Nuprl Lemma : b-almost-full-filter
∀A,B:ℕ ⟶ ℕ ⟶ ℙ.
((b-almost-full(n,m.A[n;m]) ⇒ b-almost-full(n,m.B[n;m]) ⇒ b-almost-full(n,m.A[n;m] ∧ B[n;m]))
∧ ((∀n,m:ℕ. (A[n;m] ⇒ B[n;m])) ⇒ b-almost-full(n,m.A[n;m]) ⇒ b-almost-full(n,m.B[n;m]))
∧ b-almost-full(n,m.True))
Proof
Definitions occuring in Statement :
b-almost-full: b-almost-full(n,m.R[n; m]),
nat: ℕ,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
so_apply: x[s],
b-almost-full: b-almost-full(n,m.R[n; m]),
nat: ℕ,
strict-inc: StrictInc,
subtype_rel: A ⊆r B,
guard: {T},
int_upper: {i...},
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,
le: A ≤ B,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
quotient-member-eq,
false_wf,
equiv_rel_true,
true_wf,
strict-inc_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
le_wf,
int_upper_properties,
int_upper_subtype_nat,
int_upper_wf,
exists_wf,
implies-quotient-true,
intuitionistic-Ramsey,
all_wf,
nat_wf,
b-almost-full_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality,
applyEquality,
hypothesisEquality,
hypothesis,
independent_pairFormation,
independent_functionElimination,
because_Cache,
functionEquality,
cumulativity,
universeEquality,
dependent_functionElimination,
addEquality,
setElimination,
rename,
natural_numberEquality,
dependent_set_memberEquality,
setEquality,
intEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
productElimination,
introduction,
dependent_pairEquality,
axiomEquality
Latex:
\mforall{}A,B:\mBbbN{} {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}.
((b-almost-full(n,m.A[n;m]) {}\mRightarrow{} b-almost-full(n,m.B[n;m]) {}\mRightarrow{} b-almost-full(n,m.A[n;m] \mwedge{} B[n;m]))
\mwedge{} ((\mforall{}n,m:\mBbbN{}. (A[n;m] {}\mRightarrow{} B[n;m])) {}\mRightarrow{} b-almost-full(n,m.A[n;m]) {}\mRightarrow{} b-almost-full(n,m.B[n;m]))
\mwedge{} b-almost-full(n,m.True))
Date html generated:
2016_05_14-PM-09_53_53
Last ObjectModification:
2016_01_15-PM-10_56_04
Theory : continuity
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