Nuprl Lemma : equal-rat-cube-complexes

∀k:ℕ
  ∀[n:ℕ]
    ∀K,L:n-dim-complex.
      uiff(permutation(ℚCube(k);K;L);∀c:{c:ℚCube(k)| (↑Inhabited(c)) ∧ (dim(c) = n ∈ ℤ)} . ((c ∈ K) ⇐⇒ (c ∈ L)))


Proof



Definitions occuring in Statement :  rational-cube-complex: n-dim-complex,  rat-cube-dimension: dim(c),  inhabited-rat-cube: Inhabited(c),  rational-cube: ℚCube(k),  permutation: permutation(T;L1;L2),  l_member: (x ∈ l),  nat: ℕ,  assert: ↑b,  uiff: uiff(P;Q),  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x],  iff: P ⇐⇒ Q,  and: P ∧ Q,  set: {x:A| B[x]} ,  int: ℤ,  equal: s = t ∈ T
Definitions unfolded in proof :  top: Top,  false: False,  exists: ∃x:A. B[x],  satisfiable_int_formula: satisfiable_int_formula(fmla),  not: ¬A,  ge: i ≥ j ,  bfalse: ff,  btrue: tt,  ifthenelse: if b then t else f fi ,  sq_type: SQType(T),  or: P ∨ Q,  rat-cube-dimension: dim(c),  guard: {T},  squash: ↓T,  sq_stable: SqStable(P),  so_apply: x[s],  nat: ℕ,  so_lambda: λ2x.t[x],  int_seg: {i..j-},  subtype_rel: A ⊆r B,  rev_implies: P ⇐ Q,  prop: ℙ,  rational-cube-complex: n-dim-complex,  member: t ∈ T,  implies: P ⇒ Q,  iff: P ⇐⇒ Q,  uimplies: b supposing a,  and: P ∧ Q,  uiff: uiff(P;Q),  uall: ∀[x:A]. B[x],  all: ∀x:A. B[x]
Lemmas referenced :  sq_stable__no_repeats,  int_formula_prop_wf,  int_formula_prop_le_lemma,  int_term_value_var_lemma,  int_term_value_constant_lemma,  int_formula_prop_eq_lemma,  istype-void,  int_formula_prop_and_lemma,  intformle_wf,  itermVar_wf,  itermConstant_wf,  intformeq_wf,  intformand_wf,  full-omega-unsat,  nat_properties,  assert_of_bnot,  eqff_to_assert,  eqtt_to_assert,  bool_subtype_base,  bool_wf,  subtype_base_sq,  bool_cases,  l_all_iff,  sq_stable__iff,  sq_stable__all,  iff_wf,  equal-wf-base,  assert_wf,  permutation-when-no_repeats,  permutation_inversion,  l_member_functionality_wrt_permutation,  decidable__equal_rc,  sq_stable__l_member,  istype-nat,  rational-cube-complex_wf,  permutation_wf,  le_wf,  int_subtype_base,  lelt_wf,  set_subtype_base,  rat-cube-dimension_wf,  istype-int,  inhabited-rat-cube_wf,  istype-assert,  rational-cube_wf,  l_member_wf
Rules used in proof :  voidElimination,  isect_memberEquality_alt,  int_eqEquality,  dependent_pairFormation_alt,  approximateComputation,  dependent_set_memberEquality_alt,  equalityTransitivity,  cumulativity,  instantiate,  unionElimination,  productElimination,  closedConclusion,  baseApply,  productEquality,  setEquality,  imageElimination,  baseClosed,  imageMemberEquality,  independent_functionElimination,  dependent_functionElimination,  inhabitedIsType,  functionIsType,  equalitySymmetry,  sqequalBase,  independent_isectElimination,  addEquality,  natural_numberEquality,  minusEquality,  lambdaEquality_alt,  intEquality,  applyEquality,  equalityIstype,  productIsType,  sqequalRule,  setIsType,  because_Cache,  rename,  setElimination,  hypothesis,  hypothesisEquality,  thin,  isectElimination,  sqequalHypSubstitution,  extract_by_obid,  introduction,  cut,  universeIsType,  independent_pairFormation,  isect_memberFormation_alt,  lambdaFormation_alt,  sqequalReflexivity,  computationStep,  sqequalTransitivity,  sqequalSubstitution
Latex:
\mforall{}k:\mBbbN{}
    \mforall{}[n:\mBbbN{}]
        \mforall{}K,L:n-dim-complex.
            uiff(permutation(\mBbbQ{}Cube(k);K;L);\mforall{}c:\{c:\mBbbQ{}Cube(k)|  (\muparrow{}Inhabited(c))  \mwedge{}  (dim(c)  =  n)\} 
                                                                              ((c  \mmember{}  K)  \mLeftarrow{}{}\mRightarrow{}  (c  \mmember{}  L)))


Date html generated: 2019_10_29-AM-07_59_51
Last ObjectModification: 2019_10_22-AM-10_23_51

Theory : rationals


Home Index