Nuprl Lemma : csm-canonical-section-face-type-0
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[phi:𝔽(I)].
((canonical-section(();𝔽;I+i;⋅;s(phi)))<(i0)> = canonical-section(();𝔽;I;⋅;phi) ∈ {formal-cube(I) ⊢ _:𝔽})
Proof
Definitions occuring in Statement :
face-type: 𝔽,
csm-ap-term: (t)s,
canonical-section: canonical-section(Gamma;A;I;rho;a),
cubical-term: {X ⊢ _:A},
face-presheaf: 𝔽,
context-map: <rho>,
trivial-cube-set: (),
formal-cube: formal-cube(I),
cube-set-restriction: f(s),
I_cube: A(I),
nc-0: (i0),
nc-s: s,
add-name: I+i,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
it: ⋅,
uall: ∀[x:A]. B[x],
not: ¬A,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat: ℕ,
so_apply: x[s],
all: ∀x:A. B[x],
unit: Unit,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
trivial-cube-set: (),
face-presheaf: 𝔽,
lattice-point: Point(l),
record-select: r.x,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
cubical-type-at: A(a),
face-type: 𝔽,
constant-cubical-type: (X),
top: Top,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
bdd-distributive-lattice: BoundedDistributiveLattice,
not: ¬A,
false: False,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2)
Lemmas referenced :
uall_wf,
squash_wf,
true_wf,
fset_wf,
nat_wf,
set_wf,
not_wf,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
I_cube_wf,
face-presheaf_wf,
small_cubical_set_subtype,
equal_wf,
cubical-term_wf,
formal-cube_wf,
face-type_wf,
csm-canonical-section-face-type,
add-name_wf,
nc-0_wf,
nc-s_wf,
f-subset-add-name,
canonical-section_wf,
trivial-cube-set_wf,
it_wf,
subtype_rel_self,
cubical-type-at_wf_face-type,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-type_wf,
context-map_wf,
csm-face-type,
iff_weakening_equal,
face-type-at,
Error :cube_set_restriction_pair_lemma,
lattice-point_wf,
face_lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
fl-morph-comp2,
fl-morph_wf,
names-hom_wf,
s-comp-nc-0,
bounded-lattice-hom_wf,
bdd-distributive-lattice_wf,
fl-morph-id
Rules used in proof :
cut,
applyEquality,
instantiate,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
functionEquality,
cumulativity,
universeEquality,
sqequalRule,
because_Cache,
intEquality,
independent_isectElimination,
natural_numberEquality,
setElimination,
rename,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
imageMemberEquality,
baseClosed,
setEquality,
productElimination,
independent_functionElimination,
isect_memberFormation,
axiomEquality,
productEquality,
lambdaFormation
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ]. \mforall{}[phi:\mBbbF{}(I)].
((canonical-section(();\mBbbF{};I+i;\mcdot{};s(phi)))<(i0)> = canonical-section(();\mBbbF{};I;\mcdot{};phi))
Date html generated:
2018_05_23-AM-09_24_36
Last ObjectModification:
2017_11_12-PM-05_48_00
Theory : cubical!type!theory
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