Nuprl Lemma : l-ordered-inst
∀[T:Type]. ∀L:T List. ∀R:T ⟶ T ⟶ ℙ. ∀i:ℕ||L||. ∀j:ℕi. (l-ordered(T;x,y.R[x;y];L) ⇒ R[L[j];L[i]])
Proof
Definitions occuring in Statement :
l-ordered: l-ordered(T;x,y.R[x; y];L),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
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,
l-ordered: l-ordered(T;x,y.R[x; y];L),
member: t ∈ T,
uimplies: b supposing a,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
less_than: a < b,
squash: ↓T,
l_before: x before y ∈ l,
sublist: L1 ⊆ L2,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
le: A ≤ B,
bfalse: ff,
cand: A c∧ B,
increasing: increasing(f;k),
subtract: n - m,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
select: L[n],
cons: [a / b],
eq_int: (i =z j),
less_than': less_than'(a;b),
nat: ℕ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
ge: i ≥ j ,
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
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,
length_of_cons_lemma,
length_of_nil_lemma,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
length_wf,
lelt_wf,
equal_wf,
int_seg_wf,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
decidable__equal_int,
int_subtype_base,
int_seg_subtype,
false_wf,
int_seg_cases,
increasing_wf,
le_wf,
all_wf,
cons_wf,
nil_wf,
non_neg_length,
length_wf_nat,
nat_properties,
l-ordered_wf,
list_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
independent_isectElimination,
hypothesisEquality,
setElimination,
rename,
productElimination,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
imageElimination,
independent_functionElimination,
equalityElimination,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
cumulativity,
addEquality,
promote_hyp,
instantiate,
hypothesis_subsumption,
productEquality,
functionExtensionality,
applyEquality,
applyLambdaEquality,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}L:T List. \mforall{}R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}. \mforall{}i:\mBbbN{}||L||. \mforall{}j:\mBbbN{}i. (l-ordered(T;x,y.R[x;y];L) {}\mRightarrow{} R[L[j];L[i]])
Date html generated:
2018_05_21-PM-07_37_32
Last ObjectModification:
2017_07_26-PM-05_11_42
Theory : general
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