Nuprl Lemma : rel_le_refl_cl_sp
∀[T:Type]. ∀[r:T ⟶ T ⟶ ℙ]. (dec_binrel(T;x,y:T. x = y ∈ T) ⇒ r ≡>{T} (r\\00B8) supposing anti_sym(T;r))
Proof
Definitions occuring in Statement :
s_part: E\,
refl_cl: Eo,
xxanti_sym: anti_sym(T;R),
dec_binrel: dec_binrel(T;r),
ab_binrel: x,y:T. E[x; y],
binrel_le: E ≡>{T} E',
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
s_part: E\,
refl_cl: Eo,
binrel_le: E ≡>{T} E',
xxanti_sym: anti_sym(T;R),
dec_binrel: dec_binrel(T;r),
anti_sym: AntiSym(T;x,y.R[x; y]),
ab_binrel: x,y:T. E[x; y],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
or: P ∨ Q,
and: P ∧ Q,
cand: A c∧ B,
not: ¬A,
false: False,
guard: {T}
Lemmas referenced :
all_wf,
equal_wf,
decidable_wf,
and_wf,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
hypothesis,
applyEquality,
universeEquality,
because_Cache,
rename,
lemma_by_obid,
isectElimination,
functionEquality,
cumulativity,
unionElimination,
inlFormation,
inrFormation,
independent_pairFormation,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[r:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}]. (dec\_binrel(T;x,y:T. x = y) {}\mRightarrow{} r \mequiv{}>\{T\} (r\mbackslash{}\msupzero{}) supposing anti\_sym(T;r))
Date html generated:
2016_05_15-PM-00_01_57
Last ObjectModification:
2015_12_26-PM-11_25_55
Theory : gen_algebra_1
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