Nuprl Lemma : tree-secures_wf
∀[T:Type]. ∀[p:wfd-tree(T)]. ∀[A:n:ℕ ⟶ (ℕn ⟶ T) ⟶ ℙ]. (tree-secures(T;A;p) ∈ ℙ)
Proof
Definitions occuring in Statement :
tree-secures: tree-secures(T;A;p),
wfd-tree: wfd-tree(T),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
member: t ∈ T,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
wfd-tree: wfd-tree(T),
all: ∀x:A. B[x],
nat: ℕ,
prop: ℙ,
tree-secures: tree-secures(T;A;p),
Wsup: Wsup(a;b),
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
so_lambda: λ2x.t[x],
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_apply: x[s],
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
subtype_rel: A ⊆r B,
pcw-pp-barred: Barred(pp),
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
cw-step: cw-step(A;a.B[a]),
pcw-step: pcw-step(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b]),
spreadn: spread3,
less_than: a < b,
true: True,
squash: ↓T,
isr: isr(x),
ext-eq: A ≡ B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
ext-family: F ≡ G,
pi1: fst(t),
nat_plus: ℕ+,
W-rel: W-rel(A;a.B[a];w),
param-W-rel: param-W-rel(P;p.A[p];p,a.B[p; a];p,a,b.C[p; a; b];par;w),
pcw-steprel: StepRel(s1;s2),
pi2: snd(t),
isl: isl(x),
pcw-step-agree: StepAgree(s;p1;w),
cand: A c∧ B
Lemmas referenced :
nat_wf,
int_seg_wf,
wfd-tree_wf,
bool_wf,
eqtt_to_assert,
all_wf,
false_wf,
le_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
predicate-or-shift_wf,
W-elimination-facts,
ifthenelse_wf,
subtract_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_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_less_lemma,
int_formula_prop_wf,
decidable__lt,
lelt_wf,
top_wf,
less_than_wf,
true_wf,
add-subtract-cancel,
itermAdd_wf,
int_term_value_add_lemma,
W-ext,
param-co-W-ext,
unit_wf2,
it_wf,
param-co-W_wf,
ext-eq_inversion,
subtype_rel_weakening,
assert_wf,
btrue_wf,
bfalse_wf,
pcw-steprel_wf,
subtype_rel_dep_function,
set_subtype_base,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
int_seg_subtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
sqequalRule,
axiomEquality,
functionEquality,
cumulativity,
extract_by_obid,
isectElimination,
natural_numberEquality,
setElimination,
rename,
universeEquality,
isect_memberEquality,
because_Cache,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
lambdaEquality,
applyEquality,
functionExtensionality,
dependent_set_memberEquality,
independent_pairFormation,
dependent_pairFormation,
promote_hyp,
instantiate,
independent_functionElimination,
voidElimination,
voidEquality,
strong_bar_Induction,
int_eqEquality,
intEquality,
computeAll,
lessCases,
sqequalAxiom,
imageMemberEquality,
baseClosed,
imageElimination,
addEquality,
int_eqReduceTrueSq,
hypothesis_subsumption,
dependent_pairEquality,
productEquality,
inlEquality,
unionEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[T:Type]. \mforall{}[p:wfd-tree(T)]. \mforall{}[A:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} \mBbbP{}]. (tree-secures(T;A;p) \mmember{} \mBbbP{})
Date html generated:
2017_04_17-AM-09_36_11
Last ObjectModification:
2017_02_27-PM-05_34_42
Theory : fan-theorem
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