Nuprl Lemma : connection-bound
∀[T:Type]
((∀x,y:T. Dec(x = y ∈ T))
⇒ (∀k:ℕ. (|T| ≤ k ⇒ (∀f:T ⟶ T. ∀a,b:T. (∃n:ℕ. (b = (f^n a) ∈ T) ⇐⇒ ∃n:ℕk. (b = (f^n a) ∈ T))))))
Proof
Definitions occuring in Statement :
cardinality-le: |T| ≤ n,
fun_exp: f^n,
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
finite-type: finite-type(T),
cardinality-le: |T| ≤ n,
l_member: (x ∈ l),
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
cardinality-le-no_repeats-length,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_properties,
length_wf,
lelt_wf,
decidable_wf,
all_wf,
cardinality-le_wf,
false_wf,
int_seg_subtype_nat,
int_seg_wf,
fun_exp_wf,
equal_wf,
nat_wf,
exists_wf,
orbit-exists
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
independent_functionElimination,
independent_pairFormation,
sqequalRule,
lambdaEquality,
applyEquality,
productElimination,
dependent_pairFormation,
because_Cache,
natural_numberEquality,
setElimination,
rename,
cumulativity,
independent_isectElimination,
functionEquality,
universeEquality,
dependent_functionElimination,
dependent_set_memberEquality,
equalityTransitivity,
unionElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}[T:Type]
((\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} (\mforall{}k:\mBbbN{}. (|T| \mleq{} k {}\mRightarrow{} (\mforall{}f:T {}\mrightarrow{} T. \mforall{}a,b:T. (\mexists{}n:\mBbbN{}. (b = (f\^{}n a)) \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}k. (b = (f\^{}n a)))))))
Date html generated:
2016_05_15-PM-04_11_42
Last ObjectModification:
2016_01_16-AM-11_06_27
Theory : general
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