Nuprl Lemma : mFOL-definition
∀[A:Type]. ∀[R:A ⟶ mFOL() ⟶ ℙ].
((∀name:Atom. ∀vars:ℤ List. {x:A| R[x;name(vars)]} )
⇒ (∀knd:Atom. ∀left,right:mFOL(). ({x:A| R[x;left]} ⇒ {x:A| R[x;right]} ⇒ {x:A| R[x;mFOconnect(knd;left;right)]}\000C ))
⇒ (∀isall:𝔹. ∀var:ℤ. ∀body:mFOL(). ({x:A| R[x;body]} ⇒ {x:A| R[x;mFOquant(isall;var;body)]} ))
⇒ {∀v:mFOL(). {x:A| R[x;v]} })
Proof
Definitions occuring in Statement :
mFOquant: mFOquant(isall;var;body),
mFOconnect: mFOconnect(knd;left;right),
mFOatomic: name(vars),
mFOL: mFOL(),
list: T List,
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,
atom: Atom,
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,
all: ∀x:A. B[x]
Lemmas referenced :
mFOL-induction,
set_wf,
mFOL_wf,
all_wf,
bool_wf,
mFOquant_wf,
mFOconnect_wf,
list_wf,
mFOatomic_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,
intEquality,
functionEquality,
universeEquality,
atomEquality,
setEquality,
setElimination,
rename,
cumulativity
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} mFOL() {}\mrightarrow{} \mBbbP{}].
((\mforall{}name:Atom. \mforall{}vars:\mBbbZ{} List. \{x:A| R[x;name(vars)]\} )
{}\mRightarrow{} (\mforall{}knd:Atom. \mforall{}left,right:mFOL().
(\{x:A| R[x;left]\} {}\mRightarrow{} \{x:A| R[x;right]\} {}\mRightarrow{} \{x:A| R[x;mFOconnect(knd;left;right)]\} ))
{}\mRightarrow{} (\mforall{}isall:\mBbbB{}. \mforall{}var:\mBbbZ{}. \mforall{}body:mFOL(). (\{x:A| R[x;body]\} {}\mRightarrow{} \{x:A| R[x;mFOquant(isall;var;body)]\} ))
{}\mRightarrow{} \{\mforall{}v:mFOL(). \{x:A| R[x;v]\} \})
Date html generated:
2016_05_15-PM-10_13_57
Last ObjectModification:
2015_12_27-PM-06_33_30
Theory : minimal-first-order-logic
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