Nuprl Lemma : rng_before_imp_before_all

g:OCMon. ∀r:CDRng. ∀k:|g|. ∀ps:|omral(g;r)|.
  ((↑before(k;map(λz.(fst(z));ps)))  (↑(∀bx(:|g|) ∈ map(λz.(fst(z));ps). (x <b k))))


Proof




Definitions occuring in Statement :  omralist: omral(g;r) before: before(u;ps) ball: ball map: map(f;as) assert: b pi1: fst(t) all: x:A. B[x] implies:  Q lambda: λx.A[x] cdrng: CDRng grp_blt: a <b b oset_of_ocmon: g↓oset ocmon: OCMon grp_car: |g| set_car: |p|
Definitions unfolded in proof :  all: x:A. B[x] member: t ∈ T uall: [x:A]. B[x] subtype_rel: A ⊆B ocmon: OCMon omon: OMon so_lambda: λ2x.t[x] prop: and: P ∧ Q abmonoid: AbMon mon: Mon so_lambda: λ2y.t[x; y] so_apply: x[s1;s2] implies:  Q bool: 𝔹 unit: Unit it: btrue: tt band: p ∧b q ifthenelse: if then else fi  uiff: uiff(P;Q) uimplies: supposing a bfalse: ff infix_ap: y so_apply: x[s] cand: c∧ B oset_of_ocmon: g↓oset dset_of_mon: g↓set set_car: |p| pi1: fst(t) omralist: omral(g;r) grp_blt: a <b b
Lemmas referenced :  before_imp_before_all oset_of_ocmon_wf subtype_rel_sets abmonoid_wf ulinorder_wf grp_car_wf assert_wf infix_ap_wf bool_wf grp_le_wf equal_wf grp_eq_wf eqtt_to_assert cancel_wf grp_op_wf uall_wf monot_wf cdrng_wf ocmon_wf cdrng_is_abdmonoid
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin isectElimination hypothesisEquality applyEquality sqequalRule instantiate hypothesis because_Cache lambdaEquality productEquality setElimination rename cumulativity universeEquality functionEquality unionElimination equalityElimination productElimination independent_isectElimination equalityTransitivity equalitySymmetry independent_functionElimination setEquality independent_pairFormation

Latex:
\mforall{}g:OCMon.  \mforall{}r:CDRng.  \mforall{}k:|g|.  \mforall{}ps:|omral(g;r)|.
    ((\muparrow{}before(k;map(\mlambda{}z.(fst(z));ps)))  {}\mRightarrow{}  (\muparrow{}(\mforall{}\msubb{}x(:|g|)  \mmember{}  map(\mlambda{}z.(fst(z));ps).  (x  <\msubb{}  k))))



Date html generated: 2017_10_01-AM-10_04_58
Last ObjectModification: 2017_03_03-PM-01_09_49

Theory : polynom_3


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