### Nuprl Lemma : trans_rel_func_wrt_sym_self

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].`
`  (Trans(T;x,y.R[x;y])`
`  `` {∀a,a',b,b':T.  (Symmetrize(x,y.R[x;y];a;b) `` Symmetrize(x,y.R[x;y];a';b') `` (R[a;a'] `⇐⇒` R[b;b']))})`

Proof

Definitions occuring in Statement :  symmetrize: `Symmetrize(x,y.R[x; y];a;b)` trans: `Trans(T;x,y.E[x; y])` uall: `∀[x:A]. B[x]` prop: `ℙ` guard: `{T}` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  symmetrize: `Symmetrize(x,y.R[x; y];a;b)` guard: `{T}` uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` member: `t ∈ T` prop: `ℙ` so_apply: `x[s1;s2]` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` so_lambda: `λ2x y.t[x; y]`
Lemmas referenced :  subtype_rel_self trans_wf trans_rel_self_functionality
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep Error :isect_memberFormation_alt,  lambdaFormation independent_pairFormation sqequalHypSubstitution productElimination thin applyEquality hypothesisEquality productEquality cut hypothesis instantiate introduction extract_by_obid isectElimination universeEquality because_Cache lambdaEquality Error :functionIsType,  Error :universeIsType,  Error :inhabitedIsType,  independent_functionElimination dependent_functionElimination

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
(Trans(T;x,y.R[x;y])
{}\mRightarrow{}  \{\mforall{}a,a',b,b':T.
(Symmetrize(x,y.R[x;y];a;b)  {}\mRightarrow{}  Symmetrize(x,y.R[x;y];a';b')  {}\mRightarrow{}  (R[a;a']  \mLeftarrow{}{}\mRightarrow{}  R[b;b']))\})

Date html generated: 2019_06_20-PM-00_29_00
Last ObjectModification: 2018_09_26-AM-11_46_40

Theory : rel_1

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