update in ground_2, static_2, basic_2, apps_2, alpha_1

[helm.git] / matita / matita / contribs / lambdadelta / apps_2 / models / model_props.ma

diff --git a/matita/matita/contribs/lambdadelta/apps_2/models/model_props.ma b/matita/matita/contribs/lambdadelta/apps_2/models/model_props.ma
index 863e3d063f60121290266416ee6167600dadcd32..f6c8d4c585f2b4c57ea7565be49c29275d939435 100644 (file)
--- a/matita/matita/contribs/lambdadelta/apps_2/models/model_props.ma
+++ b/matita/matita/contribs/lambdadelta/apps_2/models/model_props.ma
@@ -12,36 +12,105 @@
 (*                                                                        *)
 (**************************************************************************)
 
-include "apps_2/models/model_vlift.ma".
+include "apps_2/models/model_vpush.ma".
 
 (* MODEL ********************************************************************)
 
 record is_model (M): Prop ≝ {
-   mq: reflexive … (sq M);
-   mr: replace_2 … (sq M) (sq M) (sq M);
-   mc: compatible_3 … (ap M) (sq M) (sq M) (sq M);
-   ms: ∀gv,lv,s. ⟦⋆s⟧{M}[gv, lv] ≗ sv M s;
-   ml: ∀gv,lv,i. ⟦#i⟧{M}[gv, lv] ≗ lv i;
-   mg: ∀gv,lv,l. ⟦§l⟧{M}[gv, lv] ≗ gv l;
-   md: ∀gv,lv,p,V,T. ⟦ⓓ{p}V.T⟧{M}[gv, lv] ≗ ⟦T⟧[gv, ⫯[⟦V⟧[gv, lv]]lv];
-   mi: ∀gv,lv1,lv2,p,W,T. ⟦W⟧{M}[gv, lv1] ≗ ⟦W⟧{M}[gv, lv2] →
-       (∀d. ⟦T⟧{M}[gv, ⫯[d]lv1] ≗ ⟦T⟧{M}[gv, ⫯[d]lv2]) →
-       ⟦ⓛ{p}W.T⟧[gv, lv1] ≗ ⟦ⓛ{p}W.T⟧[gv, lv2];
-   ma: ∀gv,lv,V,T. ⟦ⓐV.T⟧{M}[gv, lv] ≗ ⟦V⟧[gv, lv] @ ⟦T⟧[gv, lv];
-   me: ∀gv,lv,W,T. ⟦ⓝW.T⟧{M}[gv, lv] ≗ ⟦T⟧[gv, lv];
-   mb: ∀gv,lv,d,p,W,T. d @ ⟦ⓛ{p}W.T⟧{M}[gv, lv] ≗ ⟦T⟧[gv, ⫯[d]lv]
+(* Note: equivalence: reflexivity *)
+   mr: reflexive … (sq M);
+(* Note: equivalence: compatibility *)
+   mq: replace_2 … (sq M) (sq M) (sq M);
+(* Note: conjunction: compatibility *)
+   mc: ∀p. compatible_3 … (co M p) (sq M) (sq M) (sq M);
+(* Note: application: compatibility *)
+   mp: compatible_3 … (ap M) (sq M) (sq M) (sq M);
+(* Note: interpretation: sort *)
+   ms: ∀gv,lv,s. ⟦⋆s⟧{M}[gv,lv] ≗ sv M s;
+(* Note: interpretation: local reference *)
+   ml: ∀gv,lv,i. ⟦#i⟧{M}[gv,lv] ≗ lv i;
+(* Note: interpretation: global reference *)
+   mg: ∀gv,lv,l. ⟦§l⟧{M}[gv,lv] ≗ gv l;
+(* Note: interpretation: intensional binder *)
+   mi: ∀p,gv1,gv2,lv1,lv2,W,T. ⟦W⟧{M}[gv1,lv1] ≗ ⟦W⟧{M}[gv2,lv2] →
+       (∀d. ⟦T⟧{M}[gv1,⫯[0←d]lv1] ≗ ⟦T⟧{M}[gv2,⫯[0←d]lv2]) →
+       ⟦ⓛ[p]W.T⟧[gv1,lv1] ≗ ⟦ⓛ[p]W.T⟧[gv2,lv2];
+(* Note: interpretation: abbreviation *)
+   md: ∀p,gv,lv,V,T. ⟦ⓓ[p]V.T⟧{M}[gv,lv] ≗ ⟦V⟧[gv,lv] ⊕[p] ⟦T⟧[gv,⫯[0←⟦V⟧[gv,lv]]lv];
+(* Note: interpretation: application *)
+   ma: ∀gv,lv,V,T. ⟦ⓐV.T⟧{M}[gv,lv] ≗ ⟦V⟧[gv,lv] @ ⟦T⟧[gv,lv];
+(* Note: interpretation: ζ-equivalence *)
+   mz: ∀d1,d2. d1 ⊕{M}[Ⓣ] d2 ≗ d2;
+(* Note: interpretation: ϵ-equivalence *)
+   me: ∀gv,lv,W,T. ⟦ⓝW.T⟧{M}[gv,lv] ≗ ⟦T⟧[gv,lv];
+(* Note: interpretation: β-requivalence *)
+   mb: ∀p,gv,lv,d,W,T. d @ ⟦ⓛ[p]W.T⟧{M}[gv,lv] ≗ d ⊕[p] ⟦T⟧[gv,⫯[0←d]lv];
+(* Note: interpretation: θ-requivalence *)
+   mh: ∀p,d1,d2,d3. d1 @ (d2 ⊕{M}[p] d3) ≗ d2 ⊕[p] (d1 @ d3)
 }.
 
