Nuprl Lemma : lg-acyclic-well-founded
∀[T:Type]. ∀g:LabeledGraph(T). (lg-acyclic(g) ⇐⇒ SWellFounded(lg-edge(g;a;b)))
Proof
Definitions occuring in Statement :
lg-acyclic: lg-acyclic(g),
lg-edge: lg-edge(g;a;b),
lg-size: lg-size(g),
labeled-graph: LabeledGraph(T),
strongwellfounded: SWellFounded(R[x; y]),
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n,
universe: Type
Lemmas :
lg-size_wf,
lg-acyclic_wf,
less_than_wf,
labeled-graph_wf,
all_wf,
subtract_wf,
strongwellfounded_wf,
int_seg_wf,
lg-edge_wf,
set_wf,
primrec-wf2,
nat_wf,
decidable__lt,
lg-acyclic-has-source,
assert-lg-is-source,
false_wf,
int_seg_subtype-nat,
lg-remove_wf,
le-add-cancel-alt,
add_functionality_wrt_le,
less-iff-le,
zero-add,
add-commutes,
add-swap,
minus-one-mul,
minus-minus,
minus-add,
add-associates,
condition-implies-le,
iff_weakening_equal,
lg-size-remove,
true_wf,
squash_wf,
lg-acyclic-remove,
add-zero,
sq_stable__le,
not-equal-2,
not-le-2,
decidable__le,
neg_assert_of_eq_int,
assert_of_eq_int,
eq_int_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
le_wf,
zero-le-nat,
lelt_wf,
le-add-cancel,
le_antisymmetry_iff,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
less_than_irreflexivity,
le_weakening,
less_than_transitivity1,
lg-edge-remove,
le-add-cancel2,
int_subtype_base,
lg-connected_wf,
zero-mul,
add-mul-special,
rel_plus_strongwellfounded
Latex:
\mforall{}[T:Type]. \mforall{}g:LabeledGraph(T). (lg-acyclic(g) \mLeftarrow{}{}\mRightarrow{} SWellFounded(lg-edge(g;a;b)))
Date html generated:
2015_07_22-PM-00_29_43
Last ObjectModification:
2015_07_16-AM-09_39_07
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