### Nuprl Lemma : permutation-generators2

`∀n:ℕ`
`  ∀[P:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)}  ⟶ ℙ]`
`    (P[λx.x]`
`    `` ∀f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} . (P[f] `` P[f o (0, 1)]) supposing 1 < n`
`    `` (∀f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} . (P[f] `` P[f o rot(n)]))`
`    `` (∀f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} . P[f]))`

Proof

Definitions occuring in Statement :  flip: `(i, j)` rotate: `rot(n)` inject: `Inj(A;B;f)` compose: `f o g` int_seg: `{i..j-}` nat: `ℕ` less_than: `a < b` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s]` all: `∀x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` natural_number: `\$n`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` nat: `ℕ` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` implies: `P `` Q` uimplies: `b supposing a` compose: `f o g` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` le: `A ≤ B` less_than': `less_than'(a;b)` false: `False` not: `¬A` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` squash: `↓T` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` sq_type: `SQType(T)` nat_plus: `ℕ+` uiff: `uiff(P;Q)` subtract: `n - m`
Lemmas referenced :  permutation-generators identity-injection int_seg_wf inject_wf funinv_wf2 nat_wf funinv-unique isect_wf less_than_wf all_wf compose-injections flip-injection false_wf nat_properties decidable__lt satisfiable-full-omega-tt intformand_wf intformnot_wf intformless_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf lelt_wf flip_wf member-less_than set_wf equal_wf squash_wf true_wf funinv-compose iff_weakening_equal flip_inverse rotate-injection rotate_wf decidable__equal_int subtype_base_sq int_subtype_base subtype_rel_sets subtype_rel_set subtype_rel_dep_function subtype_rel_self subtype_rel_wf int_seg_properties intformle_wf intformeq_wf int_formula_prop_le_lemma int_formula_prop_eq_lemma subtract_wf decidable__le itermSubtract_wf int_term_value_subtract_lemma le_wf fun_exp0_lemma primrec-wf2 inject-compose fun_exp_wf fun_exp_add_apply1 subtract-add-cancel fun_exp-injection rotate-inverse not-lt-2 not-equal-2 add_functionality_wrt_le add-associates add-zero zero-add le-add-cancel condition-implies-le add-commutes minus-add minus-zero funinv-funinv
Rules used in proof :  cut introduction extract_by_obid sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality isectElimination natural_numberEquality setElimination rename dependent_set_memberEquality lambdaEquality because_Cache functionExtensionality applyEquality isect_memberFormation sqequalRule setEquality functionEquality independent_functionElimination cumulativity universeEquality addLevel hyp_replacement equalitySymmetry levelHypothesis equalityTransitivity independent_isectElimination independent_pairFormation unionElimination dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll imageElimination imageMemberEquality baseClosed productElimination applyLambdaEquality instantiate addEquality minusEquality

Latex:
\mforall{}n:\mBbbN{}
\mforall{}[P:\{f:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}n|  Inj(\mBbbN{}n;\mBbbN{}n;f)\}    {}\mrightarrow{}  \mBbbP{}]
(P[\mlambda{}x.x]
{}\mRightarrow{}  \mforall{}f:\{f:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}n|  Inj(\mBbbN{}n;\mBbbN{}n;f)\}  .  (P[f]  {}\mRightarrow{}  P[f  o  (0,  1)])  supposing  1  <  n
{}\mRightarrow{}  (\mforall{}f:\{f:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}n|  Inj(\mBbbN{}n;\mBbbN{}n;f)\}  .  (P[f]  {}\mRightarrow{}  P[f  o  rot(n)]))
{}\mRightarrow{}  (\mforall{}f:\{f:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}n|  Inj(\mBbbN{}n;\mBbbN{}n;f)\}  .  P[f]))

Date html generated: 2017_04_17-AM-08_22_14
Last ObjectModification: 2017_02_27-PM-04_45_54

Theory : list_1

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