Nuprl Lemma : markov-streamless-function
(∀P:ℕ ⟶ ℙ. ((∀m:ℕ. ((P m) ∨ (¬(P m)))) ⇒ (¬(∀m:ℕ. (¬(P m)))) ⇒ (∃m:ℕ. (P m))))
⇒ (∀A,B:Type. ((∃a:ℕ. A ~ ℕa) ⇒ streamless(B) ⇒ streamless(A ⟶ B)))
Proof
Definitions occuring in Statement :
streamless: streamless(T),
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s],
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
guard: {T},
equipollent: A ~ B,
decidable: Dec(P),
or: P ∨ Q,
biject: Bij(A;B;f),
surject: Surj(A;B;f),
squash: ↓T,
true: True,
uimplies: b supposing a,
not: ¬A,
false: False
Lemmas referenced :
markov-streamless-iff,
streamless_wf,
exists_wf,
nat_wf,
equipollent_wf,
int_seg_wf,
all_wf,
or_wf,
not_wf,
equipollent_inversion,
decidable__all_int_seg,
equal_wf,
squash_wf,
true_wf,
iff_weakening_equal,
and_wf,
exp_wf4,
exp_wf2,
equipollent_functionality_wrt_equipollent2,
equipollent-exp,
equipollent_functionality_wrt_equipollent,
function_functionality_wrt_equipollent_right,
equipollent_weakening_ext-eq,
ext-eq_weakening,
function_functionality_wrt_equipollent_left
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
independent_functionElimination,
thin,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
productElimination,
functionEquality,
cumulativity,
independent_pairFormation,
isectElimination,
sqequalRule,
lambdaEquality,
natural_numberEquality,
setElimination,
rename,
universeEquality,
instantiate,
applyEquality,
functionExtensionality,
because_Cache,
unionElimination,
inlFormation,
inrFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
independent_isectElimination,
dependent_set_memberEquality,
applyLambdaEquality,
voidElimination,
dependent_pairFormation
Latex:
(\mforall{}P:\mBbbN{} {}\mrightarrow{} \mBbbP{}. ((\mforall{}m:\mBbbN{}. ((P m) \mvee{} (\mneg{}(P m)))) {}\mRightarrow{} (\mneg{}(\mforall{}m:\mBbbN{}. (\mneg{}(P m)))) {}\mRightarrow{} (\mexists{}m:\mBbbN{}. (P m))))
{}\mRightarrow{} (\mforall{}A,B:Type. ((\mexists{}a:\mBbbN{}. A \msim{} \mBbbN{}a) {}\mRightarrow{} streamless(B) {}\mRightarrow{} streamless(A {}\mrightarrow{} B)))
Date html generated:
2018_05_21-PM-09_03_34
Last ObjectModification:
2017_07_26-PM-06_26_21
Theory : general
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