Nuprl Lemma : rm-zeros_wf
∀[n:ℕ+]. ∀[p:polynom(n - 1) List]. (rm-zeros(n - 1;p) ∈ polynom(n))
Proof
Definitions occuring in Statement :
rm-zeros: rm-zeros(n;p),
polynom: polynom(n),
list: T List,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtract: n - m,
natural_number: $n
Definitions unfolded in proof :
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
polyform: polyform(n),
polyform-lead-nonzero: polyform-lead-nonzero(n;p),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
unit: Unit,
bool: 𝔹,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
it: ⋅,
nil: [],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
colength: colength(L),
cons: [a / b],
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
rm-zeros: rm-zeros(n;p),
guard: {T},
subtype_rel: A ⊆r B,
ge: i ≥ j ,
prop: ℙ,
and: P ∧ Q,
top: Top,
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
nat_plus: ℕ+,
nat: ℕ,
member: t ∈ T,
polynom: polynom(n),
uall: ∀[x:A]. B[x]
Lemmas referenced :
length_wf,
reduce_hd_cons_lemma,
length_of_cons_lemma,
cons_wf,
poly-zero_wf,
polynom_subtype_polyform,
polyform_wf,
subtype_rel_list,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
length_of_nil_lemma,
nat_plus_subtype_nat,
polyform-lead-nonzero_wf,
nil_wf,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
iff_transitivity,
assert_of_eq_int,
eqtt_to_assert,
uiff_transitivity,
list_ind_cons_lemma,
decidable__equal_int,
int_subtype_base,
set_subtype_base,
subtype_base_sq,
equal_wf,
int_term_value_add_lemma,
itermAdd_wf,
spread_cons_lemma,
product_subtype_list,
list_ind_nil_lemma,
list-cases,
less_than_irreflexivity,
less_than_transitivity1,
colength_wf_list,
nat_wf,
less_than_wf,
ge_wf,
nat_properties,
not_wf,
bnot_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
assert_wf,
equal-wf-T-base,
bool_wf,
eq_int_wf,
nat_plus_wf,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties,
subtract_wf,
polynom_wf,
list_wf
Rules used in proof :
impliesFunctionality,
equalityElimination,
imageElimination,
cumulativity,
instantiate,
addEquality,
productElimination,
hypothesis_subsumption,
promote_hyp,
applyLambdaEquality,
applyEquality,
axiomEquality,
independent_functionElimination,
intWeakElimination,
lambdaFormation,
because_Cache,
baseClosed,
equalitySymmetry,
equalityTransitivity,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
natural_numberEquality,
hypothesisEquality,
rename,
setElimination,
dependent_set_memberEquality,
thin,
isectElimination,
extract_by_obid,
introduction,
hypothesis,
sqequalHypSubstitution,
cut,
sqequalRule,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[p:polynom(n - 1) List]. (rm-zeros(n - 1;p) \mmember{} polynom(n))
Date html generated:
2017_04_17-AM-09_06_11
Last ObjectModification:
2017_04_13-PM-02_12_08
Theory : list_1
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