<section9 name="foreword">Foreword</section9>
<body>
The formal systems of the λδ (\lambda\delta) family are typed λ-calculi aiming to support
- the foundational frameworks for Mathematics that require an underlying specification language
- (for example the <link to="http://www.math.unipd.it/~maietti/">Minimalist Foundation</link>
- and its predecessors).
+ the foundational frameworks for Mathematics that require an underlying specification language,
+ for example the
+ <link to="http://www.math.unipd.it/~maietti/">Minimalist Foundation (MF)</link>
+ and its predecessors.
</body>
<body>
The λδ family is developed within the
- <link to="http://helm.cs.unibo.it/">Hypertextual Electronic Library of Mathematics</link>
+ <link to="http://helm.cs.unibo.it/">Hypertextual Electronic Library of Mathematics (HELM)</link>
as a set of machine-checked digital specifications.
</body>
<body>
(revised <notice class="alpha" text="2012-09"/>).
</body>
<body>
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<topitem name="C10">
- Matthias Weber:
+ M. Weber:
<notice class="alpha">An extended type system with lambda-typed lambda-expressions</notice>
(2018). Technical report. Faculty of Computer Science, Technical University of Berlin.
</topitem>
<topitem name="C9">
- Matthias Weber:
+ M. Weber:
<notice class="alpha">An extended type system with lambda-typed lambda-expressions (extended version)</notice>
(2018). Technical report. Faculty of Computer Science, Technical University of Berlin.
</topitem>
(* Properties with evaluation evaluation lift *******************************)
+theorem vlift_swap (M): ∀i1,i2. i1 ≤ i2 →
+ ∀lv,d1,d2. ⫯[i1←d1] ⫯[i2←d2] lv ≐{?,dd M} ⫯[↑i2←d2] ⫯[i1←d1] lv.
+#M #i1 #i2 #Hi12 #lv #d1 #d2 #j
+elim (lt_or_eq_or_gt j i1) #Hji1 destruct
+[ >vlift_lt // >vlift_lt /2 width=3 by lt_to_le_to_lt/
+ >vlift_lt /3 width=3 by lt_S, lt_to_le_to_lt/ >vlift_lt //
+| >vlift_eq >vlift_lt /2 width=1 by monotonic_le_plus_l/ >vlift_eq //
+| >vlift_gt // elim (lt_or_eq_or_gt (↓j) i2) #Hji2 destruct
+ [ >vlift_lt // >vlift_lt /2 width=1 by lt_minus_to_plus/ >vlift_gt //
+ | >vlift_eq <(lt_succ_pred … Hji1) >vlift_eq //
+ | >vlift_gt // >vlift_gt /2 width=1 by lt_minus_to_plus_r/ >vlift_gt /2 width=3 by le_to_lt_to_lt/
+ ]
+]
+qed-.
+
lemma vlift_comp (M): ∀i. compatible_3 … (vlift M i) (sq M) (veq M) (veq M).
#m #i #d1 #d2 #Hd12 #lv1 #lv2 #HLv12 #j
elim (lt_or_eq_or_gt j i) #Hij destruct
-[ >(vlift_lt … Hij) >(vlift_lt … Hij) //
-| >(vlift_eq …) >(vlift_eq …) //
-| >(vlift_gt … Hij) >(vlift_gt … Hij) //
+[ >vlift_lt // >vlift_lt //
+| >vlift_eq >vlift_eq //
+| >vlift_gt // >vlift_gt //
]
-qed.
+qed-.
(* Properies with term interpretation ***************************************)
| /4 width=5 by seq_sym, me, mr/
]
qed.
+
+lemma ti_ext_l (M): is_model M →
+ ∀T,gv,lv1,lv2. lv1 ≐ lv2 →
+ ⟦T⟧[gv, lv1] ≗{M} ⟦T⟧[gv, lv2].
+/3 width=1 by ti_comp_l, ext_veq/ qed.
--- /dev/null
+(**************************************************************************)
+(* ___ *)
+(* ||M|| *)
+(* ||A|| A project by Andrea Asperti *)
+(* ||T|| *)
+(* ||I|| Developers: *)
+(* ||T|| The HELM team. *)
+(* ||A|| http://helm.cs.unibo.it *)
+(* \ / *)
+(* \ / This file is distributed under the terms of the *)
+(* v GNU General Public License Version 2 *)
+(* *)
+(**************************************************************************)
+
+include "basic_2/relocation/lifts.ma".
+include "apps_2/models/veq.ma".
+
+(* EVALUATION EQUIVALENCE **************************************************)
+
+lemma pippo (M) (gv): is_model M → is_extensional M →
+ ∀f,T1,T2. ⬆*[f] T1 ≘ T2 → ∀n. ⫯*[n] 𝐔❴1❵ = f →
+ ∀lv,d. ⟦T1⟧[gv, lv] ≗{M} ⟦T2⟧[gv, ⫯[n←d]lv].
+#M #gv #H1M #H2M #f #T1 #T2 #H elim H -f -T1 -T2
+[ /4 width=3 by seq_trans, seq_sym, ms/
+| #f #i1 #i2 #Hi12 #n #Hn #lv #d
+ @(mr … H1M) [4,5: @(seq_sym … H1M) @(ml … H1M) |1,2: skip ]
+| /4 width=3 by seq_trans, seq_sym, mg/
+| #f #p * #V1 #V2 #T1 #T2 #_ #_ #IHV #IHT #n #Hn #lv #d
+ [ @(mr … H1M) [4,5: @(seq_sym … H1M) @(md … H1M) |1,2: skip ]
+ @(seq_trans … H1M)
+ [2: @(ti_comp_l … H1M) | skip ]
+ [2: @(vlift_comp … lv lv) | skip ]
+ [3: /2 width=1 by veq_refl/ ]
+ [2: @(IHV … d) // | skip ]
+ @(seq_trans … H1M) [2: @(IHT … d) // | skip ]
+ /4 width=1 by seq_sym, ti_ext_l, vlift_swap/
+ | @mx /2 width=1 by/ #d0 @(seq_trans … H1M)
+ [3: @(seq_sym … H1M) @(ti_ext_l … H1M) | skip ]
+ [2: @vlift_swap // | skip ]
+ /2 width=1 by/
+ ]
+| #f * #V1 #v2 #T1 #T2 #_ #_ #IHV #IHT #n #Hn #lv #d
+ [ /4 width=5 by seq_sym, ma, mc, mr/
+ | /4 width=5 by seq_sym, me, mr/
+ ]
+]
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