Nuprl Lemma : wellfounded-acyclic-rel
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. (SWellFounded(x R y) ⇒ acyclic-rel(T;R))
Proof
Definitions occuring in Statement :
acyclic-rel: acyclic-rel(T;R),
strongwellfounded: SWellFounded(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],
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
acyclic-rel: acyclic-rel(T;R),
all: ∀x:A. B[x],
not: ¬A,
false: False,
strongwellfounded: SWellFounded(R[x; y]),
exists: ∃x:A. B[x],
less_than: a < b,
squash: ↓T,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
itermVar_wf,
intformless_wf,
satisfiable-full-omega-tt,
rel_plus_wf,
strongwellfounded_wf,
rel_plus_strongwellfounded
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
universeEquality,
dependent_functionElimination,
because_Cache,
functionEquality,
cumulativity,
isect_memberEquality,
voidElimination,
productElimination,
imageElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
voidEquality,
computeAll
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. (SWellFounded(x R y) {}\mRightarrow{} acyclic-rel(T;R))
Date html generated:
2016_05_14-PM-03_53_30
Last ObjectModification:
2016_01_14-PM-11_10_38
Theory : relations2
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