Nuprl Lemma : csm-comp_term
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[u:{Gamma, phi.𝕀 ⊢ _:A}].
∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ (u)[0(𝕀)]]}]. ∀[Delta:j⊢]. ∀[s:Delta j⟶ Gamma].
((comp cA [phi ⊢→ u] a0)s = comp (cA)s+ [(phi)s ⊢→ (u)s+] (a0)s ∈ {Delta ⊢ _:((A)s+)[1(𝕀)][(phi)s |⟶ ((u)s+)[1(𝕀)]]})
Proof
Definitions occuring in Statement :
comp_term: comp cA [phi ⊢→ u] a0,
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
csm+: tau+,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
composition-structure: Gamma ⊢ Compositon(A),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
all: ∀x:A. B[x],
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
true: True,
comp_term: comp cA [phi ⊢→ u] a0,
csm-id-adjoin: [u],
csm-id: 1(X),
cubical-type: {X ⊢ _},
interval-0: 0(𝕀),
csm-ap-type: (AF)s,
interval-type: 𝕀,
csm-adjoin: (s;u),
csm-ap: (s)x,
csm+: tau+,
csm-comp: G o F,
cc-snd: q,
cc-fst: p,
constant-cubical-type: (X),
compose: f o g,
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
pi1: fst(t),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
pi2: snd(t),
type-cat: TypeCat,
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
interval-1: 1(𝕀),
csm-ap-term: (t)s,
and: P ∧ Q,
same-cubical-type: Gamma ⊢ A = B,
face-term-implies: Gamma ⊢ (phi ⇒ psi),
implies: P ⇒ Q,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
so_apply: x[s],
cubical-type-at: A(a),
face-type: 𝔽,
I_cube: A(I),
functor-ob: ob(F),
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-term-at: u(a),
csm-comp-structure: (cA)tau,
composition-function: composition-function{j:l,i:l}(Gamma;A),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]}
Lemmas referenced :
cube-context-adjoin_wf,
interval-type_wf,
context-subset-adjoin-subtype,
cubical-term_wf,
squash_wf,
true_wf,
cubical-type_wf,
context-subset_wf,
csm-ap-id-type,
cubical-type-cumulativity2,
subtype_rel_transitivity,
csm-id_wf,
cube_set_map_wf,
constrained-cubical-term_wf,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-0,
csm-ap-term_wf,
subset-cubical-type,
sub_cubical_set_functionality,
context-subset-is-subset,
composition-structure_wf,
face-type_wf,
cubical_set_wf,
thin-context-subset,
csm-face-type,
csm+_wf_interval,
context-subset-map,
csm-comp_wf,
subtype_rel_self,
csm-context-subset-subtype3,
csm-id-adjoin_wf-interval-1,
subset-cubical-term,
cc-fst_wf,
sub_cubical_set_transitivity,
sub_cubical_set_self,
context-adjoin-subset1,
csm-context-subset-subtype2,
csm-ap-term-wf-subset,
csm-id-adjoin_wf,
interval-1_wf,
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,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
cubical-term-at_wf,
lattice-1_wf,
I_cube_wf,
fset_wf,
nat_wf,
istype-universe,
subtype_rel-equal,
subset-cubical-term2,
equal_functionality_wrt_subtype_rel2,
csm-comp-type,
interval-0_wf,
context-subset-term-subtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
setElimination,
rename,
dependent_functionElimination,
applyEquality,
lambdaEquality_alt,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeIsType,
because_Cache,
sqequalRule,
cumulativity,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
inhabitedIsType,
productElimination,
Error :memTop,
independent_pairFormation,
lambdaFormation_alt,
equalityIstype,
productEquality,
isectEquality,
universeEquality,
independent_functionElimination,
dependent_set_memberEquality_alt,
applyLambdaEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} Compositon(A)].
\mforall{}[u:\{Gamma, phi.\mBbbI{} \mvdash{} \_:A\}]. \mforall{}[a0:\{Gamma \mvdash{} \_:(A)[0(\mBbbI{})][phi |{}\mrightarrow{} (u)[0(\mBbbI{})]]\}]. \mforall{}[Delta:j\mvdash{}].
\mforall{}[s:Delta j{}\mrightarrow{} Gamma].
((comp cA [phi \mvdash{}\mrightarrow{} u] a0)s = comp (cA)s+ [(phi)s \mvdash{}\mrightarrow{} (u)s+] (a0)s)
Date html generated:
2020_05_20-PM-04_37_34
Last ObjectModification:
2020_04_11-PM-00_57_55
Theory : cubical!type!theory
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