Nuprl Lemma : q-constraint-negative
∀[x:ℕ ⟶ ℚ]. ∀[r:ℤ]. ∀[k:ℕ+]. ∀[y:ℚ List].
(uiff(q-rel(r;q-linear(k;j.x j;y));q-rel(r;-(y[k - 1]) + ((-1/x k) * q-linear(k - 1;j.x j;y))))) supposing
(x k < 0 and
(k ≤ ||y||))
Proof
Definitions occuring in Statement :
q-rel: q-rel(r;x),
q-linear: q-linear(k;i.X[i];y),
qless: r < s,
qdiv: (r/s),
qmul: r * s,
qadd: r + s,
rationals: ℚ,
select: L[n],
length: ||as||,
list: T List,
nat_plus: ℕ+,
nat: ℕ,
uiff: uiff(P;Q),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
minus: -n,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
nat: ℕ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
qless: r < s,
grp_lt: a < b,
set_lt: a <p b,
assert: ↑b,
ifthenelse: if b then t else f fi ,
set_blt: a <b b,
band: p ∧b q,
infix_ap: x f y,
set_le: ≤b,
pi2: snd(t),
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
grp_le: ≤b,
pi1: fst(t),
qadd_grp: <ℚ+>,
q_le: q_le(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
bor: p ∨bq,
qpositive: qpositive(r),
qsub: r - s,
qadd: r + s,
qmul: r * s,
btrue: tt,
lt_int: i <z j,
bnot: ¬bb,
bfalse: ff,
qeq: qeq(r;s),
eq_int: (i =z j),
true: True,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
q-rel: q-rel(r;x),
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T},
iff: P ⇐⇒ Q,
bool: 𝔹,
le: A ≤ B,
rev_implies: P ⇐ Q,
unit: Unit,
it: ⋅,
qge: a ≥ b,
qgt: a > b
Lemmas referenced :
qmul_reverses_qless,
qmul_wf,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
istype-le,
q-rel_wf,
squash_wf,
true_wf,
rationals_wf,
q-linear-unroll,
istype-nat,
iff_weakening_equal,
eq_int_wf,
qadd_wf,
select_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
decidable__lt,
length_wf,
qdiv_wf,
nat_plus_subtype_nat,
q-linear_wf,
qle_witness,
qle_wf,
qless_witness,
qless_wf,
ifthenelse_wf,
int-subtype-rationals,
list_wf,
nat_plus_wf,
qless_transitivity_2_qorder,
qle_weakening_eq_qorder,
qless_irreflexivity,
equal-wf-base,
bool_wf,
int_subtype_base,
assert_wf,
bnot_wf,
not_wf,
istype-assert,
istype-void,
qmul_zero_qrng,
subtype_rel_self,
qinv_inv_q,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
qmul-preserves-eq,
equal_wf,
istype-universe,
qmul_over_minus_qrng,
qmul_over_plus_qrng,
qadd_comm_q,
qmul-qdiv-cancel3,
qmul_assoc,
qmul_one_qrng,
qmul_preserves_qle,
qmul_preserves_qle2,
qle-minus,
qinv_id_q,
qle_functionality_wrt_implies,
qle_weakening_lt_qorder,
qmul_preserves_qless
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
minusEquality,
natural_numberEquality,
hypothesis,
applyEquality,
sqequalRule,
hypothesisEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
productElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
inhabitedIsType,
lambdaFormation_alt,
axiomEquality,
equalityIstype,
closedConclusion,
sqequalBase,
instantiate,
universeEquality,
promote_hyp,
independent_pairEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
functionIsType,
baseApply,
intEquality,
equalityElimination,
applyLambdaEquality
Latex:
\mforall{}[x:\mBbbN{} {}\mrightarrow{} \mBbbQ{}]. \mforall{}[r:\mBbbZ{}]. \mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[y:\mBbbQ{} List].
(uiff(q-rel(r;q-linear(k;j.x j;y));q-rel(r;-(y[k - 1])
+ ((-1/x k) * q-linear(k - 1;j.x j;y))))) supposing
(x k < 0 and
(k \mleq{} ||y||))
Date html generated:
2020_05_20-AM-09_27_28
Last ObjectModification:
2020_01_31-AM-11_08_04
Theory : rationals
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