Nuprl Lemma : integer-induction
∀[P:ℤ ⟶ ℙ]. (P[0] ⇒ (∀y:{x:ℤ| 0 < x} . (P[y - 1] ⇒ P[y])) ⇒ (∀y:{x:ℤ| x < 0} . (P[y + 1] ⇒ P[y])) ⇒ (∀x:ℤ. P[x]))
Proof
Definitions occuring in Statement :
less_than: a < b,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
subtype_rel: A ⊆r B,
top: Top,
subtract: n - m,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
le: A ≤ B,
decidable: Dec(P),
gt: i > j
Lemmas referenced :
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
iff_transitivity,
assert_wf,
bnot_wf,
not_wf,
less_than_wf,
iff_weakening_uiff,
assert_of_bnot,
all_wf,
subtract_wf,
primrec-wf2,
nat_wf,
int_subtype_base,
minus-zero,
minus-one-mul,
subtype_rel-equal,
add-commutes,
minus-one-mul-top,
minus-add,
minus-minus,
le_weakening2,
le_wf,
less-iff-le,
add_functionality_wrt_le,
le_reflexive,
add-associates,
zero-add,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
add-zero,
not-lt-2,
mul-associates,
omega-shadow,
false_wf,
add-swap,
decidable__lt,
subtype_rel_dep_function,
subtype_rel_self,
not-gt-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
hypothesis,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
because_Cache,
voidElimination,
independent_pairFormation,
impliesFunctionality,
intEquality,
setEquality,
lambdaEquality,
functionEquality,
applyEquality,
functionExtensionality,
addEquality,
setElimination,
universeEquality,
minusEquality,
dependent_set_memberEquality,
isect_memberEquality,
voidEquality,
addLevel,
multiplyEquality,
levelHypothesis,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[P:\mBbbZ{} {}\mrightarrow{} \mBbbP{}]
(P[0]
{}\mRightarrow{} (\mforall{}y:\{x:\mBbbZ{}| 0 < x\} . (P[y - 1] {}\mRightarrow{} P[y]))
{}\mRightarrow{} (\mforall{}y:\{x:\mBbbZ{}| x < 0\} . (P[y + 1] {}\mRightarrow{} P[y]))
{}\mRightarrow{} (\mforall{}x:\mBbbZ{}. P[x]))
Date html generated:
2017_04_14-AM-07_25_52
Last ObjectModification:
2017_02_27-PM-02_55_36
Theory : call!by!value_2
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