Nuprl Lemma : bag_remove1_aux_property
∀[T:Type]
∀eq:EqDecider(T). ∀x:T. ∀L,checked:T List.
((∃as,bs:T List
((L = ((as @ [x]) @ bs) ∈ (T List))
∧ (bag_remove1_aux(eq;checked;x;L) = (inl ((rev(as) @ checked) @ bs)) ∈ (T List?))))
∨ ((¬(x ∈ L)) ∧ (bag_remove1_aux(eq;checked;x;L) = (inr ⋅ ) ∈ (T List?))))
Proof
Definitions occuring in Statement :
bag_remove1_aux: bag_remove1_aux(eq;checked;a;as),
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],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
top: Top,
so_apply: x[s],
implies: P ⇒ Q,
bag_remove1_aux: bag_remove1_aux(eq;checked;a;as),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
deq: EqDecider(T),
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
eqof: eqof(d),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
cand: A c∧ B,
append: as @ bs,
list_ind: list_ind,
reverse: rev(as),
rev-append: rev(as) + bs,
list_accum: list_accum,
nil: [],
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
all_wf,
list_wf,
or_wf,
exists_wf,
equal_wf,
append_wf,
cons_wf,
nil_wf,
length_wf,
length-append,
not_wf,
l_member_wf,
equal-wf-T-base,
unit_wf2,
bag_remove1_aux_wf,
list_ind_nil_lemma,
list_ind_cons_lemma,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
deq_wf,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
btrue_neq_bfalse,
it_wf,
equal-wf-base-T,
length_of_nil_lemma,
reverse_wf,
and_wf,
length_of_cons_lemma,
squash_wf,
true_wf,
iff_weakening_equal,
reverse-cons,
append_assoc,
cons_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesisEquality,
hypothesis,
productEquality,
applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
unionEquality,
baseClosed,
independent_functionElimination,
dependent_functionElimination,
rename,
applyEquality,
setElimination,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
universeEquality,
inrFormation,
independent_pairFormation,
inrEquality,
inlEquality,
inlFormation,
dependent_set_memberEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
hyp_replacement
Latex:
\mforall{}[T:Type]
\mforall{}eq:EqDecider(T). \mforall{}x:T. \mforall{}L,checked:T List.
((\mexists{}as,bs:T List
((L = ((as @ [x]) @ bs))
\mwedge{} (bag\_remove1\_aux(eq;checked;x;L) = (inl ((rev(as) @ checked) @ bs)))))
\mvee{} ((\mneg{}(x \mmember{} L)) \mwedge{} (bag\_remove1\_aux(eq;checked;x;L) = (inr \mcdot{} ))))
Date html generated:
2018_05_21-PM-09_48_03
Last ObjectModification:
2017_07_26-PM-06_30_31
Theory : bags_2
Home
Index