Nuprl Lemma : nh-comp-nc-m-eq3
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ I+i} ]. (s,i=j ⋅ (j0) = m(i;j) ⋅ (j0) ∈ I+i ⟶ I+i)
Proof
Definitions occuring in Statement :
nc-e': g,i=j,
nc-m: m(i;j),
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,
squash: ↓T,
prop: ℙ,
uimplies: b supposing a,
all: ∀x:A. B[x],
true: True,
subtype_rel: A ⊆r B,
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]
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
names-hom_wf,
add-name_wf,
nc-m-nc-0,
nh-comp_wf,
nc-0_wf,
nc-e'_wf,
nc-s_wf,
f-subset-add-name,
iff_weakening_equal,
nc-e'-lemma2,
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,
equalitySymmetry,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
universeEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
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,i=j \mcdot{} (j0) = m(i;j) \mcdot{} (j0))
Date html generated:
2017_10_05-AM-01_04_44
Last ObjectModification:
2017_07_28-AM-09_27_09
Theory : cubical!type!theory
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