### 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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