Nuprl Lemma : remove1_functionality_wrt_permr
∀s:DSet. ∀a,a':|s|. ∀bs,bs':|s| List. ((a = a' ∈ |s|) ⇒ (bs ≡(|s|) bs') ⇒ ((bs \ a) ≡(|s|) (bs' \ a')))
Proof
Definitions occuring in Statement :
remove1: as \ a,
permr: as ≡(T) bs,
list: T List,
all: ∀x:A. B[x],
implies: P ⇒ Q,
equal: s = t ∈ T,
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],
dset: DSet,
prop: ℙ,
true: True,
decidable: Dec(P),
or: P ∨ Q,
squash: ↓T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
not: ¬A,
false: False
Lemmas referenced :
permr_wf,
set_car_wf,
list_wf,
dset_wf,
remove1_wf,
decidable__assert,
mem_wf,
assert_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
assert_functionality_wrt_uiff,
mem_functionality_wrt_permr,
iff_weakening_uiff,
permr_hd_cancel,
cons_wf,
permr_functionality_wrt_permr,
cons_remove1_permr,
not_mem_remove1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
setElimination,
rename,
hypothesisEquality,
hypothesis,
equalityIsType1,
inhabitedIsType,
because_Cache,
natural_numberEquality,
unionElimination,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
instantiate,
independent_isectElimination,
productElimination,
independent_functionElimination,
voidElimination
Latex:
\mforall{}s:DSet. \mforall{}a,a':|s|. \mforall{}bs,bs':|s| List. ((a = a') {}\mRightarrow{} (bs \mequiv{}(|s|) bs') {}\mRightarrow{} ((bs \mbackslash{} a) \mequiv{}(|s|) (bs' \mbackslash{} a')))
Date html generated:
2019_10_16-PM-01_03_51
Last ObjectModification:
2018_10_08-AM-11_44_33
Theory : list_2
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