Nuprl Lemma : combinations_aux_rem_wf
∀[k:ℕ+]. ∀[n,b,m:ℕ]. (combinations_aux_rem(b;n;m;k) ∈ ℕ)
Proof
Definitions occuring in Statement :
combinations_aux_rem: combinations_aux_rem(b;n;m;k),
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
member: 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,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
combinations_aux_rem: combinations_aux_rem(b;n;m;k),
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
has-value: (a)↓,
sq_type: SQType(T),
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
nequal: a ≠ b ∈ T ,
squash: ↓T,
true: True,
le: A ≤ B,
less_than': less_than'(a;b)
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
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,
less_than_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
nat_plus_wf,
eq_int_wf,
bool_wf,
equal-wf-base,
int_subtype_base,
assert_wf,
bnot_wf,
not_wf,
value-type-has-value,
decidable__equal_int,
subtype_base_sq,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
mul-zero,
zero-rem,
subtype_rel_sets,
nequal_wf,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
int-value-type,
zero-mul,
squash_wf,
true_wf,
iff_weakening_equal,
le_wf,
false_wf,
remainder_wf,
mul_bounds_1a
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
callbyvalueReduce,
sqleReflexivity,
unionElimination,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
instantiate,
cumulativity,
equalityElimination,
productElimination,
impliesFunctionality,
setEquality,
applyLambdaEquality,
remainderEquality,
imageElimination,
universeEquality,
imageMemberEquality,
dependent_set_memberEquality,
multiplyEquality
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[n,b,m:\mBbbN{}]. (combinations\_aux\_rem(b;n;m;k) \mmember{} \mBbbN{})
Date html generated:
2018_05_21-PM-08_10_43
Last ObjectModification:
2017_07_26-PM-05_46_13
Theory : general
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