Nuprl Lemma : l_all_exists_max
∀[A:Type]. ∀[R:A ⟶ ℤ ⟶ ℙ].
((∀x:A. ∀n,m:ℤ. (R[x;n] ⇒ R[x;m] supposing n ≤ m)) ⇒ (∀L:A List. ((∀x∈L.∃n:ℤ. R[x;n]) ⇒ (∃n:ℤ. (∀x∈L.R[x;n])))))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
uimplies: b supposing a,
exists: ∃x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
or: P ∨ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
guard: {T},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T
Lemmas referenced :
int_seg_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_term_value_constant_lemma,
int_formula_prop_and_lemma,
itermConstant_wf,
intformand_wf,
length_wf,
int_seg_properties,
select_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
intformle_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__le,
imax_ub,
imax_wf,
cons_wf,
l_all_cons,
and_wf,
l_all_wf_nil,
l_all_nil,
le_wf,
isect_wf,
all_wf,
list_wf,
l_member_wf,
exists_wf,
l_all_wf,
list_induction
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
intEquality,
applyEquality,
setElimination,
rename,
hypothesis,
setEquality,
independent_functionElimination,
because_Cache,
dependent_functionElimination,
cumulativity,
universeEquality,
dependent_pairFormation,
natural_numberEquality,
isect_memberEquality,
voidElimination,
voidEquality,
productElimination,
addLevel,
impliesFunctionality,
existsFunctionality,
productEquality,
independent_pairFormation,
independent_isectElimination,
inlFormation,
unionElimination,
int_eqEquality,
computeAll,
imageElimination,
inrFormation
Latex:
\mforall{}[A:Type]. \mforall{}[R:A {}\mrightarrow{} \mBbbZ{} {}\mrightarrow{} \mBbbP{}].
((\mforall{}x:A. \mforall{}n,m:\mBbbZ{}. (R[x;n] {}\mRightarrow{} R[x;m] supposing n \mleq{} m))
{}\mRightarrow{} (\mforall{}L:A List. ((\mforall{}x\mmember{}L.\mexists{}n:\mBbbZ{}. R[x;n]) {}\mRightarrow{} (\mexists{}n:\mBbbZ{}. (\mforall{}x\mmember{}L.R[x;n])))))
Date html generated:
2016_05_14-PM-02_45_46
Last ObjectModification:
2016_01_15-AM-07_35_48
Theory : list_1
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