Nuprl Lemma : satisfies-gcd-reduce-eq-constraints
∀[n:ℕ+]. ∀[eqs,sat:{L:ℤ List| ||L|| = n ∈ ℤ} List]. ∀[xs:{L:ℤ List| ||L|| = n ∈ ℤ} ].
uiff((∀as∈eqs.xs ⋅ as =0);(↑isl(gcd-reduce-eq-constraints(sat;eqs)))
∧ (∀as∈outl(gcd-reduce-eq-constraints(sat;eqs)).xs ⋅ as =0))
supposing (∀as∈sat.xs ⋅ as =0) ∧ 0 < ||xs|| ∧ (hd(xs) = 1 ∈ ℤ)
Proof
Definitions occuring in Statement :
gcd-reduce-eq-constraints: gcd-reduce-eq-constraints(sat;LL),
satisfies-integer-equality: xs ⋅ as =0,
l_all: (∀x∈L.P[x]),
length: ||as||,
hd: hd(l),
list: T List,
nat_plus: ℕ+,
outl: outl(x),
assert: ↑b,
isl: isl(x),
less_than: a < b,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
l_all: (∀x∈L.P[x]),
all: ∀x:A. B[x],
satisfies-integer-equality: xs ⋅ as =0,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
isl: isl(x),
outl: outl(x),
not: ¬A,
false: False,
listp: A List+,
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
guard: {T},
cons: [a / b],
or: P ∨ Q,
less_than': less_than'(a;b),
length: ||as||,
list_ind: list_ind,
nil: [],
it: ⋅,
sq_type: SQType(T),
top: Top,
true: True,
subtract: n - m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
decidable: Dec(P),
sq_stable: SqStable(P),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
colength: colength(L),
ge: i ≥ j ,
nat: ℕ,
gcd-reduce-eq-constraints: gcd-reduce-eq-constraints(sat;LL),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
eager-map: eager-map(f;as),
bfalse: ff,
exposed-bfalse: exposed-bfalse,
bool: 𝔹,
unit: Unit,
exists: ∃x:A. B[x],
bnot: ¬bb,
nequal: a ≠ b ∈ T ,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
outr: outr(x),
evalall: evalall(t),
rev_uimplies: rev_uimplies(P;Q),
int_nzero: ℤ-o,
pi1: fst(t),
hd: hd(l),
int-vec-mul: a * as,
compose: f o g,
gt: i > j,
int_lower: {...i},
l_member: (x ∈ l),
select: L[n]
Lemmas referenced :
assert_witness,
member-less_than,
l_all_wf,
list_wf,
equal-wf-base,
list_subtype_base,
int_subtype_base,
set_subtype_base,
less_than_wf,
satisfies-integer-equality_wf,
l_member_wf,
istype-assert,
gcd-reduce-eq-constraints_wf,
btrue_wf,
bfalse_wf,
listp_wf,
assert_elim,
btrue_neq_bfalse,
subtype_rel_list,
istype-int,
subtype_rel_sets_simple,
length_wf,
less_than_transitivity1,
le_weakening,
istype-less_than,
equal_wf,
nat_plus_wf,
product_subtype_list,
list-cases,
subtype_base_sq,
reduce_hd_cons_lemma,
istype-void,
length_of_cons_lemma,
istype-nat,
le_weakening2,
minus-minus,
le-add-cancel,
zero-add,
add-commutes,
add_functionality_wrt_le,
le_antisymmetry_iff,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
add-associates,
condition-implies-le,
not-equal-2,
istype-false,
subtract_wf,
decidable__equal_int,
sq_stable__le,
spread_cons_lemma,
le_wf,
nat_wf,
subtract-1-ge-0,
istype-le,
colength_wf_list,
colength-cons-not-zero,
set_wf,
ge_wf,
less_than_irreflexivity,
nat_properties,
accumulate_abort_nil_lemma,
l_all_wf_nil,
cons_wf,
length_of_nil_lemma,
eager_map_nil_lemma,
istype-true,
null_nil_lemma,
null_cons_lemma,
null_wf,
decidable__assert,
eager_map_cons_lemma,
cons-listp,
accumulate_abort_cons_lemma,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
valueall-type-has-valueall,
union-valueall-type,
list-valueall-type,
int-valueall-type,
void-valueall-type,
