Nuprl Lemma : poly_int_val_cons_cons
∀n:ℕ. ∀p:polyform(n) List. ∀l:{l:ℤ List| ||l|| = n ∈ ℤ} . ∀a:ℤ. ∀u:polyform(n).
([u / p]@[a / l] = ((u@l * a^||p||) + p@[a / l]) ∈ ℤ)
Proof
Definitions occuring in Statement :
poly-int-val: p@l,
polyform: polyform(n),
exp: i^n,
length: ||as||,
cons: [a / b],
list: T List,
nat: ℕ,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
multiply: n * m,
add: n + m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
subtract: n - m,
cons: [a / b],
select: L[n],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
uiff: uiff(P;Q),
less_than: a < b,
lelt: i ≤ j < k,
guard: {T},
int_seg: {i..j-},
not: ¬A,
implies: P ⇒ Q,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
and: P ∧ Q,
le: A ≤ B,
ge: i ≥ j ,
nat_plus: ℕ+,
squash: ↓T,
top: Top,
so_apply: x[s],
nat: ℕ,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
poly_int_val_cons,
int_term_value_mul_lemma,
itermMultiply_wf,
minus-one-mul,
minus-add,
subtract-is-int-iff,
add-subtract-cancel,
select-cons-tl,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
zero-add,
add-commutes,
add-swap,
add-zero,
mul-commutes,
minus-zero,
add-associates,
iff_weakening_equal,
sum_wf,
length_wf_nat,
int_seg_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
exp_wf2,
false_wf,
add-is-int-iff,
le_wf,
decidable__le,
int_seg_properties,
select_wf,
poly-int-val_wf,
less_than_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_properties,
non_neg_length,
length_of_cons_lemma,
cons_wf,
length_wf,
sum_split_first,
true_wf,
squash_wf,
equal_wf,
nat_wf,
int_subtype_base,
list_subtype_base,
equal-wf-base-T,
list_wf,
set_wf,
polyform_wf
Rules used in proof :
minusEquality,
independent_functionElimination,
imageMemberEquality,
promote_hyp,
pointwiseFunctionality,
multiplyEquality,
computeAll,
independent_pairFormation,
int_eqEquality,
dependent_pairFormation,
unionElimination,
productElimination,
addEquality,
natural_numberEquality,
dependent_functionElimination,
dependent_set_memberEquality,
because_Cache,
universeEquality,
equalitySymmetry,
equalityTransitivity,
imageElimination,
voidEquality,
voidElimination,
isect_memberEquality,
rename,
setElimination,
independent_isectElimination,
applyEquality,
baseClosed,
closedConclusion,
baseApply,
lambdaEquality,
sqequalRule,
intEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
hypothesis,
cut,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}n:\mBbbN{}. \mforall{}p:polyform(n) List. \mforall{}l:\{l:\mBbbZ{} List| ||l|| = n\} . \mforall{}a:\mBbbZ{}. \mforall{}u:polyform(n).
([u / p]@[a / l] = ((u@l * a\^{}||p||) + p@[a / l]))
Date html generated:
2017_04_20-AM-07_08_45
Last ObjectModification:
2017_04_17-AM-11_47_38
Theory : list_1
Home
Index