### Nuprl Lemma : restrict_perm_using_txpose

`∀n:{1...}. ∀p:Sym(n).  ∃q:Sym(n - 1). ∃i,j:ℕn. (p = txpose_perm(i;j) O ↑{n - 1}(q) ∈ Sym(n))`

Proof

Definitions occuring in Statement :  extend_perm: `↑{n}(p)` txpose_perm: txpose_perm sym_grp: `Sym(n)` comp_perm: comp_perm int_upper: `{i...}` int_seg: `{i..j-}` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` subtract: `n - m` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` sym_grp: `Sym(n)` uall: `∀[x:A]. B[x]` int_upper: `{i...}` exists: `∃x:A. B[x]` subtype_rel: `A ⊆r B` guard: `{T}` uimplies: `b supposing a` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` int_seg: `{i..j-}` nat: `ℕ` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` comp_perm: comp_perm mk_perm: `mk_perm(f;b)` perm_f: `p.f` pi1: `fst(t)` txpose_perm: txpose_perm compose: `f o g` swap: `swap(i;j)` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` subtract: `n - m` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` perm: `Perm(T)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` bfalse: `ff` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` true: `True` squash: `↓T`
Lemmas referenced :  perm_wf int_seg_wf int_upper_wf restrict_perm_wf int_seg_subtype_nat false_wf comp_perm_wf txpose_perm_wf subtract_wf subtract-add-cancel int_seg_properties int_upper_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf le_wf subtype_base_sq set_subtype_base int_subtype_base add-associates add-swap add-commutes zero-add perm_f_wf decidable__equal_int intformeq_wf itermSubtract_wf int_formula_prop_eq_lemma int_term_value_subtract_lemma decidable__lt lelt_wf equal_wf int_upper_subtype_nat extend_perm_wf subtype_rel_self exists_wf eq_int_wf bool_wf uiff_transitivity equal-wf-T-base assert_wf eqtt_to_assert assert_of_eq_int iff_transitivity bnot_wf not_wf iff_weakening_uiff eqff_to_assert assert_of_bnot squash_wf true_wf extend_restrict_perm_cancel iff_weakening_equal perm_assoc txpose_perm_order_2 perm_ident
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin isectElimination natural_numberEquality setElimination rename hypothesisEquality hypothesis dependent_pairFormation equalityTransitivity equalitySymmetry applyEquality independent_isectElimination sqequalRule independent_pairFormation addEquality lambdaEquality because_Cache dependent_set_memberEquality applyLambdaEquality productElimination unionElimination int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll instantiate cumulativity independent_functionElimination equalityElimination baseClosed impliesFunctionality imageElimination universeEquality imageMemberEquality

Latex:
\mforall{}n:\{1...\}.  \mforall{}p:Sym(n).    \mexists{}q:Sym(n  -  1).  \mexists{}i,j:\mBbbN{}n.  (p  =  txpose\_perm(i;j)  O  \muparrow{}\{n  -  1\}(q))

Date html generated: 2017_10_01-AM-09_53_36
Last ObjectModification: 2017_03_03-PM-00_48_33

Theory : perms_1

Home Index