Nuprl Lemma : nc-r'-to-e'
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[j:{i:ℕ| ¬i ∈ J} ]. (f,i=1-j = f,i=j ⋅ r_j ∈ J+j ⟶ I+i)
Proof
Definitions occuring in Statement :
nc-r': g,i=1-j,
nc-e': g,i=j,
nc-r: r_i,
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,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
set_wf,
nat_wf,
not_wf,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
names-hom_wf,
fset_wf,
add-name_wf,
nc-r'_wf,
nc-r_wf,
trivial-member-add-name1,
equal_wf,
nh-id-left,
iff_weakening_equal,
squash_wf,
true_wf,
nh-comp_wf,
nc-r'-r,
nh-comp-assoc,
r-comp-r
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
applyEquality,
intEquality,
independent_isectElimination,
because_Cache,
natural_numberEquality,
hypothesisEquality,
isect_memberEquality,
axiomEquality,
setElimination,
rename,
dependent_functionElimination,
imageElimination,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination,
universeEquality
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ]. \mforall{}[J:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I]. \mforall{}[j:\{i:\mBbbN{}| \mneg{}i \mmember{} J\} ].
(f,i=1-j = f,i=j \mcdot{} r\_j)
Date html generated:
2017_10_05-AM-01_06_06
Last ObjectModification:
2017_07_28-AM-09_27_45
Theory : cubical!type!theory
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