Nuprl Lemma : formula-definition
∀[A:Type]. ∀[R:A ⟶ formula() ⟶ ℙ].
((∀name:Atom. {x:A| R[x;pvar(name)]} )
⇒ (∀sub:formula(). ({x:A| R[x;sub]} ⇒ {x:A| R[x;pnot(sub)]} ))
⇒ (∀left,right:formula(). ({x:A| R[x;left]} ⇒ {x:A| R[x;right]} ⇒ {x:A| R[x;pand(left;right)]} ))
⇒ (∀left,right:formula(). ({x:A| R[x;left]} ⇒ {x:A| R[x;right]} ⇒ {x:A| R[x;por(left;right)]} ))
⇒ (∀left,right:formula(). ({x:A| R[x;left]} ⇒ {x:A| R[x;right]} ⇒ {x:A| R[x;pimp(left;right)]} ))
⇒ {∀v:formula(). {x:A| R[x;v]} })
Proof
Definitions occuring in Statement :
pimp: pimp(left;right),
por: por(left;right),
pand: pand(left;right),
pnot: pnot(sub),
pvar: pvar(name),
formula: formula(),
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],
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
Lemmas referenced :
formula-induction,
set_wf,
formula_wf,
all_wf,
pimp_wf,
por_wf,
pand_wf,
pnot_wf,
pvar_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,
functionEquality,
universeEquality,
atomEquality,
cumulativity
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} formula() {}\mrightarrow{} \mBbbP{}].
((\mforall{}name:Atom. \{x:A| R[x;pvar(name)]\} )
{}\mRightarrow{} (\mforall{}sub:formula(). (\{x:A| R[x;sub]\} {}\mRightarrow{} \{x:A| R[x;pnot(sub)]\} ))
{}\mRightarrow{} (\mforall{}left,right:formula().
(\{x:A| R[x;left]\} {}\mRightarrow{} \{x:A| R[x;right]\} {}\mRightarrow{} \{x:A| R[x;pand(left;right)]\} ))
{}\mRightarrow{} (\mforall{}left,right:formula().
(\{x:A| R[x;left]\} {}\mRightarrow{} \{x:A| R[x;right]\} {}\mRightarrow{} \{x:A| R[x;por(left;right)]\} ))
{}\mRightarrow{} (\mforall{}left,right:formula().
(\{x:A| R[x;left]\} {}\mRightarrow{} \{x:A| R[x;right]\} {}\mRightarrow{} \{x:A| R[x;pimp(left;right)]\} ))
{}\mRightarrow{} \{\mforall{}v:formula(). \{x:A| R[x;v]\} \})
Date html generated:
2016_05_15-PM-07_11_39
Last ObjectModification:
2015_12_27-AM-11_32_34
Theory : general
Home
Index