Nuprl Lemma : int_lt_to_int_upper
∀i:ℤ. ∀[A:{i + 1...} ⟶ ℙ]. ({∀j:ℤ. A[j] supposing i < j} ⇐⇒ {∀j:{i + 1...}. A[j]})
Proof
Definitions occuring in Statement :
int_upper: {i...},
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
guard: {T},
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
int_upper: {i...},
uimplies: b supposing a,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
decidable__lt,
false_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
le-add-cancel,
int_upper_wf,
all_wf,
isect_wf,
less_than_wf,
decidable__le,
not-le-2,
less-iff-le,
add-associates,
zero-add,
le-add-cancel2,
le_wf,
member-less_than
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
isect_memberFormation,
independent_pairFormation,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
independent_isectElimination,
productElimination,
lemma_by_obid,
unionElimination,
voidElimination,
independent_functionElimination,
isectElimination,
addEquality,
natural_numberEquality,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
intEquality,
because_Cache,
minusEquality,
dependent_set_memberEquality,
introduction,
functionEquality,
cumulativity,
universeEquality
Latex:
\mforall{}i:\mBbbZ{}. \mforall{}[A:\{i + 1...\} {}\mrightarrow{} \mBbbP{}]. (\{\mforall{}j:\mBbbZ{}. A[j] supposing i < j\} \mLeftarrow{}{}\mRightarrow{} \{\mforall{}j:\{i + 1...\}. A[j]\})
Date html generated:
2016_05_13-PM-04_02_39
Last ObjectModification:
2015_12_26-AM-10_56_45
Theory : int_1
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