Nuprl Lemma : remove-combine-implies-member2
∀[T:Type]. ∀cmp:T ⟶ ℤ. ∀x:T. ∀l:T List. ((¬((cmp x) = 0 ∈ ℤ)) ⇒ (x ∈ l) ⇒ (x ∈ remove-combine(cmp;l)))
Proof
Definitions occuring in Statement :
remove-combine: remove-combine(cmp;l),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
false: False,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
or: P ∨ Q,
not: ¬A,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b
Lemmas referenced :
list_induction,
not_wf,
equal-wf-T-base,
l_member_wf,
remove-combine_wf,
list_wf,
false_wf,
nil_member,
remove-combine-nil,
nil_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
and_wf,
equal_wf,
or_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
lt_int_wf,
assert_of_lt_int,
cons_member,
less_than_wf,
remove-combine-cons,
cons_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
functionEquality,
intEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
cumulativity,
baseClosed,
hypothesis,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
addLevel,
impliesFunctionality,
productElimination,
isect_memberEquality,
voidEquality,
rename,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
setElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
inlFormation,
inrFormation,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}cmp:T {}\mrightarrow{} \mBbbZ{}. \mforall{}x:T. \mforall{}l:T List. ((\mneg{}((cmp x) = 0)) {}\mRightarrow{} (x \mmember{} l) {}\mRightarrow{} (x \mmember{} remove-combine(cmp;l)))
Date html generated:
2017_04_17-AM-08_30_28
Last ObjectModification:
2017_02_27-PM-04_51_54
Theory : list_1
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