### Nuprl Lemma : rel_exp_monotone

`∀n:ℕ. ∀[T:Type]. ∀[R1,R2:T ⟶ T ⟶ ℙ].  (R1 => R2 `` R1^n => R2^n)`

Proof

Definitions occuring in Statement :  rel_implies: `R1 => R2` rel_exp: `R^n` nat: `ℕ` uall: `∀[x:A]. B[x]` prop: `ℙ` all: `∀x:A. B[x]` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` subtype_rel: `A ⊆r B` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` and: `P ∧ Q` not: `¬A` rev_implies: `P `` Q` false: `False` uiff: `uiff(P;Q)` uimplies: `b supposing a` subtract: `n - m` top: `Top` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True` so_apply: `x[s]` rel_implies: `R1 => R2` rel_exp: `R^n` eq_int: `(i =z j)` ifthenelse: `if b then t else f fi ` btrue: `tt` infix_ap: `x f y` guard: `{T}` exists: `∃x:A. B[x]` cand: `A c∧ B` bool: `𝔹` unit: `Unit` it: `⋅` bfalse: `ff`
Lemmas referenced :  uall_wf rel_implies_wf rel_exp_wf subtract_wf decidable__le false_wf not-le-2 less-iff-le condition-implies-le minus-one-mul zero-add minus-one-mul-top minus-add minus-minus add-associates add-swap add-commutes add_functionality_wrt_le add-zero le-add-cancel le_wf set_wf less_than_wf primrec-wf2 nat_wf equal_wf all_wf eq_int_wf bool_wf equal-wf-base int_subtype_base assert_wf less_than_transitivity1 le_weakening less_than_irreflexivity bnot_wf not_wf infix_ap_wf subtype_rel_self exists_wf uiff_transitivity eqtt_to_assert assert_of_eq_int eqff_to_assert assert_of_bnot not_functionality_wrt_uiff
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation cut thin rename setElimination instantiate introduction extract_by_obid sqequalHypSubstitution isectElimination universeEquality sqequalRule lambdaEquality functionEquality cumulativity hypothesisEquality because_Cache functionExtensionality applyEquality hypothesis dependent_set_memberEquality natural_numberEquality dependent_functionElimination unionElimination independent_pairFormation voidElimination productElimination independent_functionElimination independent_isectElimination addEquality isect_memberEquality voidEquality intEquality minusEquality Error :isect_memberFormation_alt,  axiomEquality Error :inhabitedIsType,  Error :functionIsType,  Error :universeIsType,  baseApply closedConclusion baseClosed equalityTransitivity equalitySymmetry dependent_pairFormation productEquality equalityElimination

Latex:
\mforall{}n:\mBbbN{}.  \mforall{}[T:Type].  \mforall{}[R1,R2:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    (R1  =>  R2  {}\mRightarrow{}  rel\_exp(T;  R1;  n)  =>  rel\_exp(T;  R2;  n))

Date html generated: 2019_06_20-PM-00_30_27
Last ObjectModification: 2018_09_26-PM-00_49_25

Theory : relations

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