Nuprl Lemma : imax_nat
∀[a:ℤ]. ∀[b:ℕ]. (imax(a;b) ∈ ℕ)
Proof
Definitions occuring in Statement :
imax: imax(a;b),
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
le: A ≤ B,
sq_stable: SqStable(P),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
decidable: Dec(P),
subtract: n - m,
top: Top,
less_than': less_than'(a;b)
Lemmas referenced :
imax_wf,
le_wf,
squash_wf,
true_wf,
imax_unfold,
subtype_rel_self,
iff_weakening_equal,
le_int_wf,
eqtt_to_assert,
assert_of_le_int,
sq_stable__le,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
istype-le,
decidable__le,
istype-false,
not-le-2,
condition-implies-le,
minus-add,
istype-void,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
zero-add,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
istype-nat,
istype-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
Error :dependent_set_memberEquality_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
applyEquality,
Error :lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
Error :universeIsType,
Error :inhabitedIsType,
because_Cache,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
independent_isectElimination,
productElimination,
independent_functionElimination,
Error :lambdaFormation_alt,
unionElimination,
equalityElimination,
Error :dependent_pairFormation_alt,
Error :equalityIstype,
promote_hyp,
dependent_functionElimination,
cumulativity,
voidElimination,
independent_pairFormation,
addEquality,
Error :isect_memberEquality_alt,
minusEquality,
axiomEquality,
Error :isectIsTypeImplies
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[b:\mBbbN{}]. (imax(a;b) \mmember{} \mBbbN{})
Date html generated:
2019_06_20-AM-11_32_45
Last ObjectModification:
2019_01_02-PM-03_55_26
Theory : bool_1
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