update in apps_2

[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 346917ff80f7ffddc9248790e3666f707980de4e..b3acfc26241778918dd58a40b33a411e3c3bdd55 100644 (file)
--- a/matita/matita/contribs/lambdadelta/apps_2/models/model_props.ma
+++ b/matita/matita/contribs/lambdadelta/apps_2/models/model_props.ma
@@ -21,6 +21,8 @@ record is_model (M): Prop ≝ {
    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 *)
@@ -29,27 +31,36 @@ record is_model (M): Prop ≝ {
    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: local δ-equivalence *)
-   md: ∀gv,lv,p,V,T. ⟦ⓓ{p}V.T⟧{M}[gv, lv] ≗ ⟦T⟧[gv, ⫯[0←⟦V⟧[gv, lv]]lv];
-(* Note: interpretation: intensional abstraction *)
-   mi: ∀gv1,gv2,lv1,lv2,p,W,T. ⟦W⟧{M}[gv1, lv1] ≗ ⟦W⟧{M}[gv2, lv2] →
+(* 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: ∀gv,lv,d,p,W,T. d @ ⟦ⓛ{p}W.T⟧{M}[gv, lv] ≗ ⟦T⟧[gv, ⫯[0←d]lv]
+   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 ≝ {
 (* Note: interpretation: extensional abstraction *)
-   mx: ∀gv1,gv2,lv1,lv2,p,W1,W2,T1,T2. ⟦W1⟧{M}[gv1, lv1] ≗ ⟦W2⟧{M}[gv2, lv2] →
+   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).
@@ -64,6 +75,12 @@ lemma seq_canc_sn (M): is_model M → left_cancellable … (sq M).
 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
@@ -80,10 +97,20 @@ qed.
 
 (* Basic Forward lemmas *****************************************************)
 
-lemma ti_fwd_mx_dx (M): is_model M →
-                        ∀gv1,gv2,lv1,lv2,p,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 #HM #gv1 #gv2 #lv1 #lv2 #p #W1 #W2 #T1 #T2 #H12 #d
-@(mq … HM) /3 width=5 by mb, mp, mr/
+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-.