Nuprl Lemma : mul-polynom-val
∀[n:ℕ]. ∀[p,q:polyform(n)]. ∀[l:{l:ℤ List| n ≤ ||l||} ]. (mul-polynom(p;q)@l = (p@l * q@l) ∈ ℤ)
Proof
Definitions occuring in Statement :
mul-polynom: mul-polynom(p;q),
poly-int-val: p@l,
polyform: polyform(n),
length: ||as||,
list: T List,
nat: ℕ,
uall: ∀[x:A]. B[x],
le: A ≤ B,
set: {x:A| B[x]} ,
multiply: n * m,
int: ℤ,
equal: s = 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,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
polyform: polyform(n),
ext-eq: A ≡ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
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),
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
polyconst: polyconst(k),
has-value: (a)↓,
nequal: a ≠ b ∈ T ,
band: p ∧b q,
ispolyform: ispolyform(p),
tree_ind: tree_ind,
cons: [a / b],
cand: A c∧ B,
poly-int-val: p@l,
poly-val-fun: poly-val-fun(p),
le: A ≤ B,
less_than': less_than'(a;b),
polynom: polynom(n)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
istype-less_than,
int_seg_properties,
int_seg_wf,
subtract-1-ge-0,
decidable__equal_int,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
decidable__le,
decidable__lt,
istype-le,
subtype_rel_self,
tree-ext,
eq_atom_wf,
eqtt_to_assert,
assert_of_eq_atom,
atom_subtype_base,
ispolyform_leaf_lemma,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
ispolyform_node_lemma,
list_wf,
length_wf,
tree_size_wf,
polyform_wf,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
add_nat_wf,
add-is-int-iff,
false_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
poly-int-val_wf,
iff_weakening_equal,
value-type-has-value,
int-value-type,
polyconst_val_lemma,
eq_int_wf,
assert_of_eq_int,
itermMultiply_wf,
int_term_value_mul_lemma,
neg_assert_of_eq_int,
assert_wf,
ispolyform_wf,
bool_cases,
band_wf,
btrue_wf,
bfalse_wf,
lt_int_wf,
less_than_wf,
assert_of_band,
istype-assert,
mul-polynom_wf,
tree_leaf_wf,
value-type-polyform,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
iff_transitivity,
iff_weakening_uiff,
assert_of_lt_int,
tree_node_wf,
cons_wf,
reduce_tl_cons_lemma,
reduce_hd_cons_lemma,
list-value-type,
subtract_nat_wf,
non_neg_length,
length_wf_nat,
nat_wf,
le_wf,
istype-false,
mul-zero,
zero-mul,
int_entire_a,
multiply-is-int-iff,
le_weakening2,
assert_elim,
add-polynom-val,
add-polynom_wf,
zero_ann_a,
polyconst_wf,
mul_preserves_eq
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
productElimination,
because_Cache,
unionElimination,
applyEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
dependent_set_memberEquality_alt,
productIsType,
hypothesis_subsumption,
isect_memberFormation_alt,
intEquality,
promote_hyp,
tokenEquality,
equalityElimination,
cumulativity,
atomEquality,
equalityIstype,
setIsType,
addEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
baseClosed,
imageElimination,
universeEquality,
multiplyEquality,
imageMemberEquality,
callbyvalueReduce,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
productEquality,
sqleReflexivity,
inrFormation_alt,
sqequalBase
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[p,q:polyform(n)]. \mforall{}[l:\{l:\mBbbZ{} List| n \mleq{} ||l||\} ]. (mul-polynom(p;q)@l = (p@l * q@l))
Date html generated:
2019_10_15-AM-10_52_55
Last ObjectModification:
2018_11_28-PM-11_21_47
Theory : integer!polynomial!trees
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