Nuprl Lemma : divide-and-conquer
∀[Q:a:ℤ ⟶ {a...} ⟶ ℙ]
∀s:{2...}
((∀a:ℤ. ∀b:{a..a + s-}. Q[a;b])
⇒ (∀a,b,c:ℤ. (Q[a;c] ⇒ Q[a;b]) ∨ (Q[c;b] ⇒ Q[a;b]) supposing a < c ∧ c < b)
⇒ (∀a:ℤ. ∀b:{a...}. Q[a;b]))
Proof
Definitions occuring in Statement :
int_upper: {i...},
int_seg: {i..j-},
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],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
and: P ∧ Q,
so_apply: x[s1;s2],
int_upper: {i...},
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
subtype_rel: A ⊆r B,
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
nat: ℕ,
ge: i ≥ j ,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
nat_plus: ℕ+,
le: A ≤ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
true: True,
int_nzero: ℤ-o,
subtract: n - m,
less_than: a < b,
squash: ↓T
Lemmas referenced :
all_wf,
isect_wf,
less_than_wf,
or_wf,
int_upper_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
int_seg_wf,
subtype_rel_sets,
lelt_wf,
int_upper_wf,
uniform-comp-nat-induction,
nat_wf,
uall_wf,
decidable__lt,
subtract_wf,
nat_properties,
itermAdd_wf,
itermSubtract_wf,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_seg_properties,
intformeq_wf,
itermConstant_wf,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
equal-wf-base,
int_subtype_base,
equal_wf,
set-value-type,
int-value-type,
subtype_base_sq,
mul_cancel_in_lt,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
div_rem_sum2,
nequal_wf,
rem_bounds_1,
itermMinus_wf,
int_term_value_minus_lemma,
mul-distributes,
minus-one-mul,
mul-commutes,
mul_bounds_1b,
condition-implies-le,
minus-add,
minus-minus,
minus-one-mul-top,
add-associates,
add-swap,
less-iff-le,
decidable__equal_int,
itermMultiply_wf,
int_term_value_mul_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
sqequalRule,
lambdaEquality,
productEquality,
hypothesisEquality,
hypothesis,
functionEquality,
applyEquality,
functionExtensionality,
productElimination,
dependent_set_memberEquality,
natural_numberEquality,
setElimination,
rename,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
universeEquality,
addEquality,
setEquality,
cumulativity,
independent_functionElimination,
divideEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
cutEval,
instantiate,
multiplyEquality,
minusEquality,
applyLambdaEquality,
imageElimination
Latex:
\mforall{}[Q:a:\mBbbZ{} {}\mrightarrow{} \{a...\} {}\mrightarrow{} \mBbbP{}]
\mforall{}s:\{2...\}
((\mforall{}a:\mBbbZ{}. \mforall{}b:\{a..a + s\msupminus{}\}. Q[a;b])
{}\mRightarrow{} (\mforall{}a,b,c:\mBbbZ{}. (Q[a;c] {}\mRightarrow{} Q[a;b]) \mvee{} (Q[c;b] {}\mRightarrow{} Q[a;b]) supposing a < c \mwedge{} c < b)
{}\mRightarrow{} (\mforall{}a:\mBbbZ{}. \mforall{}b:\{a...\}. Q[a;b]))
Date html generated:
2017_04_14-AM-09_17_14
Last ObjectModification:
2017_02_27-PM-03_54_55
Theory : int_2
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