Nuprl Lemma : cons_member
∀[T:Type]. ∀l:T List. ∀a,x:T. ((x ∈ [a / l]) ⇐⇒ (x = a ∈ T) ∨ (x ∈ l))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
or: P ∨ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
sq_stable: SqStable(P),
uimplies: b supposing a,
prop: ℙ,
nat: ℕ,
cand: A c∧ B,
member: t ∈ T,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
l_member: (x ∈ l),
subtract: n - m,
true: True,
less_than': less_than'(a;b),
top: Top,
subtype_rel: A ⊆r B,
false: False,
not: ¬A,
nequal: a ≠ b ∈ T ,
le: A ≤ B,
sq_type: SQType(T),
guard: {T},
so_apply: x[s],
so_lambda: λ2x.t[x],
uiff: uiff(P;Q),
decidable: Dec(P),
cons: [a / b],
select: L[n],
less_than: a < b,
nat_plus: ℕ+
Lemmas referenced :
list_wf,
sq_stable__le,
select_wf,
cons_wf,
length_wf,
less_than_wf,
nat_wf,
minus-zero,
minus-add,
add-commutes,
condition-implies-le,
le-add-cancel,
zero-add,
add-zero,
istype-int,
istype-void,
add-associates,
add_functionality_wrt_le,
not-equal-2,
not-lt-2,
istype-false,
decidable__lt,
neg_assert_of_eq_int,
le_weakening,
int_subtype_base,
le_wf,
set_subtype_base,
subtype_base_sq,
assert_of_eq_int,
eq_int_wf,
decidable__assert,
length_of_cons_lemma,
le_weakening2,
iff_weakening_equal,
subtype_rel_self,
le-add-cancel2,
select_cons_tl,
true_wf,
squash_wf,
equal_wf,
add-swap,
minus-minus,
minus-one-mul-top,
minus-one-mul,
less-iff-le,
not-le-2,
decidable__le,
subtract_wf,
length_wf_nat,
add_nat_plus,
add-subtract-cancel,
select-cons-tl
Rules used in proof :
universeEquality,
Error :unionIsType,
imageElimination,
baseClosed,
imageMemberEquality,
independent_functionElimination,
natural_numberEquality,
independent_isectElimination,
because_Cache,
cumulativity,
Error :inhabitedIsType,
Error :equalityIsType1,
hypothesisEquality,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
extract_by_obid,
introduction,
cut,
Error :universeIsType,
Error :productIsType,
independent_pairFormation,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
computationStep,
sqequalTransitivity,
sqequalReflexivity,
sqequalRule,
sqequalSubstitution,
minusEquality,
Error :isect_memberEquality_alt,
applyEquality,
addEquality,
voidElimination,
Error :dependent_set_memberEquality_alt,
equalitySymmetry,
equalityTransitivity,
Error :lambdaEquality_alt,
intEquality,
instantiate,
unionElimination,
dependent_functionElimination,
productElimination,
Error :inlFormation_alt,
Error :dependent_pairFormation_alt,
Error :inrFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}l:T List. \mforall{}a,x:T. ((x \mmember{} [a / l]) \mLeftarrow{}{}\mRightarrow{} (x = a) \mvee{} (x \mmember{} l))
Date html generated:
2019_06_20-PM-00_41_03
Last ObjectModification:
2019_02_27-PM-04_21_33
Theory : list_0
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