### Nuprl Lemma : type-monotone_fun_exp_top

`∀[F:Type ⟶ Type]. ∀[n,m:ℕ].  (F^n Top) ⊆r (F^m Top) supposing m ≤ n supposing Monotone(T.F[T])`

Proof

Definitions occuring in Statement :  type-monotone: `Monotone(T.F[T])` fun_exp: `f^n` nat: `ℕ` uimplies: `b supposing a` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` top: `Top` so_apply: `x[s]` le: `A ≤ B` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` nat: `ℕ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` le: `A ≤ B` and: `P ∧ Q` subtract: `n - m` subtype_rel: `A ⊆r B` top: `Top` prop: `ℙ` sq_type: `SQType(T)` all: `∀x:A. B[x]` implies: `P `` Q` guard: `{T}` squash: `↓T` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` uiff: `uiff(P;Q)` nat_plus: `ℕ+` less_than: `a < b` less_than': `less_than'(a;b)` not: `¬A` false: `False` decidable: `Dec(P)` or: `P ∨ Q` ge: `i ≥ j ` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` bnot: `¬bb` assert: `↑b` compose: `f o g` type-monotone: `Monotone(T.F[T])` nequal: `a ≠ b ∈ T `
Lemmas referenced :  subtype_base_sq nat_wf set_subtype_base le_wf int_subtype_base minus-one-mul add-commutes minus-one-mul-top add-associates add-mul-special zero-mul zero-add subtract_wf subtype_rel_wf squash_wf true_wf fun_exp_add_apply top_wf fun_exp_wf iff_weakening_equal equal_wf type-monotone_wf add_functionality_wrt_le le_reflexive one-mul two-mul mul-distributes-right add-zero not-le-2 minus-zero add-swap omega-shadow less_than_wf mul-distributes mul-associates mul-commutes le-add-cancel-alt nat_properties decidable__le less_than_transitivity1 less_than_irreflexivity ge_wf fun_exp0_lemma false_wf not-ge-2 less-iff-le condition-implies-le minus-add minus-minus le-add-cancel le_weakening2 eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int eqff_to_assert bool_cases_sqequal bool_subtype_base assert-bnot neg_assert_of_eq_int not-equal-2 fun_exp_unroll
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin instantiate extract_by_obid sqequalHypSubstitution isectElimination cumulativity hypothesis independent_isectElimination sqequalRule intEquality lambdaEquality natural_numberEquality hypothesisEquality setElimination rename dependent_set_memberEquality productElimination applyEquality isect_memberEquality voidElimination voidEquality minusEquality because_Cache dependent_functionElimination equalityTransitivity equalitySymmetry independent_functionElimination lambdaFormation imageElimination universeEquality functionExtensionality imageMemberEquality baseClosed axiomEquality functionEquality multiplyEquality addEquality independent_pairFormation unionElimination intWeakElimination equalityElimination dependent_pairFormation promote_hyp

Latex:
\mforall{}[F:Type  {}\mrightarrow{}  Type].  \mforall{}[n,m:\mBbbN{}].    (F\^{}n  Top)  \msubseteq{}r  (F\^{}m  Top)  supposing  m  \mleq{}  n  supposing  Monotone(T.F[T])

Date html generated: 2017_04_14-AM-07_37_25
Last ObjectModification: 2017_02_27-PM-03_09_44

Theory : subtype_1

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