### 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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