Nuprl Lemma : FOLRule-definition
∀[A:Type]. ∀[R:A ⟶ FOLRule() ⟶ ℙ].
({x:A| R[x;andI]}
⇒ {x:A| R[x;impI]}
⇒ (∀var:ℤ. {x:A| R[x;allI with var]} )
⇒ (∀var:ℤ. {x:A| R[x;existsI with var]} )
⇒ (∀left:𝔹. {x:A| R[x;fRuleorI(left)]} )
⇒ {x:A| R[x;hyp]}
⇒ (∀hypnum:ℕ. {x:A| R[x;andE on hypnum]} )
⇒ (∀hypnum:ℕ. {x:A| R[x;orE on hypnum]} )
⇒ (∀hypnum:ℕ. {x:A| R[x;impE on hypnum]} )
⇒ (∀hypnum:ℕ. ∀var:ℤ. {x:A| R[x;allE on hypnum with var]} )
⇒ (∀hypnum:ℕ. ∀var:ℤ. {x:A| R[x;existsE on hypnum with var]} )
⇒ (∀hypnum:ℕ. {x:A| R[x;falseE on hypnum]} )
⇒ {∀v:FOLRule(). {x:A| R[x;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[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
guard: {T},
so_lambda: λ2x.t[x],
member: t ∈ T,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
so_apply: x[s],
prop: ℙ
Lemmas referenced :
FOLRule-induction,
set_wf,
FOLRule_wf,
all_wf,
nat_wf,
fRulefalseE_wf,
fRuleexistsE_wf,
fRuleallE_wf,
fRuleimpE_wf,
fRuleorE_wf,
fRuleandE_wf,
fRulehyp_wf,
bool_wf,
fRuleorI_wf,
fRuleexistsI_wf,
fRuleallI_wf,
fRuleimpI_wf,
fRuleandI_wf
Rules used in proof :
cut,
lemma_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
hypothesisEquality,
applyEquality,
because_Cache,
independent_functionElimination,
universeEquality,
intEquality,
functionEquality,
cumulativity
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} FOLRule() {}\mrightarrow{} \mBbbP{}].
(\{x:A| R[x;andI]\}
{}\mRightarrow{} \{x:A| R[x;impI]\}
{}\mRightarrow{} (\mforall{}var:\mBbbZ{}. \{x:A| R[x;allI with var]\} )
{}\mRightarrow{} (\mforall{}var:\mBbbZ{}. \{x:A| R[x;existsI with var]\} )
{}\mRightarrow{} (\mforall{}left:\mBbbB{}. \{x:A| R[x;fRuleorI(left)]\} )
{}\mRightarrow{} \{x:A| R[x;hyp]\}
{}\mRightarrow{} (\mforall{}hypnum:\mBbbN{}. \{x:A| R[x;andE on hypnum]\} )
{}\mRightarrow{} (\mforall{}hypnum:\mBbbN{}. \{x:A| R[x;orE on hypnum]\} )
{}\mRightarrow{} (\mforall{}hypnum:\mBbbN{}. \{x:A| R[x;impE on hypnum]\} )
{}\mRightarrow{} (\mforall{}hypnum:\mBbbN{}. \mforall{}var:\mBbbZ{}. \{x:A| R[x;allE on hypnum with var]\} )
{}\mRightarrow{} (\mforall{}hypnum:\mBbbN{}. \mforall{}var:\mBbbZ{}. \{x:A| R[x;existsE on hypnum with var]\} )
{}\mRightarrow{} (\mforall{}hypnum:\mBbbN{}. \{x:A| R[x;falseE on hypnum]\} )
{}\mRightarrow{} \{\mforall{}v:FOLRule(). \{x:A| R[x;v]\} \})
Date html generated:
2016_05_15-PM-10_25_32
Last ObjectModification:
2015_12_27-PM-06_27_32
Theory : minimal-first-order-logic
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