### Nuprl Lemma : strong-continuity2-no-inner-squash-unique-bound

`∀F:(ℕ ⟶ ℕ) ⟶ ℕ`
`  ⇃(∃M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕn?)`
`     ∀f:ℕ ⟶ ℕ. ∃n:ℕ. (F f < n ∧ ((M n f) = (inl (F f)) ∈ (ℕ?)) ∧ (∀m:ℕ. ((↑isl(M m f)) `` (m = n ∈ ℕ)))))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` int_seg: `{i..j-}` nat: `ℕ` assert: `↑b` isl: `isl(x)` less_than: `a < b` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` true: `True` unit: `Unit` apply: `f a` function: `x:A ⟶ B[x]` inl: `inl x` union: `left + right` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` nat: `ℕ` so_lambda: `λ2x.t[x]` prop: `ℙ` and: `P ∧ Q` subtype_rel: `A ⊆r B` uimplies: `b supposing a` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` so_apply: `x[s]` exists: `∃x:A. B[x]` guard: `{T}` unit: `Unit` int_seg: `{i..j-}` lelt: `i ≤ j < k` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` sq_type: `SQType(T)` isl: `isl(x)` assert: `↑b` ifthenelse: `if b then t else f fi ` bfalse: `ff` btrue: `tt` true: `True` outl: `outl(x)`
Lemmas referenced :  strong-continuity2-no-inner-squash-bound implies-quotient-true exists_wf nat_wf int_seg_wf unit_wf2 all_wf less_than_wf equal_wf subtype_rel_function int_seg_subtype_nat false_wf subtype_rel_self subtype_rel_union assert_wf isl_wf strong-continuity-test-bound_wf decidable__assert nat_properties decidable__le full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformnot_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_wf lelt_wf unit_subtype_base int_subtype_base le_wf set_subtype_base union_subtype_base subtype_base_sq satisfiable-full-omega-tt subtype_rel_dep_function strong-continuity-test-bound-prop1 and_wf btrue_wf bool_wf bool_subtype_base outl_wf int_seg_properties intformeq_wf int_formula_prop_eq_lemma decidable__lt intformless_wf int_formula_prop_less_lemma strong-continuity-test-bound-prop3 not-isl-assert-isr strong-continuity-test-bound-prop4 decidable__equal_int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality isectElimination functionEquality hypothesis natural_numberEquality setElimination rename unionEquality because_Cache sqequalRule lambdaEquality productEquality applyEquality independent_isectElimination independent_pairFormation inlEquality independent_functionElimination productElimination dependent_pairFormation unionElimination functionExtensionality inrEquality axiomEquality equalityTransitivity equalitySymmetry dependent_set_memberEquality applyLambdaEquality approximateComputation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality cumulativity instantiate computeAll hyp_replacement promote_hyp

Latex:
\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbN{}
\00D9(\mexists{}M:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  (\mBbbN{}n?)
\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \mexists{}n:\mBbbN{}.  (F  f  <  n  \mwedge{}  ((M  n  f)  =  (inl  (F  f)))  \mwedge{}  (\mforall{}m:\mBbbN{}.  ((\muparrow{}isl(M  m  f))  {}\mRightarrow{}  (m  =  n)))))

Date html generated: 2019_06_20-PM-02_53_52
Last ObjectModification: 2018_08_22-AM-00_06_08

Theory : continuity

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