Nuprl Lemma : l_tree_covariant
∀[A,B,T:Type]. l_tree(A;T) ⊆r l_tree(B;T) supposing A ⊆r B
Proof
Definitions occuring in Statement :
l_tree: l_tree(L;T),
uimplies: b supposing a,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
ext-eq: A ≡ B,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
sq_type: SQType(T),
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
l_tree_leaf: l_tree_leaf(val),
l_tree_size: l_tree_size(p),
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
l_tree_node: l_tree_node(val;left_subtree;right_subtree),
spreadn: spread3,
cand: A c∧ B,
less_than: a < b,
squash: ↓T
Lemmas referenced :
l_tree_node_wf,
nat_wf,
int_term_value_add_lemma,
itermAdd_wf,
decidable__lt,
neg_assert_of_eq_atom,
assert-bnot,
bool_subtype_base,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
l_tree_leaf_wf,
atom_subtype_base,
subtype_base_sq,
assert_of_eq_atom,
eqtt_to_assert,
bool_wf,
eq_atom_wf,
l_tree-ext,
int_formula_prop_eq_lemma,
intformeq_wf,
lelt_wf,
false_wf,
int_seg_subtype,
decidable__equal_int,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
subtract_wf,
decidable__le,
int_seg_properties,
int_seg_wf,
l_tree_size_wf,
le_wf,
less_than_wf,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
satisfiable-full-omega-tt,
nat_properties,
l_tree_wf,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
axiomEquality,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
applyEquality,
productElimination,
unionElimination,
setEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
promote_hyp,
tokenEquality,
equalityElimination,
instantiate,
cumulativity,
atomEquality,
imageElimination,
equalityEquality,
addEquality
Latex:
\mforall{}[A,B,T:Type]. l\_tree(A;T) \msubseteq{}r l\_tree(B;T) supposing A \msubseteq{}r B
Date html generated:
2016_05_16-AM-08_43_47
Last ObjectModification:
2016_01_17-AM-00_05_29
Theory : labeled!trees
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