Nuprl Lemma : permutation-s-group-sep-or
∀rv:SeparationSpace. ∀sepw:x:Point ⟶ y:{y:Point| x # y} ⟶ x # y. ∀x,x',y,y':Point.
(let f,g = x
in let f',g' = y
in <f o f', g' o g> # let f,g = x'
in let f',g' = y'
in <f o f', g' o g>
⇒ (x # x' ∨ y # y'))
Proof
Definitions occuring in Statement :
permutation-ss: permutation-ss(ss),
ss-sep: x # y,
ss-point: Point,
separation-space: SeparationSpace,
compose: f o g,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
spread: spread def,
pair: <a, b>
Definitions unfolded in proof :
all: ∀x:A. B[x],
sq_stable: SqStable(P),
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
squash: ↓T,
prop: ℙ,
top: Top,
fun-sep: fun-sep(ss;A;f;g),
compose: f o g,
or: P ∨ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
guard: {T}
Lemmas referenced :
ss-point_wf,
ss-sep_wf,
squash_wf,
permutation-ss-point,
permutation-ss_wf,
separation-space_wf,
or_wf,
exists_wf,
permutation-ss-sep,
ss-sep-or,
ss-sep-symmetry
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
applyEquality,
functionExtensionality,
hypothesisEquality,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
dependent_functionElimination,
imageElimination,
dependent_set_memberEquality,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
productElimination,
sqequalRule,
because_Cache,
functionEquality,
setEquality,
unionElimination,
lambdaEquality,
independent_functionElimination,
inrFormation,
inlFormation,
dependent_pairFormation,
imageMemberEquality,
baseClosed
Latex:
\mforall{}rv:SeparationSpace. \mforall{}sepw:x:Point {}\mrightarrow{} y:\{y:Point| x \# y\} {}\mrightarrow{} x \# y. \mforall{}x,x',y,y':Point.
(let f,g = x
in let f',g' = y
in <f o f', g' o g> \# let f,g = x'
in let f',g' = y'
in <f o f', g' o g>
{}\mRightarrow{} (x \# x' \mvee{} y \# y'))
Date html generated:
2017_10_02-PM-03_25_18
Last ObjectModification:
2017_07_12-PM-01_51_45
Theory : constructive!algebra
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