 record is_extensional (M): Prop ≝ {
-   mx: ∀gv,lv1,lv2,p,W1,W2,T1,T2. ⟦W1⟧{M}[gv, lv1] ≗ ⟦W2⟧{M}[gv, lv2] →
-       (∀d. ⟦T1⟧{M}[gv, ⫯[d]lv1] ≗ ⟦T2⟧{M}[gv, ⫯[d]lv2]) →
-       ⟦ⓛ{p}W1.T1⟧[gv, lv1] ≗ ⟦ⓛ{p}W2.T2⟧[gv, lv2]
+(* Note: interpretation: extensional abstraction *)
+   mx: ∀p,gv1,gv2,lv1,lv2,W1,W2,T1,T2. ⟦W1⟧{M}[gv1,lv1] ≗ ⟦W2⟧{M}[gv2,lv2] →
+       (∀d. ⟦T1⟧{M}[gv1,⫯[0←d]lv1] ≗ ⟦T2⟧{M}[gv2,⫯[0←d]lv2]) →
+       ⟦ⓛ[p]W1.T1⟧[gv1,lv1] ≗ ⟦ⓛ[p]W2.T2⟧[gv2,lv2]
+}.
+
+record is_injective (M): Prop ≝ {
+(* Note: conjunction: injectivity *)
+   mj: ∀d1,d3,d2,d4. d1 ⊕[Ⓕ] d2 ≗{M} d3 ⊕[Ⓕ] d4 → d2 ≗ d4
 }.
 
 (* Basic properties *********************************************************)
 
 lemma seq_sym (M): is_model M → symmetric … (sq M).
-/3 width=5 by mr, mq/ qed-.
+/3 width=5 by mq, mr/ qed-.
 
 lemma seq_trans (M): is_model M → Transitive … (sq M).
-/3 width=5 by mr, mq/ qed-.
+/3 width=5 by mq, mr/ qed-.
+
+lemma seq_canc_sn (M): is_model M → left_cancellable … (sq M).
+/3 width=3 by seq_trans, seq_sym/ qed-.
+
+lemma seq_canc_dx (M): is_model M → right_cancellable … (sq M).
+/3 width=3 by seq_trans, seq_sym/ qed-.
+
+lemma co_inv_dx (M): is_model M → is_injective M →
+                     ∀p,d1,d3,d2,d4. d1⊕[p]d2 ≗{M} d3⊕[p]d4 → d2 ≗ d4.
+#M #H1M #H2M * #d1 #d3 #d2 #d4 #Hd
+/3 width=5 by mj, mz, mq/
+qed-.
+
+lemma ti_lref_vpush_eq (M): is_model M →
+                            ∀gv,lv,d,i. ⟦#i⟧[gv,⫯[i←d]lv] ≗{M} d.
+#M #HM #gv #lv #d #i
+@(seq_trans … HM) [2: @ml // | skip ]
+>vpush_eq /2 width=1 by mr/
+qed.
+
+lemma ti_lref_vpush_gt (M): is_model M →
+                            ∀gv,lv,d,i. ⟦#(↑i)⟧[gv,⫯[0←d]lv] ≗{M} ⟦#i⟧[gv,lv].
+#M #HM #gv #lv #d #i
+@(mq … HM) [4,5: @(seq_sym … HM) /2 width=2 by ml/ |1,2: skip ]
+>vpush_gt /2 width=1 by mr/
+qed.
+
+(* Basic Forward lemmas *****************************************************)
+
+lemma ti_fwd_mx_dx (M): is_model M → is_injective M →
+                        ∀p,gv1,gv2,lv1,lv2,W1,W2,T1,T2.
+                        ⟦ⓛ[p]W1.T1⟧[gv1,lv1] ≗ ⟦ⓛ[p]W2.T2⟧[gv2,lv2] →
+                        ∀d. ⟦T1⟧{M}[gv1,⫯[0←d]lv1] ≗ ⟦T2⟧{M}[gv2,⫯[0←d]lv2].
+#M #H1M #H2M #p #gv1 #gv2 #lv1 #lv2 #W1 #W2 #T1 #T2 #H12 #d
+@(co_inv_dx … p d d)
+/4 width=5 by mb, mp, mq, mr/
+qed-.
+
+lemma ti_fwd_abbr_dx (M): is_model M → is_injective M →
+                          ∀p,gv1,gv2,lv1,lv2,V1,V2,T1,T2.
+                          ⟦ⓓ[p]V1.T1⟧[gv1,lv1] ≗ ⟦ⓓ[p]V2.T2⟧[gv2,lv2] →
+                          ⟦T1⟧{M}[gv1,⫯[0←⟦V1⟧[gv1,lv1]]lv1] ≗ ⟦T2⟧{M}[gv2,⫯[0←⟦V2⟧[gv2,lv2]]lv2].
+#M #H1M #H2M #p #gv1 #gv2 #lv1 #lv2 #V1 #V2 #T1 #T2 #H12
+@(co_inv_dx … p (⟦V1⟧[gv1,lv1]) (⟦V2⟧[gv2,lv2]))
+/3 width=5 by md, mq/
+qed-.