Nuprl Lemma : nc-s-i1-j1
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[j:{j:ℕ| ¬j ∈ I} ]. ((i1) = s ⋅ (i1) ⋅ (j1) ∈ I ⟶ I+i)
Proof
Definitions occuring in Statement :
nc-1: (i1),
nc-s: s,
add-name: I+i,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
not: ¬A,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x]
Lemmas referenced :
nc-1_wf,
add-name_wf,
equal_wf,
squash_wf,
true_wf,
names-hom_wf,
fset_wf,
nat_wf,
add-name-com,
iff_weakening_equal,
set_wf,
not_wf,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
nh-comp-assoc,
nc-s_wf,
f-subset-add-name,
nh-comp_wf,
nc-1-s-commute,
nh-id-left,
s-comp-nc-1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
instantiate,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
because_Cache,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
hyp_replacement,
intEquality,
isect_memberEquality,
axiomEquality,
dependent_functionElimination
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\mBbbN{}]. \mforall{}[j:\{j:\mBbbN{}| \mneg{}j \mmember{} I\} ]. ((i1) = s \mcdot{} (i1) \mcdot{} (j1))
Date html generated:
2017_10_05-AM-01_06_41
Last ObjectModification:
2017_07_28-AM-09_28_00
Theory : cubical!type!theory
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