Nuprl Lemma : add-polynom_wf1
∀[n:ℕ]. ∀[p,q:polyform(n)]. ∀[rmz:𝔹]. (add-polynom(n;rmz;p;q) ∈ polyform(n))
Proof
Definitions occuring in Statement :
add-polynom: add-polynom(n;rmz;p;q),
polyform: polyform(n),
nat: ℕ,
bool: 𝔹,
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,
le: A ≤ B,
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),
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,
less_than': less_than'(a;b),
int_upper: {i...},
add-polynom: add-polynom(n;rmz;p;q),
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nil: [],
less_than: a < b,
squash: ↓T,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
length: ||as||,
list_ind: list_ind,
has-value: (a)↓,
has-valueall: has-valueall(a),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rm-zeros: rm-zeros(n;p),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
true: True
Lemmas referenced :
nat_properties,
full-omega-unsat,
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,
bool_wf,
polyform_wf,
le_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,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__lt,
lelt_wf,
subtype_rel_self,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
upper_subtype_nat,
false_wf,
nequal-le-implies,
zero-add,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
int_upper_properties,
equal-wf-T-base,
colength_wf_list,
list-cases,
product_subtype_list,
spread_cons_lemma,
list_wf,
assert_wf,
bnot_wf,
not_wf,
null_nil_lemma,
length_of_nil_lemma,
valueall-type-has-valueall,
list-valueall-type,
valueall-type-polyform,
evalall-reduce,
uiff_transitivity,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
cons_wf,
null_cons_lemma,
length_of_cons_lemma,
value-type-has-value,
int-value-type,
length_wf,
bfalse_wf,
btrue_wf,
list_ind_cons_lemma,
poly-zero_wf,
list_ind_wf,
nil_wf,
top_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
productElimination,
because_Cache,
unionElimination,
applyEquality,
instantiate,
applyLambdaEquality,
hypothesis_subsumption,
equalityElimination,
promote_hyp,
cumulativity,
addEquality,
baseClosed,
imageElimination,
callbyvalueReduce,
sqleReflexivity,
impliesFunctionality,
lessCases,
imageMemberEquality,
axiomSqEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[p,q:polyform(n)]. \mforall{}[rmz:\mBbbB{}]. (add-polynom(n;rmz;p;q) \mmember{} polyform(n))
Date html generated:
2019_06_20-PM-01_52_11
Last ObjectModification:
2018_08_20-PM-09_32_27
Theory : integer!polynomials
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