Nuprl Lemma : weakly-infinite-cases
∀[S:ℕ ⟶ ℙ]. (w∃∞x.S[x] ⇒ (∀[A:ℕ ⟶ ℙ]. ((∀n:ℕ. (A[n] ⇒ S[n])) ⇒ (¬¬(w∃∞n.A[n] ∨ w∃∞n.S[n] ∧ (¬A[n]))))))
Proof
Definitions occuring in Statement :
weakly-infinite: w∃∞p.S[p],
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
not: ¬A,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
subtype_rel: A ⊆r B,
or: P ∨ Q,
all: ∀x:A. B[x],
nat: ℕ,
exists: ∃x:A. B[x],
uiff: uiff(P;Q),
uimplies: b supposing a,
weakly-infinite: w∃∞p.S[p],
iff: P ⇐⇒ Q,
guard: {T},
decidable: Dec(P),
cand: A c∧ B,
le: A ≤ B,
rev_implies: P ⇐ Q,
subtract: n - m,
top: Top,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
not_wf,
or_wf,
weakly-infinite_wf,
nat_wf,
all_wf,
false_wf,
exists_wf,
less_than_wf,
not_over_exists,
minimal-double-negation-hyp-elim,
minimal-not-not-excluded-middle,
dneg_elim_a,
double-negation-hyp-elim,
decidable__le,
le_weakening,
le_wf,
le_reflexive,
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,
less_than_transitivity2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
hypothesis,
sqequalHypSubstitution,
independent_functionElimination,
voidElimination,
extract_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
productEquality,
universeEquality,
functionEquality,
dependent_functionElimination,
because_Cache,
cumulativity,
isect_memberEquality,
inlFormation,
setElimination,
rename,
productElimination,
independent_isectElimination,
unionElimination,
dependent_pairFormation,
addLevel,
impliesFunctionality,
levelHypothesis,
inrFormation,
independent_pairFormation,
addEquality,
natural_numberEquality,
voidEquality,
intEquality,
minusEquality,
promote_hyp
Latex:
\mforall{}[S:\mBbbN{} {}\mrightarrow{} \mBbbP{}]
(w\mexists{}\minfty{}x.S[x] {}\mRightarrow{} (\mforall{}[A:\mBbbN{} {}\mrightarrow{} \mBbbP{}]. ((\mforall{}n:\mBbbN{}. (A[n] {}\mRightarrow{} S[n])) {}\mRightarrow{} (\mneg{}\mneg{}(w\mexists{}\minfty{}n.A[n] \mvee{} w\mexists{}\minfty{}n.S[n] \mwedge{} (\mneg{}A[n]))))))
Date html generated:
2017_09_29-PM-05_47_37
Last ObjectModification:
2017_07_26-PM-01_25_15
Theory : bar-induction
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