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

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

Proof

Definitions occuring in Statement :  strong-continuity-test-bound: `strong-continuity-test-bound(M;n;f;b)` int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` isl: `isl(x)` less_than: `a < b` uall: `∀[x:A]. B[x]` implies: `P `` Q` unit: `Unit` apply: `f a` function: `x:A ⟶ B[x]` union: `left + right` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  member: `t ∈ T` uall: `∀[x:A]. B[x]` nat: `ℕ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` 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]` int_seg: `{i..j-}` lelt: `i ≤ j < k` less_than: `a < b` decidable: `Dec(P)` or: `P ∨ Q` guard: `{T}` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` assert: `↑b` ifthenelse: `if b then t else f fi ` bfalse: `ff`
Lemmas referenced :  le_wf int_term_value_constant_lemma int_formula_prop_le_lemma itermConstant_wf intformle_wf decidable__le int_formula_prop_eq_lemma intformeq_wf decidable__equal_int isr-not-isl int_formula_prop_wf int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt nat_properties int_seg_properties strong-continuity-test-bound-prop2 decidable__lt less_than_wf lelt_wf subtype_rel_self false_wf int_seg_subtype_nat subtype_rel_dep_function nat_wf strong-continuity-test-bound_wf unit_wf2 int_seg_wf isl_wf assert_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut lemma_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename because_Cache hypothesis functionExtensionality applyEquality hypothesisEquality sqequalRule lambdaEquality independent_isectElimination independent_pairFormation lambdaFormation dependent_set_memberEquality productElimination functionEquality unionEquality isect_memberFormation introduction dependent_functionElimination axiomEquality isect_memberEquality unionElimination independent_functionElimination dependent_pairFormation int_eqEquality intEquality voidElimination voidEquality computeAll equalityTransitivity equalitySymmetry

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

Date html generated: 2016_05_19-AM-11_59_55
Last ObjectModification: 2016_05_17-PM-05_02_57

Theory : continuity

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