Nuprl Lemma : absval_elim
∀[P:ℤ ⟶ ℙ]. (∀x:ℤ. P[|x|] ⇐⇒ ∀x:ℕ. P[x])
Proof
Definitions occuring in Statement :
absval: |i|,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
so_apply: x[s],
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
sq_stable: SqStable(P),
le: A ≤ B
Lemmas referenced :
nat_wf,
all_wf,
absval_wf,
absval_unfold,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
uiff_transitivity,
assert_wf,
bnot_wf,
not_wf,
assert_of_bnot,
not_functionality_wrt_uiff,
sq_stable_from_decidable,
le_wf,
decidable__le,
not-lt-2,
add_functionality_wrt_le,
add-zero,
le-add-cancel-alt
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
Error :isect_memberFormation_alt,
independent_pairFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
intEquality,
lambdaEquality,
applyEquality,
hypothesisEquality,
setElimination,
rename,
Error :functionIsType,
Error :universeIsType,
universeEquality,
dependent_functionElimination,
minusEquality,
natural_numberEquality,
because_Cache,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
lessCases,
axiomSqEquality,
Error :inhabitedIsType,
isect_memberEquality,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
independent_functionElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[P:\mBbbZ{} {}\mrightarrow{} \mBbbP{}]. (\mforall{}x:\mBbbZ{}. P[|x|] \mLeftarrow{}{}\mRightarrow{} \mforall{}x:\mBbbN{}. P[x])
Date html generated:
2019_06_20-AM-11_24_35
Last ObjectModification:
2018_09_26-AM-10_58_23
Theory : arithmetic
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