Nuprl Lemma : not-assert-bdd-all
∀n:ℕ. ∀P:ℕn ⟶ 𝔹. (¬↑bdd-all(n;i.P[i]) ⇐⇒ ∃i:ℕn. (¬↑P[i]))
Proof
Definitions occuring in Statement :
bdd-all: bdd-all(n;i.P[i]),
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
bool: 𝔹,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
guard: {T},
or: P ∨ Q,
decidable: Dec(P),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
decidable__assert,
decidable__not,
decidable__exists_int_seg,
iff_wf,
bdd-all_wf,
assert-bdd-all,
nat_wf,
bool_wf,
exists_wf,
assert_wf,
int_seg_wf,
all_wf,
not_wf
Rules used in proof :
dependent_pairFormation,
unionElimination,
instantiate,
impliesLevelFunctionality,
dependent_functionElimination,
impliesFunctionality,
productElimination,
allFunctionality,
addLevel,
functionEquality,
voidElimination,
independent_functionElimination,
because_Cache,
functionExtensionality,
applyEquality,
lambdaEquality,
sqequalRule,
hypothesis,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
independent_pairFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
cut
Latex:
\mforall{}n:\mBbbN{}. \mforall{}P:\mBbbN{}n {}\mrightarrow{} \mBbbB{}. (\mneg{}\muparrow{}bdd-all(n;i.P[i]) \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}n. (\mneg{}\muparrow{}P[i]))
Date html generated:
2017_09_29-PM-05_47_59
Last ObjectModification:
2017_09_06-AM-11_35_31
Theory : bool_1
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