Nuprl Lemma : greatest-lower-bound-assoc
∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].
∀[a,b,c,x,y,u1,u2:T].
(u1 = u2 ∈ T) supposing
(greatest-lower-bound(T;x,y.R[x;y];a;b;x) and
greatest-lower-bound(T;x,y.R[x;y];x;c;u1) and
greatest-lower-bound(T;x,y.R[x;y];b;c;y) and
greatest-lower-bound(T;x,y.R[x;y];a;y;u2))
supposing Order(T;x,y.R[x;y])
Proof
Definitions occuring in Statement :
greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c),
order: Order(T;x,y.R[x; y]),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
greatest-lower-bound: greatest-lower-bound(T;x,y.R[x; y];a;b;c),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
order: Order(T;x,y.R[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y]),
cand: A c∧ B,
trans: Trans(T;x,y.E[x; y])
Lemmas referenced :
greatest-lower-bound_wf,
order_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
applyEquality,
isect_memberEquality,
axiomEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
productElimination,
dependent_functionElimination,
independent_functionElimination,
independent_pairFormation,
lambdaFormation
Latex:
\mforall{}[T:Type]. \mforall{}[R:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbP{}].
\mforall{}[a,b,c,x,y,u1,u2:T].
(u1 = u2) supposing
(greatest-lower-bound(T;x,y.R[x;y];a;b;x) and
greatest-lower-bound(T;x,y.R[x;y];x;c;u1) and
greatest-lower-bound(T;x,y.R[x;y];b;c;y) and
greatest-lower-bound(T;x,y.R[x;y];a;y;u2))
supposing Order(T;x,y.R[x;y])
Date html generated:
2016_05_13-PM-04_18_37
Last ObjectModification:
2015_12_26-AM-11_27_55
Theory : rel_1
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