Nuprl Lemma : iseg_product_rem_property
∀[k,j:ℕ]. ∀[i:ℕj + 1]. iseg_product_rem(i;j;k) = (iseg_product(i;j) rem k) ∈ ℤ supposing 1 < k
Proof
Definitions occuring in Statement :
iseg_product_rem: iseg_product_rem(i;j;k),
iseg_product: iseg_product(i;j),
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
remainder: n rem m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
iseg_product: iseg_product(i;j),
iseg_product_rem: iseg_product_rem(i;j;k),
combinations: C(n;m),
prop: ℙ,
nat: ℕ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
and: P ∧ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
top: Top,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
int_seg: {i..j-},
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
squash: ↓T
Lemmas referenced :
less_than_wf,
int_seg_wf,
nat_wf,
combinations_aux_rem_wf,
decidable__lt,
false_wf,
not-lt-2,
less-iff-le,
add_functionality_wrt_le,
add-swap,
add-commutes,
add-associates,
zero-add,
le-add-cancel,
subtract_wf,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
one-rem,
equal_wf,
squash_wf,
true_wf,
combinations_aux_rem_property,
iff_weakening_equal,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
addEquality,
intEquality,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
lambdaFormation,
voidElimination,
productElimination,
independent_functionElimination,
independent_isectElimination,
applyEquality,
lambdaEquality,
voidEquality,
dependent_pairFormation,
int_eqEquality,
computeAll,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[k,j:\mBbbN{}]. \mforall{}[i:\mBbbN{}j + 1]. iseg\_product\_rem(i;j;k) = (iseg\_product(i;j) rem k) supposing 1 < k
Date html generated:
2018_05_21-PM-08_11_14
Last ObjectModification:
2017_07_26-PM-05_46_40
Theory : general
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