Nuprl Lemma : test-rel-connected
∀T:Type. ∀R:T ⟶ T ⟶ ℙ. ∀x,y,z,w:T. ((x (R^*) y) ⇒ (y = z ∈ T) ⇒ (z (R^*) w) ⇒ (x (R^*) w))
Proof
Definitions occuring in Statement :
rel_star: R^*,
prop: ℙ,
infix_ap: x f y,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
infix_ap: x f y,
uall: ∀[x:A]. B[x],
rel-connected: x──R⟶y,
guard: {T},
uimplies: b supposing a
Lemmas referenced :
rel_star_wf,
equal_wf,
rel-connected_transitivity,
rel-connected_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
applyEquality,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
functionEquality,
cumulativity,
universeEquality,
independent_functionElimination,
independent_isectElimination
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. \mforall{}x,y,z,w:T.
((x (R\^{}*) y) {}\mRightarrow{} (y = z) {}\mRightarrow{} (z (R\^{}*) w) {}\mRightarrow{} (x rel\_star(T; R) w))
Date html generated:
2016_05_13-PM-04_19_21
Last ObjectModification:
2015_12_26-AM-11_33_36
Theory : relations
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