Nuprl Lemma : fset-max_wf
∀[T:Type]. ∀[f:T ⟶ ℕ]. ∀[s:fset(T)]. (fset-max(f;s) ∈ ℕ) supposing ∀x,y:T. Dec(x = y ∈ T)
Proof
Definitions occuring in Statement :
fset-max: fset-max(f;s),
fset: fset(T),
nat: ℕ,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fset: fset(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
fset-max: fset-max(f;s),
imax-list: imax-list(L),
combine-list: combine-list(x,y.f[x; y];L),
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
nat: ℕ,
implies: P ⇒ Q,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
so_apply: x[s1;s2],
le: A ≤ B,
less_than': less_than'(a;b),
comm: Comm(T;op),
infix_ap: x f y,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assoc: Assoc(T;op),
sq_type: SQType(T)
Lemmas referenced :
nat_wf,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
equal-wf-base,
list_wf,
set-equal_wf,
fset_wf,
all_wf,
decidable_wf,
equal_wf,
mk_deq_wf,
list_accum-set-equal-idemp,
imax_wf,
imax_nat,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
le_wf,
imax-idempotent,
false_wf,
squash_wf,
true_wf,
imax_com,
iff_weakening_equal,
imax_assoc,
subtype_base_sq,
int_subtype_base,
list_induction,
list_accum_wf,
map_wf,
map_nil_lemma,
list_accum_nil_lemma,
map_cons_lemma,
list_accum_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
extract_by_obid,
hypothesis,
sqequalRule,
pertypeElimination,
productElimination,
thin,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
productEquality,
isectElimination,
cumulativity,
hypothesisEquality,
because_Cache,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
lambdaEquality,
universeEquality,
rename,
dependent_set_memberEquality,
setElimination,
lambdaFormation,
applyLambdaEquality,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
applyEquality,
imageElimination,
imageMemberEquality,
baseClosed,
instantiate,
promote_hyp,
functionExtensionality
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} \mBbbN{}]. \mforall{}[s:fset(T)]. (fset-max(f;s) \mmember{} \mBbbN{}) supposing \mforall{}x,y:T. Dec(x = y)
Date html generated:
2017_04_17-AM-09_20_00
Last ObjectModification:
2017_02_27-PM-05_23_29
Theory : finite!sets
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