Nuprl Lemma : bag-remove1-non-member
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:T]. ∀[bs:bag(T)]. bag-remove1(eq;bs;x) = (inr ⋅ ) ∈ (bag(T)?) supposing ¬x ↓∈ bs
Proof
Definitions occuring in Statement :
bag-remove1: bag-remove1(eq;bs;a),
bag-member: x ↓∈ bs,
bag: bag(T),
deq: EqDecider(T),
it: ⋅,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
unit: Unit,
inr: inr x ,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
false: False,
prop: ℙ,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_or: a ↓∨ b,
uiff: uiff(P;Q)
Lemmas referenced :
bag-remove1-property,
bag-member_wf,
istype-void,
bag_wf,
istype-universe,
deq_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
bag-member-append,
single-bag_wf,
bag-member-single
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
unionElimination,
independent_functionElimination,
productElimination,
voidElimination,
hypothesis,
sqequalRule,
functionIsType,
universeIsType,
isect_memberEquality_alt,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
universeEquality,
applyEquality,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination,
inlFormation_alt
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:T]. \mforall{}[bs:bag(T)].
bag-remove1(eq;bs;x) = (inr \mcdot{} ) supposing \mneg{}x \mdownarrow{}\mmember{} bs
Date html generated:
2019_10_16-AM-11_30_49
Last ObjectModification:
2018_10_11-AM-09_35_53
Theory : bags_2
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