Nuprl Lemma : es-cut-induction-ordered
∀[Info:Type]
∀es:EO+(Info). ∀X:EClass(Top). ∀f:sys-antecedent(es;X).
∀[P:Cut(X;f) ─→ ℙ]
((∃R:E(X) ─→ E(X) ─→ ℙ. (Linorder(E(X);x,y.R[x;y]) ∧ (∀x,y:E(X). Dec(R[x;y]))))
⇒ P[{}]
⇒ (∀c:Cut(X;f). ∀e:E(X).
(P[c]
⇒ (P[c+e]) supposing (prior(X)(e) ∈ c supposing ↑e ∈b prior(X) and f e ∈ c supposing ¬((f e) = e ∈ E(X)))))
⇒ (∀c:Cut(X;f). P[c]))
Proof
Definitions occuring in Statement :
es-cut-add: c+e,
es-cut: Cut(X;f),
es-prior-interface: prior(X),
sys-antecedent: sys-antecedent(es;Sys),
es-E-interface: E(X),
eclass-val: X(e),
in-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-eq: es-eq(es),
empty-fset: {},
fset-member: a ∈ s,
linorder: Linorder(T;x,y.R[x; y]),
assert: ↑b,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ─→ B[x],
universe: Type,
equal: s = t ∈ T
Lemmas :
es-eq_wf-interface,
less_than_transitivity1,
less_than_irreflexivity,
int_seg_wf,
decidable__equal_int,
subtype_rel-int_seg,
false_wf,
le_weakening,
subtract_wf,
int_seg_properties,
le_wf,
fset-size_wf,
es-E-interface_wf,
es-cut_wf,
all_wf,
nat_wf,
decidable__lt,
not-equal-2,
condition-implies-le,
minus-add,
minus-minus,
minus-one-mul,
add-swap,
add-commutes,
add-associates,
add_functionality_wrt_le,
zero-add,
le-add-cancel-alt,
less-iff-le,
le-add-cancel,
lelt_wf,
set_wf,
less_than_wf,
primrec-wf2,
decidable__le,
not-le-2,
sq_stable__le,
add-zero,
add-mul-special,
zero-mul,
empty-fset_wf,
decidable__equal_fset,
decidable__equal_es-E-interface,
equal-wf-T-base,
fset_wf,
fset-closed_wf,
cons_wf,
es-interface-pred_wf2,
nil_wf,
empty-fset-closed,
fset-member_wf-cut,
fset-add-remove,
iff_weakening_equal,
sq_stable_from_decidable,
decidable__fset-closed,
fset-member_witness,
not_wf,
equal_wf,
eclass-val_wf2,
es-prior-interface_wf,
assert_wf,
in-eclass_wf,
es-prior-interface_wf0,
es-interface-subtype_rel2,
es-E_wf,
event-ordering+_subtype,
event-ordering+_wf,
top_wf,
subtype_top,
es-cut-remove-1,
subtype_base_sq,
int_subtype_base,
fset-size-empty,
squash_wf,
true_wf,
fset-size-remove,
es-eq_wf,
subtype_rel_fset,
strong-subtype-set3,
strong-subtype-self,
fset-member_wf,
fset-remove_wf,
decidable__fset-member,
member-fset-remove,
l_all_iff,
sys-antecedent_wf,
es-interface-pred_wf,
l_member_wf,
isect_wf,
cons_member,
bool_wf,
bnot_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
es-prior-interface-locl,
es-locl_wf,
es-locl_transitivity2,
es-le_weakening_eq,
es-locl_irreflexivity,
fset-to-list,
fset-extensionality,
subtype_rel_quotient_trivial,
list_wf,
set-equal_wf,
set-equal-equiv,
assert-deq-member,
iff_wf,
es-cut-add_wf,
cut-list-maximal-exists
Latex:
\mforall{}[Info:Type]
\mforall{}es:EO+(Info). \mforall{}X:EClass(Top). \mforall{}f:sys-antecedent(es;X).
\mforall{}[P:Cut(X;f) {}\mrightarrow{} \mBbbP{}]
((\mexists{}R:E(X) {}\mrightarrow{} E(X) {}\mrightarrow{} \mBbbP{}. (Linorder(E(X);x,y.R[x;y]) \mwedge{} (\mforall{}x,y:E(X). Dec(R[x;y]))))
{}\mRightarrow{} P[\{\}]
{}\mRightarrow{} (\mforall{}c:Cut(X;f). \mforall{}e:E(X).
(P[c]
{}\mRightarrow{} (P[c+e]) supposing
(prior(X)(e) \mmember{} c supposing \muparrow{}e \mmember{}\msubb{} prior(X) and
f e \mmember{} c supposing \mneg{}((f e) = e))))
{}\mRightarrow{} (\mforall{}c:Cut(X;f). P[c]))
Date html generated:
2015_07_21-PM-04_02_49
Last ObjectModification:
2015_02_04-PM-06_08_55
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