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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