Nuprl Lemma : decidable__wellfound-bounded-exists
∀[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)
⇒ (∀y:T. Dec(∃x:T. ((R+ x y) ∧ P[x]))))
Proof
Definitions occuring in Statement :
rel_plus: R+,
l_member: (x ∈ l),
list: T List,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
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,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
so_lambda: λ2x.t[x],
member: t ∈ T,
prop: ℙ,
subtype_rel: A ⊆r B,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
guard: {T},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
and: P ∧ Q,
cand: A c∧ B,
decidable: Dec(P),
or: P ∨ Q,
l_exists: (∃x∈L. P[x]),
int_seg: {i..j-},
uimplies: b supposing a,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
less_than: a < b,
squash: ↓T,
infix_ap: x f y,
trans: Trans(T;x,y.E[x; y]),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
l_exists_iff,
rel_plus_implies,
rel_plus_trans,
rel-rel-plus,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
length_wf,
int_seg_properties,
select_wf,
not_wf,
decidable__cand,
decidable__l_exists_better-extract,
decidable__and,
decidable__l_exists,
l_member_wf,
list_wf,
wellfounded_wf,
all_wf,
rel_plus_wf,
and_wf,
exists_wf,
decidable_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
cut,
hypothesis,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
lemma_by_obid,
hypothesisEquality,
applyEquality,
because_Cache,
independent_functionElimination,
rename,
dependent_functionElimination,
productElimination,
functionEquality,
cumulativity,
universeEquality,
isect_memberEquality,
unionElimination,
inlFormation,
dependent_pairFormation,
setElimination,
independent_isectElimination,
natural_numberEquality,
int_eqEquality,
intEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageElimination,
inrFormation,
productEquality,
setEquality
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{}y:T. Dec(\mexists{}x:T. ((R\msupplus{} x y) \mwedge{} P[x]))))
Date html generated:
2016_05_15-PM-04_52_03
Last ObjectModification:
2016_01_16-AM-11_29_16
Theory : general
Home
Index