### Nuprl Lemma : omral_scale_dom_bound

`∀g:OCMon. ∀r:CDRng. ∀bound,k:|g|. ∀v:|r|. ∀ps:(|g| × |r|) List.`
`  ((↑(∀bx(:|g|) ∈ map(λz.(fst(z));ps)`
`         (x <b bound)))`
`  `` (↑(∀bx(:|g|) ∈ map(λz.(fst(z));<k,v>* ps)`
`           (x <b (k * bound)))))`

Proof

Definitions occuring in Statement :  omral_scale: `<k,v>* ps` ball: ball map: `map(f;as)` list: `T List` assert: `↑b` infix_ap: `x f y` pi1: `fst(t)` all: `∀x:A. B[x]` implies: `P `` Q` lambda: `λx.A[x]` product: `x:A × B[x]` cdrng: `CDRng` rng_car: `|r|` grp_blt: `a <b b` ocmon: `OCMon` grp_op: `*` grp_car: `|g|`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` so_lambda: `λ2x.t[x]` uall: `∀[x:A]. B[x]` ocmon: `OCMon` abmonoid: `AbMon` mon: `Mon` infix_ap: `x f y` so_apply: `x[s]` prop: `ℙ` cdrng: `CDRng` crng: `CRng` rng: `Rng` pi1: `fst(t)` oset_of_ocmon: `g↓oset` dset_of_mon: `g↓set` set_car: `|p|` subtype_rel: `A ⊆r B` omon: `OMon` and: `P ∧ Q` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` band: `p ∧b q` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` uimplies: `b supposing a` bfalse: `ff` cand: `A c∧ B` squash: `↓T` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  omral_scale_dom_pred grp_blt_wf grp_op_wf grp_car_wf assert_wf ball_wf map_wf rng_car_wf list_wf cdrng_wf ocmon_wf iff_transitivity infix_ap_wf all_wf mem_wf oset_of_ocmon_wf subtype_rel_sets abmonoid_wf ulinorder_wf bool_wf grp_le_wf equal_wf grp_eq_wf eqtt_to_assert cancel_wf uall_wf monot_wf iff_weakening_uiff assert_functionality_wrt_uiff squash_wf true_wf ball_char set_car_wf oset_of_ocmon_wf0 pi1_wf assert_of_grp_blt grp_op_preserves_lt
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality sqequalRule lambdaEquality isectElimination setElimination rename hypothesis applyEquality because_Cache independent_functionElimination productEquality productElimination equalityTransitivity equalitySymmetry functionEquality instantiate cumulativity universeEquality unionElimination equalityElimination independent_isectElimination setEquality independent_pairFormation imageElimination natural_numberEquality imageMemberEquality baseClosed independent_pairEquality

Latex:
\mforall{}g:OCMon.  \mforall{}r:CDRng.  \mforall{}bound,k:|g|.  \mforall{}v:|r|.  \mforall{}ps:(|g|  \mtimes{}  |r|)  List.
((\muparrow{}(\mforall{}\msubb{}x(:|g|)  \mmember{}  map(\mlambda{}z.(fst(z));ps)
(x  <\msubb{}  bound)))
{}\mRightarrow{}  (\muparrow{}(\mforall{}\msubb{}x(:|g|)  \mmember{}  map(\mlambda{}z.(fst(z));<k,v>*  ps)
(x  <\msubb{}  (k  *  bound)))))

Date html generated: 2017_10_01-AM-10_05_32
Last ObjectModification: 2017_03_03-PM-01_11_56

Theory : polynom_3

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