Nuprl Lemma : bag-remove1-property1
∀[T:Type]
∀eq:EqDecider(T). ∀x:T. ∀L:T List.
((∃as,bs:T List. ((L = ((as @ [x]) @ bs) ∈ (T List)) ∧ (bag-remove1(eq;L;x) = (inl (rev(as) @ bs)) ∈ (T List?))))
∨ ((¬(x ∈ L)) ∧ (bag-remove1(eq;L;x) = (inr ⋅ ) ∈ (T List?))))
Proof
Definitions occuring in Statement :
bag-remove1: bag-remove1(eq;bs;a),
l_member: (x ∈ l),
reverse: rev(as),
append: as @ bs,
cons: [a / b],
nil: [],
list: T List,
deq: EqDecider(T),
it: ⋅,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
unit: Unit,
inr: inr x ,
inl: inl x,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
bag-remove1: bag-remove1(eq;bs;a),
or: P ∨ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
prop: ℙ,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
bag_remove1_aux_property,
nil_wf,
append-nil,
reverse_wf,
subtype_rel_list,
top_wf,
equal_wf,
list_wf,
append_wf,
cons_wf,
length_wf,
length-append,
exists_wf,
not_wf,
l_member_wf,
equal-wf-T-base,
unit_wf2,
bag_remove1_aux_wf,
deq_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
cumulativity,
unionElimination,
inlFormation,
productElimination,
dependent_pairFormation,
independent_pairFormation,
promote_hyp,
sqequalRule,
applyEquality,
independent_isectElimination,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
productEquality,
applyLambdaEquality,
unionEquality,
baseClosed,
inrFormation,
universeEquality
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}x:T. \mforall{}L:T List.
((\mexists{}as,bs:T List. ((L = ((as @ [x]) @ bs)) \mwedge{} (bag-remove1(eq;L;x) = (inl (rev(as) @ bs)))))
\mvee{} ((\mneg{}(x \mmember{} L)) \mwedge{} (bag-remove1(eq;L;x) = (inr \mcdot{} ))))
Date html generated:
2018_05_21-PM-09_48_06
Last ObjectModification:
2017_07_26-PM-06_30_33
Theory : bags_2
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