Nuprl Lemma : b-almost-full-intersection-lemma
∀R,T:ℕ ⟶ ℕ ⟶ ℙ.
(b-almost-full(n,m.R[n;m])
⇒ b-almost-full(n,m.T[n;m])
⇒ (∀s:StrictInc. ⇃(∃m:ℕ. ∃n,p:{m + 1...}. (R[s m;s n] ∧ T[s m;s p]))))
Proof
Definitions occuring in Statement :
b-almost-full: b-almost-full(n,m.R[n; m]),
strict-inc: StrictInc,
quotient: x,y:A//B[x; y],
int_upper: {i...},
nat: ℕ,
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_apply: x[s1;s2],
strict-inc: StrictInc,
subtype_rel: A ⊆r B,
guard: {T},
int_upper: {i...},
prop: ℙ,
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,
and: P ∧ Q,
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
b-almost-full: b-almost-full(n,m.R[n; m]),
compose: f o g,
le: A ≤ B,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b)
Lemmas referenced :
false_wf,
int_seg_subtype_nat,
less_than_wf,
all_wf,
int_seg_wf,
int_seg_properties,
lelt_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
implies-quotient-true,
compose-strict-inc,
b-almost-full_wf,
strict-inc_wf,
nat_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,
intuitionistic-pigeonhole
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
lambdaEquality,
isectElimination,
addEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
natural_numberEquality,
applyEquality,
because_Cache,
dependent_set_memberEquality,
setEquality,
intEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
functionEquality,
cumulativity,
universeEquality,
productElimination,
equalityTransitivity,
equalitySymmetry,
imageElimination,
productEquality
Latex:
\mforall{}R,T:\mBbbN{} {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}.
(b-almost-full(n,m.R[n;m])
{}\mRightarrow{} b-almost-full(n,m.T[n;m])
{}\mRightarrow{} (\mforall{}s:StrictInc. \00D9(\mexists{}m:\mBbbN{}. \mexists{}n,p:\{m + 1...\}. (R[s m;s n] \mwedge{} T[s m;s p]))))
Date html generated:
2016_05_14-PM-09_51_16
Last ObjectModification:
2016_01_15-PM-10_58_12
Theory : continuity
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