Nuprl Lemma : special-mod4-decomp-unique
∀m:ℤ. ∃!k:ℤ. ∃b:{-2..3-}. ((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,
exists!: ∃!x:T. P[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
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],
so_lambda: λ2x.t[x],
and: P ∧ Q,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
exists!: ∃!x:T. P[x],
exists: ∃x:A. B[x],
cand: A c∧ B,
divides: b | a,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
sq_type: SQType(T),
subtract: n - m,
absval: |i|,
uiff: uiff(P;Q),
same-parity: same-parity(n;m),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
isEven: isEven(n),
modulus: a mod n,
eq_int: (i =z j),
assert: ↑b,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q
Lemmas referenced :
set_wf,
int_seg_wf,
equal-wf-base,
int_subtype_base,
set_subtype_base,
lelt_wf,
assert_wf,
isEven_wf,
special-mod4-decomp_wf,
set-value-type,
equal_wf,
product-value-type,
istype-int,
assert_witness,
subtract_wf,
int_seg_properties,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermSubtract_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_formula_prop_wf,
intformless_wf,
intformle_wf,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
subtype_base_sq,
isEven-add,
bool_cases,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
eqff_to_assert,
assert_of_bnot,
decidable__le,
decidable__lt,
le_wf,
less_than_wf,
divides_iff_rem_zero,
nequal_wf,
int_seg_subtype_special,
int_seg_cases
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productEquality,
intEquality,
minusEquality,
natural_numberEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
productElimination,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
inhabitedIsType,
independent_isectElimination,
functionEquality,
because_Cache,
productIsType,
universeIsType,
cutEval,
dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
equalityIsType1,
setElimination,
rename,
dependent_pairFormation_alt,
independent_pairFormation,
independent_functionElimination,
equalityIsType4,
functionIsType,
dependent_functionElimination,
unionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
instantiate,
cumulativity,
promote_hyp,
callbyvalueReduce,
sqleReflexivity,
hypothesis_subsumption,
applyLambdaEquality
Latex:
\mforall{}m:\mBbbZ{}. \mexists{}!k:\mBbbZ{}. \mexists{}b:\{-2..3\msupminus{}\}. ((m = ((4 * k) + b)) \mwedge{} ((|b| = 2) {}\mRightarrow{} (\muparrow{}isEven(k))))
Date html generated:
2019_10_15-AM-11_26_53
Last ObjectModification:
2018_10_09-PM-00_14_32
Theory : general
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