Nuprl Lemma : mul-polynom_wf
∀[n:ℕ]. ∀[p,q:polyform(n)]. (mul-polynom(p;q) ∈ polyform(n))
Proof
Definitions occuring in Statement :
mul-polynom: mul-polynom(p;q),
polyform: polyform(n),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
polyform: polyform(n),
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
ext-eq: A ≡ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
sq_type: SQType(T),
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
tree_leaf: tree_leaf(value),
assert: ↑b,
tree_size: tree_size(p),
mul-polynom: mul-polynom(p;q),
tree_leaf?: tree_leaf?(v),
pi1: fst(t),
tree_leaf-value: tree_leaf-value(v),
tree_node-left: tree_node-left(v),
pi2: snd(t),
tree_node-right: tree_node-right(v),
bfalse: ff,
bnot: ¬bb,
tree_node: tree_node(left;right),
less_than: a < b,
polyconst: polyconst(k),
ispolyform: ispolyform(p),
tree_ind: tree_ind,
true: True,
polynom: polynom(n),
has-value: (a)↓,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
cand: A c∧ B
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
le_wf,
tree_size_wf,
polyform_wf,
nat_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
tree-ext,
eq_atom_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
atom_subtype_base,
ispolyform_leaf_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
ispolyform_node_lemma,
decidable__lt,
lelt_wf,
itermAdd_wf,
int_term_value_add_lemma,
add_nat_wf,
add-is-int-iff,
polynom_wf,
assert_wf,
ispolyform_wf,
value-type-has-value,
int-value-type,
band_wf,
lt_int_wf,
tree_leaf_wf,
set_subtype_base,
int_subtype_base,
iff_transitivity,
iff_weakening_uiff,
assert_of_band,
assert_of_lt_int,
tree_node_wf,
tree_wf,
valuetype__tree,
le_weakening2,
value-type-polyform,
polyconst_wf,
assert_elim,
add-polynom_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
addEquality,
applyEquality,
because_Cache,
productElimination,
unionElimination,
applyLambdaEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
isect_memberFormation,
promote_hyp,
tokenEquality,
equalityElimination,
instantiate,
cumulativity,
atomEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
callbyvalueReduce,
multiplyEquality,
productEquality,
imageMemberEquality,
imageElimination,
addLevel,
levelHypothesis
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[p,q:polyform(n)]. (mul-polynom(p;q) \mmember{} polyform(n))
Date html generated:
2017_10_01-AM-08_33_09
Last ObjectModification:
2017_05_03-AM-11_13_41
Theory : integer!polynomial!trees
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