Nuprl Lemma : decidable__all_fun
∀[A,B:Type]. ∀[P:(A ⟶ B) ⟶ ℙ]. ((∃a:ℕ. A ~ ℕa) ⇒ (∃b:ℕ. B ~ ℕb) ⇒ (∀f:A ⟶ B. Dec(P[f])) ⇒ Dec(∀f:A ⟶ B. P[f]))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
prop: ℙ,
so_apply: x[s],
nat: ℕ,
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
ext-eq_weakening,
equipollent_weakening_ext-eq,
equipollent_functionality_wrt_equipollent,
equipollent-exp,
exp_wf2,
istype-universe,
equipollent_wf,
istype-nat,
decidable_wf,
int_seg_wf,
indep-function_functionality_wrt_equipollent,
exp_wf4,
decidable__all_finite
Rules used in proof :
independent_isectElimination,
instantiate,
Error :inhabitedIsType,
universeEquality,
Error :productIsType,
applyEquality,
Error :universeIsType,
Error :functionIsType,
sqequalRule,
because_Cache,
rename,
setElimination,
natural_numberEquality,
independent_functionElimination,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
functionEquality,
isectElimination,
extract_by_obid,
introduction,
cut,
thin,
productElimination,
sqequalHypSubstitution,
Error :lambdaFormation_alt,
Error :isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[A,B:Type]. \mforall{}[P:(A {}\mrightarrow{} B) {}\mrightarrow{} \mBbbP{}].
((\mexists{}a:\mBbbN{}. A \msim{} \mBbbN{}a) {}\mRightarrow{} (\mexists{}b:\mBbbN{}. B \msim{} \mBbbN{}b) {}\mRightarrow{} (\mforall{}f:A {}\mrightarrow{} B. Dec(P[f])) {}\mRightarrow{} Dec(\mforall{}f:A {}\mrightarrow{} B. P[f]))
Date html generated:
2019_06_20-PM-02_19_33
Last ObjectModification:
2019_06_06-PM-00_23_51
Theory : equipollence!!cardinality!
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