Nuprl Lemma : set-equal-cons
∀[T:Type]
∀u:T. ∀v,bs:T List.
(set-equal(T;[u / v];bs) ⇐⇒ ∃cs,ds:T List. ((bs = (cs @ [u / ds]) ∈ (T List)) ∧ set-equal(T;v;cs @ ds))) supposing
(no_repeats(T;bs) and
no_repeats(T;[u / v]))
Proof
Definitions occuring in Statement :
set-equal: set-equal(T;x;y),
no_repeats: no_repeats(T;l),
append: as @ bs,
cons: [a / b],
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
set-equal: set-equal(T;x;y),
or: P ∨ Q,
exists: ∃x:A. B[x],
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cand: A c∧ B,
guard: {T},
uiff: uiff(P;Q),
not: ¬A,
false: False
Lemmas referenced :
no_repeats_witness,
cons_wf,
set-equal_wf,
exists_wf,
list_wf,
equal_wf,
append_wf,
length_wf,
length-append,
no_repeats_wf,
cons_member,
l_member_wf,
l_member_decomp,
list_ind_cons_lemma,
list_ind_nil_lemma,
member_append,
or_wf,
iff_wf,
no_repeats_cons,
no_repeats-append,
l_disjoint_cons,
length_wf_nat,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
cumulativity,
hypothesis,
independent_functionElimination,
rename,
because_Cache,
independent_pairFormation,
sqequalRule,
lambdaEquality,
productEquality,
applyLambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
universeEquality,
dependent_functionElimination,
productElimination,
inlFormation,
dependent_pairFormation,
equalitySymmetry,
hyp_replacement,
promote_hyp,
addLevel,
orFunctionality,
impliesFunctionality,
inrFormation,
unionElimination,
independent_isectElimination,
dependent_set_memberEquality,
setElimination
Latex:
\mforall{}[T:Type]
\mforall{}u:T. \mforall{}v,bs:T List.
(set-equal(T;[u / v];bs)
\mLeftarrow{}{}\mRightarrow{} \mexists{}cs,ds:T List. ((bs = (cs @ [u / ds])) \mwedge{} set-equal(T;v;cs @ ds))) supposing
(no\_repeats(T;bs) and
no\_repeats(T;[u / v]))
Date html generated:
2017_04_17-AM-07_37_04
Last ObjectModification:
2017_02_27-PM-04_12_23
Theory : list_1
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