Nuprl Lemma : mul-ipoly-equiv
∀[p,q:iMonomial() List]. ipolynomial-term(mul-ipoly(p;q)) ≡ ipolynomial-term(p) (*) ipolynomial-term(q)
Proof
Definitions occuring in Statement :
mul-ipoly: mul-ipoly(p;q),
ipolynomial-term: ipolynomial-term(p),
iMonomial: iMonomial(),
equiv_int_terms: t1 ≡ t2,
itermMultiply: left (*) right,
list: T List,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
or: P ∨ Q,
mul-ipoly: mul-ipoly(p;q),
uimplies: b supposing a,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
iMonomial: iMonomial(),
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
implies: P ⇒ Q,
int_nzero: ℤ-o,
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
bfalse: ff,
equiv_int_terms: t1 ≡ t2,
itermMultiply: left (*) right,
int_term_ind: int_term_ind,
itermConstant: "const",
int_term_value: int_term_value(f;t),
ipolynomial-term: ipolynomial-term(p),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
subtype_rel: A ⊆r B,
rev_uimplies: rev_uimplies(P;Q),
itermAdd: left (+) right
Lemmas referenced :
iMonomial_wf,
list-cases,
valueall-type-has-valueall,
list_wf,
list-valueall-type,
void-valueall-type,
nil_wf,
evalall-reduce,
null_nil_lemma,
product_subtype_list,
product-valueall-type,
int_nzero_wf,
sorted_wf,
subtype_rel_self,
set-valueall-type,
nequal_wf,
int-valueall-type,
cons_wf,
null_cons_lemma,
spread_cons_lemma,
ipolynomial-term_wf,
int_term_value_wf,
zero-mul,
null_wf,
bool_wf,
eqtt_to_assert,
assert_of_null,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-wf-T-base,
btrue_neq_bfalse,
bfalse_wf,
and_wf,
btrue_wf,
itermAdd_wf,
imonomial-term_wf,
subtype_rel_product,
subtype_rel_list,
int_term_wf,
list_accum_wf,
mul-commutes,
eager-accum_wf,
mul-mono-poly_wf1,
add-ipoly_wf1,
itermMultiply_wf,
equiv_int_terms_functionality,
equiv_int_terms_weakening,
itermMultiply_functionality,
ipolynomial-term-cons,
mul-distributes-right,
list_accum_cons_lemma,
list_accum_nil_lemma,
equiv_int_terms_wf,
all_wf,
list_induction,
add-zero,
mul-mono-poly-equiv,
add-ipoly-equiv,
itermAdd_functionality,
equiv_int_terms_transitivity,
add-associates,
eager-accum-list_accum
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
sqequalRule,
voidEquality,
independent_isectElimination,
callbyvalueReduce,
promote_hyp,
hypothesis_subsumption,
productElimination,
lambdaEquality,
setEquality,
intEquality,
because_Cache,
independent_functionElimination,
natural_numberEquality,
lambdaFormation,
isect_memberEquality,
voidElimination,
axiomEquality,
functionEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
instantiate,
cumulativity,
baseClosed,
applyLambdaEquality,
independent_pairFormation,
dependent_set_memberEquality,
rename,
setElimination,
applyEquality,
productEquality,
independent_pairEquality,
multiplyEquality,
addEquality
Latex:
\mforall{}[p,q:iMonomial() List].
ipolynomial-term(mul-ipoly(p;q)) \mequiv{} ipolynomial-term(p) (*) ipolynomial-term(q)
Date html generated:
2017_09_29-PM-05_53_40
Last ObjectModification:
2017_07_26-PM-01_42_53
Theory : omega
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