Nuprl Lemma : least-equiv-cases
∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ].
∀a,b:A.
((least-equiv(A;R) a b)
⇒ ((a = b ∈ A) ∨ ((R a b) ∨ (R b a)) ∨ (∃c:A. (((R c b) ∨ (R b c)) ∧ (least-equiv(A;R) a c)))))
Proof
Definitions occuring in Statement :
least-equiv: least-equiv(A;R),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
guard: {T},
equiv_rel: EquivRel(T;x,y.E[x; y]),
and: P ∧ Q,
sym: Sym(T;x,y.E[x; y]),
least-equiv: least-equiv(A;R),
transitive-reflexive-closure: R^*,
or: P ∨ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
infix_ap: x f y,
exists: ∃x:A. B[x],
cand: A c∧ B
Lemmas referenced :
least-equiv-is-equiv,
least-equiv_wf,
or_wf,
exists_wf,
subtype_rel_self,
equal_wf,
transitive-closure-cases
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
applyEquality,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
productElimination,
dependent_functionElimination,
independent_functionElimination,
sqequalRule,
unionElimination,
inlFormation,
equalitySymmetry,
lambdaEquality,
productEquality,
instantiate,
inrFormation,
dependent_pairFormation,
independent_pairFormation
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
\mforall{}a,b:A.
((least-equiv(A;R) a b)
{}\mRightarrow{} ((a = b) \mvee{} ((R a b) \mvee{} (R b a)) \mvee{} (\mexists{}c:A. (((R c b) \mvee{} (R b c)) \mwedge{} (least-equiv(A;R) a c)))))
Date html generated:
2018_05_21-PM-00_52_02
Last ObjectModification:
2018_05_04-AM-00_45_50
Theory : relations2
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