Nuprl Lemma : trivial-tree-secures
∀T:Type. ∀p:wfd-tree(T). ∀[A:n:ℕ ⟶ (ℕn ⟶ T) ⟶ ℙ]. ((A 0 (λx.⊥)) ⇒ tree-secures(T;A;p))
Proof
Definitions occuring in Statement :
tree-secures: tree-secures(T;A;p),
wfd-tree: wfd-tree(T),
int_seg: {i..j-},
nat: ℕ,
bottom: ⊥,
uall: ∀[x:A]. B[x],
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
prop: ℙ,
implies: P ⇒ Q,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
so_apply: x[s],
tree-secures: tree-secures(T;A;p),
Wsup: Wsup(a;b),
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
predicate-or-shift: A[x],
or: P ∨ Q,
decidable: Dec(P)
Lemmas referenced :
wfd-tree-induction,
uall_wf,
nat_wf,
int_seg_wf,
false_wf,
le_wf,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
tree-secures_wf,
wfd-tree_wf,
predicate-or-shift_wf,
predicate-shift_wf,
all_wf,
decidable__equal_int,
intformnot_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
cumulativity,
hypothesisEquality,
dependent_functionElimination,
sqequalRule,
lambdaEquality,
functionEquality,
hypothesis,
applyEquality,
universeEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
functionExtensionality,
dependent_set_memberEquality,
independent_pairFormation,
productElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_functionElimination,
isect_memberFormation,
inlFormation,
addLevel,
hyp_replacement,
equalitySymmetry,
unionElimination,
equalityTransitivity,
levelHypothesis
Latex:
\mforall{}T:Type. \mforall{}p:wfd-tree(T). \mforall{}[A:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} T) {}\mrightarrow{} \mBbbP{}]. ((A 0 (\mlambda{}x.\mbot{})) {}\mRightarrow{} tree-secures(T;A;p))
Date html generated:
2016_10_21-AM-11_02_57
Last ObjectModification:
2016_07_12-AM-05_58_29
Theory : fan-theorem
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