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: prop: so_apply: x[s1;s2] all: x:A. B[x] exists: x:A. B[x] implies:  Q and: P ∧ Q true: True apply: a function: x:A ⟶ B[x] add: m natural_number: $n
Definitions unfolded in proof :  all: x:A. B[x] implies:  Q so_lambda: λ2x.t[x] member: t ∈ T uall: [x:A]. B[x] nat: so_apply: x[s1;s2] strict-inc: StrictInc subtype_rel: A ⊆B guard: {T} int_upper: {i...} prop: ge: i ≥  decidable: Dec(P) or: P ∨ Q uimplies: 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: λ2y.t[x; y] b-almost-full: b-almost-full(n,m.R[n; m]) compose: 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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