Nuprl Lemma : max-fst-property
∀[Info,A,T:Type].
∀es:EO+(Info). ∀X:EClass(T × A). ∀e:E.
{(fst(MaxFst(X)(e)) ~ imax-list(map(λe.(fst(X(e)));≤(X)(e))))
∧ (∃mxe:E(X)
(mxe ≤loc e
∧ (MaxFst(X)(e) = X(mxe) ∈ (T × A))
∧ (∀e':E(X). (e' ≤loc e ⇒ ((fst(X(e'))) ≤ (fst(X(mxe))))))))}
supposing ↑e ∈b MaxFst(X)
supposing T ⊆r ℤ
Proof
Definitions occuring in Statement :
max-fst-class: MaxFst(X),
es-interface-predecessors: ≤(X)(e),
es-E-interface: E(X),
eclass-val: X(e),
in-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-le: e ≤loc e' ,
es-E: E,
imax-list: imax-list(L),
map: map(f;as),
assert: ↑b,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
guard: {T},
pi1: fst(t),
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
lambda: λx.A[x],
product: x:A × B[x],
int: ℤ,
universe: Type,
sqequal: s ~ t,
equal: s = t ∈ T
Lemmas :
max-fst-val,
assert_wf,
in-eclass_wf,
max-fst-class_wf,
subtype_rel_self,
es-interface-subtype_rel2,
es-E_wf,
event-ordering+_subtype,
event-ordering+_wf,
top_wf,
subtype_top,
eclass_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
equal-wf-T-base,
colength_wf_list,
list-cases,
list_accum_nil_lemma,
map_nil_lemma,
product_subtype_list,
spread_cons_lemma,
sq_stable__le,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
nat_wf,
decidable__le,
false_wf,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
add-commutes,
le_wf,
subtract_wf,
not-ge-2,
less-iff-le,
minus-minus,
add-swap,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
list_accum_cons_lemma,
map_cons_lemma,
list_wf,
es-E-interface_wf,
subtype_rel_product,
pi1_wf_top,
list_accum_wf,
lt_int_wf,
eclass-val_wf,
assert_elim,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
assert-bnot,
imax_wf,
le_int_wf,
map_wf,
assert_of_le_int,
iseg_wf,
append_wf,
cons_wf,
nil_wf,
decidable__equal_int,
not-equal-2,
list_accum_functionality,
imax_unfold,
iff_weakening_equal,
es-interface-predecessors_wf,
subtype_rel_list,
Id_wf,
es-loc_wf,
reduce_tl_nil_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
is-max-fst,
es-interface-predecessors-nonempty,
length_wf_nat,
length_wf,
length_of_nil_lemma,
list_induction,
all_wf,
exists_wf,
or_wf,
l_member_wf,
cons_member,
accum_list_cons_lemma,
es-le_wf,
member-interface-predecessors,
non_neg_length,
length_cons,
accum_list_wf,
le_weakening,
length_of_cons_lemma,
decidable__lt,
squash_wf,
true_wf,
imax-list-ub,
map_length_nat,
set_wf,
cons_neq_nil,
l_exists_iff,
member_map,
es-le-loc,
sq_stable__assert,
member-interface-predecessors-subtype,
assert_witness,
subtype_rel_wf
Latex:
\mforall{}[Info,A,T:Type].
\mforall{}es:EO+(Info). \mforall{}X:EClass(T \mtimes{} A). \mforall{}e:E.
\{(fst(MaxFst(X)(e)) \msim{} imax-list(map(\mlambda{}e.(fst(X(e)));\mleq{}(X)(e))))
\mwedge{} (\mexists{}mxe:E(X)
(mxe \mleq{}loc e
\mwedge{} (MaxFst(X)(e) = X(mxe))
\mwedge{} (\mforall{}e':E(X). (e' \mleq{}loc e {}\mRightarrow{} ((fst(X(e'))) \mleq{} (fst(X(mxe))))))))\}
supposing \muparrow{}e \mmember{}\msubb{} MaxFst(X)
supposing T \msubseteq{}r \mBbbZ{}
Date html generated:
2015_07_21-PM-03_38_43
Last ObjectModification:
2015_02_04-PM-06_19_23
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