### Nuprl Lemma : strong-continuity-test-prop3

`∀[T:Type]. ∀[M:n:ℕ ⟶ (ℕn ⟶ T) ⟶ (ℕ?)]. ∀[n,m:ℕ]. ∀[f:ℕ ⟶ T].`
`  ((↑isl(strong-continuity-test(M;n;f;M n f))) `` (↑isl(strong-continuity-test(M;m;f;M m f))) `` (n = m ∈ ℤ))`

Proof

Definitions occuring in Statement :  strong-continuity-test: `strong-continuity-test(M;n;f;b)` int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` isl: `isl(x)` uall: `∀[x:A]. B[x]` implies: `P `` Q` unit: `Unit` apply: `f a` function: `x:A ⟶ B[x]` union: `left + right` natural_number: `\$n` int: `ℤ` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  member: `t ∈ T` uall: `∀[x:A]. B[x]` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat: `ℕ` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` guard: `{T}` top: `Top` assert: `↑b` ifthenelse: `if b then t else f fi ` bfalse: `ff` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]`
Lemmas referenced :  int_formula_prop_wf int_formula_prop_eq_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma intformeq_wf itermVar_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt nat_properties top_wf subtype_rel_union isr-not-isl decidable__lt decidable__equal_int strong-continuity-test-prop1 false_wf int_seg_subtype_nat int_seg_wf subtype_rel_dep_function strong-continuity-test_wf unit_wf2 nat_wf isl_wf assert_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesis cumulativity hypothesisEquality functionExtensionality applyEquality because_Cache sqequalRule lambdaEquality natural_numberEquality setElimination rename independent_isectElimination independent_pairFormation lambdaFormation functionEquality unionEquality universeEquality isect_memberFormation introduction dependent_functionElimination axiomEquality isect_memberEquality independent_functionElimination productElimination unionElimination equalityTransitivity equalitySymmetry voidElimination voidEquality dependent_pairFormation int_eqEquality intEquality computeAll

Latex:
\mforall{}[T:Type].  \mforall{}[M:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  T)  {}\mrightarrow{}  (\mBbbN{}?)].  \mforall{}[n,m:\mBbbN{}].  \mforall{}[f:\mBbbN{}  {}\mrightarrow{}  T].
((\muparrow{}isl(strong-continuity-test(M;n;f;M  n  f)))
{}\mRightarrow{}  (\muparrow{}isl(strong-continuity-test(M;m;f;M  m  f)))
{}\mRightarrow{}  (n  =  m))

Date html generated: 2016_05_19-AM-11_59_28
Last ObjectModification: 2016_05_16-PM-05_42_13

Theory : continuity

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