Nuprl Lemma : wfd-subtrees_wf
∀[A:Type]. ∀[w:wfd-tree(A)]. wfd-subtrees(w) ∈ A ⟶ wfd-tree(A) supposing ¬↑co-w-null(w)
Proof
Definitions occuring in Statement :
wfd-subtrees: wfd-subtrees(w),
wfd-tree2: wfd-tree(A),
co-w-null: co-w-null(w),
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
wfd-tree2: wfd-tree(A),
subtype_rel: A ⊆r B,
guard: {T},
all: ∀x:A. B[x],
implies: P ⇒ Q,
wfd-subtrees: wfd-subtrees(w),
co-w-null: co-w-null(w),
isl: isl(x),
outr: outr(x),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
not: ¬A,
true: True,
false: False,
prop: ℙ,
bfalse: ff,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
and: P ∧ Q,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
le: A ≤ B,
less_than': less_than'(a;b),
ge: i ≥ j ,
int_upper: {i...},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
w-bars: w-bars(w;p),
squash: ↓T,
upto: upto(n),
from-upto: [n, m),
lt_int: i <z j,
co-w-select: w@s,
nequal: a ≠ b ∈ T ,
bor: p ∨bq,
nat_plus: ℕ+,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
eq_int: (i =z j),
compose: f o g,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
nil: [],
less_than: a < b
Lemmas referenced :
co-w-ext,
subtype_rel_weakening,
co-w_wf,
unit_wf2,
not_wf,
true_wf,
nat_wf,
all_wf,
w-bars_wf,
subtype_rel_union,
ext-eq_inversion,
subtype_rel_transitivity,
false_wf,
equal_wf,
assert_wf,
co-w-null_wf,
wfd-tree2_wf,
eq_int_wf,
bool_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
le_wf,
nat_properties,
nequal-le-implies,
zero-add,
subtract_wf,
int_upper_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_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_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__equal_int,
int_subtype_base,
map_nil_lemma,
co_w_select_nil_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
co-w-select_wf,
map_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
upto_wf,
null-map,
null-upto,
eqtt_to_assert,
assert_of_eq_int,
decidable__lt,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
less_than_wf,
map_cons_lemma,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
map-map,
upto_decomp2,
subtype_rel_list,
list_wf,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
equal-wf-T-base,
colength_wf_list,
less_than_transitivity1,
less_than_irreflexivity,
list-cases,
product_subtype_list,
spread_cons_lemma,
itermAdd_wf,
int_term_value_add_lemma,
set_subtype_base,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
hypothesis,
cumulativity,
unionEquality,
functionEquality,
independent_isectElimination,
sqequalRule,
lambdaFormation,
unionElimination,
independent_functionElimination,
natural_numberEquality,
voidElimination,
functionExtensionality,
dependent_set_memberEquality,
lambdaEquality,
inrEquality,
because_Cache,
voidEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
axiomEquality,
isect_memberEquality,
universeEquality,
equalityElimination,
productElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
hypothesis_subsumption,
independent_pairFormation,
int_eqEquality,
intEquality,
computeAll,
imageElimination,
imageMemberEquality,
baseClosed,
addEquality,
minusEquality,
intWeakElimination,
sqequalAxiom,
applyLambdaEquality
Latex:
\mforall{}[A:Type]. \mforall{}[w:wfd-tree(A)]. wfd-subtrees(w) \mmember{} A {}\mrightarrow{} wfd-tree(A) supposing \mneg{}\muparrow{}co-w-null(w)
Date html generated:
2018_05_21-PM-10_18_07
Last ObjectModification:
2017_07_26-PM-06_36_33
Theory : bar!induction
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