Nuprl Lemma : s-comp-1-e'
∀[I:fset(ℕ)]. ∀[i:ℕ]. ∀[J:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[j:{j:ℕ| ¬j ∈ I} ]. ∀[k:{j:ℕ| ¬j ∈ J} ].
(s ⋅ (i1) ⋅ f,j=k = (i1) ⋅ f ⋅ s ∈ J+k ⟶ I+i)
Proof
Definitions occuring in Statement :
nc-e': g,i=j,
nc-1: (i1),
nc-s: s,
add-name: I+i,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
not: ¬A,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat: ℕ,
so_apply: x[s],
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
set_wf,
nat_wf,
not_wf,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
names-hom_wf,
add-name_wf,
nc-e'_wf,
nc-1_wf,
squash_wf,
true_wf,
equal_wf,
add-name-com,
iff_weakening_equal,
nc-s_wf,
f-subset-add-name,
nh-comp_wf,
nh-comp-assoc,
nc-1-s-commute,
nc-e'-lemma3
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
setElimination,
thin,
rename,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
sqequalRule,
lambdaEquality,
applyEquality,
intEquality,
independent_isectElimination,
because_Cache,
natural_numberEquality,
hypothesisEquality,
isect_memberEquality,
axiomEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
independent_functionElimination,
hyp_replacement,
dependent_functionElimination,
dependent_set_memberEquality
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\mBbbN{}]. \mforall{}[J:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I]. \mforall{}[j:\{j:\mBbbN{}| \mneg{}j \mmember{} I\} ]. \mforall{}[k:\{j:\mBbbN{}| \mneg{}j \mmember{} J\} ].
(s \mcdot{} (i1) \mcdot{} f,j=k = (i1) \mcdot{} f \mcdot{} s)
Date html generated:
2017_10_05-AM-01_07_04
Last ObjectModification:
2017_07_28-AM-09_28_11
Theory : cubical!type!theory
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