Nuprl Lemma : polynom-equal-iff
∀[n:ℕ]. ∀[p,q:polynom(n)]. uiff(p = q ∈ polynom(n);∀l:{l:ℤ List| ||l|| = n ∈ ℤ} . (l@p = l@q ∈ ℤ))
Proof
Definitions occuring in Statement :
poly-int-val: l@p,
polynom: polynom(n),
length: ||as||,
list: T List,
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
squash: ↓T,
true: True,
top: Top,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
le: A ≤ B,
less_than': less_than'(a;b),
polynom: polynom(n),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
btrue: tt,
polyform: polyform(n),
poly-zero: poly-zero(n;p),
add-polynom: add-polynom(n;rmz;p;q),
minus-polynom: Error :minus-polynom,
subtract: n - m,
bool: 𝔹,
unit: Unit,
it: ⋅,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
polyform-lead-nonzero: polyform-lead-nonzero(n;p),
nat_plus: ℕ+,
has-valueall: has-valueall(a),
has-value: (a)↓,
callbyvalueall: callbyvalueall,
cons: [a / b],
less_than: a < b,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
minus-polynom: minus-polynom(n;p),
rm-zeros: rm-zeros(n;p),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
ext-eq: A ≡ B
Lemmas referenced :
and_wf,
equal_wf,
polynom_wf,
poly-int-val_wf2,
set_wf,
list_wf,
equal-wf-base-T,
all_wf,
int_subtype_base,
list_subtype_base,
minus-polynom_wf2,
add-polynom_wf,
assert-poly-zero,
iff_weakening_equal,
minus-polynom-val,
add_functionality_wrt_eq,
add-polynom-int-val,
true_wf,
squash_wf,
btrue_wf,
polynom_subtype_polyform,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_minus_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
itermConstant_wf,
itermMinus_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__equal_int,
nat_properties,
intformand_wf,
intformle_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_formula_prop_less_lemma,
ge_wf,
less_than_wf,
assert_wf,
poly-zero_wf,
add-polynom_wf1,
less_than_transitivity1,
less_than_irreflexivity,
minus-polynom_wf,
false_wf,
le_wf,
subtype_rel_self,
decidable__le,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
assert_of_eq_int,
eq_int_wf,
bnot_wf,
not_wf,
equal-wf-base,
bool_wf,
polyform-lead-nonzero_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
subtype_rel_list,
polyform_wf,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
uiff_transitivity,
length-minus-polynom,
valueall-type-polyform,
evalall-reduce,
valueall-type-polynom,
valueall-type-has-valueall,
polynom-subtype-list,
length_wf,
assert_of_null,
equal-wf-T-base,
null_wf,
length_of_cons_lemma,
null_cons_lemma,
product_subtype_list,
length_of_null_list,
length_of_nil_lemma,
null_nil_lemma,
list-cases,
reduce_hd_cons_lemma,
length_wf_nat,
int-value-type,
set-value-type,
nat_wf,
value-type-has-value,
top_wf,
decidable__lt,
spread_cons_lemma,
bfalse_wf,
assert_of_ff,
nil_wf,
btrue_neq_bfalse,
assert_elim,
non_neg_length,
le_weakening2,
list-valueall-type,
less_than_anti-reflexive,
cons_wf,
polyform-value-type,
map_cons_lemma,
map-rev-sq-map,
list_ind_cons_lemma,
list_ind_wf,
list_induction,
subtype_rel_transitivity,
map-length,
add-is-int-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
lambdaFormation,
equalitySymmetry,
dependent_set_memberEquality,
hypothesis,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
applyLambdaEquality,
setElimination,
rename,
productElimination,
intEquality,
sqequalRule,
lambdaEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
because_Cache,
dependent_functionElimination,
axiomEquality,
setEquality,
equalityTransitivity,
independent_pairEquality,
isect_memberEquality,
independent_isectElimination,
independent_functionElimination,
imageMemberEquality,
universeEquality,
imageElimination,
natural_numberEquality,
minusEquality,
computeAll,
voidEquality,
voidElimination,
int_eqEquality,
dependent_pairFormation,
unionElimination,
intWeakElimination,
addEquality,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
impliesFunctionality,
callbyvalueReduce,
levelHypothesis,
addLevel,
hypothesis_subsumption,
sqequalAxiom,
lessCases,
functionEquality,
equalityUniverse,
int_eqReduceFalseSq,
pointwiseFunctionality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[p,q:polynom(n)]. uiff(p = q;\mforall{}l:\{l:\mBbbZ{} List| ||l|| = n\} . (l@p = l@q))
Date html generated:
2017_09_29-PM-06_03_47
Last ObjectModification:
2017_07_26-PM-02_52_58
Theory : integer!polynomials
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