Nuprl Lemma : sparse-signed-rep-lemma1-ext
∀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 :
member: t ∈ T,
subtract: n - m,
isEven: isEven(n),
eq_int: (i =z j),
modulus: a mod n,
absval: |i|,
btrue: tt,
it: ⋅,
bfalse: ff,
sparse-signed-rep-lemma1,
decidable__equal_int,
decidable__assert,
decidable__int_equal,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
or: P ∨ Q,
squash: ↓T
Lemmas referenced :
sparse-signed-rep-lemma1,
lifting-strict-int_eq,
istype-void,
strict4-decide,
lifting-strict-decide,
lifting-strict-callbyvalue,
value-type-has-value,
int-value-type,
has-value_wf_base,
istype-base,
is-exception_wf,
istype-universe,
lifting-strict-less,
decidable__equal_int,
decidable__assert,
decidable__int_equal
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination,
independent_pairFormation,
lambdaFormation_alt,
callbyvalueIntEq,
baseApply,
closedConclusion,
hypothesisEquality,
productElimination,
intEquality,
universeIsType,
int_eqExceptionCases,
inrFormation_alt,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inlFormation_alt
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_33
Last ObjectModification:
2019_06_26-PM-04_35_33
Theory : general
Home
Index