### Nuprl Lemma : values-for-distinct_wf

`∀[A,V:Type]. ∀[eq:EqDecider(A)]. ∀[L:(A × V) List].  (values-for-distinct(eq;L) ∈ V List)`

Proof

Definitions occuring in Statement :  values-for-distinct: `values-for-distinct(eq;L)` list: `T List` deq: `EqDecider(T)` uall: `∀[x:A]. B[x]` member: `t ∈ T` product: `x:A × B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` values-for-distinct: `values-for-distinct(eq;L)` prop: `ℙ` all: `∀x:A. B[x]` implies: `P `` Q` outl: `outl(x)` uimplies: `b supposing a` isl: `isl(x)` and: `P ∧ Q` not: `¬A` false: `False` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat: `ℕ` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` guard: `{T}` or: `P ∨ Q` cons: `[a / b]` colength: `colength(L)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` decidable: `Dec(P)` nil: `[]` it: `⋅` sq_type: `SQType(T)` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` eqof: `eqof(d)` deq: `EqDecider(T)` bool: `𝔹` unit: `Unit` btrue: `tt` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` assert: `↑b` true: `True` bfalse: `ff` bnot: `¬bb`
Lemmas referenced :  list_wf deq_wf assert_wf isl_wf unit_wf2 apply-alist_wf map_wf assert_elim bfalse_wf and_wf equal_wf btrue_neq_bfalse remove-repeats_wf strong-subtype-deq-subtype strong-subtype-set2 nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list-cases map_nil_lemma product_subtype_list spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int apply_alist_cons_lemma map_cons_lemma nil_wf cons_wf ifthenelse_wf pi1_wf pi2_wf bool_wf eqtt_to_assert safe-assert-deq eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot subtype_rel_list_set
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution hypothesis sqequalRule axiomEquality equalityTransitivity equalitySymmetry extract_by_obid isectElimination thin productEquality cumulativity hypothesisEquality isect_memberEquality because_Cache universeEquality setEquality lambdaFormation lambdaEquality setElimination rename unionEquality unionElimination addLevel independent_isectElimination levelHypothesis dependent_set_memberEquality independent_pairFormation applyLambdaEquality productElimination independent_functionElimination voidElimination dependent_functionElimination applyEquality intWeakElimination natural_numberEquality dependent_pairFormation int_eqEquality intEquality voidEquality computeAll promote_hyp hypothesis_subsumption addEquality baseClosed instantiate imageElimination independent_pairEquality inlEquality equalityElimination

Latex:
\mforall{}[A,V:Type].  \mforall{}[eq:EqDecider(A)].  \mforall{}[L:(A  \mtimes{}  V)  List].    (values-for-distinct(eq;L)  \mmember{}  V  List)

Date html generated: 2017_04_17-AM-09_11_36
Last ObjectModification: 2017_02_27-PM-05_19_14

Theory : decidable!equality

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