Nuprl Lemma : xxconnex_iff_trichot
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
((∀a,b:T. Dec(R a b)) ⇒ (connex(T;R) ⇐⇒ {∀a,b:T. (((R\) a b) ∨ ((R↔) a b) ∨ ((R\) b a))}))
Proof
Definitions occuring in Statement :
s_part: E\,
sym_cl: E↔,
xxconnex: connex(T;R),
decidable: Dec(P),
uall: ∀[x:A]. B[x],
prop: ℙ,
guard: {T},
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
so_apply: x[s1;s2],
strict_part: strict_part(x,y.R[x; y];a;b),
symmetrize: Symmetrize(x,y.R[x; y];a;b),
s_part: E\,
sym_cl: E↔,
xxconnex: connex(T;R)
Lemmas referenced :
connex_iff_trichot
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
hypothesis
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
((\mforall{}a,b:T. Dec(R a b)) {}\mRightarrow{} (connex(T;R) \mLeftarrow{}{}\mRightarrow{} \{\mforall{}a,b:T. (((R\mbackslash{}) a b) \mvee{} ((R\mrightleftharpoons{}) a b) \mvee{} ((R\mbackslash{}) b a))\}))
Date html generated:
2016_05_15-PM-00_01_51
Last ObjectModification:
2015_12_26-PM-11_25_48
Theory : gen_algebra_1
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