Nuprl Lemma : sq_stable__is_accum_splitting
∀[T,A:Type]. ∀[L:T List]. ∀[LL:(T List × A) List]. ∀[L2:T List × A]. ∀[f:(T List × A) ⟶ 𝔹]. ∀[x:A].
∀[g:(T List × A) ⟶ A].
SqStable(is_accum_splitting(T;A;L;LL;L2;f;g;x))
Proof
Definitions occuring in Statement :
is_accum_splitting: is_accum_splitting(T;A;L;LL;L2;f;g;x),
list: T List,
bool: 𝔹,
sq_stable: SqStable(P),
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
is_accum_splitting: is_accum_splitting(T;A;L;LL;L2;f;g;x),
l_all: (∀x∈L.P[x]),
top: Top,
prop: ℙ,
and: P ∧ Q,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
less_than: a < b,
squash: ↓T,
subtype_rel: A ⊆r B,
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
nat_plus: ℕ+,
less_than': less_than'(a;b),
true: True,
uiff: uiff(P;Q),
listp: A List+,
sq_stable: SqStable(P)
Lemmas referenced :
sq_stable__and,
equal_wf,
list_wf,
append_wf,
concat_wf,
map_wf,
pi1_wf_top,
all_wf,
int_seg_wf,
length_wf,
assert_wf,
select_wf,
int_seg_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
not_wf,
null_wf3,
subtype_rel_list,
top_wf,
iseg_wf,
pi2_wf,
hd_wf,
listp_properties,
length-append,
length_of_cons_lemma,
length_of_nil_lemma,
add_nat_plus,
length_wf_nat,
less_than_wf,
nat_plus_wf,
nat_plus_properties,
add-is-int-iff,
itermAdd_wf,
intformeq_wf,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
false_wf,
cons_wf,
nil_wf,
sq_stable__equal,
assert_of_null,
equal-wf-T-base,
sq_stable__all,
sq_stable__assert,
sq_stable__not,
assert_witness,
squash_wf,
is_accum_splitting_wf,
bool_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
cumulativity,
hypothesisEquality,
hypothesis,
productEquality,
lambdaEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
sqequalRule,
applyEquality,
functionExtensionality,
because_Cache,
setElimination,
rename,
independent_isectElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
imageElimination,
lambdaFormation,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
functionEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
addEquality,
allFunctionality,
impliesFunctionality,
axiomEquality,
universeEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}[L:T List]. \mforall{}[LL:(T List \mtimes{} A) List]. \mforall{}[L2:T List \mtimes{} A]. \mforall{}[f:(T List \mtimes{} A) {}\mrightarrow{} \mBbbB{}]. \mforall{}[x:A].
\mforall{}[g:(T List \mtimes{} A) {}\mrightarrow{} A].
SqStable(is\_accum\_splitting(T;A;L;LL;L2;f;g;x))
Date html generated:
2018_05_21-PM-08_06_12
Last ObjectModification:
2017_07_26-PM-05_42_13
Theory : general
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