Nuprl Lemma : cons_remove1_permr
∀s:DSet. ∀a:|s|. ∀bs:|s| List. ((↑(a ∈b bs)) ⇒ ([a / (bs \ a)] ≡(|s|) bs))
Proof
Definitions occuring in Statement :
remove1: as \ a,
mem: a ∈b as,
permr: as ≡(T) bs,
cons: [a / b],
list: T List,
assert: ↑b,
all: ∀x:A. B[x],
implies: P ⇒ Q,
dset: DSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
prop: ℙ,
dset: DSet,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
false: False,
infix_ap: x f y,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
bor: p ∨bq
Lemmas referenced :
assert_wf,
mem_wf,
list_wf,
set_car_wf,
dset_wf,
list_induction,
permr_wf,
cons_wf,
remove1_wf,
mem_nil_lemma,
istype-void,
remove1_nil_lemma,
mem_cons_lemma,
remove1_cons_lemma,
set_eq_wf,
uiff_transitivity,
equal-wf-T-base,
bool_wf,
equal_wf,
eqtt_to_assert,
assert_of_dset_eq,
testxxx_lemma,
true_wf,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
permr_weakening,
permr_functionality_wrt_permr,
cons_functionality_wrt_permr,
permr_inversion,
hd_two_swap_permr
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
lambdaEquality_alt,
functionEquality,
applyEquality,
because_Cache,
independent_functionElimination,
isect_memberEquality_alt,
voidElimination,
functionIsType,
inhabitedIsType,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
productElimination,
independent_isectElimination,
independent_pairFormation,
equalityIsType1,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}s:DSet. \mforall{}a:|s|. \mforall{}bs:|s| List. ((\muparrow{}(a \mmember{}\msubb{} bs)) {}\mRightarrow{} ([a / (bs \mbackslash{} a)] \mequiv{}(|s|) bs))
Date html generated:
2019_10_16-PM-01_03_46
Last ObjectModification:
2018_10_08-AM-11_15_18
Theory : list_2
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