Nuprl Lemma : split_tail_correct
∀[A:Type]. ∀[f:A ⟶ 𝔹]. ∀[L:A List]. (∀b∈snd(split_tail(L | ∀x.f[x])).↑f[b])
Proof
Definitions occuring in Statement :
split_tail: split_tail(L | ∀x.f[x]),
l_all: (∀x∈L.P[x]),
list: T List,
assert: ↑b,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
pi2: snd(t),
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
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,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
l_all: (∀x∈L.P[x]),
guard: {T},
or: P ∨ Q,
split_tail: split_tail(L | ∀x.f[x]),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
pi2: snd(t),
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
so_apply: x[s],
int_seg: {i..j-},
so_lambda: λ2x.t[x],
lelt: i ≤ j < k,
decidable: Dec(P),
less_than: a < b,
squash: ↓T,
colength: colength(L),
nil: [],
it: ⋅,
sq_type: SQType(T),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
bool: 𝔹,
unit: Unit,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
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,
assert_witness,
intformeq_wf,
int_formula_prop_eq_lemma,
list-cases,
list_ind_nil_lemma,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
le_wf,
select_wf,
int_seg_properties,
length_wf,
split_tail_wf,
istype-universe,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
subtract-1-ge-0,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
itermSubtract_wf,
itermAdd_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
list_ind_cons_lemma,
nat_wf,
list_wf,
bool_wf,
l_member_wf,
btrue_neq_bfalse,
member-implies-null-eq-bfalse,
btrue_wf,
null_nil_lemma,
assert_wf,
nil_wf,
l_all_iff,
l_all_cons,
l_all_wf,
list_ind_wf,
cons_wf,
equal-wf-T-base,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
thin,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
functionIsTypeImplies,
inhabitedIsType,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
equalityIsType1,
because_Cache,
dependent_set_memberEquality_alt,
applyEquality,
imageElimination,
instantiate,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
intEquality,
functionIsType,
universeEquality,
lambdaFormation,
setEquality,
cumulativity,
functionExtensionality,
lambdaEquality,
voidEquality,
setIsType,
productEquality,
independent_pairEquality,
productIsType,
equalityElimination
Latex:
\mforall{}[A:Type]. \mforall{}[f:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[L:A List]. (\mforall{}b\mmember{}snd(split\_tail(L | \mforall{}x.f[x])).\muparrow{}f[b])
Date html generated:
2019_10_15-AM-10_54_49
Last ObjectModification:
2018_10_09-AM-10_28_13
Theory : list!
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