Nuprl Lemma : int-eq-constraint-factor-sym
∀[a:ℤ]. ∀[g:ℤ-o]. ∀[xs,L:ℤ List].
uiff([a / g * L] ⋅ [1 / xs] = 0 ∈ ℤ;((a rem g) = 0 ∈ ℤ) ∧ ([a ÷ g / L] ⋅ [1 / xs] = 0 ∈ ℤ))
Proof
Definitions occuring in Statement :
int-vec-mul: a * as,
integer-dot-product: as ⋅ bs,
cons: [a / b],
list: T List,
int_nzero: ℤ-o,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
remainder: n rem m,
divide: n ÷ m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
sq_type: SQType(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
guard: {T},
int_nzero: ℤ-o,
prop: ℙ,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
squash: ↓T,
true: True
Lemmas referenced :
int-eq-constraint-factor,
subtype_base_sq,
int_subtype_base,
integer-dot-product-comm,
cons_wf,
int-vec-mul_wf,
equal-wf-T-base,
integer-dot-product_wf,
equal_wf,
list_wf,
int_nzero_wf,
squash_wf,
true_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_pairFormation,
productElimination,
independent_isectElimination,
instantiate,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
sqequalRule,
intEquality,
natural_numberEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
independent_pairEquality,
axiomEquality,
baseClosed,
productEquality,
remainderEquality,
lambdaFormation,
voidElimination,
divideEquality,
hyp_replacement,
applyEquality,
lambdaEquality,
imageElimination,
universeEquality,
imageMemberEquality
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[g:\mBbbZ{}\msupminus{}\msupzero{}]. \mforall{}[xs,L:\mBbbZ{} List].
uiff([a / g * L] \mcdot{} [1 / xs] = 0;((a rem g) = 0) \mwedge{} ([a \mdiv{} g / L] \mcdot{} [1 / xs] = 0))
Date html generated:
2016_10_21-AM-09_52_16
Last ObjectModification:
2016_07_12-AM-05_10_54
Theory : omega
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