Nuprl Lemma : s-comp-nc-0-alt
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ I+i} ]. (s ⋅ (i0) = s ∈ I+j ⟶ I)
Proof
Definitions occuring in Statement :
nc-0: (i0),
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,
prop: ℙ,
squash: ↓T,
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,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s]
Lemmas referenced :
s-comp-nc-0',
equal_wf,
nh-comp_wf,
add-name-com,
add-name_wf,
iff_weakening_equal,
nc-s_wf,
f-subset-add-name1,
f-subset-add-name,
set_wf,
nat_wf,
not_wf,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
hypothesis,
hyp_replacement,
equalitySymmetry,
sqequalRule,
applyEquality,
lambdaEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
setElimination,
rename,
intEquality,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ]. \mforall{}[j:\{j:\mBbbN{}| \mneg{}j \mmember{} I+i\} ]. (s \mcdot{} (i0) = s)
Date html generated:
2017_10_05-AM-01_02_56
Last ObjectModification:
2017_07_28-AM-09_26_29
Theory : cubical!type!theory
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