Nuprl Lemma : rel_star_order
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. (WellFnd{i}(T;x,y.x R y) ⇒ Order(T;x,y.x (R^*) y))
Proof
Definitions occuring in Statement :
rel_star: R^*,
order: Order(T;x,y.R[x; y]),
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
order: Order(T;x,y.R[x; y]),
and: P ∧ Q,
refl: Refl(T;x,y.E[x; y]),
all: ∀x:A. B[x],
member: t ∈ T,
uimplies: b supposing a,
cand: A c∧ B,
trans: Trans(T;x,y.E[x; y]),
prop: ℙ,
infix_ap: x f y,
anti_sym: AntiSym(T;x,y.R[x; y]),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
or: P ∨ Q,
not: ¬A,
false: False,
exists: ∃x:A. B[x]
Lemmas referenced :
rel_star_weakening,
rel_star_transitivity,
rel_star_wf,
wellfounded_wf,
rel_plus_irreflexive,
rel_star_iff,
rel-star-rel-plus2,
rel-star-rel-plus3
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
independent_isectElimination,
hypothesis,
independent_functionElimination,
applyEquality,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
productElimination,
unionElimination,
voidElimination
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. (WellFnd\{i\}(T;x,y.x R y) {}\mRightarrow{} Order(T;x,y.x rel\_star(T; R) y))
Date html generated:
2016_05_14-PM-03_53_58
Last ObjectModification:
2015_12_26-PM-06_56_17
Theory : relations2
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