Nuprl Lemma : l-first-when-none
∀[T:Type]. ∀[f:T ⟶ 𝔹]. ∀[L:T List]. l-first(x.f[x];L) ~ inr (λx.Ax) supposing (∀x∈L.¬↑f[x])
Proof
Definitions occuring in Statement :
l-first: l-first(x.f[x];L),
l_all: (∀x∈L.P[x]),
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
not: ¬A,
lambda: λx.A[x],
function: x:A ⟶ B[x],
inr: inr x ,
universe: Type,
sqequal: s ~ t,
axiom: Ax
Definitions unfolded in proof :
l-first: l-first(x.f[x];L),
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
guard: {T},
or: P ∨ Q,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P),
nil: [],
it: ⋅,
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
nat_plus: ℕ+,
true: True,
select: L[n],
subtract: n - m
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
l_all_wf,
not_wf,
assert_wf,
l_member_wf,
equal-wf-T-base,
nat_wf,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
list_ind_nil_lemma,
l_all_wf_nil,
product_subtype_list,
spread_cons_lemma,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
le_wf,
equal_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
list_ind_cons_lemma,
bool_wf,
eqtt_to_assert,
cons_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
list_wf,
length_of_cons_lemma,
false_wf,
add_nat_plus,
length_wf_nat,
nat_plus_wf,
nat_plus_properties,
decidable__lt,
add-is-int-iff,
lelt_wf,
length_wf,
add-member-int_seg2,
non_neg_length,
select-cons-tl,
int_seg_properties,
add-subtract-cancel,
int_seg_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
sqequalAxiom,
cumulativity,
applyEquality,
functionExtensionality,
setEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
applyLambdaEquality,
dependent_set_memberEquality,
addEquality,
baseClosed,
instantiate,
imageElimination,
equalityElimination,
functionEquality,
universeEquality,
imageMemberEquality,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:T List]. l-first(x.f[x];L) \msim{} inr (\mlambda{}x.Ax) supposing (\mforall{}x\mmember{}L.\mneg{}\muparrow{}f[x])
Date html generated:
2017_04_17-AM-07_24_54
Last ObjectModification:
2017_02_27-PM-04_03_48
Theory : list_1
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