Nuprl Lemma : glue-comp_wf
∀G:j⊢. ∀A:{G ⊢ _}. ∀cA:G +⊢ Compositon(A). ∀psi:{G ⊢ _:𝔽}. ∀T:{G, psi ⊢ _}. ∀cT:G, psi +⊢ Compositon(T).
∀f:{G, psi ⊢ _:Equiv(T;A)}.
(comp(Glue [psi ⊢→ (T, f)] A) ∈ composition-function{j:l,i:l}(G;Glue [psi ⊢→ (T;equiv-fun(f))] A))
Proof
Definitions occuring in Statement :
glue-comp: comp(Glue [phi ⊢→ (T, f)] A) ,
glue-type: Glue [phi ⊢→ (T;w)] A,
composition-structure: Gamma ⊢ Compositon(A),
composition-function: composition-function{j:l,i:l}(Gamma;A),
equiv-fun: equiv-fun(f),
cubical-equiv: Equiv(T;A),
context-subset: Gamma, phi,
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
all: ∀x:A. B[x],
member: t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
composition-function: composition-function{j:l,i:l}(Gamma;A),
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
guard: {T},
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
csm-id-adjoin: [u],
csm-id: 1(X),
and: P ∧ Q,
true: True,
squash: ↓T,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
csm-ap-term: (t)s,
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-comp: G o F,
csm-adjoin: (s;u),
csm-ap: (s)x,
compose: f o g,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
cubical-fun: (A ⟶ B),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
same-cubical-type: Gamma ⊢ A = B,
glue-comp: comp(Glue [phi ⊢→ (T, f)] A) ,
partial-term-0: u[0],
so_lambda: λ2x.t[x],
so_apply: x[s],
csm-comp-structure: (cA)tau,
partial-term-1: u[1],
let: let,
composition-structure: Gamma ⊢ Compositon(A),
glue-type: Glue [phi ⊢→ (T;w)] A,
interval-1: 1(𝕀),
or: P ∨ Q,
same-cubical-term: X ⊢ u=v:A,
cc-fst: p,
pi1: fst(t),
face-term-implies: Gamma ⊢ (phi ⇒ psi),
bdd-distributive-lattice: BoundedDistributiveLattice,
cubical-type-at: A(a),
face-type: 𝔽,
constant-cubical-type: (X),
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,
face-or: (a ∨ b),
face-and: (a ∧ b),
cubical-term-at: u(a),
pres-c1: pres-c1(G;phi;f;t;t0;cA),
pres-c2: pres-c2(G;phi;f;t;t0;cT),
cc-snd: q,
fiber-point: fiber-point(t;c),
fiber-path: fiber-path(p),
cubical-pair: cubical-pair(u;v),
cubical-snd: p.2,
pi2: snd(t),
cand: A c∧ B,
fiber-member: fiber-member(p),
cubical-fst: p.1,
respects-equality: respects-equality(S;T)
Lemmas referenced :
glue-type_wf,
equiv-fun_wf,
context-subset_wf,
thin-context-subset,
csm-glue-type,
cube-context-adjoin_wf,
interval-type_wf,
subset-cubical-type,
sub_cubical_set_functionality,
context-subset-is-subset,
equal_functionality_wrt_subtype_rel2,
cubical-type_wf,
cubical-term-eqcd,
subset-cubical-term,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
cc-fst_wf_interval,
context-adjoin-subset1,
thin-context-subset-adjoin,
csm-ap-type_wf,
csm-id-adjoin_wf,
interval-0_wf,
csm-context-subset-subtype3,
csm-id-adjoin_wf-interval-0,
constrained-cubical-term-eqcd,
cube_set_map_wf,
cubical_set_wf,
istype-cubical-term,
cubical-equiv_wf,
composition-structure_wf,
cubical_set_cumulativity-i-j,
cubical-fun_wf,
context-subset-map,
subtype_rel_transitivity,
face-and_wf,
context-subset-term-subtype,
face-term-implies-subset,
face-term-and-implies1,
context-iterated-subset,
equal_wf,
squash_wf,
true_wf,
istype-universe,
csm-cubical-fun,
subtype_rel_self,
iff_weakening_equal,
csm-comp-type,
csm-comp-term,
csm-comp_wf,
unglue-term_wf2,
glue-type-subset,
context-iterated-subset1,
cubical-fun-subset,
glue-type-constraint,
csm-context-subset-subtype2,
sub_cubical_set_transitivity,
context-subset-swap,
sub_cubical_set_functionality2,
cubical-app_wf_fun,
csm-unglue,
unglue-term_wf,
cubical-term_wf,
cubical-type-cumulativity2,
partial-term-0_wf,
subtype_rel_set,
