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: m natural_number: $n equal: 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 ⊆B guard: {T} uimplies: supposing a le: A ≤ B and: P ∧ Q less_than': less_than'(a;b) false: False not: ¬A implies:  Q prop: int_seg: {i..j-} nat: 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: g swap: swap(i;j) ifthenelse: if then else fi  eq_int: (i =z j) subtract: 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: ⇐⇒ Q rev_implies:  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


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