### Nuprl Lemma : finite-double-negation-shift

`∀[A:ℙ]. ∀[B:ℕ ⟶ ℙ].  ∀n:ℕ. ((∀i:ℕn. (((B i) `` A) `` A)) `` ((∀i:ℕn. (B i)) `` A) `` A)`

Proof

Definitions occuring in Statement :  int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` prop: `ℙ` all: `∀x:A. B[x]` implies: `P `` Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` sq_type: `SQType(T)` guard: `{T}` prop: `ℙ` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` so_apply: `x[s]` int_seg: `{i..j-}` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` ge: `i ≥ j ` lelt: `i ≤ j < k` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` less_than: `a < b` label: `...\$L... t` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  iff_weakening_equal le_wf lelt_wf decidable__lt int_term_value_subtract_lemma int_formula_prop_not_lemma itermSubtract_wf intformnot_wf decidable__le subtract_wf int_formula_prop_wf int_formula_prop_eq_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_and_lemma intformeq_wf itermConstant_wf intformle_wf itermVar_wf intformless_wf intformand_wf satisfiable-full-omega-tt nat_properties int_seg_properties natrec_wf nat_wf false_wf int_seg_subtype_nat int_seg_wf all_wf int_subtype_base subtype_base_sq decidable__equal_int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut thin lemma_by_obid sqequalHypSubstitution dependent_functionElimination setElimination rename hypothesisEquality natural_numberEquality hypothesis unionElimination instantiate isectElimination cumulativity intEquality independent_isectElimination because_Cache independent_functionElimination functionEquality sqequalRule lambdaEquality applyEquality independent_pairFormation introduction universeEquality equalityTransitivity equalitySymmetry productElimination dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality computeAll dependent_set_memberEquality

Latex:
\mforall{}[A:\mBbbP{}].  \mforall{}[B:\mBbbN{}  {}\mrightarrow{}  \mBbbP{}].    \mforall{}n:\mBbbN{}.  ((\mforall{}i:\mBbbN{}n.  (((B  i)  {}\mRightarrow{}  A)  {}\mRightarrow{}  A))  {}\mRightarrow{}  ((\mforall{}i:\mBbbN{}n.  (B  i))  {}\mRightarrow{}  A)  {}\mRightarrow{}  A)

Date html generated: 2016_05_15-PM-03_20_38
Last ObjectModification: 2016_01_16-AM-10_48_07

Theory : general

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