Nuprl Lemma : add-ipoly_wf
∀[p,q:iPolynomial()]. (add-ipoly(p;q) ∈ iPolynomial())
Proof
Definitions occuring in Statement :
add-ipoly: add-ipoly(p;q),
iPolynomial: iPolynomial(),
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
pi2: snd(t),
iMonomial: iMonomial(),
guard: {T},
lelt: i ≤ j < k,
sq_stable: SqStable(P),
uimplies: b supposing a,
int_seg: {i..j-},
false: False,
not: ¬A,
and: P ∧ Q,
imonomial-less: imonomial-less(m1;m2),
squash: ↓T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
iPolynomial: iPolynomial(),
member: t ∈ T,
uall: ∀[x:A]. B[x],
true: True,
less_than': less_than'(a;b),
top: Top,
subtype_rel: A ⊆r B,
subtract: n - m,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
le: A ≤ B,
ge: i ≥ j ,
nat: ℕ,
less_than: a < b,
nat_plus: ℕ+,
exists: ∃x:A. B[x],
cons: [a / b],
sq_type: SQType(T),
int_nzero: ℤ-o,
ifthenelse: if b then t else f fi ,
btrue: tt,
null: null(as),
has-value: (a)↓,
it: ⋅,
nil: [],
add-ipoly: add-ipoly(p;q),
bfalse: ff,
assert: ↑b,
bnot: ¬bb,
unit: Unit,
bool: 𝔹,
has-valueall: has-valueall(a),
callbyvalueall: callbyvalueall,
pi1: fst(t),
cand: A c∧ B,
nequal: a ≠ b ∈ T ,
imonomial-le: imonomial-le(m1;m2),
select: L[n]
Lemmas referenced :
iPolynomial_wf,
imonomial-less_wf,
all_wf,
int_seg_wf,
imonomial-le_wf,
assert_witness,
le_weakening2,
length_wf,
less_than_transitivity2,
sq_stable__le,
iMonomial_wf,
select_wf,
list_wf,
equal_wf,
add-ipoly_wf1,
zero-mul,
add-mul-special,
not-lt-2,
decidable__lt,
le_wf,
length_wf_nat,
add_nat_wf,
nat_wf,
le-add-cancel,
add-zero,
add_functionality_wrt_le,
add-commutes,
add-swap,
add-associates,
minus-minus,
minus-add,
minus-one-mul-top,
zero-add,
minus-one-mul,
condition-implies-le,
less-iff-le,
not-ge-2,
false_wf,
subtract_wf,
decidable__le,
less_than_wf,
ge_wf,
less_than_irreflexivity,
less_than_transitivity1,
nat_properties,
int_seg_properties,
mul-swap,
mul-associates,
mul-commutes,
mul-distributes,
omega-shadow,
minus-zero,
mul-distributes-right,
two-mul,
one-mul,
le_reflexive,
int_subtype_base,
set_subtype_base,
non_neg_length,
product_subtype_list,
list-cases,
nequal_wf,
subtype_rel_self,
sorted_wf,
int_nzero_wf,
product_subtype_base,
list_subtype_base,
subtype_base_sq,
btrue_wf,
bool_subtype_base,
bool_wf,
list-value-type,
value-type-has-value,
cons_wf,
assert-bnot,
bool_cases_sqequal,
eqff_to_assert,
eqtt_to_assert,
bfalse_wf,
evalall-reduce,
int-valueall-type,
set-valueall-type,
product-valueall-type,
list-valueall-type,
valueall-type-has-valueall,
int-value-type,
eq_int_wf,
assert_of_eq_int,
length_of_cons_lemma,
istype-void,
istype-sqequal,
add-member-int_seg2,
istype-less_than,
squash_wf,
true_wf,
select-cons-tl,
istype-false,
not-equal-2,
add-is-int-iff,
not-le-2,
le-add-cancel2,
istype-le,
neg_assert_of_eq_int,
not-equal-implies-less,
int_nzero_properties,
intlex-antisym,
intlex_wf,
add-poly-lemma1,
add-subtract-cancel,
lelt_wf,
le-add-cancel-alt,
iff_weakening_equal,
select_cons_tl,
trivial-equal,
decidable__equal_int,
le_antisymmetry_iff,
imonomial-less-transitive,
not-imonomial-le
Rules used in proof :
isect_memberEquality,
axiomEquality,
equalitySymmetry,
equalityTransitivity,
lambdaFormation,
baseClosed,
imageMemberEquality,
natural_numberEquality,
independent_isectElimination,
because_Cache,
intEquality,
voidElimination,
independent_pairEquality,
productElimination,
lambdaEquality,
sqequalRule,
imageElimination,
independent_functionElimination,
dependent_functionElimination,
hypothesis,
hypothesisEquality,
isectElimination,
extract_by_obid,
dependent_set_memberEquality,
rename,
thin,
setElimination,
sqequalHypSubstitution,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
multiplyEquality,
voidEquality,
minusEquality,
applyEquality,
independent_pairFormation,
unionElimination,
addEquality,
intWeakElimination,
levelHypothesis,
addLevel,
promote_hyp,
sqequalIntensionalEquality,
dependent_pairFormation,
hypothesis_subsumption,
setEquality,
cumulativity,
instantiate,
sqleReflexivity,
callbyvalueReduce,
equalityElimination,
Error :inhabitedIsType,
Error :lambdaFormation_alt,
int_eqReduceTrueSq,
Error :isect_memberEquality_alt,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
Error :equalityIstype,
Error :dependent_set_memberEquality_alt,
Error :productIsType,
hyp_replacement,
Error :universeIsType,
closedConclusion,
int_eqReduceFalseSq,
baseApply,
universeEquality
Latex:
\mforall{}[p,q:iPolynomial()]. (add-ipoly(p;q) \mmember{} iPolynomial())
Date html generated:
2019_06_20-PM-00_45_35
Last ObjectModification:
2019_01_20-PM-01_14_57
Theory : omega
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