Nuprl Lemma : int_term-induction
∀[P:int_term() ⟶ ℙ]
((∀const:ℤ. P["const"])
⇒ (∀var:ℤ. P[vvar])
⇒ (∀left,right:int_term(). (P[left] ⇒ P[right] ⇒ P[left (+) right]))
⇒ (∀left,right:int_term(). (P[left] ⇒ P[right] ⇒ P[left (-) right]))
⇒ (∀left,right:int_term(). (P[left] ⇒ P[right] ⇒ P[left (*) right]))
⇒ (∀num:int_term(). (P[num] ⇒ P["-"num]))
⇒ {∀v:int_term(). P[v]})
Proof
Definitions occuring in Statement :
itermMinus: "-"num,
itermMultiply: left (*) right,
itermSubtract: left (-) right,
itermAdd: left (+) right,
itermVar: vvar,
itermConstant: "const",
int_term: int_term(),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
so_apply: x[s],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
so_lambda: λ2x.t[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
ext-eq: A ≡ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
sq_type: SQType(T),
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
itermConstant: "const",
int_term_size: int_term_size(p),
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
itermVar: vvar,
itermAdd: left (+) right,
pi1: fst(t),
pi2: snd(t),
cand: A c∧ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
top: Top,
less_than': less_than'(a;b),
true: True,
itermSubtract: left (-) right,
itermMultiply: left (*) right,
itermMinus: "-"num,
ge: i ≥ j ,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T
Lemmas referenced :
uniform-comp-nat-induction,
all_wf,
int_term_wf,
isect_wf,
le_wf,
int_term_size_wf,
nat_wf,
less_than'_wf,
int_term-ext,
eq_atom_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
atom_subtype_base,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
set_subtype_base,
int_subtype_base,
add-is-int-iff,
nat_properties,
subtract_wf,
decidable__le,
false_wf,
not-le-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-one-mul-top,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
decidable__lt,
not-lt-2,
add-mul-special,
zero-mul,
le-add-cancel-alt,
lelt_wf,
uall_wf,
int_seg_wf,
le_reflexive,
itermMinus_wf,
itermMultiply_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
one-mul,
two-mul,
mul-distributes-right,
omega-shadow,
less_than_wf,
mul-distributes,
mul-commutes,
mul-associates,
mul-swap,
minus-zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
hypothesis,
hypothesisEquality,
applyEquality,
because_Cache,
setElimination,
rename,
functionExtensionality,
independent_functionElimination,
productElimination,
independent_pairEquality,
dependent_functionElimination,
voidElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
hypothesis_subsumption,
tokenEquality,
unionElimination,
equalityElimination,
independent_isectElimination,
instantiate,
cumulativity,
atomEquality,
dependent_pairFormation,
independent_pairFormation,
sqequalIntensionalEquality,
intEquality,
natural_numberEquality,
baseClosed,
baseApply,
closedConclusion,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
isect_memberEquality,
voidEquality,
minusEquality,
functionEquality,
universeEquality,
multiplyEquality,
imageMemberEquality
Latex:
\mforall{}[P:int\_term() {}\mrightarrow{} \mBbbP{}]
((\mforall{}const:\mBbbZ{}. P["const"])
{}\mRightarrow{} (\mforall{}var:\mBbbZ{}. P[vvar])
{}\mRightarrow{} (\mforall{}left,right:int\_term(). (P[left] {}\mRightarrow{} P[right] {}\mRightarrow{} P[left (+) right]))
{}\mRightarrow{} (\mforall{}left,right:int\_term(). (P[left] {}\mRightarrow{} P[right] {}\mRightarrow{} P[left (-) right]))
{}\mRightarrow{} (\mforall{}left,right:int\_term(). (P[left] {}\mRightarrow{} P[right] {}\mRightarrow{} P[left (*) right]))
{}\mRightarrow{} (\mforall{}num:int\_term(). (P[num] {}\mRightarrow{} P["-"num]))
{}\mRightarrow{} \{\mforall{}v:int\_term(). P[v]\})
Date html generated:
2017_04_14-AM-08_57_23
Last ObjectModification:
2017_02_27-PM-03_41_26
Theory : omega
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