Nuprl Lemma : transitive-reflexive-closure_transitivity
∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ]. ∀x,y,z:A. ((x R^* y) ⇒ (y R^* z) ⇒ (x R^* z))
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],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
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,
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
utrans: UniformlyTrans(T;x,y.E[x; y])
Lemmas referenced :
transitive-reflexive-closure_wf,
iff_weakening_equal,
equal_wf,
transitive-closure-transitive
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesis,
addLevel,
sqequalHypSubstitution,
sqequalRule,
unionElimination,
thin,
applyEquality,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
functionExtensionality,
because_Cache,
lambdaEquality,
universeEquality,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
productElimination,
independent_functionElimination,
levelHypothesis,
functionEquality,
inrFormation
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. \mforall{}x,y,z:A. ((x R\^{}* y) {}\mRightarrow{} (y R\^{}* z) {}\mRightarrow{} (x R\^{}* z))
Date html generated:
2017_01_19-PM-02_17_45
Last ObjectModification:
2017_01_14-PM-04_59_10
Theory : relations2
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