### Nuprl Lemma : apply-alist-no_repeats

[A,T:Type]. ∀[eq:EqDecider(T)]. ∀[L:(T × A) List].
∀[x:T]. ∀[a:A].  apply-alist(eq;L;x) (inl a) ∈ (A?) supposing (<x, a> ∈ L)
supposing no_repeats(T;map(λp.(fst(p));L))

Proof

Definitions occuring in Statement :  apply-alist: apply-alist(eq;L;x) no_repeats: no_repeats(T;l) l_member: (x ∈ l) map: map(f;as) list: List deq: EqDecider(T) uimplies: supposing a uall: [x:A]. B[x] pi1: fst(t) unit: Unit lambda: λx.A[x] pair: <a, b> product: x:A × B[x] inl: inl x union: left right universe: Type equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a subtype_rel: A ⊆B so_lambda: λ2x.t[x] so_apply: x[s] all: x:A. B[x] top: Top l_member: (x ∈ l) exists: x:A. B[x] cand: c∧ B and: P ∧ Q nat: int_seg: {i..j-} lelt: i ≤ j < k le: A ≤ B prop: not: ¬A implies:  Q false: False guard: {T} ge: i ≥  decidable: Dec(P) or: P ∨ Q satisfiable_int_formula: satisfiable_int_formula(fmla) less_than: a < b squash: T pi1: fst(t) no_repeats: no_repeats(T;l) less_than': less_than'(a;b) true: True iff: ⇐⇒ Q pi2: snd(t)
Lemmas referenced :  apply-alist-cases subtype_rel_list top_wf subtype_rel_product lelt_wf length_wf equal_wf select_wf int_seg_properties nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma pi1_wf int_seg_wf and_wf l_member_wf no_repeats_wf map_wf list_wf deq_wf int_seg_subtype_nat false_wf map-length intformeq_wf int_formula_prop_eq_lemma nat_wf not_wf squash_wf true_wf map_select iff_weakening_equal pi2_wf unit_wf2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesisEquality applyEquality productEquality cumulativity hypothesis independent_isectElimination sqequalRule lambdaEquality lambdaFormation isect_memberEquality voidElimination voidEquality productElimination setElimination rename dependent_set_memberEquality independent_pairFormation independent_functionElimination dependent_functionElimination unionElimination natural_numberEquality dependent_pairFormation int_eqEquality intEquality computeAll imageElimination independent_pairEquality equalityTransitivity equalitySymmetry applyLambdaEquality axiomEquality universeEquality imageMemberEquality baseClosed hyp_replacement inlEquality

Latex:
\mforall{}[A,T:Type].  \mforall{}[eq:EqDecider(T)].  \mforall{}[L:(T  \mtimes{}  A)  List].
\mforall{}[x:T].  \mforall{}[a:A].    apply-alist(eq;L;x)  =  (inl  a)  supposing  (<x,  a>  \mmember{}  L)
supposing  no\_repeats(T;map(\mlambda{}p.(fst(p));L))

Date html generated: 2017_09_29-PM-06_04_26
Last ObjectModification: 2017_07_26-PM-02_53_04

Theory : decidable!equality

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