Nuprl Lemma : qsquash_ex
∀P:ℕ ⟶ ℙ. ((∀n:ℕ. Dec(P[n])) ⇒ ⇃∃n:ℕ. P[n] ⇒ (∃n:ℕ. P[n]))
Proof
Definitions occuring in Statement :
qsquash: ⇃T,
nat: ℕ,
decidable: Dec(P),
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
and: P ∧ Q,
guard: {T},
all: ∀x:A. B[x],
uimplies: b supposing a,
prop: ℙ,
squash: ↓T,
implies: P ⇒ Q,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
qsquash: ⇃T,
member: t ∈ T,
exists: ∃x:A. B[x]
Lemmas referenced :
mu-property2,
mu-wf2,
nat_wf,
exists_wf,
squash-from-quotient,
decidable_wf,
all_wf,
qsquash_wf
Rules used in proof :
productElimination,
dependent_functionElimination,
independent_isectElimination,
imageElimination,
independent_functionElimination,
functionExtensionality,
sqequalRule,
extract_by_obid,
introduction,
sqequalHypSubstitution,
dependent_pairFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
rename,
Error :direct_computation,
lambdaFormation,
Error :reverse_direct_computation,
cut,
lemma_by_obid,
Error :direct_computation_hypothesis,
isectElimination,
thin,
hypothesis,
lambdaEquality,
applyEquality,
hypothesisEquality,
because_Cache,
functionEquality,
cumulativity,
universeEquality,
Error :reverse_direct_computation_hypothesis
Latex:
\mforall{}P:\mBbbN{} {}\mrightarrow{} \mBbbP{}. ((\mforall{}n:\mBbbN{}. Dec(P[n])) {}\mRightarrow{} \00D9\mexists{}n:\mBbbN{}. P[n] {}\mRightarrow{} (\mexists{}n:\mBbbN{}. P[n]))
Date html generated:
2017_09_29-PM-05_57_35
Last ObjectModification:
2017_08_30-AM-10_26_45
Theory : int_2
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