Nuprl Lemma : decidable__rel_exp_finite
∀[T:Type]
((∀x,y:T. Dec(x = y ∈ T))
⇒ (∀[R:T ⟶ T ⟶ ℙ]. (rel_finite(T;R) ⇒ (∀x,y:T. Dec(x R y)) ⇒ (∀k:ℕ. ∀x,y:T. Dec(x R^k y)))))
Proof
Definitions occuring in Statement :
rel_finite: rel_finite(T;R),
rel_exp: R^n,
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
rel_exp: R^n,
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
infix_ap: x f y,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
so_apply: x[s],
iff: P ⇐⇒ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
rel_finite: rel_finite(T;R),
l_exists: (∃x∈L. P[x]),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T
Lemmas referenced :
all_wf,
decidable_wf,
infix_ap_wf,
rel_exp_wf,
decidable__le,
subtract_wf,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
set_wf,
less_than_wf,
primrec-wf2,
nat_wf,
rel_finite_wf,
equal_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
or_wf,
exists_wf,
equal-wf-base,
int_subtype_base,
rel_exp_iff,
iff_wf,
decidable_functionality,
decidable__l_exists,
decidable__and2,
select_wf,
int_seg_properties,
length_wf,
decidable__lt,
not_wf,
l_exists_iff,
l_member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
sqequalRule,
hypothesis,
rename,
setElimination,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
lambdaEquality,
because_Cache,
instantiate,
universeEquality,
dependent_set_memberEquality,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
functionExtensionality,
applyEquality,
functionEquality,
productElimination,
productEquality,
baseClosed,
inlFormation,
addLevel,
impliesFunctionality,
independent_functionElimination,
imageElimination,
inrFormation,
setEquality
Latex:
\mforall{}[T:Type]
((\mforall{}x,y:T. Dec(x = y))
{}\mRightarrow{} (\mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]
(rel\_finite(T;R) {}\mRightarrow{} (\mforall{}x,y:T. Dec(x R y)) {}\mRightarrow{} (\mforall{}k:\mBbbN{}. \mforall{}x,y:T. Dec(x rel\_exp(T; R; k) y)))))
Date html generated:
2017_04_17-AM-09_26_33
Last ObjectModification:
2017_02_27-PM-05_28_04
Theory : relations2
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