### Nuprl Lemma : permutation-generators3

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

Proof

Definitions occuring in Statement :  flip: `(i, j)` 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 :  subtype_rel: `A ⊆r B` int_seg: `{i..j-}` uimplies: `b supposing a` so_apply: `x[s]` so_lambda: `λ2x.t[x]` nat: `ℕ` prop: `ℙ` implies: `P `` Q` uall: `∀[x:A]. B[x]` member: `t ∈ T` all: `∀x:A. B[x]` true: `True` squash: `↓T` less_than: `a < b` top: `Top` exists: `∃x:A. B[x]` satisfiable_int_formula: `satisfiable_int_formula(fmla)` or: `P ∨ Q` decidable: `Dec(P)` ge: `i ≥ j ` not: `¬A` false: `False` less_than': `less_than'(a;b)` le: `A ≤ B` and: `P ∧ Q` lelt: `i ≤ j < k` guard: `{T}` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` compose-flips: `compose-flips(flips)` sq_stable: `SqStable(P)` compose: `f o g`
Lemmas referenced :  nat_wf identity-injection less_than_wf isect_wf all_wf inject_wf int_seg_wf set_wf permutation-generators2 lelt_wf int_formula_prop_wf int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf itermConstant_wf intformless_wf intformnot_wf intformand_wf full-omega-unsat decidable__lt nat_properties false_wf member-less_than iff_weakening_equal inject-compose rotate-injection rotate-as-flips equal_wf compose_wf list_wf compose-flips_wf compose-flips-injection reduce_cons_lemma map_cons_lemma reduce_nil_lemma map_nil_lemma list_induction sq_stable__inject flip-injection flip_wf comp_assoc true_wf squash_wf flip_symmetry int_formula_prop_eq_lemma intformeq_wf decidable__equal_int int_seg_properties flip_identity
Rules used in proof :  cumulativity instantiate universeEquality dependent_set_memberEquality setEquality applyEquality functionExtensionality lambdaEquality sqequalRule because_Cache rename setElimination natural_numberEquality functionEquality independent_functionElimination isectElimination isect_memberFormation hypothesisEquality thin dependent_functionElimination sqequalHypSubstitution hypothesis lambdaFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution extract_by_obid introduction cut baseClosed imageMemberEquality voidEquality voidElimination isect_memberEquality intEquality int_eqEquality dependent_pairFormation approximateComputation unionElimination independent_pairFormation independent_isectElimination equalitySymmetry equalityTransitivity applyLambdaEquality productElimination productEquality levelHypothesis hyp_replacement addLevel imageElimination

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)\}  .  \mforall{}i,j:\mBbbN{}n.    P[f]  {}\mRightarrow{}  P[f  o  (i,  j)]  supposing  i  <  j)
{}\mRightarrow{}  (\mforall{}f:\{f:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}n|  Inj(\mBbbN{}n;\mBbbN{}n;f)\}  .  P[f]))

Date html generated: 2018_05_21-PM-00_42_43
Last ObjectModification: 2017_12_10-PM-03_57_02

Theory : list_1

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