Nuprl Lemma : transitive-reflexive-closure-base-case
∀[A:Type]. ∀[R:A ⟶ A ⟶ ℙ]. ∀x,y:A. ((x R y) ⇒ (x 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],
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,
guard: {T},
or: P ∨ Q,
member: t ∈ T,
prop: ℙ,
rel_implies: R1 => R2
Lemmas referenced :
equal_wf,
transitive-closure-contains
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalRule,
cut,
hypothesis,
inrFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
functionEquality,
universeEquality,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. \mforall{}x,y:A. ((x R y) {}\mRightarrow{} (x R\^{}* y))
Date html generated:
2017_01_19-PM-02_17_47
Last ObjectModification:
2017_01_14-PM-05_04_26
Theory : relations2
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