Nuprl Lemma : combinations_aux_wf
∀[n,b,m:ℕ]. (combinations_aux(b;n;m) ∈ ℕ)
Proof
Definitions occuring in Statement :
combinations_aux: combinations_aux(b;n;m),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
or: P ∨ Q,
decidable: Dec(P),
btrue: tt,
ifthenelse: if b then t else f fi ,
subtract: n - m,
eq_int: (i =z j),
combinations_aux: combinations_aux(b;n;m),
prop: ℙ,
and: P ∧ Q,
top: Top,
all: ∀x:A. B[x],
not: ¬A,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
bfalse: ff,
uiff: uiff(P;Q),
it: ⋅,
unit: Unit,
bool: 𝔹,
guard: {T},
sq_type: SQType(T),
has-value: (a)↓,
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
less_than': less_than'(a;b),
le: A ≤ B
Lemmas referenced :
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
nat_wf,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
equal_wf,
assert_of_bnot,
eqff_to_assert,
iff_weakening_uiff,
iff_transitivity,
assert_of_eq_int,
eqtt_to_assert,
uiff_transitivity,
subtype_base_sq,
decidable__equal_int,
value-type-has-value,
not_wf,
bnot_wf,
assert_wf,
int_subtype_base,
equal-wf-base,
bool_wf,
eq_int_wf,
int-value-type,
full-omega-unsat,
istype-int,
istype-less_than,
subtract-1-ge-0,
istype-assert,
istype-void,
intformeq_wf,
int_formula_prop_eq_lemma,
add-commutes,
zero-mul,
istype-le,
mul-zero,
le_wf,
mul_bounds_1a
Rules used in proof :
unionElimination,
sqleReflexivity,
callbyvalueReduce,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
independent_functionElimination,
computeAll,
independent_pairFormation,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_isectElimination,
natural_numberEquality,
lambdaFormation,
intWeakElimination,
sqequalRule,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
because_Cache,
impliesFunctionality,
productElimination,
equalityElimination,
cumulativity,
instantiate,
applyEquality,
baseClosed,
closedConclusion,
baseApply,
lambdaFormation_alt,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
Error :memTop,
universeIsType,
functionIsTypeImplies,
inhabitedIsType,
equalityIstype,
sqequalBase,
functionIsType,
multiplyEquality,
addEquality,
minusEquality,
imageElimination,
hyp_replacement,
dependent_set_memberEquality_alt,
productIsType,
applyLambdaEquality,
dependent_set_memberEquality
Latex:
\mforall{}[n,b,m:\mBbbN{}]. (combinations\_aux(b;n;m) \mmember{} \mBbbN{})
Date html generated:
2020_05_20-AM-08_12_50
Last ObjectModification:
2020_02_06-PM-01_37_16
Theory : general
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