Nuprl Lemma : comp-fun-to-comp-op1_wf
∀Gamma:j⊢. ∀A:{Gamma ⊢ _}.
∀[comp:composition-function{j:l,i:l}(Gamma;A)]
(comp-fun-to-comp-op1(Gamma;A;comp) ∈ I:fset(ℕ)
⟶ i:{i:ℕ| ¬i ∈ I}
⟶ rho:Gamma(I+i)
⟶ phi:𝔽(I)
⟶ u:{I+i,s(phi) ⊢ _:(A)<rho> o iota}
⟶ cubical-path-0(Gamma;A;I;i;rho;phi;u)
⟶ {formal-cube(I) ⊢ _:((A)<rho> o cube+(I;i))[1(𝕀)][canonical-section(();𝔽;I;⋅;phi) |⟶ ((u)cube+(I;i))[1(𝕀)]]})
Proof
Definitions occuring in Statement :
comp-fun-to-comp-op1: comp-fun-to-comp-op1(Gamma;A;comp),
composition-function: composition-function{j:l,i:l}(Gamma;A),
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
cube+: cube+(I;i),
interval-1: 1(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
canonical-section: canonical-section(Gamma;A;I;rho;a),
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
subset-iota: iota,
cubical-subset: I,psi,
face-presheaf: 𝔽,
csm-comp: G o F,
context-map: <rho>,
trivial-cube-set: (),
formal-cube: formal-cube(I),
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
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],
all: ∀x:A. B[x],
not: ¬A,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
comp-fun-to-comp-op1: comp-fun-to-comp-op1(Gamma;A;comp),
member: t ∈ T,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
canonical-section: canonical-section(Gamma;A;I;rho;a),
cubical-term-at: u(a),
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,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
face-presheaf: 𝔽,
composition-function: composition-function{j:l,i:l}(Gamma;A),
unit: Unit,
trivial-cube-set: (),
cubical-type-at: A(a),
face-type: 𝔽,
constant-cubical-type: (X)
Lemmas referenced :
cubical-path-0_wf,
cubical-type-cumulativity2,
cubical-term_wf,
cubical-subset_wf,
add-name_wf,
cube-set-restriction_wf,
face-presheaf_wf2,
nc-s_wf,
f-subset-add-name,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity,
csm-comp_wf,
formal-cube_wf1,
subset-iota_wf,
context-map_wf,
I_cube_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
istype-nat,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
istype-void,
fset_wf,
composition-function_wf,
cubical-type_wf,
cubical_set_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
context-subset-is-cubical-subset,
canonical-section_wf,
face-type_wf,
subtype_rel_self,
iff_weakening_equal,
fl-morph-id,
face-type-ap-morph,
cube-context-adjoin_wf,
interval-type_wf,
cube+_wf,
trivial-cube-set_wf,
it_wf,
cubical-type-at_wf_face-type,
subset-cubical-term2,
sub_cubical_set_self,
csm-face-type,
csm-ap-term-cube+,
canonical-section-cubical-path-0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
isect_memberFormation_alt,
lambdaEquality_alt,
universeIsType,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
setElimination,
rename,
independent_isectElimination,
dependent_functionElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
setIsType,
functionIsType,
intEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination
Latex:
\mforall{}Gamma:j\mvdash{}. \mforall{}A:\{Gamma \mvdash{} \_\}.
\mforall{}[comp:composition-function\{j:l,i:l\}(Gamma;A)]
(comp-fun-to-comp-op1(Gamma;A;comp) \mmember{} I:fset(\mBbbN{})
{}\mrightarrow{} i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\}
{}\mrightarrow{} rho:Gamma(I+i)
{}\mrightarrow{} phi:\mBbbF{}(I)
{}\mrightarrow{} u:\{I+i,s(phi) \mvdash{} \_:(A)<rho> o iota\}
{}\mrightarrow{} cubical-path-0(Gamma;A;I;i;rho;phi;u)
{}\mrightarrow{} \{formal-cube(I) \mvdash{} \_:((A)<rho> o cube+(I;i))[1(\mBbbI{})][canonical-section(();\mBbbF{};I;\mcdot{};phi)
|{}\mrightarrow{} ((u)cube+(I;i))[1(\mBbbI{})]]\})
Date html generated:
2020_05_20-PM-04_29_35
Last ObjectModification:
2020_04_11-AM-10_02_26
Theory : cubical!type!theory
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