Nuprl Lemma : comparison-max_wf
∀[T:Type]. ∀[cmp:comparison(T)]. ∀L:T List+. (comparison-max(cmp;L) ∈ {mx:T| (mx ∈ L) ∧ (∀x∈L.0 ≤ (cmp x mx))} ) suppos\000Cing valueall-type(T)
Proof
Definitions occuring in Statement :
comparison-max: comparison-max(cmp;L),
comparison: comparison(T),
l_all: (∀x∈L.P[x]),
l_member: (x ∈ l),
listp: A List+,
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
connex: Connex(T;x,y.R[x; y]),
sq_stable: SqStable(P),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
has-valueall: has-valueall(a),
has-value: (a)↓,
callbyvalueall: callbyvalueall,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
unit: Unit,
bool: 𝔹,
refl: Refl(T;x,y.E[x; y]),
comparison: comparison(T),
cand: A c∧ B,
squash: ↓T,
less_than: a < b,
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
it: ⋅,
nil: [],
decidable: Dec(P),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
colength: colength(L),
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
eager-accum: eager-accum(x,a.f[x; a];y;l),
guard: {T},
prop: ℙ,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
nat: ℕ,
subtype_rel: A ⊆r B,
top: Top,
comparison-max: comparison-max(cmp;L),
false: False,
implies: P ⇒ Q,
not: ¬A,
true: True,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
ge: i ≥ j ,
cons: [a / b],
or: P ∨ Q,
listp: A List+,
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
comparison-connex,
sq_stable__le,
assert_of_bnot,
iff_weakening_uiff,
iff_transitivity,
bool_cases,
l_all_cons,
cons_member,
not_wf,
bnot_wf,
assert_wf,
cons_wf,
l_all_wf,
ifthenelse_wf,
evalall-reduce,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
eqff_to_assert,
valueall-type-has-valueall,
assert_of_le_int,
eqtt_to_assert,
bool_wf,
le_int_wf,
l_all_wf_nil,
or_wf,
l_all_nil,
nil_wf,
l_member_wf,
comparison-refl,
list_ind_cons_lemma,
decidable__equal_int,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
equal_wf,
le_wf,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__le,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
spread_cons_lemma,
list_ind_nil_lemma,
list_wf,
less_than_irreflexivity,
less_than_transitivity1,
colength_wf_list,
nat_wf,
equal-wf-T-base,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
reduce_tl_cons_lemma,
reduce_hd_cons_lemma,
length_of_nil_lemma,
comparison_wf,
valueall-type_wf,
listp_wf,
product_subtype_list,
list-cases,
listp_properties
Rules used in proof :
imageMemberEquality,
impliesFunctionality,
inrFormation,
setEquality,
callbyvalueReduce,
equalityElimination,
functionExtensionality,
productEquality,
inlFormation,
imageElimination,
instantiate,
baseClosed,
addEquality,
dependent_set_memberEquality,
applyLambdaEquality,
computeAll,
independent_pairFormation,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_isectElimination,
intWeakElimination,
applyEquality,
voidEquality,
voidElimination,
natural_numberEquality,
independent_functionElimination,
universeEquality,
isect_memberEquality,
because_Cache,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
lambdaEquality,
cumulativity,
sqequalRule,
productElimination,
hypothesis_subsumption,
promote_hyp,
unionElimination,
dependent_functionElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
lambdaFormation,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[T:Type]. \mforall{}[cmp:comparison(T)].
\mforall{}L:T List\msupplus{}. (comparison-max(cmp;L) \mmember{} \{mx:T| (mx \mmember{} L) \mwedge{} (\mforall{}x\mmember{}L.0 \mleq{} (cmp x mx))\} ) supposing valueall\000C-type(T)
Date html generated:
2017_04_17-AM-08_31_42
Last ObjectModification:
2017_03_30-PM-03_44_51
Theory : list_1
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