Nuprl Lemma : values-for-distinct-nil
∀[A,V:Type]. ∀[eq:EqDecider(A)]. ∀[L:(A × V) List].
uiff(values-for-distinct(eq;L) = [] ∈ (V List);L = [] ∈ ((A × V) List))
Proof
Definitions occuring in Statement :
values-for-distinct: values-for-distinct(eq;L),
nil: [],
list: T List,
deq: EqDecider(T),
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
pi1: fst(t),
deq: EqDecider(T),
ge: i ≥ j ,
le: A ≤ B,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
values-for-distinct: values-for-distinct(eq;L)
Lemmas referenced :
values-for-distinct-property,
equal-wf-T-base,
list_wf,
values-for-distinct_wf,
length_wf_nat,
equal_wf,
nat_wf,
length_wf,
remove-repeats_wf,
map_wf,
pi1_wf,
length_of_nil_lemma,
list_induction,
equal-wf-base-T,
map_nil_lemma,
remove_repeats_nil_lemma,
nil_wf,
equal-wf-base,
map_cons_lemma,
remove_repeats_cons_lemma,
length_of_cons_lemma,
non_neg_length,
filter_wf5,
l_member_wf,
bnot_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_formula_prop_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
productElimination,
independent_pairFormation,
cumulativity,
baseClosed,
productEquality,
because_Cache,
sqequalRule,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_set_memberEquality,
hyp_replacement,
Error :applyLambdaEquality,
intEquality,
setElimination,
rename,
lambdaEquality,
functionEquality,
independent_functionElimination,
voidElimination,
voidEquality,
lambdaFormation,
applyEquality,
setEquality,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
addEquality
Latex:
\mforall{}[A,V:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[L:(A \mtimes{} V) List]. uiff(values-for-distinct(eq;L) = [];L = [])
Date html generated:
2016_10_21-AM-10_39_41
Last ObjectModification:
2016_07_12-AM-05_49_54
Theory : decidable!equality
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