Nuprl Lemma : sparse-signed-rep-lemma1
∀m:ℤ. (∃p:ℤ × {-2..3-} [let k,b = p in (m = ((4 * k) + b) ∈ ℤ) ∧ ((|b| = 2 ∈ ℤ) ⇒ (↑isEven(k)))])
Proof
Definitions occuring in Statement :
isEven: isEven(n),
absval: |i|,
int_seg: {i..j-},
assert: ↑b,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
implies: P ⇒ Q,
and: P ∧ Q,
spread: spread def,
product: x:A × B[x],
multiply: n * m,
add: n + m,
minus: -n,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
sq_type: SQType(T),
guard: {T},
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
pi2: snd(t),
pi1: fst(t),
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
and: P ∧ Q,
sq_exists: ∃x:A [B[x]],
int_seg: {i..j-},
lelt: i ≤ j < k,
absval: |i|,
cand: A c∧ B,
subtype_rel: A ⊆r B,
nat: ℕ,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bfalse: ff,
same-parity: same-parity(n;m),
bool: 𝔹,
unit: Unit,
it: ⋅,
bnot: ¬bb,
assert: ↑b,
isOdd: isOdd(n),
eq_int: (i =z j),
modulus: a mod n,
remainder: n rem m
Lemmas referenced :
divrem_wf,
subtype_base_sq,
int_subtype_base,
nequal_wf,
set-value-type,
equal_wf,
product-value-type,
divrem-sq,
pi2_wf,
pi1_wf_top,
istype-void,
div_rem_sum,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermAdd_wf,
itermMultiply_wf,
itermConstant_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
istype-le,
istype-less_than,
set_subtype_base,
lelt_wf,
istype-assert,
isEven_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
rem_bounds_absval,
absval_ifthenelse,
decidable__assert,
lt_int_wf,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
itermMinus_wf,
int_term_value_minus_lemma,
absval_wf,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
int_seg_properties,
isEven-add,
bool_cases_sqequal,
assert-bnot,
subtract-elim,
equal-wf-base,
le_int_wf,
le_wf,
uiff_transitivity,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
productEquality,
intEquality,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
dependent_set_memberEquality_alt,
natural_numberEquality,
instantiate,
cumulativity,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
equalityIstype,
inhabitedIsType,
baseClosed,
sqequalBase,
universeIsType,
cutEval,
sqequalRule,
lambdaEquality_alt,
setElimination,
rename,
productElimination,
applyLambdaEquality,
independent_pairEquality,
isect_memberEquality_alt,
because_Cache,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
independent_pairFormation,
dependent_set_memberFormation_alt,
addEquality,
minusEquality,
productIsType,
callbyvalueReduce,
sqleReflexivity,
applyEquality,
baseApply,
closedConclusion,
functionIsType,
hyp_replacement,
equalityElimination,
promote_hyp
Latex:
\mforall{}m:\mBbbZ{}. (\mexists{}p:\mBbbZ{} \mtimes{} \{-2..3\msupminus{}\} [let k,b = p in (m = ((4 * k) + b)) \mwedge{} ((|b| = 2) {}\mRightarrow{} (\muparrow{}isEven(k)))])
Date html generated:
2019_10_15-AM-11_26_17
Last ObjectModification:
2019_06_26-PM-04_34_09
Theory : general
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