Nuprl Lemma : tree-big-least
∀[T:Type]. ∀[A:(T List) ⟶ ℙ].
((∃k:ℕ. T ~ ℕk)
⇒ Decidable(A)
⇒ (¬(A []))
⇒ (∀n:ℕ
(tree-big(T;upwd-closure(T;A);n)
⇒ (∃k:ℕn. ((¬tree-big(T;upwd-closure(T;A);k)) ∧ tree-big(T;upwd-closure(T;A);k + 1))))))
Proof
Definitions occuring in Statement :
tree-big: tree-big(T;A;n),
upwd-closure: upwd-closure(T;A),
dec-predicate: Decidable(X),
equipollent: A ~ B,
nil: [],
list: T List,
int_seg: {i..j-},
nat: ℕ,
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],
add: n + m,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
so_apply: x[s],
ge: i ≥ j ,
tree-big: tree-big(T;A;n),
upwd-closure: upwd-closure(T;A),
iff: P ⇐⇒ Q,
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
cons: [a / b],
bfalse: ff,
dec-predicate: Decidable(X),
less_than: a < b,
cand: A c∧ B
Lemmas referenced :
subtract-add-cancel,
lelt_wf,
decidable__lt,
decidable__upwd-closure,
decidable__tree-big,
null_cons_lemma,
product_subtype_list,
null_nil_lemma,
list-cases,
iseg_nil,
length_of_nil_lemma,
equipollent_wf,
list_wf,
dec-predicate_wf,
nil_wf,
nat_wf,
nat_properties,
primrec-wf2,
less_than_wf,
set_wf,
int_term_value_add_lemma,
itermAdd_wf,
int_seg_properties,
int_seg_subtype_nat,
not_wf,
int_seg_wf,
exists_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
subtract_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
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,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
le_wf,
false_wf,
upwd-closure_wf,
tree-big_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
because_Cache,
hypothesis,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
rename,
setElimination,
productElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
functionEquality,
productEquality,
applyEquality,
addEquality,
instantiate,
introduction,
universeEquality,
independent_functionElimination,
promote_hyp,
hypothesis_subsumption
Latex:
\mforall{}[T:Type]. \mforall{}[A:(T List) {}\mrightarrow{} \mBbbP{}].
((\mexists{}k:\mBbbN{}. T \msim{} \mBbbN{}k)
{}\mRightarrow{} Decidable(A)
{}\mRightarrow{} (\mneg{}(A []))
{}\mRightarrow{} (\mforall{}n:\mBbbN{}
(tree-big(T;upwd-closure(T;A);n)
{}\mRightarrow{} (\mexists{}k:\mBbbN{}n. ((\mneg{}tree-big(T;upwd-closure(T;A);k)) \mwedge{} tree-big(T;upwd-closure(T;A);k + 1))))))
Date html generated:
2016_05_14-PM-04_10_14
Last ObjectModification:
2016_01_14-PM-10_58_13
Theory : fan-theorem
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