Nuprl Lemma : decidable__connection
∀[T:Type]. ((∀x,y:T. Dec(x = y ∈ T)) ⇒ finite-type(T) ⇒ (∀f:T ⟶ T. ∀a,b:T. Dec(∃n:ℕ. (b = (f^n a) ∈ T))))
Proof
Definitions occuring in Statement :
finite-type: finite-type(T),
fun_exp: f^n,
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
orbit-exists,
finite-type_wf,
all_wf,
decidable_wf,
equal_wf,
decidable_functionality,
exists_wf,
nat_wf,
fun_exp_wf,
l_member_wf,
decidable__l_member
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
dependent_functionElimination,
productElimination,
functionEquality,
sqequalRule,
lambdaEquality,
universeEquality,
applyEquality,
independent_pairFormation,
because_Cache
Latex:
\mforall{}[T:Type]
((\mforall{}x,y:T. Dec(x = y)) {}\mRightarrow{} finite-type(T) {}\mRightarrow{} (\mforall{}f:T {}\mrightarrow{} T. \mforall{}a,b:T. Dec(\mexists{}n:\mBbbN{}. (b = (f\^{}n a)))))
Date html generated:
2016_05_15-PM-04_11_49
Last ObjectModification:
2015_12_27-PM-03_00_19
Theory : general
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