### Nuprl Lemma : restrict_perm_wf

`∀n:ℕ. ∀p:Sym(n + 1).  (((p.f n) = n ∈ ℕn + 1) `` (restrict_perm(p;n) ∈ Sym(n)))`

Proof

Definitions occuring in Statement :  restrict_perm: `restrict_perm(p;n)` sym_grp: `Sym(n)` perm_f: `p.f` int_seg: `{i..j-}` nat: `ℕ` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` apply: `f a` add: `n + m` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  sym_grp: `Sym(n)` all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` uall: `∀[x:A]. B[x]` nat: `ℕ` perm: `Perm(T)` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` top: `Top` prop: `ℙ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` so_apply: `x[s]` restrict_perm: `restrict_perm(p;n)` ge: `i ≥ j ` biject: `Bij(A;B;f)` inject: `Inj(A;B;f)` guard: `{T}` inv_funs: `InvFuns(A;B;f;g)` compose: `f o g` tidentity: `Id{T}` identity: `Id` sq_type: `SQType(T)` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` perm_sig: `perm_sig(T)` perm_f: `p.f` pi1: `fst(t)` perm_b: `p.b` pi2: `snd(t)` le: `A ≤ B` less_than': `less_than'(a;b)` rev_implies: `P `` Q` uiff: `uiff(P;Q)` subtract: `n - m`
Lemmas referenced :  int_seg_wf perm_f_wf decidable__lt full-omega-unsat intformnot_wf intformless_wf itermVar_wf itermAdd_wf itermConstant_wf istype-int int_formula_prop_not_lemma istype-void int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_constant_lemma int_formula_prop_wf le_wf less_than_wf set_subtype_base lelt_wf int_subtype_base perm_wf nat_wf nat_properties intformand_wf int_formula_prop_and_lemma perm_properties perm_b_wf fun_with_inv_is_bij int_seg_properties decidable__equal_int intformeq_wf int_formula_prop_eq_lemma decidable__le intformle_wf int_formula_prop_le_lemma not_wf equal_wf subtype_base_sq squash_wf true_wf istype-universe subtype_rel_self iff_weakening_equal subtype_rel_dep_function int_seg_subtype istype-false not-le-2 condition-implies-le minus-add minus-one-mul add-swap minus-one-mul-top add-mul-special zero-mul add-zero add-associates add-commutes le-add-cancel inv_funs_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep lambdaFormation_alt cut sqequalHypSubstitution hypothesis equalityIsType4 universeIsType introduction extract_by_obid isectElimination thin natural_numberEquality addEquality setElimination rename hypothesisEquality applyEquality dependent_functionElimination because_Cache dependent_set_memberEquality_alt independent_pairFormation unionElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation_alt lambdaEquality_alt int_eqEquality isect_memberEquality_alt voidElimination productIsType intEquality baseApply closedConclusion baseClosed functionExtensionality_alt productElimination equalityTransitivity equalitySymmetry applyLambdaEquality inhabitedIsType equalityIsType1 instantiate cumulativity imageElimination universeEquality imageMemberEquality dependent_pairEquality_alt minusEquality multiplyEquality

Latex:
\mforall{}n:\mBbbN{}.  \mforall{}p:Sym(n  +  1).    (((p.f  n)  =  n)  {}\mRightarrow{}  (restrict\_perm(p;n)  \mmember{}  Sym(n)))

Date html generated: 2019_10_16-PM-01_00_13
Last ObjectModification: 2018_10_08-AM-09_12_46

Theory : perms_1

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