Nuprl Lemma : remove-repeats-length-one
∀T:Type. ∀eq:EqDecider(T). ∀L:T List.
(||remove-repeats(eq;L)|| = 1 ∈ ℤ ⇐⇒ ∃x:T. ((x ∈ L) ∧ (∀y:T. y = x ∈ T supposing (y ∈ L))))
Proof
Definitions occuring in Statement :
remove-repeats: remove-repeats(eq;L),
l_member: (x ∈ l),
length: ||as||,
list: T List,
deq: EqDecider(T),
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
so_apply: x[s],
exists: ∃x:A. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
cand: A c∧ B,
set-equal: set-equal(T;x;y),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
sq_type: SQType(T),
guard: {T},
true: True,
cons: [a / b],
bfalse: ff,
uiff: uiff(P;Q),
less_than: a < b,
squash: ↓T,
nat: ℕ,
le: A ≤ B,
subtract: n - m,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b)
Lemmas referenced :
set-equal-remove-repeats,
equal-wf-T-base,
length_wf,
remove-repeats_wf,
exists_wf,
l_member_wf,
all_wf,
isect_wf,
equal_wf,
list_wf,
deq_wf,
hd_wf,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_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_eq_lemma,
int_formula_prop_wf,
hd_member,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
subtype_base_sq,
int_subtype_base,
false_wf,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
member-remove-repeats,
length-one-iff,
intformless_wf,
int_formula_prop_less_lemma,
remove-repeats-no_repeats,
nil_member,
length_wf_nat,
nat_wf,
decidable__lt,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
hypothesis,
intEquality,
cumulativity,
baseClosed,
sqequalRule,
lambdaEquality,
productEquality,
universeEquality,
dependent_pairFormation,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
unionElimination,
natural_numberEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
productElimination,
independent_functionElimination,
addLevel,
instantiate,
levelHypothesis,
promote_hyp,
hypothesis_subsumption,
rename,
isect_memberFormation,
imageElimination,
axiomEquality,
setElimination,
addEquality,
applyEquality,
minusEquality
Latex:
\mforall{}T:Type. \mforall{}eq:EqDecider(T). \mforall{}L:T List.
(||remove-repeats(eq;L)|| = 1 \mLeftarrow{}{}\mRightarrow{} \mexists{}x:T. ((x \mmember{} L) \mwedge{} (\mforall{}y:T. y = x supposing (y \mmember{} L))))
Date html generated:
2017_04_17-AM-09_10_58
Last ObjectModification:
2017_02_27-PM-05_19_07
Theory : decidable!equality
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