### Nuprl Lemma : replace-seq-from-succ

`∀T:Type. ∀f:ℕ ⟶ T. ∀m:ℕ. ∀k:T.`
`  (0 < m `` (replace-seq-from(f;m;k) = (λx.if x=m - 1  then f x  else (replace-seq-from(f;m - 1;k) x)) ∈ (ℕ ⟶ T)))`

Proof

Definitions occuring in Statement :  replace-seq-from: `replace-seq-from(s;n;k)` nat: `ℕ` less_than: `a < b` all: `∀x:A. B[x]` implies: `P `` Q` int_eq: `if a=b  then c  else d` apply: `f a` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` subtract: `n - m` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` replace-seq-from: `replace-seq-from(s;n;k)` member: `t ∈ T` uall: `∀[x:A]. B[x]` nat: `ℕ` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` less_than: `a < b` less_than': `less_than'(a;b)` top: `Top` true: `True` squash: `↓T` not: `¬A` false: `False` prop: `ℙ` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` nequal: `a ≠ b ∈ T ` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)`
Lemmas referenced :  lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int top_wf less_than_wf eq_int_wf subtract_wf assert_of_eq_int nat_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int nat_properties satisfiable-full-omega-tt intformand_wf intformless_wf itermVar_wf intformnot_wf intformeq_wf itermSubtract_wf itermConstant_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_subtract_lemma int_term_value_constant_lemma int_formula_prop_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut sqequalRule functionExtensionality introduction extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis because_Cache unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination lessCases isect_memberFormation sqequalAxiom isect_memberEquality independent_pairFormation voidElimination voidEquality natural_numberEquality imageMemberEquality baseClosed imageElimination independent_functionElimination int_eqReduceTrueSq applyEquality dependent_pairFormation promote_hyp dependent_functionElimination instantiate cumulativity int_eqReduceFalseSq lambdaEquality int_eqEquality intEquality computeAll functionEquality universeEquality

Latex:
\mforall{}T:Type.  \mforall{}f:\mBbbN{}  {}\mrightarrow{}  T.  \mforall{}m:\mBbbN{}.  \mforall{}k:T.
(0  <  m
{}\mRightarrow{}  (replace-seq-from(f;m;k)  =  (\mlambda{}x.if  x=m  -  1    then  f  x    else  (replace-seq-from(f;m  -  1;k)  x))))

Date html generated: 2017_04_20-AM-07_22_51
Last ObjectModification: 2017_02_27-PM-05_58_17

Theory : continuity

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