nil_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
unit_wf2,
equal-valueall-type,
it_wf,
evalall-reduce,
l_all_cons,
int_dot_cons_lemma,
int_dot_nil_left_lemma,
mul-commutes,
add-zero,
one-mul,
non_neg_length,
length_wf_nat,
istype-sqequal,
le-add-cancel2,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
nat_plus_properties,
int-value-type,
value-type-has-value,
not-lt-2,
decidable__lt,
gcd-list_wf,
absval_wf,
istype-top,
top_wf,
nequal_wf,
less-iff-le,
not-le-2,
decidable__le,
remainder_wfa,
accumulate_abort-aborted,
eager-map_wf,
list-value-type,
divide_wfa,
eager-map-is-map,
map-length,
map_wf,
gcd-list-property,
minus-zero,
assert_wf,
iff_weakening_uiff,
assert_of_lt_int,
lt_int_wf,
absval_unfold2,
int-eq-constraint-factor-sym,
absval-positive,
squash_wf,
true_wf,
istype-universe,
integer-dot-product_wf,
subtype_rel_self,
iff_weakening_equal,
absval_pos,
map-map,
list_extensionality,
map_length,
length-map,
select-map,
divide-exact,
select_wf,
not-gt-2,
absval_neg,
div_anti_sym,
mul-minus-1,
map_cons_lemma,
int-dot-mul-left,
zero_ann_a,
rem_sym,
l_all_nil,
equal-wf-base-T,
subtype_rel_sets,
isl_wf,
and_wf,
length-singleton,
int_dot_cons_nil_lemma,
add-is-int-iff,
trivial-equal,
l_all_iff,
member-gcd-reduce-constraints,
add_nat_plus,
mul_preserves_eq,
int-vec-mul_wf,
div_rem_sum,
add_functionality_wrt_eq,
absval_lbound,
absval-non-neg,
mul-associates,
mul-swap
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
extract_by_obid,
isectElimination,
because_Cache,
independent_functionElimination,
hypothesis,
lambdaEquality_alt,
dependent_functionElimination,
hypothesisEquality,
axiomEquality,
independent_isectElimination,
functionIsTypeImplies,
inhabitedIsType,
universeIsType,
setEquality,
intEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
setIsType,
productIsType,
lambdaFormation_alt,
unionElimination,
equalityIstype,
dependent_set_memberEquality_alt,
applyLambdaEquality,
voidElimination,
imageElimination,
sqequalBase,
isect_memberEquality_alt,
isectIsTypeImplies,
hypothesis_subsumption,
promote_hyp,
cumulativity,
instantiate,
minusEquality,
addEquality,
imageMemberEquality,
intWeakElimination,
functionIsType,
sqleReflexivity,
callbyvalueReduce,
equalityElimination,
int_eqReduceTrueSq,
unionEquality,
voidEquality,
inlEquality_alt,
dependent_pairFormation_alt,
int_eqReduceFalseSq,
inrEquality_alt,
int_eqEquality,
multiplyEquality,
lessCases,
axiomSqEquality,
inrFormation_alt,
inlFormation_alt,
universeEquality,
hyp_replacement,
Error :memTop,
lambdaFormation,
isect_memberEquality,
lambdaEquality,
addLevel,
levelHypothesis,
dependent_set_memberEquality
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[eqs,sat:\{L:\mBbbZ{} List| ||L|| = n\} List]. \mforall{}[xs:\{L:\mBbbZ{} List| ||L|| = n\} ].
uiff((\mforall{}as\mmember{}eqs.xs \mcdot{} as =0);(\muparrow{}isl(gcd-reduce-eq-constraints(sat;eqs)))
\mwedge{} (\mforall{}as\mmember{}outl(gcd-reduce-eq-constraints(sat;eqs)).xs \mcdot{} as =0))
supposing (\mforall{}as\mmember{}sat.xs \mcdot{} as =0) \mwedge{} 0 < ||xs|| \mwedge{} (hd(xs) = 1)
Date html generated:
2020_05_19-PM-09_38_47
Last ObjectModification:
2020_01_01-AM-10_05_12
Theory : omega
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