.ma b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_etc.ma
index a6c77b5c9..a10376a89 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_etc.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/rt_transition/cpx_etc.ma
@@ -23,13 +23,14 @@ include "basic_2/rt_transition/cpx.ma".
 (* Properties with context-sensitive free variables *************************)
 
 (* Note: "⦃L2, T1⦄ ⊆ ⦃L0, T1⦄" may not hold *)
-axiom cpx_lfxs_conf_fle: ∀R. c_reflexive … R →
-                         ∀h,G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
+axiom cpx_lfxs_conf_fle: ∀R,h. c_reflexive … R →
+                         (∨∨ (∀G. (cpx h G) = R) | R_fle_compatible R) →
+                         ∀G,L0,T0,T1. ⦃G, L0⦄ ⊢ T0 ⬈[h] T1 →
                          ∀L2. L0 ⪤*[R, T0] L2 →
                          ∧∧ ⦃L2, T0⦄ ⊆ ⦃L0, T0⦄ & ⦃L2, T1⦄ ⊆ ⦃L2, T0⦄
                           & ⦃L0, T1⦄ ⊆ ⦃L0, T0⦄.
 (*
-#R #HR #h #G #L0 #T0 @(fqup_wf_ind_eq (Ⓕ) … G L0 T0) -G -L0 -T0
+#R #h #H1R #H2R #G #L0 #T0 @(fqup_wf_ind_eq (Ⓕ) … G L0 T0) -G -L0 -T0
 #G #L #T #IH #G0 #L0 * *
 [ #s #HG #HL #HT #X #HX #Y #HY destruct -IH
   lapply (lfxs_fwd_length … HY) -HY #H0
@@ -41,14 +42,33 @@ axiom cpx_lfxs_conf_fle: ∀R. c_reflexive … R →
       elim (lfxs_inv_zero … HY) -HY *
       [ #H1 #H2 destruct -IH /2 width=1 by and3_intro/
       | #I #K0 #K2 #V0 #V2 #HK02 #HV02 #H1 #H2 destruct
-        elim (IH … HK02) [4,5: /2 width=2 by fqu_fqup, fqu_lref_O/ ]
-   
-    lapply (lfxs_fwd_length … HY) -HY #H0
-    /3 width=1 by fle_lref_length, and3_intro/
-  | * #I0 #K0 #V0 #V1 #HLK0 #HV01 #HX  
-
-  elim (lfxs_inv_lref … HY) -HY // #HV0 #HT0
-
+        elim H2R -H2R #H2R
+        [ <(H2R G0) in HV02; -H2R #HV02
+          elim (IH … HV02 … HK02) /2 width=2 by fqu_fqup, fqu_lref_O/ -IH -HV02 -HK02 #H1V #H2V #H3V
+        | lapply (H2R … HV02) -H2R -HV02 #HV20
+          elim (IH … V0 … HK02) [|*: /2 width=4 by fqu_fqup, fqu_lref_O/ ] -IH -HK02 #H1V #_ #_
+        ]
+      | #f #I #K0 #K2 #Hf #HK02 #H1 #H2 destruct
+      ]
+    | * #I0 #K0 #V0 #V1 #HV01 #HV1X #H destruct
+      elim (lfxs_inv_zero_pair_sn … HY) -HY #K2 #V2 #HK02 #HV02 #H destruct
+    ]
+  | elim (cpx_inv_lref1 … HX) -HX
+    [ #H destruct
+      elim (lfxs_inv_lref … HY) -HY *
+      [ #H0 #H1 destruct /2 width=1 by and3_intro/
+      | #I0 #I2 #K0 #K2 #HK02 #H1 #H2 destruct
+        lapply (lfxs_fwd_length … HK02) #HK
+        elim (IH … HK02) [|*: /2 width=4 by fqu_fqup/ ] -IH -HK02
+        /3 width=5 by and3_intro, fle_lifts_SO/
+      ]
+    | * #I0 #K0 #V1 #HV1 #HV1X #H0 destruct
+      elim (lfxs_inv_lref_bind_sn … HY) -HY #I2 #K2 #HK02 #H destruct
+      lapply (lfxs_fwd_length … HK02) #HK
+      elim (IH … HK02) [|*: /2 width=4 by fqu_fqup/ ] -IH -HV1 -HK02
+      /3 width=5 by fle_lifts_SO, and3_intro/
+    ]
+  ]
 | #l #HG #HL #HT #X #HX #Y #HY destruct -IH
   lapply (lfxs_fwd_length … HY) -HY #H0
   >(cpx_inv_gref1 … HX) -X
@@ -104,4 +124,4 @@ axiom cpx_lfxs_conf_fle: ∀R. c_reflexive … R →
     ]
   ]
 ]
-*)
\ No newline at end of file
+*)
diff --git a/matita/matita/contribs/lambdadelta/basic_2/static/fle_drops.ma b/matita/matita/contribs/lambdadelta/basic_2/static/fle_drops.ma
index 39bf29385..c64e197ef 100644
--- a/matita/matita/contribs/lambdadelta/basic_2/static/fle_drops.ma
+++ b/matita/matita/contribs/lambdadelta/basic_2/static/fle_drops.ma
@@ -30,6 +30,16 @@ lapply (frees_lifts_SO (Ⓣ) (L1.ⓧ) … HTU1 … Hf)
 @(ex4_4_intro … Hf Hg) /2 width=4 by lveq_void_sn/ (**) (* explict constructor *)
 qed-.
 
+lemma fle_lifts_SO: ∀K1,K2. |K1| = |K2| → ∀T1,T2. ⦃K1, T1⦄ ⊆ ⦃K2, T2⦄ →
+                    ∀U1,U2. ⬆*[1] T1 ≡ U1 → ⬆*[1] T2 ≡ U2 →
+                    ∀I1,I2.  ⦃K1.ⓘ{I1}, U1⦄ ⊆ ⦃K2.ⓘ{I2}, U2⦄.
+#K1 #K2 #HK #T1 #T2
+* #n1 #n2 #f1 #f2 #Hf1 #Hf2 #HK12 #Hf12
+#U1 #U2 #HTU1 #HTU2 #I1 #I2
+elim (lveq_inj_length … HK12) // -HK #H1 #H2 destruct
+/5 width=12 by frees_lifts_SO, drops_refl, drops_drop, lveq_bind, sle_push, ex4_4_intro/
+qed.
+
 (* Advanced inversion lemmas ************************************************)
 
 lemma fle_inv_lifts_sn: ∀T1,U1. ⬆*[1] T1 ≡ U1 →
-- 
2.39.2