Nuprl Lemma : bs_tree_max_wf
∀[E:Type]. ∀[cmp:comparison(E)]. ∀[tr:ordered_bs_tree(E;cmp)]. ∀[d:E].
(bs_tree_max(tr;d) ∈ {p:E × ordered_bs_tree(E;cmp)|
let m,t = p
in (∀x:E. (x ∈ tr ⇒ (x ∈ t ∨ (x = m ∈ E))))
∧ ((¬↑bst_null?(tr)) ⇒ m ∈ tr)
∧ (∀x:E. (x ∈ t ⇒ (x ∈ tr ∧ 0 < cmp x m)))} )
Proof
Definitions occuring in Statement :
bs_tree_max: bs_tree_max(tr;d),
ordered_bs_tree: ordered_bs_tree(E;cmp),
member_bs_tree: x ∈ tr,
bst_null?: bst_null?(v),
comparison: comparison(T),
assert: ↑b,
less_than: a < b,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
spread: spread def,
product: x:A × B[x],
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
ordered_bs_tree: ordered_bs_tree(E;cmp),
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
or: P ∨ Q,
guard: {T},
not: ¬A,
false: False,
true: True,
cand: A c∧ B,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
eq_atom: x =a y,
pi1: fst(t),
bst_null?: bst_null?(v),
bs_tree_ind: bs_tree_ind,
bst_null: bst_null(),
bs_tree_max: bs_tree_max(tr;d),
bs_tree_ordered: bs_tree_ordered(E;cmp;tr),
member_bs_tree: x ∈ tr,
bfalse: ff,
bst_leaf: bst_leaf(value),
comparison: comparison(T),
bst_node: bst_node(left;value;right),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
uimplies: b supposing a,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb,
ext-eq: A ≡ B,
subtype_rel: A ⊆r B,
squash: ↓T,
trans: Trans(T;x,y.E[x; y]),
iff: P ⇐⇒ Q,
less_than: a < b,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top
Lemmas referenced :
istype-universe,
ordered_bs_tree_wf,
comparison_wf,
bs_tree-induction,
bs_tree_ordered_wf,
all_wf,
bs_tree_wf,
true_wf,
not_wf,
false_wf,
equal_wf,
bst_node_wf,
bs_tree_max_wf1,
member_bs_tree_wf,
or_wf,
assert_wf,
bst_null?_wf,
less_than_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
bs_tree-ext,
eq_atom_wf,
assert_of_eq_atom,
atom_subtype_base,
unit_wf2,
unit_subtype_base,
it_wf,
bst_null_wf,
neg_assert_of_eq_atom,
assert_elim,
bst_leaf_wf,
bfalse_wf,
btrue_neq_bfalse,
istype-void,
squash_wf,
istype-int,
strict-comparison-trans,
comparison-anti,
subtype_rel_self,
iff_weakening_equal,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermConstant_wf,
itermMinus_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_minus_lemma,
int_term_value_var_lemma,
int_formula_prop_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
hypothesisEquality,
isect_memberEquality_alt,
universeIsType,
because_Cache,
dependent_functionElimination,
universeEquality,
independent_functionElimination,
lambdaEquality_alt,
functionEquality,
inhabitedIsType,
lambdaFormation_alt,
productElimination,
productEquality,
equalityIsType1,
voidElimination,
independent_pairFormation,
natural_numberEquality,
lambdaFormation,
cumulativity,
inrFormation,
functionIsType,
closedConclusion,
applyEquality,
unionElimination,
equalityElimination,
independent_isectElimination,
dependent_pairFormation_alt,
promote_hyp,
instantiate,
hyp_replacement,
applyLambdaEquality,
hypothesis_subsumption,
tokenEquality,
atomEquality,
equalityIsType2,
baseApply,
baseClosed,
inrFormation_alt,
inlFormation_alt,
unionIsType,
productIsType,
dependent_set_memberEquality_alt,
imageElimination,
imageMemberEquality,
approximateComputation,
int_eqEquality,
setEquality,
lambdaEquality,
independent_pairEquality,
dependent_set_memberEquality
Latex:
\mforall{}[E:Type]. \mforall{}[cmp:comparison(E)]. \mforall{}[tr:ordered\_bs\_tree(E;cmp)]. \mforall{}[d:E].
(bs\_tree\_max(tr;d) \mmember{} \{p:E \mtimes{} ordered\_bs\_tree(E;cmp)|
let m,t = p
in (\mforall{}x:E. (x \mmember{} tr {}\mRightarrow{} (x \mmember{} t \mvee{} (x = m))))
\mwedge{} ((\mneg{}\muparrow{}bst\_null?(tr)) {}\mRightarrow{} m \mmember{} tr)
\mwedge{} (\mforall{}x:E. (x \mmember{} t {}\mRightarrow{} (x \mmember{} tr \mwedge{} 0 < cmp x m)))\} )
Date html generated:
2019_10_15-AM-10_47_26
Last ObjectModification:
2018_10_11-PM-11_23_50
Theory : tree_1
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