Nuprl Lemma : combine-list-rel-and
∀[T:Type]
∀f:T ⟶ T ⟶ T. ∀R:T ⟶ T ⟶ ℙ.
((∀x,y,z:T. (R[x;f[y;z]] ⇐⇒ R[x;y] ∧ R[x;z]))
⇒ (∀L:T List. ∀a:T. R[a;combine-list(x,y.f[x;y];L)] ⇐⇒ (∀b∈L.R[a;b]) supposing 0 < ||L|| ∧ Assoc(T;λx,y. f[x;y]))\000C)
Proof
Definitions occuring in Statement :
combine-list: combine-list(x,y.f[x; y];L),
l_all: (∀x∈L.P[x]),
length: ||as||,
list: T List,
assoc: Assoc(T;op),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
guard: {T},
int_seg: {i..j-},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
lelt: i ≤ j < k,
and: P ∧ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j ,
less_than: a < b,
squash: ↓T,
assoc: Assoc(T;op),
cons: [a / b],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
combine-list: combine-list(x,y.f[x; y];L),
list_accum: list_accum,
tl: tl(l),
pi2: snd(t),
nil: [],
hd: hd(l),
pi1: fst(t),
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
bfalse: ff,
subtract: n - m,
cand: A c∧ B
Lemmas referenced :
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
int_seg_subtype,
false_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
le_wf,
length_wf,
non_neg_length,
nat_properties,
decidable__lt,
lelt_wf,
less_than_wf,
member-less_than,
list-cases,
length_of_nil_lemma,
product_subtype_list,
length_of_cons_lemma,
null_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
list_wf,
eqtt_to_assert,
assert_of_null,
combine-list_wf,
cons_wf,
length-singleton,
l_all_single,
equal_wf,
l_all_wf,
nil_wf,
l_member_wf,
iff_wf,
itermAdd_wf,
int_term_value_add_lemma,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
length-nil,
length_wf_nat,
nat_wf,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
squash_wf,
true_wf,
combine-list-cons,
iff_weakening_equal,
l_all_cons,
assoc_wf,
all_wf,
isect_wf,
set_wf,
primrec-wf2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
because_Cache,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
productElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
unionElimination,
addLevel,
applyEquality,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
levelHypothesis,
hypothesis_subsumption,
dependent_set_memberEquality,
cumulativity,
imageElimination,
independent_functionElimination,
independent_pairEquality,
axiomEquality,
promote_hyp,
equalityElimination,
baseClosed,
functionExtensionality,
imageMemberEquality,
impliesFunctionality,
setEquality,
hyp_replacement,
addEquality,
minusEquality,
universeEquality,
productEquality,
functionEquality,
andLevelFunctionality
Latex:
\mforall{}[T:Type]
\mforall{}f:T {}\mrightarrow{} T {}\mrightarrow{} T. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}.
((\mforall{}x,y,z:T. (R[x;f[y;z]] \mLeftarrow{}{}\mRightarrow{} R[x;y] \mwedge{} R[x;z]))
{}\mRightarrow{} (\mforall{}L:T List. \mforall{}a:T.
R[a;combine-list(x,y.f[x;y];L)] \mLeftarrow{}{}\mRightarrow{} (\mforall{}b\mmember{}L.R[a;b])
supposing 0 < ||L|| \mwedge{} Assoc(T;\mlambda{}x,y. f[x;y])))
Date html generated:
2017_04_14-AM-09_23_52
Last ObjectModification:
2017_02_27-PM-03_58_59
Theory : list_1
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