Nuprl Lemma : decidable-exists-int_seg-subtype
∀[i:ℤ]. ∀[j:{i + 1...}]. ∀[P:{i..j-} ⟶ ℙ]. Dec(∃k:{i + 1..j-}. P[k]) ⊆r Dec(∃k:{i..j-}. P[k]) supposing ¬P[i]
Proof
Definitions occuring in Statement :
int_upper: {i...},
int_seg: {i..j-},
decidable: Dec(P),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
exists: ∃x:A. B[x],
not: ¬A,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
true: True,
top: Top,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
decidable-subtype,
exists_wf,
int_seg_wf,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
le-add-cancel,
and_wf,
le_wf,
less_than_wf,
not_wf,
le_reflexive,
decidable__lt,
not-lt-2,
int_upper_wf,
decidable__int_equal,
not-equal-2,
zero-add,
le-add-cancel2,
subtype_base_sq,
int_subtype_base
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
addEquality,
hypothesisEquality,
natural_numberEquality,
setElimination,
rename,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
because_Cache,
independent_isectElimination,
productElimination,
dependent_pairEquality,
dependent_set_memberEquality,
independent_pairFormation,
dependent_functionElimination,
unionElimination,
lambdaFormation,
voidElimination,
independent_functionElimination,
minusEquality,
dependent_pairFormation,
axiomEquality,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
voidEquality,
intEquality,
instantiate
Latex:
\mforall{}[i:\mBbbZ{}]. \mforall{}[j:\{i + 1...\}]. \mforall{}[P:\{i..j\msupminus{}\} {}\mrightarrow{} \mBbbP{}].
Dec(\mexists{}k:\{i + 1..j\msupminus{}\}. P[k]) \msubseteq{}r Dec(\mexists{}k:\{i..j\msupminus{}\}. P[k]) supposing \mneg{}P[i]
Date html generated:
2016_05_13-PM-03_47_38
Last ObjectModification:
2015_12_26-AM-09_58_39
Theory : call!by!value_2
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