Nuprl Lemma : add-ipoly-ringeq
∀r:Rng. ∀p,q:iMonomial() List. ipolynomial-term(add-ipoly(p;q)) ≡ ipolynomial-term(p) (+) ipolynomial-term(q)
Proof
Definitions occuring in Statement :
ringeq_int_terms: t1 ≡ t2,
rng: Rng,
add-ipoly: add-ipoly(p;q),
ipolynomial-term: ipolynomial-term(p),
iMonomial: iMonomial(),
itermAdd: left (+) right,
list: T List,
all: ∀x:A. B[x]
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],
and: P ∧ Q,
prop: ℙ,
ringeq_int_terms: t1 ≡ t2,
le: A ≤ B,
less_than': less_than'(a;b),
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
uiff: uiff(P;Q),
add-ipoly: add-ipoly(p;q),
has-value: (a)↓,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bfalse: ff,
ipolynomial-term: ipolynomial-term(p),
ring_term_value: ring_term_value(f;t),
itermAdd: left (+) right,
int_term_ind: int_term_ind,
itermConstant: "const",
rng: Rng,
bool: 𝔹,
unit: Unit,
it: ⋅,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
iMonomial: iMonomial(),
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
int_nzero: ℤ-o,
callbyvalueall: callbyvalueall,
has-valueall: has-valueall(a),
pi1: fst(t),
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q),
imonomial-le: imonomial-le(m1;m2),
pi2: snd(t),
squash: ↓T,
label: ...$L... t,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
infix_ap: x f y
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
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,
istype-less_than,
length_wf,
iMonomial_wf,
subtract-1-ge-0,
istype-nat,
add_nat_wf,
length_wf_nat,
istype-void,
istype-le,
decidable__le,
add-is-int-iff,
intformnot_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
decidable__lt,
list_wf,
rng_wf,
non_neg_length,
list-cases,
value-type-has-value,
list-value-type,
nil_wf,
null_nil_lemma,
product_subtype_list,
cons_wf,
null_cons_lemma,
spread_cons_lemma,
ipolynomial-term_wf,
int-to-ring-zero,
rng_plus_zero,
ring_term_value_wf,
ring_term_value_add_lemma,
ring_term_value_const_lemma,
rng_car_wf,
imonomial-le_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
add-ipoly_wf1,
valueall-type-has-valueall,
list-valueall-type,
product-valueall-type,
int_nzero_wf,
sorted_wf,
set-valueall-type,
nequal_wf,
int-valueall-type,
evalall-reduce,
ipolynomial-term-cons-ringeq,
int-value-type,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
imonomial-term_wf,
length_of_cons_lemma,
int_nzero_properties,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
ringeq_int_terms_functionality,
itermAdd_functionality_wrt_ringeq,
ringeq_int_terms_transitivity,
ringeq_int_terms_weakening,
intlex_wf,
intlex-antisym,
subtype_rel_universe1,
set_subtype_base,
list_subtype_base,
int_subtype_base,
equal_wf,
squash_wf,
true_wf,
istype-universe,
set_wf,
member_wf,
subtype_rel_self,
iff_weakening_equal,
imonomial-term-add-ringeq,
rng_zero_wf,
imonomial-term-linear-ringeq,
rng_times_zero,
rng_plus_wf,
rng_properties,
rng_plus_assoc,
rng_plus_ac_1,
rng_plus_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
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,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
axiomEquality,
functionIsTypeImplies,
inhabitedIsType,
addEquality,
because_Cache,
dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
unionElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
productElimination,
equalityIstype,
callbyvalueReduce,
voidEquality,
hypothesis_subsumption,
functionIsType,
equalityElimination,
instantiate,
cumulativity,
setEquality,
intEquality,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
independent_pairEquality,
applyEquality,
imageElimination,
universeEquality,
hyp_replacement,
imageMemberEquality
Latex:
\mforall{}r:Rng. \mforall{}p,q:iMonomial() List.
ipolynomial-term(add-ipoly(p;q)) \mequiv{} ipolynomial-term(p) (+) ipolynomial-term(q)
Date html generated:
2020_05_19-PM-10_08_08
Last ObjectModification:
2020_01_08-PM-06_06_09
Theory : rings_1
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