Nuprl Lemma : cardinality-le-int_seg

`∀[x,y:ℤ]. ∀[n:ℕ].  (y - x) ≤ n supposing |{x..y-}| ≤ n`

Proof

Definitions occuring in Statement :  cardinality-le: `|T| ≤ n` int_seg: `{i..j-}` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` le: `A ≤ B` subtract: `n - m` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` cardinality-le: `|T| ≤ n` exists: `∃x:A. B[x]` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` nat: `ℕ` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` implies: `P `` Q` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` le: `A ≤ B` iff: `P `⇐⇒` Q` int_seg: `{i..j-}` uiff: `uiff(P;Q)` lelt: `i ≤ j < k` inject: `Inj(A;B;f)` guard: `{T}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` subtype_rel: `A ⊆r B` squash: `↓T` true: `True`
Lemmas referenced :  decidable__lt nat_properties decidable__le subtract_wf satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermSubtract_wf itermVar_wf intformless_wf itermConstant_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_term_value_constant_lemma int_formula_prop_wf less_than'_wf cardinality-le_wf int_seg_wf nat_wf surject-inverse pigeon-hole le_wf add-member-int_seg1 lelt_wf equal_wf int_seg_properties subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int_seg itermAdd_wf int_term_value_add_lemma intformeq_wf int_formula_prop_eq_lemma decidable__equal_int squash_wf true_wf iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalHypSubstitution productElimination thin extract_by_obid dependent_functionElimination hypothesisEquality hypothesis unionElimination isectElimination setElimination rename natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_pairEquality because_Cache axiomEquality equalityTransitivity equalitySymmetry independent_functionElimination dependent_set_memberEquality applyEquality functionExtensionality lambdaFormation promote_hyp instantiate cumulativity addEquality imageElimination universeEquality applyLambdaEquality imageMemberEquality baseClosed

Latex:
\mforall{}[x,y:\mBbbZ{}].  \mforall{}[n:\mBbbN{}].    (y  -  x)  \mleq{}  n  supposing  |\{x..y\msupminus{}\}|  \mleq{}  n

Date html generated: 2017_04_17-AM-07_46_22
Last ObjectModification: 2017_02_27-PM-04_18_36

Theory : list_1

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