Nuprl Lemma : wellfounded-minimal
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ]. ∀[P:T ⟶ ℙ].
((∀x,y:T. Dec(R x y))
⇒ (∀x:T. Dec(P x))
⇒ (∀y:T. ∃L:T List. ∀x:T. ((R x y) ⇒ (x ∈ L)))
⇒ WellFnd{i}(T;x,y.R x y)
⇒ (∀z:T. ((P z) ⇒ (∃y:T. ((P y) ∧ ((R^*) y z) ∧ (∀x:T. ((R+ x y) ⇒ (¬(P x)))))))))
Proof
Definitions occuring in Statement :
rel_plus: R+,
l_member: (x ∈ l),
list: T List,
rel_star: R^*,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
so_apply: x[s],
all: ∀x:A. B[x],
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
guard: {T},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
or: P ∨ Q,
infix_ap: x f y,
cand: A c∧ B,
uimplies: b supposing a,
not: ¬A,
false: False,
iff: P ⇐⇒ Q
Lemmas referenced :
decidable__wellfound-bounded-exists,
exists_wf,
rel_star_wf,
all_wf,
rel_plus_wf,
not_wf,
wellfounded_wf,
list_wf,
l_member_wf,
decidable_wf,
rel_plus_implies,
rel_star_transitivity,
rel_star_weakening,
rel-plus-rel-star,
rel_rel_star,
rel-star-iff-rel-plus-or
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_functionElimination,
hypothesis,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
productEquality,
applyEquality,
because_Cache,
functionExtensionality,
rename,
universeEquality,
dependent_functionElimination,
unionElimination,
productElimination,
dependent_pairFormation,
independent_pairFormation,
promote_hyp,
independent_isectElimination,
voidElimination,
equalitySymmetry,
hyp_replacement,
Error :applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. \mforall{}[P:T {}\mrightarrow{} \mBbbP{}].
((\mforall{}x,y:T. Dec(R x y))
{}\mRightarrow{} (\mforall{}x:T. Dec(P x))
{}\mRightarrow{} (\mforall{}y:T. \mexists{}L:T List. \mforall{}x:T. ((R x y) {}\mRightarrow{} (x \mmember{} L)))
{}\mRightarrow{} WellFnd\{i\}(T;x,y.R x y)
{}\mRightarrow{} (\mforall{}z:T. ((P z) {}\mRightarrow{} (\mexists{}y:T. ((P y) \mwedge{} (rel\_star(T; R) y z) \mwedge{} (\mforall{}x:T. ((R\msupplus{} x y) {}\mRightarrow{} (\mneg{}(P x)))))))))
Date html generated:
2016_10_25-AM-11_00_33
Last ObjectModification:
2016_07_12-AM-07_07_41
Theory : general
Home
Index