Nuprl Lemma : mFOLRule-induction
∀[P:mFOLRule() ─→ ℙ]
(P[andI]
⇒ P[impI]
⇒ (∀var:ℤ. P[allI with var])
⇒ (∀var:ℤ. P[existsI with var])
⇒ (∀left:𝔹. P[mRuleorI(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])
⇒ {∀v:mFOLRule(). P[v]})
Proof
Definitions occuring in Statement :
mRuleexistsE: existsE on hypnum with var,
mRuleallE: allE on hypnum with var,
mRuleimpE: impE on hypnum,
mRuleorE: orE on hypnum,
mRuleandE: andE on hypnum,
mRulehyp: hyp,
mRuleorI: mRuleorI(left),
mRuleexistsI: existsI with var,
mRuleallI: allI with var,
mRuleimpI: impI,
mRuleandI: andI,
mFOLRule: mFOLRule(),
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
Lemmas :
mFOLRule-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,
mFOLRule_wf
\mforall{}[P:mFOLRule() {}\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[mRuleorI(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{}v:mFOLRule(). P[v]\})
Date html generated:
2015_07_17-AM-07_56_13
Last ObjectModification:
2015_01_27-AM-10_05_58
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