Nuprl Lemma : rel_star_closure
∀[T:Type]. ∀[R,R2:T ⟶ T ⟶ ℙ].
(Trans(T)(R2[_1;_2]) ⇒ (∀x,y:T. ((x R y) ⇒ (x R2 y))) ⇒ (∀x,y:T. ((x (R^*) y) ⇒ ((x R2 y) ∨ (x = y ∈ T)))))
Proof
Definitions occuring in Statement :
rel_star: R^*,
trans: Trans(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
bfalse: ff,
it: ⋅,
unit: Unit,
bool: 𝔹,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
so_apply: x[s],
true: True,
top: Top,
subtype_rel: A ⊆r B,
subtract: n - m,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
decidable: Dec(P),
so_lambda: λ2x.t[x],
uimplies: b supposing a,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
prop: ℙ,
member: t ∈ T,
btrue: tt,
eq_int: (i =z j),
ifthenelse: if b then t else f fi ,
rel_exp: R^n,
or: P ∨ Q,
guard: {T},
exists: ∃x:A. B[x],
infix_ap: x f y,
rel_star: R^*,
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
trans: Trans(T;x,y.E[x; y])
Lemmas referenced :
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
iff_transitivity,
assert_of_eq_int,
eqtt_to_assert,
uiff_transitivity,
exists_wf,
not_wf,
bnot_wf,
less_than_irreflexivity,
le_weakening,
less_than_transitivity1,
assert_wf,
int_subtype_base,
equal-wf-base,
bool_wf,
eq_int_wf,
trans_wf,
rel_star_wf,
nat_wf,
primrec-wf2,
less_than_wf,
set_wf,
equal_wf,
or_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-le-2,
decidable__le,
subtract_wf,
all_wf,
le_weakening2,
le_wf,
false_wf,
rel_exp_wf,
infix_ap_wf
Rules used in proof :
impliesFunctionality,
equalityElimination,
productEquality,
equalitySymmetry,
equalityTransitivity,
baseClosed,
closedConclusion,
baseApply,
minusEquality,
intEquality,
voidEquality,
isect_memberEquality,
addEquality,
voidElimination,
unionElimination,
functionEquality,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
setElimination,
rename,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality,
universeEquality,
because_Cache,
isectElimination,
extract_by_obid,
introduction,
instantiate,
cumulativity,
hypothesisEquality,
functionExtensionality,
applyEquality,
inrFormation,
hypothesis,
cut,
thin,
productElimination,
sqequalRule,
sqequalHypSubstitution,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
inlFormation,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}[T:Type]. \mforall{}[R,R2:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
(Trans(T)(R2[$_{1}$;$_{2}$])
{}\mRightarrow{} (\mforall{}x,y:T. ((x R y) {}\mRightarrow{} (x R2 y)))
{}\mRightarrow{} (\mforall{}x,y:T. ((x rel\_star(T; R) y) {}\mRightarrow{} ((x R2 y) \mvee{} (x = y)))))
Date html generated:
2019_06_20-PM-00_30_44
Last ObjectModification:
2018_08_03-PM-05_27_20
Theory : relations
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