Nuprl Lemma : FOLRule-induction
∀[P:FOLRule() ⟶ ℙ]
(P[andI]
⇒ P[impI]
⇒ (∀var:ℤ. P[allI with var])
⇒ (∀var:ℤ. P[existsI with var])
⇒ (∀left:𝔹. P[fRuleorI(left)])
⇒ P[hyp]
⇒ (∀hypnum:ℕ. P[andE on hypnum])
⇒ (∀hypnum:ℕ. P[orE on hypnum])
⇒ (∀hypnum:ℕ. P[impE on hypnum])
⇒ (∀hypnum:ℕ. ∀var:ℤ. P[allE on hypnum with var])
⇒ (∀hypnum:ℕ. ∀var:ℤ. P[existsE on hypnum with var])
⇒ (∀hypnum:ℕ. P[falseE on hypnum])
⇒ {∀v:FOLRule(). P[v]})
Proof
Definitions occuring in Statement :
fRulefalseE: falseE on hypnum,
fRuleexistsE: existsE on hypnum with var,
fRuleallE: allE on hypnum with var,
fRuleimpE: impE on hypnum,
fRuleorE: orE on hypnum,
fRuleandE: andE on hypnum,
fRulehyp: hyp,
fRuleorI: fRuleorI(left),
fRuleexistsI: existsI with var,
fRuleallI: allI with var,
fRuleimpI: impI,
fRuleandI: andI,
FOLRule: FOLRule(),
nat: ℕ,
bool: 𝔹,
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},
all: ∀x:A. B[x],
ext-eq: A ≡ B,
and: P ∧ Q,
member: t ∈ T,
subtype_rel: A ⊆r B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
sq_type: SQType(T),
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
fRuleandI: andI,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
bnot: ¬bb,
assert: ↑b,
false: False,
fRuleimpI: impI,
fRuleallI: allI with var,
fRuleexistsI: existsI with var,
fRuleorI: fRuleorI(left),
fRulehyp: hyp,
fRuleandE: andE on hypnum,
fRuleorE: orE on hypnum,
fRuleimpE: impE on hypnum,
fRuleallE: allE on hypnum with var,
fRuleexistsE: existsE on hypnum with var,
fRulefalseE: falseE on hypnum
Lemmas referenced :
FOLRule-ext,
eq_atom_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_atom,
subtype_base_sq,
atom_subtype_base,
unit_wf2,
unit_subtype_base,
it_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_atom,
FOLRule_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
promote_hyp,
sqequalHypSubstitution,
productElimination,
thin,
hypothesis_subsumption,
hypothesis,
hypothesisEquality,
applyEquality,
sqequalRule,
isectElimination,
tokenEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
instantiate,
cumulativity,
atomEquality,
dependent_functionElimination,
independent_functionElimination,
because_Cache,
dependent_pairFormation,
voidElimination
Latex:
\mforall{}[P:FOLRule() {}\mrightarrow{} \mBbbP{}]
(P[andI]
{}\mRightarrow{} P[impI]
{}\mRightarrow{} (\mforall{}var:\mBbbZ{}. P[allI with var])
{}\mRightarrow{} (\mforall{}var:\mBbbZ{}. P[existsI with var])
{}\mRightarrow{} (\mforall{}left:\mBbbB{}. P[fRuleorI(left)])
{}\mRightarrow{} P[hyp]
{}\mRightarrow{} (\mforall{}hypnum:\mBbbN{}. P[andE on hypnum])
{}\mRightarrow{} (\mforall{}hypnum:\mBbbN{}. P[orE on hypnum])
{}\mRightarrow{} (\mforall{}hypnum:\mBbbN{}. P[impE on hypnum])
{}\mRightarrow{} (\mforall{}hypnum:\mBbbN{}. \mforall{}var:\mBbbZ{}. P[allE on hypnum with var])
{}\mRightarrow{} (\mforall{}hypnum:\mBbbN{}. \mforall{}var:\mBbbZ{}. P[existsE on hypnum with var])
{}\mRightarrow{} (\mforall{}hypnum:\mBbbN{}. P[falseE on hypnum])
{}\mRightarrow{} \{\mforall{}v:FOLRule(). P[v]\})
Date html generated:
2018_05_21-PM-10_29_12
Last ObjectModification:
2017_07_26-PM-06_41_24
Theory : minimal-first-order-logic
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