### Nuprl Lemma : rel_exp_one

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].  ∀x,y:T.  (x R^1 y `⇐⇒` x R y)`

Proof

Definitions occuring in Statement :  rel_exp: `R^n` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` function: `x:A ⟶ B[x]` natural_number: `\$n` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` subtract: `n - m` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` or: `P ∨ Q` exists: `∃x:A. B[x]` cand: `A c∧ B` member: `t ∈ T` uimplies: `b supposing a` sq_type: `SQType(T)` guard: `{T}` true: `True` false: `False` prop: `ℙ` so_lambda: `λ2x.t[x]` nat: `ℕ` le: `A ≤ B` less_than': `less_than'(a;b)` not: `¬A` subtype_rel: `A ⊆r B` so_apply: `x[s]` rev_implies: `P `` Q` infix_ap: `x f y` rel_exp: `R^n` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` btrue: `tt`
Lemmas referenced :  subtype_base_sq int_subtype_base false_wf or_wf exists_wf less_than_wf infix_ap_wf rel_exp_wf le_wf equal-wf-base equal_wf rel_exp_iff iff_wf subtract_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut sqequalRule independent_pairFormation sqequalHypSubstitution unionElimination thin productElimination hypothesis addLevel instantiate introduction extract_by_obid isectElimination cumulativity intEquality independent_isectElimination dependent_functionElimination equalityTransitivity equalitySymmetry independent_functionElimination natural_numberEquality voidElimination levelHypothesis promote_hyp hypothesisEquality lambdaEquality productEquality because_Cache universeEquality dependent_set_memberEquality functionExtensionality applyEquality functionEquality baseClosed impliesFunctionality hyp_replacement inlFormation dependent_pairFormation

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].    \mforall{}x,y:T.    (x  rel\_exp(T;  R;  1)  y  \mLeftarrow{}{}\mRightarrow{}  x  R  y)

Date html generated: 2017_04_17-AM-09_26_49
Last ObjectModification: 2017_02_27-PM-05_27_48

Theory : relations2

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