### Nuprl Lemma : strong-continuity2-no-inner-squash-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 m f) = (inl (F f)) ∈ (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` implies: `P `` Q` exists: `∃x:A. B[x]` uall: `∀[x:A]. B[x]` nat: `ℕ` 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` isl: `isl(x)` prop: `ℙ` so_lambda: `λ2x.t[x]` outl: `outl(x)` so_apply: `x[s]` ge: `i ≥ j ` pi1: `fst(t)` int_seg: `{i..j-}` sq_stable: `SqStable(P)` lelt: `i ≤ j < k` guard: `{T}` squash: `↓T` assert: `↑b` ifthenelse: `if b then t else f fi ` btrue: `tt` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` bfalse: `ff` uiff: `uiff(P;Q)` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` true: `True` cand: `A c∧ B` sq_type: `SQType(T)` istype: `istype(T)` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` quotient: `x,y:A//B[x; y]`
Lemmas referenced :  strong-continuity2-no-inner-squash nat_wf int_seg_wf unit_wf2 subtype_rel_function int_seg_subtype_nat istype-void subtype_rel_self assert_wf btrue_wf bfalse_wf decidable__exists_int_seg less_than_wf decidable__and2 decidable__assert decidable__lt nat_properties le_wf it_wf istype-false int_seg_subtype sq_stable__le le_weakening2 int_seg_properties decidable__le full-omega-unsat intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf istype-int int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf true_wf imax_wf add_nat_wf add-is-int-iff itermAdd_wf intformeq_wf int_term_value_add_lemma int_formula_prop_eq_lemma false_wf decidable_wf exists_wf imax_ub intformless_wf int_formula_prop_less_lemma equal_wf squash_wf istype-universe iff_weakening_equal isl_wf subtype_base_sq bool_wf bool_subtype_base assert_elim btrue_neq_bfalse union_subtype_base set_subtype_base int_subtype_base unit_subtype_base subtype_rel_union quotient_wf all_wf equiv_rel_true quotient-member-eq member_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality Error :functionIsType,  Error :universeIsType,  hypothesis Error :inhabitedIsType,  sqequalRule Error :productIsType,  isectElimination natural_numberEquality setElimination rename Error :unionIsType,  because_Cache Error :equalityIsType1,  applyEquality independent_isectElimination independent_pairFormation Error :inlEquality_alt,  Error :isectIsType,  unionElimination equalityTransitivity equalitySymmetry independent_functionElimination productElimination instantiate Error :lambdaEquality_alt,  productEquality Error :isect_memberEquality_alt,  Error :dependent_pairFormation_alt,  functionExtensionality Error :dependent_set_memberEquality_alt,  Error :inrEquality_alt,  imageMemberEquality baseClosed imageElimination approximateComputation int_eqEquality voidElimination addEquality applyLambdaEquality pointwiseFunctionality promote_hyp baseApply closedConclusion functionEquality Error :inrFormation_alt,  universeEquality unionEquality cumulativity Error :inlFormation_alt,  intEquality pertypeElimination Error :equalityIsType4

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  m  f)  =  (inl  (F  f)))))))

Date html generated: 2019_06_20-PM-02_53_46
Last ObjectModification: 2018_10_06-PM-11_55_25

Theory : continuity

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