comp_term_wf,
csm-comp-structure-composition-function,
composition-structure-cumulativity,
context-adjoin-subset4,
face-forall_wf,
context-adjoin-subset2,
face-forall-implies,
face-forall-type-subtype,
subset-comp-structure,
csm-comp-structure_wf,
cube_set_map_cumulativity-i-j,
composition-structure-subset,
face-forall-implies-0,
cube_set_map_subtype3,
sub_cubical_set_self,
composition-function-subset,
pres_wf2,
cubical-fun-subset-adjoin,
context-adjoin-subset0,
pres-c1_wf,
composition-function-cumulativity,
pres-c2_wf,
partial-term-1_wf,
interval-1_wf,
cube_set_map_subtype,
csm-face-and,
csm_id_adjoin_fst_term_lemma,
csm-ap-id-term,
sub_cubical_set_wf,
face-term-and-implies2,
subset-constrained-cubical-term,
face-forall-implies-1,
case-term_wf2,
face-or_wf,
csm-equiv-fun,
csm-id-adjoin_wf-interval-1,
face-term-or-implies,
subset-cubical-term2,
face-term-implies_wf,
face-term-implies-and,
face-term-implies-or,
face-term-implies-or1,
path-type_wf,
csm-cubical-app,
csm-cubical-equiv,
face-term-implies-or2,
case-term-equal-right,
context-iterated-subset0,
term-to-path-is-refl,
term-to-path-subset,
path-type-subset,
lattice-point_wf,
face_lattice_wf,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
cubical-term-at_wf,
lattice-1_wf,
I_cube_wf,
fset_wf,
nat_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
iff_weakening_uiff,
lattice-meet-eq-1,
face_lattice-1-join-irreducible,
iff_transitivity,
constrained-cubical-term_wf,
composition-function_wf,
comp_term-subset,
case-term-equal-left,
context-subset-subtype-simple,
subtype_rel_wf,
context-subset-subtype-or2,
fiber-comp_wf,
subtype_rel-equal,
equiv-term_wf,
fiber-member_wf,
fiber-path_wf,
cubical-path-app_wf,
csm-path-type,
cc-snd_wf,
cubical-fiber_wf,
cubical-fiber-subset,
fiber-member-fiber-point,
context-subset-subtype-or,
face-and-com,
term-to-path-app-snd,
partial-term-1-subset,
csm_id_ap_term_lemma,
csm-face-or,
context-adjoin-subset3,
csm_id_adjoin_fst_type_lemma,
csm-case-term,
cubical-path-ap-id-adjoin2,
cubical-path-app-0,
case-term-same2,
csm-id_wf,
glue-term_wf,
cubical-path-app-1,
glue-unglue,
glue-term-subset,
cubical-term-equal,
case-term-equality-right,
csm-constrained-cubical-term,
respects-equality-context-subset-term
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
functionExtensionality,
rename,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
because_Cache,
instantiate,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
applyEquality,
lambdaEquality_alt,
hyp_replacement,
universeIsType,
sqequalRule,
Error :memTop,
productElimination,
natural_numberEquality,
imageElimination,
universeEquality,
dependent_functionElimination,
imageMemberEquality,
baseClosed,
setElimination,
inhabitedIsType,
independent_pairFormation,
equalityIstype,
dependent_set_memberEquality_alt,
applyLambdaEquality,
cumulativity,
inrFormation_alt,
productEquality,
isectEquality,
unionEquality,
productIsType,
unionIsType,
unionElimination,
inlFormation_alt,
promote_hyp,
setEquality
Latex:
\mforall{}G:j\mvdash{}. \mforall{}A:\{G \mvdash{} \_\}. \mforall{}cA:G +\mvdash{} Compositon(A). \mforall{}psi:\{G \mvdash{} \_:\mBbbF{}\}. \mforall{}T:\{G, psi \mvdash{} \_\}.
\mforall{}cT:G, psi +\mvdash{} Compositon(T). \mforall{}f:\{G, psi \mvdash{} \_:Equiv(T;A)\}.
(comp(Glue [psi \mvdash{}\mrightarrow{} (T, f)] A)
\mmember{} composition-function\{j:l,i:l\}(G;Glue [psi \mvdash{}\mrightarrow{} (T;equiv-fun(f))] A))
Date html generated:
2020_05_20-PM-06_12_43
Last ObjectModification:
2020_04_25-AM-10_54_50
Theory : cubical!type!theory
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