Nuprl Lemma : minus-polynom_wf2
∀[n:ℕ]. ∀[p:polynom(n)]. (minus-polynom(n;p) ∈ polynom(n))
Proof
Definitions occuring in Statement :
minus-polynom: minus-polynom(n;p),
polynom: polynom(n),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
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],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
polynom: polynom(n),
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
minus-polynom: minus-polynom(n;p),
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
polyform-lead-nonzero: polyform-lead-nonzero(n;p),
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
squash: ↓T,
less_than: a < b,
less_than': less_than'(a;b),
le: A ≤ B,
poly-zero: poly-zero(n;p),
polyform: polyform(n)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
polynom_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
subtract-1-ge-0,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
istype-nat,
map-rev_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
value-type-polynom,
polyform-lead-nonzero_wf,
map-rev-sq-map,
le_wf,
length-map,
length_wf,
hd-map,
subtype_rel_list,
top_wf,
null_wf,
assert_of_null,
length_of_nil_lemma,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
list_wf,
polynom_subtype_polyform,
hd_wf,
poly-zero_wf,
not_wf,
bnot_wf,
int_subtype_base,
int_term_value_minus_lemma,
itermMinus_wf,
decidable__equal_int,
iff_imp_equal_bool,
equal-wf-base,
nat_wf,
istype-false,
polyform_wf,
less_than_wf,
uiff_transitivity,
iff_transitivity,
assert_of_bnot,
trivial-equal,
null-map,
polyform-value-type,
minus-polynom_wf,
assert_functionality_wrt_uiff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
Error :lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
Error :dependent_pairFormation_alt,
Error :lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
Error :universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
Error :functionIsTypeImplies,
Error :dependent_set_memberEquality_alt,
unionElimination,
because_Cache,
equalityElimination,
productElimination,
Error :equalityIstype,
promote_hyp,
instantiate,
cumulativity,
minusEquality,
int_eqReduceTrueSq,
int_eqReduceFalseSq,
applyEquality,
applyLambdaEquality,
baseClosed,
sqequalBase,
imageElimination,
Error :equalityIsType1,
Error :equalityIsType2,
closedConclusion,
baseApply,
Error :equalityIsType4,
intEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[p:polynom(n)]. (minus-polynom(n;p) \mmember{} polynom(n))
Date html generated:
2019_06_20-PM-01_52_56
Last ObjectModification:
2018_12_30-PM-10_14_55
Theory : integer!polynomials
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