Nuprl Lemma : tree-big-monotone
∀[T:Type]. ∀[A:(T List) ⟶ ℙ]. ∀a,b:ℕ. ((a ≤ b) ⇒ tree-big(T;upwd-closure(T;A);a) ⇒ tree-big(T;upwd-closure(T;A);b))
Proof
Definitions occuring in Statement :
tree-big: tree-big(T;A;n),
upwd-closure: upwd-closure(T;A),
list: T List,
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
upwd-closure: upwd-closure(T;A),
tree-big: tree-big(T;A;n),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
int_iseg: {i...j},
and: P ∧ Q,
cand: A c∧ B,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B
Lemmas referenced :
iseg_transitivity,
lelt_wf,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermAdd_wf,
intformless_wf,
decidable__lt,
firstn_is_iseg,
nat_wf,
exists_wf,
all_wf,
list_wf,
equal_wf,
iseg_wf,
length_wf,
le_wf,
and_wf,
int_formula_prop_eq_lemma,
intformeq_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
length_firstn,
firstn_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
lemma_by_obid,
isectElimination,
hypothesisEquality,
setElimination,
rename,
independent_functionElimination,
because_Cache,
dependent_set_memberEquality,
natural_numberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
equalityTransitivity,
equalitySymmetry,
productElimination,
productEquality,
applyEquality,
universeEquality,
functionEquality,
cumulativity,
addEquality
Latex:
\mforall{}[T:Type]. \mforall{}[A:(T List) {}\mrightarrow{} \mBbbP{}].
\mforall{}a,b:\mBbbN{}. ((a \mleq{} b) {}\mRightarrow{} tree-big(T;upwd-closure(T;A);a) {}\mRightarrow{} tree-big(T;upwd-closure(T;A);b))
Date html generated:
2016_05_14-PM-04_10_11
Last ObjectModification:
2016_01_14-PM-10_58_11
Theory : fan-theorem
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