Nuprl Lemma : add-polynom_wf
∀[n:ℕ]. ∀[p,q:polynom(n)]. (add-polynom(n;tt;p;q) ∈ polynom(n))
Proof
Definitions occuring in Statement :
add-polynom: add-polynom(n;rmz;p;q),
polynom: polynom(n),
nat: ℕ,
btrue: tt,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
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),
polynom: polynom(n),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
less_than': less_than'(a;b),
int_upper: {i...},
add-polynom: add-polynom(n;rmz;p;q),
squash: ↓T,
less_than: a < b,
nil: [],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
colength: colength(L),
cons: [a / b],
has-valueall: has-valueall(a),
has-value: (a)↓,
list_ind: list_ind,
length: ||as||,
evalall: evalall(t),
callbyvalueall: callbyvalueall,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
true: True,
polyform: polyform(n),
nequal: a ≠ b ∈ T ,
polyform-lead-nonzero: polyform-lead-nonzero(n;p),
sq_stable: SqStable(P)
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,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
istype-false,
nequal-le-implies,
zero-add,
itermAdd_wf,
int_term_value_add_lemma,
istype-nat,
int_upper_properties,
le_wf,
false_wf,
equal_wf,
list_wf,
spread_cons_lemma,
product_subtype_list,
list-cases,
polynom_wf,
colength_wf_list,
nat_wf,
equal-wf-T-base,
less_than_wf,
evalall-reduce,
valueall-type-polynom,
list-valueall-type,
valueall-type-has-valueall,
length_of_nil_lemma,
null_nil_lemma,
not_wf,
bnot_wf,
assert_wf,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
length_wf,
int-value-type,
value-type-has-value,
length_of_cons_lemma,
null_cons_lemma,
cons_wf,
bfalse_wf,
polynom-subtype-list,
le-add-cancel,
add-commutes,
add_functionality_wrt_le,
not-lt-2,
rm-zeros_wf,
top_wf,
btrue_wf,
polyform-lead-nonzero_wf,
subtype_rel_list,
polyform_wf,
polynom_subtype_polyform,
equal-wf-base,
istype-assert,
lt_int_wf,
assert_of_lt_int,
istype-top,
reduce_hd_cons_lemma,
add_nat_plus,
length_wf_nat,
nat_plus_properties,
add-is-int-iff,
poly-zero_wf,
sq_stable__polyform-lead-nonzero,
bool_cases
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
thin,
Error :lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
productElimination,
because_Cache,
unionElimination,
applyEquality,
instantiate,
applyLambdaEquality,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
hypothesis_subsumption,
equalityElimination,
Error :equalityIstype,
promote_hyp,
cumulativity,
addEquality,
voidEquality,
isect_memberEquality,
intEquality,
lambdaEquality,
dependent_set_memberEquality,
dependent_pairFormation,
lambdaFormation,
isect_memberFormation,
imageElimination,
baseClosed,
sqleReflexivity,
callbyvalueReduce,
impliesFunctionality,
axiomSqEquality,
imageMemberEquality,
lessCases,
Error :setIsType,
baseApply,
closedConclusion,
sqequalBase,
Error :functionIsType,
pointwiseFunctionality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[p,q:polynom(n)]. (add-polynom(n;tt;p;q) \mmember{} polynom(n))
Date html generated:
2019_06_20-PM-01_52_19
Last ObjectModification:
2019_01_17-PM-04_36_33
Theory : integer!polynomials
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