Nuprl Lemma : flip-conjugation

`∀[n:ℕ]. ∀[k:ℕn]. ∀[j:ℕk]. ∀[l:ℕn - k].  ((j, k + l) = ((k, k + l) o ((j, k) o (k, k + l))) ∈ (ℕn ⟶ ℕn))`

Proof

Definitions occuring in Statement :  flip: `(i, j)` compose: `f o g` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` function: `x:A ⟶ B[x]` subtract: `n - m` add: `n + m` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` int_seg: `{i..j-}` flip: `(i, j)` compose: `f o g` lelt: `i ≤ j < k` and: `P ∧ Q` uimplies: `b supposing a` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` all: `∀x:A. B[x]` top: `Top` prop: `ℙ` nequal: `a ≠ b ∈ T ` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` bfalse: `ff` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` ge: `i ≥ j ` decidable: `Dec(P)`
Lemmas referenced :  int_seg_wf subtract_wf nat_wf eq_int_wf bool_wf full-omega-unsat intformand_wf intformeq_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf itermAdd_wf intformle_wf itermConstant_wf int_term_value_add_lemma int_formula_prop_le_lemma int_term_value_constant_lemma ifthenelse_wf eqtt_to_assert assert_of_eq_int eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int intformnot_wf int_formula_prop_not_lemma int_seg_properties nat_properties decidable__le decidable__lt itermSubtract_wf int_term_value_subtract_lemma lelt_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut functionExtensionality because_Cache hypothesis extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename hypothesisEquality sqequalRule isect_memberEquality axiomEquality dependent_set_memberEquality productElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination voidElimination voidEquality independent_pairFormation addEquality lambdaFormation unionElimination equalityElimination equalityTransitivity equalitySymmetry promote_hyp instantiate cumulativity

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[k:\mBbbN{}n].  \mforall{}[j:\mBbbN{}k].  \mforall{}[l:\mBbbN{}n  -  k].    ((j,  k  +  l)  =  ((k,  k  +  l)  o  ((j,  k)  o  (k,  k  +  l))))

Date html generated: 2018_05_21-PM-00_41_12
Last ObjectModification: 2018_05_19-AM-06_48_36

Theory : list_1

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