Nuprl Lemma : nil-llex
∀[A:Type]. ∀[<:A ⟶ A ⟶ ℙ]. ∀L:A List. ((L = [] ∈ (A List)) ∨ ([] llex(A;a,b.<[a;b]) L))
Proof
Definitions occuring in Statement :
llex: llex(A;a,b.<[a; b]),
nil: [],
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
infix_ap: x f y,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
or: P ∨ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
or: P ∨ Q,
prop: ℙ,
infix_ap: x f y,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons: [a / b],
guard: {T},
llex: llex(A;a,b.<[a; b]),
select: L[n],
uimplies: b supposing a,
nil: [],
it: ⋅,
top: Top,
and: P ∧ Q,
cand: A c∧ B,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
implies: P ⇒ Q,
decidable: Dec(P),
false: False,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
int_seg: {i..j-},
lelt: i ≤ j < k,
so_lambda: λ2x.t[x],
nat: ℕ,
ge: i ≥ j ,
so_apply: x[s],
subtype_rel: A ⊆r B
Lemmas referenced :
list-cases,
nil_wf,
llex_wf,
product_subtype_list,
equal-wf-T-base,
list_wf,
cons_wf,
length_of_nil_lemma,
stuck-spread,
base_wf,
length_of_cons_lemma,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
itermAdd_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
false_wf,
equal_wf,
int_seg_properties,
intformle_wf,
int_formula_prop_le_lemma,
int_seg_wf,
exists_wf,
nat_wf,
length_wf,
all_wf,
equal-wf-base-T,
select_wf,
nat_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesisEquality,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
unionElimination,
inlFormation,
cumulativity,
applyEquality,
sqequalRule,
lambdaEquality,
functionExtensionality,
promote_hyp,
hypothesis_subsumption,
productElimination,
inrFormation,
baseClosed,
functionEquality,
universeEquality,
independent_isectElimination,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
natural_numberEquality,
independent_pairFormation,
imageMemberEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
setElimination,
rename,
pointwiseFunctionality,
baseApply,
closedConclusion,
dependent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
independent_functionElimination,
because_Cache,
productEquality,
addEquality
Latex:
\mforall{}[A:Type]. \mforall{}[<:A {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}]. \mforall{}L:A List. ((L = []) \mvee{} ([] llex(A;a,b.<[a;b]) L))
Date html generated:
2018_05_21-PM-07_17_17
Last ObjectModification:
2017_07_26-PM-05_04_22
Theory : general
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