Nuprl Lemma : bs_tree_insert_wf
∀[E:Type]. ∀[cmp:comparison(E)]. ∀[x:E]. ∀[tr:ordered_bs_tree(E;cmp)].
(bs_tree_insert(cmp;x;tr) ∈ ordered_bs_tree(E;cmp))
Proof
Definitions occuring in Statement :
bs_tree_insert: bs_tree_insert(cmp;x;tr),
ordered_bs_tree: ordered_bs_tree(E;cmp),
comparison: comparison(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type
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],
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s],
bs_tree_insert: bs_tree_insert(cmp;x;tr),
bst_null: bst_null(),
bs_tree_ind: bs_tree_ind,
bs_tree_ordered: bs_tree_ordered(E;cmp;tr),
bst_leaf: bst_leaf(value),
true: True,
has-value: (a)↓,
uimplies: b supposing a,
comparison: comparison(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
squash: ↓T,
not: ¬A,
false: False,
bst_node: bst_node(left;value;right),
member_bs_tree: x ∈ tr,
cand: A c∧ B,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
ordered_bs_tree_wf,
comparison_wf,
bs_tree_insert_wf1,
bs_tree-induction,
bs_tree_ordered_wf,
bs_tree_wf,
bst_null_wf,
value-type-has-value,
int-value-type,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
and_wf,
equal_wf,
false_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bst_leaf_wf,
member_bs_tree_wf,
bst_node_wf,
squash_wf,
true_wf,
comparison-anti,
iff_weakening_equal,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermMinus_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_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,
member_bs_tree_insert,
minus-is-int-iff,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
isect_memberEquality,
because_Cache,
dependent_functionElimination,
universeEquality,
lambdaEquality,
functionEquality,
independent_functionElimination,
lambdaFormation,
natural_numberEquality,
callbyvalueReduce,
intEquality,
independent_isectElimination,
applyEquality,
unionElimination,
equalityElimination,
productElimination,
lessCases,
sqequalAxiom,
independent_pairFormation,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
imageElimination,
hyp_replacement,
applyLambdaEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
minusEquality,
int_eqEquality,
computeAll,
pointwiseFunctionality,
baseApply,
closedConclusion,
functionExtensionality
Latex:
\mforall{}[E:Type]. \mforall{}[cmp:comparison(E)]. \mforall{}[x:E]. \mforall{}[tr:ordered\_bs\_tree(E;cmp)].
(bs\_tree\_insert(cmp;x;tr) \mmember{} ordered\_bs\_tree(E;cmp))
Date html generated:
2017_10_01-AM-08_31_15
Last ObjectModification:
2017_07_26-PM-04_25_00
Theory : tree_1
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