Nuprl Lemma : l-ordered-insert-combine
∀T:Type. ∀R:T ⟶ T ⟶ ℙ. ∀cmp:comparison(T). ∀f:T ⟶ T ⟶ T. ∀x:T.
((∀u,x,y:T. (R[u;x] ⇒ R[x;y] ⇒ R[u;y]))
⇒ (∀u,x,y:T. (((cmp x u) = 0 ∈ ℤ) ⇒ R[x;y] ⇒ R[u;y]))
⇒ (∀u,x,y:T. (((cmp y u) = 0 ∈ ℤ) ⇒ R[x;y] ⇒ R[x;u]))
⇒ (∀u,x:T. (((cmp x u) = 0 ∈ ℤ) ⇒ ((cmp u (f x u)) = 0 ∈ ℤ)))
⇒ (∀x,y:T. (0 < cmp x y ⇒ R[x;y]))
⇒ (∀L:T List. (l-ordered(T;x,y.R[x;y];L) ⇒ l-ordered(T;x,y.R[x;y];insert-combine(cmp;f;x;L)))))
Proof
Definitions occuring in Statement :
l-ordered: l-ordered(T;x,y.R[x; y];L),
insert-combine: insert-combine(cmp;f;x;l),
comparison: comparison(T),
list: T List,
less_than: a < b,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_apply: x[s],
comparison: comparison(T),
and: P ∧ Q,
cand: A c∧ B,
true: True,
false: False,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
has-value: (a)↓,
uimplies: b supposing a,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
subtype_rel: A ⊆r B,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
not: ¬A,
nequal: a ≠ b ∈ T ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
list_induction,
l-ordered_wf,
insert-combine_wf,
list_wf,
all_wf,
less_than_wf,
equal-wf-T-base,
comparison_wf,
false_wf,
true_wf,
l-ordered-nil-true,
nil_member,
l_member_wf,
nil_wf,
l-ordered-cons,
insert-combine-nil,
value-type-has-value,
int-value-type,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
lt_int_wf,
assert_of_lt_int,
cons_wf,
insert-combine-cons,
cons_member,
and_wf,
member-insert-combine,
decidable__lt,
minus-is-int-iff,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
itermMinus_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_term_value_minus_lemma,
int_formula_prop_wf,
l_exists_iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
because_Cache,
sqequalRule,
lambdaEquality,
functionEquality,
cumulativity,
hypothesisEquality,
applyEquality,
functionExtensionality,
hypothesis,
dependent_functionElimination,
independent_functionElimination,
rename,
natural_numberEquality,
setElimination,
intEquality,
baseClosed,
universeEquality,
independent_pairFormation,
voidElimination,
addLevel,
impliesFunctionality,
productElimination,
allFunctionality,
productEquality,
isect_memberEquality,
voidEquality,
callbyvalueReduce,
independent_isectElimination,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality,
pointwiseFunctionality,
baseApply,
closedConclusion,
int_eqEquality,
computeAll,
setEquality
Latex:
\mforall{}T:Type. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. \mforall{}cmp:comparison(T). \mforall{}f:T {}\mrightarrow{} T {}\mrightarrow{} T. \mforall{}x:T.
((\mforall{}u,x,y:T. (R[u;x] {}\mRightarrow{} R[x;y] {}\mRightarrow{} R[u;y]))
{}\mRightarrow{} (\mforall{}u,x,y:T. (((cmp x u) = 0) {}\mRightarrow{} R[x;y] {}\mRightarrow{} R[u;y]))
{}\mRightarrow{} (\mforall{}u,x,y:T. (((cmp y u) = 0) {}\mRightarrow{} R[x;y] {}\mRightarrow{} R[x;u]))
{}\mRightarrow{} (\mforall{}u,x:T. (((cmp x u) = 0) {}\mRightarrow{} ((cmp u (f x u)) = 0)))
{}\mRightarrow{} (\mforall{}x,y:T. (0 < cmp x y {}\mRightarrow{} R[x;y]))
{}\mRightarrow{} (\mforall{}L:T List. (l-ordered(T;x,y.R[x;y];L) {}\mRightarrow{} l-ordered(T;x,y.R[x;y];insert-combine(cmp;f;x;L)))))
Date html generated:
2018_05_21-PM-07_38_01
Last ObjectModification:
2017_07_26-PM-05_12_17
Theory : general
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