Nuprl Lemma : fset-member-add-name
∀I:fset(ℕ). ∀i,j:ℕ. uiff(j ∈ I+i;(j = i ∈ ℤ) ∨ j ∈ I)
Proof
Definitions occuring in Statement :
add-name: I+i,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
or: P ∨ Q,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
add-name: I+i,
names-deq: NamesDeq,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
nat: ℕ,
subtype_rel: A ⊆r B,
sq_stable: SqStable(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
guard: {T},
prop: ℙ,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
decidable__equal_nat,
fset_wf,
uiff_wf,
member-fset-add,
fset-add_wf,
fset-member_witness,
strong-subtype-self,
strong-subtype-set3,
strong-subtype-deq-subtype,
or_wf,
int_subtype_base,
le_wf,
set_subtype_base,
subtype_base_sq,
decidable__fset-member,
decidable__equal_int,
int-deq_wf,
nat_wf,
fset-member_wf,
equal_wf,
decidable__or,
sq_stable_from_decidable
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
independent_pairFormation,
isect_memberFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
independent_functionElimination,
intEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
dependent_functionElimination,
introduction,
unionElimination,
inlFormation,
instantiate,
cumulativity,
independent_isectElimination,
lambdaEquality,
natural_numberEquality,
inrFormation,
imageMemberEquality,
baseClosed,
imageElimination,
addLevel,
productElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}I:fset(\mBbbN{}). \mforall{}i,j:\mBbbN{}. uiff(j \mmember{} I+i;(j = i) \mvee{} j \mmember{} I)
Date html generated:
2016_05_18-PM-00_00_01
Last ObjectModification:
2016_01_18-PM-01_03_50
Theory : cubical!type!theory
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