Nuprl Lemma : finite-acyclic-rel
∀[T:Type]. ((∃n:ℕ. T ~ ℕn) ⇒ (∀[R:T ⟶ T ⟶ ℙ]. ((∀x,y:T. Dec(x R y)) ⇒ (SWellFounded(x R y) ⇐⇒ acyclic-rel(T;R)))))
Proof
Definitions occuring in Statement :
equipollent: A ~ B,
acyclic-rel: acyclic-rel(T;R),
strongwellfounded: SWellFounded(R[x; y]),
int_seg: {i..j-},
nat: ℕ,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
acyclic-rel: acyclic-rel(T;R),
all: ∀x:A. B[x],
not: ¬A,
false: False,
infix_ap: x f y,
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
exists: ∃x:A. B[x],
uimplies: b supposing a,
strongwellfounded: SWellFounded(R[x; y]),
equipollent: A ~ B,
rel_plus: R+,
nat_plus: ℕ+,
le: A ≤ B,
less_than': less_than'(a;b),
rel_exp: R^n,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
cand: A c∧ B,
pi1: fst(t),
sq_exists: ∃x:{A| B[x]},
ge: i ≥ j ,
squash: ↓T,
true: True,
compose: f o g,
subtract: n - m,
rel-path-between: rel-path-between(T;R;x;y;L),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
biject: Bij(A;B;f),
inject: Inj(A;B;f),
less_than: a < b
Lemmas referenced :
wellfounded-acyclic-rel,
rel_plus_wf,
strongwellfounded_wf,
acyclic-rel_wf,
all_wf,
decidable_wf,
exists_wf,
nat_wf,
equipollent_wf,
int_seg_wf,
subtype_rel_dep_function,
subtype_rel_self,
subtype_rel_wf,
uall_wf,
subtract_wf,
infix_ap_wf,
set_wf,
less_than_wf,
primrec-wf2,
equipollent-zero,
biject-inverse,
rel_exp_wf,
nat_plus_subtype_nat,
nat_plus_properties,
primrec-wf-nat-plus,
false_wf,
le_wf,
rel_exp_one,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
eq_int_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
assert_wf,
bnot_wf,
not_wf,
equal-wf-base,
int_subtype_base,
bool_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
bool_cases,
subtype_base_sq,
bool_subtype_base,
add-subtract-cancel,
nat_plus_wf,
decidable__exists_int_seg,
decidable__all_int_seg,
decidable__not,
pigeon-hole-implies,
decidable__lt,
fun_exp_wf,
int_seg_subtype_nat,
lelt_wf,
sq_stable__equal,
nat_properties,
subtract-add-cancel,
squash_wf,
true_wf,
fun_exp_add,
le_weakening2,
itermSubtract_wf,
int_term_value_subtract_lemma,
iff_weakening_equal,
decidable__equal_int,
fun_exp1_lemma,
add-associates,
add-swap,
add-commutes,
zero-add,
fun_exp_add1,
equipollent-general-subtract-one,
subtype_rel_transitivity,
subtype_rel_list,
length_wf,
rel-path-between_wf,
rel_plus-iff-path,
bool_cases_sqequal,
assert-bnot,
neg_assert_of_eq_int,
add_nat_wf,
and_wf,
add_nat_plus,
add-is-int-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
voidElimination,
applyEquality,
cumulativity,
functionExtensionality,
functionEquality,
universeEquality,
natural_numberEquality,
setElimination,
rename,
productElimination,
instantiate,
because_Cache,
independent_isectElimination,
intEquality,
dependent_pairFormation,
dependent_set_memberEquality,
addEquality,
unionElimination,
int_eqEquality,
isect_memberEquality,
voidEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
baseApply,
closedConclusion,
baseClosed,
equalityElimination,
impliesFunctionality,
productEquality,
promote_hyp,
imageElimination,
imageMemberEquality,
hyp_replacement,
setEquality,
addLevel,
levelHypothesis,
pointwiseFunctionality
Latex:
\mforall{}[T:Type]
((\mexists{}n:\mBbbN{}. T \msim{} \mBbbN{}n)
{}\mRightarrow{} (\mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. ((\mforall{}x,y:T. Dec(x R y)) {}\mRightarrow{} (SWellFounded(x R y) \mLeftarrow{}{}\mRightarrow{} acyclic-rel(T;R)))))
Date html generated:
2017_04_17-AM-09_35_53
Last ObjectModification:
2017_02_27-PM-05_35_42
Theory : equipollence!!cardinality!
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