Nuprl Lemma : fset-add-remove
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[s:fset(T)]. ∀[x:T]. fset-add(eq;x;fset-remove(eq;x;s)) = s ∈ fset(T) supposing x ∈ s
Proof
Definitions occuring in Statement :
fset-add: fset-add(eq;x;s),
fset-remove: fset-remove(eq;y;s),
fset-member: a ∈ s,
fset: fset(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
guard: {T},
cand: A c∧ B,
deq: EqDecider(T)
Lemmas referenced :
fset-extensionality,
fset-add_wf,
fset-remove_wf,
fset-member_witness,
or_wf,
equal_wf,
fset-member_wf,
not_wf,
member-fset-remove,
uiff_wf,
member-fset-add,
fset_wf,
deq_wf,
and_wf,
deq_property,
assert_wf,
decidable__assert,
decidable_functionality,
iff_weakening_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
cumulativity,
hypothesis,
productElimination,
independent_isectElimination,
independent_pairFormation,
independent_functionElimination,
productEquality,
rename,
addLevel,
orFunctionality,
dependent_functionElimination,
sqequalRule,
independent_pairEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
universeEquality,
unionElimination,
dependent_set_memberEquality,
applyEquality,
lambdaEquality,
setElimination,
setEquality,
hyp_replacement,
Error :applyLambdaEquality,
inlFormation,
inrFormation
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[s:fset(T)]. \mforall{}[x:T].
fset-add(eq;x;fset-remove(eq;x;s)) = s supposing x \mmember{} s
Date html generated:
2016_10_21-AM-10_44_15
Last ObjectModification:
2016_07_12-AM-05_51_07
Theory : finite!sets
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