Nuprl Lemma : combinations_aux_rem_property
∀[k:ℕ+]. ∀[n,b,m:ℕ]. (combinations_aux_rem(b rem k;n;m;k) = (combinations_aux(b;n;m) rem k) ∈ ℤ)
Proof
Definitions occuring in Statement :
combinations_aux_rem: combinations_aux_rem(b;n;m;k),
combinations_aux: combinations_aux(b;n;m),
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
remainder: n rem m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
prop: ℙ,
combinations_aux_rem: combinations_aux_rem(b;n;m;k),
combinations_aux: combinations_aux(b;n;m),
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
has-value: (a)↓,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
nequal: a ≠ b ∈ T ,
top: Top,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
cand: A c∧ B,
true: True,
squash: ↓T
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
remainder_wfa,
nat_plus_inc_int_nzero,
subtract-1-ge-0,
istype-nat,
nat_plus_wf,
eq_int_wf,
equal-wf-base,
bool_wf,
int_subtype_base,
assert_wf,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
bnot_wf,
not_wf,
istype-assert,
istype-void,
value-type-has-value,
int-value-type,
subtract_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
decidable__equal_int,
subtype_base_sq,
mul-zero,
satisfiable-full-omega-tt,
zero-rem,
subtype_rel_sets,
less_than_wf,
nequal_wf,
add-commutes,
zero-mul,
iff_weakening_equal,
subtype_rel_self,
rem-zero,
istype-universe,
true_wf,
squash_wf,
equal_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
le_wf,
rem_mul2,
mul_bounds_1a,
combinations_aux_wf_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
callbyvalueReduce,
sqleReflexivity,
applyEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
baseApply,
closedConclusion,
baseClosed,
equalityIstype,
sqequalBase,
functionIsType,
multiplyEquality,
unionElimination,
equalityElimination,
productElimination,
instantiate,
cumulativity,
intEquality,
remainderEquality,
lambdaFormation,
dependent_pairFormation,
lambdaEquality,
isect_memberEquality,
voidEquality,
computeAll,
setEquality,
applyLambdaEquality,
independent_pairEquality,
imageMemberEquality,
universeEquality,
imageElimination,
dependent_set_memberEquality
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[n,b,m:\mBbbN{}]. (combinations\_aux\_rem(b rem k;n;m;k) = (combinations\_aux(b;n;m) rem k))
Date html generated:
2020_05_20-AM-08_13_05
Last ObjectModification:
2020_02_28-PM-02_51_06
Theory : general
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