Nuprl Lemma : member-values-for-distinct
∀[A,V:Type].
∀eq:EqDecider(A). ∀L:(A × V) List. ∀a:A.
((a ∈ map(λp.(fst(p));L)) ⇒ (∃v:V. ((v ∈ values-for-distinct(eq;L)) ∧ (<a, v> ∈ L))))
Proof
Definitions occuring in Statement :
values-for-distinct: values-for-distinct(eq;L),
l_member: (x ∈ l),
map: map(f;as),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
pi1: fst(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
lambda: λx.A[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
prop: ℙ,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
nat: ℕ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
uimplies: b supposing a,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
and_wf,
iff_weakening_equal,
true_wf,
squash_wf,
select_member,
int_formula_prop_eq_lemma,
int_formula_prop_less_lemma,
intformeq_wf,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
values-for-distinct_wf,
select_wf,
lelt_wf,
values-for-distinct-property,
deq_wf,
list_wf,
l_member_wf,
pi1_wf,
map_wf,
member-remove-repeats
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
dependent_functionElimination,
hypothesisEquality,
productEquality,
lambdaEquality,
sqequalRule,
hypothesis,
productElimination,
independent_functionElimination,
universeEquality,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
cumulativity,
independent_isectElimination,
natural_numberEquality,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
applyEquality,
imageElimination,
independent_pairEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[A,V:Type].
\mforall{}eq:EqDecider(A). \mforall{}L:(A \mtimes{} V) List. \mforall{}a:A.
((a \mmember{} map(\mlambda{}p.(fst(p));L)) {}\mRightarrow{} (\mexists{}v:V. ((v \mmember{} values-for-distinct(eq;L)) \mwedge{} (<a, v> \mmember{} L))))
Date html generated:
2016_05_14-PM-03_28_01
Last ObjectModification:
2016_01_14-PM-11_21_40
Theory : decidable!equality
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