Nuprl Lemma : least-equiv-implies
∀[A:Type]. ∀[R,E:A ⟶ A ⟶ ℙ]. (R => E ⇒ EquivRel(A;x,y.E x y) ⇒ least-equiv(A;R) => E)
Proof
Definitions occuring in Statement :
least-equiv: least-equiv(A;R),
rel_implies: R1 => R2,
equiv_rel: EquivRel(T;x,y.E[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
so_apply: x[s],
so_lambda: λ2x.t[x],
trans: Trans(T;x,y.E[x; y]),
sym: Sym(T;x,y.E[x; y]),
refl: Refl(T;x,y.E[x; y]),
equiv_rel: EquivRel(T;x,y.E[x; y]),
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
member: t ∈ T,
or: P ∨ Q,
infix_ap: x f y,
transitive-reflexive-closure: R^*,
least-equiv: least-equiv(A;R),
all: ∀x:A. B[x],
rel_implies: R1 => R2,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rel_implies_wf,
equiv_rel_wf,
least-equiv_wf,
or_wf,
transitive-closure-induction,
iff_weakening_equal
Rules used in proof :
functionEquality,
cumulativity,
functionExtensionality,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
productElimination,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
isectElimination,
extract_by_obid,
introduction,
universeEquality,
lambdaEquality,
hypothesis,
hypothesisEquality,
applyEquality,
cut,
thin,
unionElimination,
sqequalRule,
sqequalHypSubstitution,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
rename
Latex:
\mforall{}[A:Type]. \mforall{}[R,E:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. (R => E {}\mRightarrow{} EquivRel(A;x,y.E x y) {}\mRightarrow{} least-equiv(A;R) => E)
Date html generated:
2018_05_21-PM-00_51_59
Last ObjectModification:
2018_01_08-AM-10_26_01
Theory : relations2
Home
Index