Nuprl Lemma : transitive-reflexive-closure-cases
∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ]. ∀x,y:A. ((x R^* y) ⇒ ((x = y ∈ A) ∨ (∃z:A. ((x R z) ∧ (z R^* y)))))
Proof
Definitions occuring in Statement :
transitive-reflexive-closure: R^*,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
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,
transitive-reflexive-closure: R^*,
infix_ap: x f y,
or: P ∨ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
so_apply: x[s],
guard: {T},
exists: ∃x:A. B[x],
cand: A c∧ B
Lemmas referenced :
exists_wf,
transitive-reflexive-closure_wf,
transitive-closure-cases,
equal_wf,
transitive-closure_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
sqequalRule,
unionElimination,
thin,
inlFormation,
hypothesis,
cut,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaEquality,
productEquality,
applyEquality,
functionExtensionality,
universeEquality,
because_Cache,
inrFormation,
dependent_functionElimination,
independent_functionElimination,
functionEquality,
dependent_pairFormation,
independent_pairFormation,
productElimination,
promote_hyp
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. \mforall{}x,y:A. ((x R\^{}* y) {}\mRightarrow{} ((x = y) \mvee{} (\mexists{}z:A. ((x R z) \mwedge{} (z R\^{}* y)))))
Date html generated:
2017_01_19-PM-02_17_41
Last ObjectModification:
2017_01_14-PM-04_28_39
Theory : relations2
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