Nuprl Lemma : stamps-example
∀n:ℕ. ∃i:ℕ. (∃j:ℕ [((n + 8) = ((3 * i) + (5 * j)) ∈ ℤ)])
Proof
Definitions occuring in Statement :
nat: ℕ,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
uimplies: b supposing a,
sq_exists: ∃x:A [B[x]],
prop: ℙ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
guard: {T},
subtract: n - m,
true: True,
sq_stable: SqStable(P),
squash: ↓T,
sq_type: SQType(T),
nat_plus: ℕ+,
less_than: a < b,
adjust_div: adjust_div(b;a)
Lemmas referenced :
nat_wf,
sq_exists_wf,
equal-wf-base,
int_subtype_base,
set_subtype_base,
le_wf,
istype-int,
less_than_wf,
primrec-wf2,
exists_wf,
istype-false,
zero-add,
mul-commutes,
istype-void,
sq_stable__equal,
decidable__le,
subtract_wf,
set-value-type,
equal_wf,
int-value-type,
not-le-2,
le_antisymmetry_iff,
condition-implies-le,
minus-add,
minus-minus,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
le-add-cancel,
sq_stable__le,
add-zero,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
one-mul,
subtype_base_sq,
decidable__int_equal,
not-equal-2,
mul-associates,
mul-distributes,
le-add-cancel2,
nat_properties,
less-iff-le,
le_reflexive,
omega-shadow,
mul-swap,
divide-le,
squash_wf,
true_wf,
minus-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
thin,
rename,
setElimination,
sqequalRule,
Error :productIsType,
Error :universeIsType,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
Error :lambdaEquality_alt,
intEquality,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
natural_numberEquality,
Error :inhabitedIsType,
independent_isectElimination,
Error :setIsType,
because_Cache,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
Error :isect_memberEquality_alt,
voidElimination,
Error :equalityIsType4,
Error :dependent_pairFormation_alt,
Error :dependent_set_memberFormation_alt,
productElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
unionElimination,
cutEval,
Error :equalityIsType1,
addEquality,
independent_functionElimination,
minusEquality,
imageMemberEquality,
imageElimination,
multiplyEquality,
promote_hyp,
instantiate,
cumulativity,
hyp_replacement
Latex:
\mforall{}n:\mBbbN{}. \mexists{}i:\mBbbN{}. (\mexists{}j:\mBbbN{} [((n + 8) = ((3 * i) + (5 * j)))])
Date html generated:
2019_06_20-PM-00_26_09
Last ObjectModification:
2018_10_09-AM-09_30_44
Theory : int_1
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