Nuprl Lemma : Longs-algorithm_wf
∀[h:ℤ ⟶ ℤ]. ∀n:ℕ ⟶ ℕ. ∀a,b,c:ℕ. (Longs-algorithm(h;n;a;b;c) ∈ ℤ)
Proof
Definitions occuring in Statement :
Longs-algorithm: Longs-algorithm(h;n;a;b;c),
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
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],
top: Top,
and: P ∧ Q,
prop: ℙ,
Longs-algorithm: Longs-algorithm(h;n;a;b;c),
subtract: n - m,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
nequal: a ≠ b ∈ T ,
has-value: (a)↓,
less_than: a < b,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
decidable: Dec(P),
subtype_rel: A ⊆r B,
le: A ≤ B
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
subtract-1-ge-0,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
value-type-has-value,
lt_int_wf,
assert_of_lt_int,
iff_weakening_uiff,
assert_wf,
less_than_wf,
istype-top,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-le,
subtract_wf,
istype-nat,
itermSubtract_wf,
int_term_value_subtract_lemma,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
because_Cache,
applyEquality,
callbyvalueReduce,
sqleReflexivity,
unionElimination,
equalityElimination,
productElimination,
int_eqReduceTrueSq,
equalityIstype,
promote_hyp,
instantiate,
cumulativity,
int_eqReduceFalseSq,
addEquality,
lessCases,
axiomSqEquality,
isectIsTypeImplies,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
closedConclusion,
functionIsType,
pointwiseFunctionality
Latex:
\mforall{}[h:\mBbbZ{} {}\mrightarrow{} \mBbbZ{}]. \mforall{}n:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \mforall{}a,b,c:\mBbbN{}. (Longs-algorithm(h;n;a;b;c) \mmember{} \mBbbZ{})
Date html generated:
2019_10_16-AM-11_37_39
Last ObjectModification:
2018_12_08-AM-11_55_08
Theory : power!series
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