### Nuprl Lemma : unsquashed-BIM-implies-unsquashed-weak-continuity-old

`(∀B,Q:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ ℙ.`
`   ((∀n:ℕ. ∀s:ℕn ⟶ ℕ.  ((∀m:ℕ. Q[n + 1;s.m@n]) `` Q[n;s]))`
`   `` (∀f:ℕ ⟶ ℕ. ⇃(∃n:ℕ. Q[n;f]))`
`   `` (∀n:ℕ. ∀s:ℕn ⟶ ℕ. ∀m:ℕ.  (B[n;s] `` B[n + 1;s.m@n]))`
`   `` (∀n:ℕ. ∀s:ℕn ⟶ ℕ.  (B[n;s] `` Q[n;s]))`
`   `` Q[0;λx.⊥]))`
` (∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ∀a:ℕ ⟶ ℕ.  ∃n:ℕ. ∀b:ℕ ⟶ ℕ. ((∀i:ℕn. ((a i) = (b i) ∈ ℕ)) `` ((F a) = (F b) ∈ ℕ)))`

Proof

Definitions occuring in Statement :  quotient: `x,y:A//B[x; y]` seq-add: `s.x@n` int_seg: `{i..j-}` nat: `ℕ` bottom: `⊥` prop: `ℙ` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` true: `True` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  implies: `P `` Q` all: `∀x:A. B[x]` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` subtype_rel: `A ⊆r B` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s1;s2]` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` and: `P ∧ Q` so_apply: `x[s]` le: `A ≤ B` less_than': `less_than'(a;b)` so_lambda: `λ2x y.t[x; y]` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` int_upper: `{i...}` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` less_than: `a < b` seq-add: `s.x@n` rep-seq-from: `rep-seq-from(s;n;f)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` nequal: `a ≠ b ∈ T `
Lemmas referenced :  nat_wf all_wf int_seg_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermAdd_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_add_lemma int_term_value_var_lemma int_formula_prop_wf le_wf seq-add_wf quotient_wf exists_wf subtype_rel_dep_function int_seg_subtype_nat false_wf subtype_rel_self true_wf equiv_rel_true int_seg_properties intformless_wf int_formula_prop_less_lemma equal_wf rep-seq-from_wf int_upper_wf int_upper_properties int_upper_subtype_int_upper rep-seq-from-prop3 squash_wf iff_weakening_equal strong-continuity2-implies-weak implies-quotient-true rep-seq-from-prop1 decidable__lt lelt_wf lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_wf eq_int_wf assert_of_eq_int decidable__equal_int intformeq_wf int_formula_prop_eq_lemma eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int rep-seq-from-0
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation functionEquality cut introduction extract_by_obid hypothesis instantiate sqequalHypSubstitution isectElimination thin applyEquality lambdaEquality cumulativity hypothesisEquality universeEquality sqequalRule natural_numberEquality setElimination rename because_Cache functionExtensionality dependent_set_memberEquality addEquality dependent_functionElimination unionElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll productElimination independent_functionElimination applyLambdaEquality hyp_replacement equalityTransitivity equalitySymmetry imageElimination imageMemberEquality baseClosed equalityElimination lessCases isect_memberFormation sqequalAxiom int_eqReduceTrueSq promote_hyp int_eqReduceFalseSq

Latex:
(\mforall{}B,Q:n:\mBbbN{}  {}\mrightarrow{}  (\mBbbN{}n  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbP{}.
((\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.    ((\mforall{}m:\mBbbN{}.  Q[n  +  1;s.m@n])  {}\mRightarrow{}  Q[n;s]))
{}\mRightarrow{}  (\mforall{}f:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  \00D9(\mexists{}n:\mBbbN{}.  Q[n;f]))
{}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.  \mforall{}m:\mBbbN{}.    (B[n;s]  {}\mRightarrow{}  B[n  +  1;s.m@n]))
{}\mRightarrow{}  (\mforall{}n:\mBbbN{}.  \mforall{}s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}.    (B[n;s]  {}\mRightarrow{}  Q[n;s]))
{}\mRightarrow{}  Q[0;\mlambda{}x.\mbot{}]))
{}\mRightarrow{}  (\mforall{}F:(\mBbbN{}  {}\mrightarrow{}  \mBbbN{})  {}\mrightarrow{}  \mBbbN{}.  \mforall{}a:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.    \mexists{}n:\mBbbN{}.  \mforall{}b:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}.  ((\mforall{}i:\mBbbN{}n.  ((a  i)  =  (b  i)))  {}\mRightarrow{}  ((F  a)  =  (F  b))))

Date html generated: 2017_04_20-AM-07_21_38
Last ObjectModification: 2017_02_27-PM-05_58_04

Theory : continuity

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