Nuprl Lemma : upper-rc-face-dimension

k:ℕ. ∀c:ℚCube(k). ∀j:ℕk.
  ((↑Inhabited(c))  (dim(upper-rc-face(c;j)) if (dim(c j) =z 0) then dim(c) else dim(c) fi  ∈ ℤ))


Proof




Definitions occuring in Statement :  upper-rc-face: upper-rc-face(c;j) rat-cube-dimension: dim(c) inhabited-rat-cube: Inhabited(c) rational-cube: Cube(k) rat-interval-dimension: dim(I) int_seg: {i..j-} nat: assert: b ifthenelse: if then else fi  eq_int: (i =z j) all: x:A. B[x] implies:  Q apply: a subtract: m natural_number: $n int: equal: t ∈ T
Definitions unfolded in proof :  inhabited-rat-interval: Inhabited(I) rat-interval-dimension: dim(I) rev_implies:  Q iff: ⇐⇒ Q nequal: a ≠ b ∈  prop: top: Top satisfiable_int_formula: satisfiable_int_formula(fmla) decidable: Dec(P) ge: i ≥  pi2: snd(t) rational-interval: Interval not: ¬A less_than': less_than'(a;b) upper-rc-face: upper-rc-face(c;j) true: True so_apply: x[s] squash: T less_than: a < b so_lambda: λ2x.t[x] nat: false: False assert: b bnot: ¬bb guard: {T} sq_type: SQType(T) or: P ∨ Q exists: x:A. B[x] bfalse: ff rat-cube-dimension: dim(c) ifthenelse: if then else fi  btrue: tt it: unit: Unit bool: 𝔹 le: A ≤ B lelt: i ≤ j < k int_seg: {i..j-} subtype_rel: A ⊆B rational-cube: Cube(k) uimplies: supposing a and: P ∧ Q uiff: uiff(P;Q) member: t ∈ T uall: [x:A]. B[x] implies:  Q all: x:A. B[x]
Lemmas referenced :  int_term_value_subtract_lemma itermSubtract_wf decidable__equal_int general_arith_equation1 int_term_value_constant_lemma itermConstant_wf qless_wf nequal_wf q_le_wf assert-q_less-eq q_less_wf istype-nat rat-interval-dimension-single iff_weakening_equal inhabited-upper-rc-face istype-universe true_wf squash_wf equal_wf subtract_wf int_formula_prop_eq_lemma intformeq_wf int_formula_prop_wf int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma istype-void int_formula_prop_and_lemma istype-int itermVar_wf intformless_wf intformnot_wf intformand_wf full-omega-unsat decidable__lt nat_properties int_seg_properties rat-point-interval_wf istype-false int_seg_subtype_nat Error :isolate_summand2,  ifthenelse_wf rational-interval_wf subtype_rel_self upper-rc-face_wf sum_wf int_seg_wf inhabited-rat-cube_wf istype-assert neg_assert_of_eq_int assert-bnot bool_subtype_base bool_wf subtype_base_sq bool_cases_sqequal eqff_to_assert assert_of_eq_int eqtt_to_assert rat-interval-dimension_wf eq_int_wf assert-inhabited-rat-cube
Rules used in proof :  functionIsType addEquality baseClosed imageMemberEquality universeEquality isect_memberEquality_alt int_eqEquality approximateComputation independent_pairFormation minusEquality imageElimination functionEquality lambdaEquality_alt intEquality natural_numberEquality universeIsType voidElimination independent_functionElimination cumulativity instantiate dependent_functionElimination promote_hyp equalityIstype dependent_pairFormation_alt equalitySymmetry equalityTransitivity equalityElimination unionElimination inhabitedIsType rename setElimination sqequalRule because_Cache applyEquality independent_isectElimination productElimination hypothesis hypothesisEquality thin isectElimination sqequalHypSubstitution extract_by_obid introduction cut lambdaFormation_alt sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}k:\mBbbN{}.  \mforall{}c:\mBbbQ{}Cube(k).  \mforall{}j:\mBbbN{}k.
    ((\muparrow{}Inhabited(c))
    {}\mRightarrow{}  (dim(upper-rc-face(c;j))  =  if  (dim(c  j)  =\msubz{}  0)  then  dim(c)  else  dim(c)  -  1  fi  ))



Date html generated: 2019_10_29-AM-07_57_00
Last ObjectModification: 2019_10_17-PM-05_16_18

Theory : rationals


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