`∀[n:ℕ]. ∀[s:ℕn ⟶ ℕ]. ∀[m:ℕ].  (s.m@n = s++m ∈ (ℕn + 1 ⟶ ℕ))`

Proof

Definitions occuring in Statement :  seq-adjoin: `s++t` seq-add: `s.x@n` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` seq-adjoin: `s++t` seq-add: `s.x@n` seq-append: `seq-append(n;m;s1;s2)` int_seg: `{i..j-}` nat: `ℕ` all: `∀x:A. B[x]` implies: `P `` Q` 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` guard: `{T}` ge: `i ≥ j ` lelt: `i ≤ j < k` subtype_rel: `A ⊆r B` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` nequal: `a ≠ b ∈ T `
Lemmas referenced :  eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int lt_int_wf assert_of_lt_int top_wf less_than_wf int_seg_properties nat_properties decidable__equal_int int_seg_wf lelt_wf full-omega-unsat intformand_wf intformless_wf itermVar_wf intformeq_wf int_formula_prop_and_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_wf le_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot intformnot_wf itermAdd_wf itermConstant_wf int_formula_prop_not_lemma int_term_value_add_lemma int_term_value_constant_lemma neg_assert_of_eq_int nat_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule lambdaEquality extract_by_obid sqequalHypSubstitution isectElimination thin setElimination rename hypothesisEquality hypothesis because_Cache lambdaFormation unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination independent_isectElimination int_eqReduceTrueSq lessCases axiomSqEquality isect_memberEquality independent_pairFormation voidElimination voidEquality natural_numberEquality imageMemberEquality baseClosed imageElimination independent_functionElimination addEquality dependent_functionElimination applyEquality functionExtensionality dependent_set_memberEquality approximateComputation dependent_pairFormation int_eqEquality intEquality promote_hyp instantiate cumulativity int_eqReduceFalseSq axiomEquality functionEquality

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[s:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}].  \mforall{}[m:\mBbbN{}].    (s.m@n  =  s++m)

Date html generated: 2019_06_20-PM-02_54_48
Last ObjectModification: 2018_08_20-PM-09_34_23

Theory : continuity

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