Nuprl Lemma : add-ipoly-equiv
∀p,q:iMonomial() List. ipolynomial-term(add-ipoly(p;q)) ≡ ipolynomial-term(p) (+) ipolynomial-term(q)
Proof
Definitions occuring in Statement :
add-ipoly: add-ipoly(p;q),
ipolynomial-term: ipolynomial-term(p),
iMonomial: iMonomial(),
equiv_int_terms: t1 ≡ t2,
itermAdd: left (+) right,
list: T List,
all: ∀x:A. B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
and: P ∧ Q,
ge: i ≥ j ,
le: A ≤ B,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
equiv_int_terms: t1 ≡ t2,
less_than': less_than'(a;b),
not: ¬A,
sq_stable: SqStable(P),
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
true: True,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
nat_plus: ℕ+,
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),
int_term_value: int_term_value(f;t),
itermAdd: left (+) right,
int_term_ind: int_term_ind,
itermConstant: "const",
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b,
iMonomial: iMonomial(),
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),
label: ...$L... t
Lemmas referenced :
list_wf,
iMonomial_wf,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
istype-less_than,
length_wf,
subtract-1-ge-0,
istype-nat,
add_nat_wf,
length_wf_nat,
istype-void,
istype-le,
sq_stable__le,
decidable__lt,
istype-false,
not-lt-2,
condition-implies-le,
add-associates,
minus-add,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-mul-special,
zero-mul,
zero-add,
add-zero,
add-commutes,
le-add-cancel,
non_neg_length,
istype-sqequal,
subtype_rel-equal,
nat_wf,
base_wf,
set_subtype_base,
le_wf,
int_subtype_base,
subtract_wf,
subtype_base_sq,
two-mul,
mul-distributes-right,
one-mul,
add_functionality_wrt_le,
le_reflexive,
less-iff-le,
minus-zero,
omega-shadow,
mul-distributes,
mul-commutes,
mul-associates,
mul-swap,
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,
add-is-int-iff,
int_term_value_wf,
imonomial-le_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
valueall-type-has-valueall,
list-valueall-type,
product-valueall-type,
int_nzero_wf,
sorted_wf,
set-valueall-type,
nequal_wf,
int-valueall-type,
add-ipoly_wf1,
evalall-reduce,
int-value-type,
eq_int_wf,
assert_of_eq_int,
neg_assert_of_eq_int,
itermAdd_wf,
imonomial-term_wf,
length_of_cons_lemma,
not-equal-2,
not-equal-implies-less,
equiv_int_terms_functionality,
itermAdd_functionality,
ipolynomial-term-cons,
int_nzero_properties,
equiv_int_terms_transitivity,
equiv_int_terms_weakening,
intlex_wf,
intlex-antisym,
subtype_rel_universe1,
list_subtype_base,
equal_wf,
squash_wf,
true_wf,
istype-universe,
set_wf,
member_wf,
subtype_rel_self,
iff_weakening_equal,
imonomial-term-add,
imonomial-term-linear,
subtype_rel_product
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
inhabitedIsType,
hypothesisEquality,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
setElimination,
rename,
intWeakElimination,
independent_pairFormation,
productElimination,
imageElimination,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
sqequalRule,
lambdaEquality_alt,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
addEquality,
because_Cache,
dependent_set_memberEquality_alt,
imageMemberEquality,
baseClosed,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
unionElimination,
Error :memTop,
minusEquality,
multiplyEquality,
dependent_pairFormation_alt,
applyEquality,
intEquality,
promote_hyp,
applyLambdaEquality,
instantiate,
cumulativity,
callbyvalueReduce,
voidEquality,
hypothesis_subsumption,
functionIsType,
equalityElimination,
setEquality,
closedConclusion,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
independent_pairEquality,
baseApply,
universeEquality,
hyp_replacement,
sqequalBase,
setIsType
Latex:
\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-09_38_20
Last ObjectModification:
2020_01_04-PM-08_47_41
Theory : omega
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