(* 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;
+ 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;
+ ml: ∀gv,lv,i. ⟦#i⟧{M}[gv,lv] ≗ lv i;
(* Note: interpretation: global reference *)
- mg: ∀gv,lv,l. ⟦§l⟧{M}[gv, lv] ≗ gv l;
+ 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];
+ 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];
+ 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];
+ 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];
+ 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];
+ 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: ∀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]
+ 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 ≝ {