Nuprl Lemma : pi-comp-property
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[B:{Gamma.A ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[cB:Gamma.A ⊢ CompOp(B)]. ∀[I:fset(ℕ)].
∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)]. ∀[phi:𝔽(I)]. ∀[mu:{I+i,s(phi) ⊢ _:(ΠA B)<rho> o iota}].
∀[lambda:cubical-path-0(Gamma;ΠA B;I;i;rho;phi;mu)]. ∀[J:fset(ℕ)]. ∀[f:I,phi(J)].
((pi-comp(Gamma;A;B;cA;cB) I i rho phi mu lambda (i1)(rho) f) = mu((i1) ⋅ f) ∈ ΠA B(f((i1)(rho))))
Proof
Definitions occuring in Statement :
pi-comp: pi-comp(Gamma;A;B;cA;cB),
composition-op: Gamma ⊢ CompOp(A),
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
cubical-pi: ΠA B,
cube-context-adjoin: X.A,
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type-ap-morph: (u a f),
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
subset-iota: iota,
cubical-subset: I,psi,
face-presheaf: 𝔽,
csm-comp: G o F,
context-map: <rho>,
formal-cube: formal-cube(I),
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nc-1: (i1),
nc-s: s,
add-name: I+i,
nh-comp: g ⋅ f,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
not: ¬A,
set: {x:A| B[x]} ,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
cubical-pi: ΠA B,
cubical-pi-family: cubical-pi-family(X;A;B;I;a),
squash: ↓T,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
pi-comp: pi-comp(Gamma;A;B;cA;cB),
has-value: (a)↓,
let: let,
composition-op: Gamma ⊢ CompOp(A),
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u),
cubical-path-condition': cubical-path-condition'(Gamma;A;I;i;rho;phi;u;a1),
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
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,
bdd-distributive-lattice: BoundedDistributiveLattice,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
name-morph-satisfies: (psi f) = 1,
compose: f o g,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
pi-comp-app: pi-comp-app(Gamma;A;I;i;rho;phi;mu;J;f;j;nu),
cubical-term-at: u(a),
canonical-section: canonical-section(Gamma;A;I;rho;a),
subset-iota: iota,
csm-ap-term: (t)s,
cubical-app: app(w; u),
csm-ap: (s)x,
subset-trans: subset-trans(I;J;f;x),
cubical-term: {X ⊢ _:A},
csm-comp: G o F,
csm-ap-type: (AF)s,
cubical-type-ap-morph: (u a f),
pi2: snd(t),
context-map: <rho>,
functor-arrow: arrow(F),
cube-set-restriction: f(s),
names-hom: I ⟶ J,
formal-cube: formal-cube(I),
cubical-type: {X ⊢ _},
cc-snd: q,
cc-fst: p
Lemmas referenced :
cubical-subset-I_cube,
cubical-type-ap-morph_wf,
cubical-pi_wf,
cube-set-restriction_wf,
add-name_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,
nc-1_wf,
pi-comp_wf3,
cubical-type-cumulativity2,
cube-context-adjoin_wf,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
names-hom_wf,
istype-cubical-type-at,
cc-adjoin-cube_wf,
nh-comp_wf,
subtype_rel-equal,
cubical-type-at_wf,
cube-set-restriction-comp,
equal_wf,
I_cube_wf,
iff_weakening_equal,
cc-adjoin-cube-restriction,
subtype_rel_self,
cubical-subset_wf,
cubical-path-0_wf,
cubical-term_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,
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-op_wf,
cubical-type_wf,
cubical_set_wf,
new-name_wf,
value-type-has-value,
not_wf,
set-value-type,
int-value-type,
nc-e'_wf,
pi-comp-nu_wf,
pi-comp-app_wf,
pi-comp-lambda_wf,
nh-id_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,
lattice-meet_wf,
lattice-join_wf,
name-morph-satisfies-comp,
squash_wf,
true_wf,
istype-universe,
fl-morph-comp,
lattice-1_wf,
fl-morph_wf,
fl-morph-1,
name-morph-satisfies_wf,
nh-id-left,
nc-e'-lemma1,
cubical-type-ap-morph-id,
nc-r_wf,
trivial-member-add-name1,
nc-r'_wf,
nc-r'-r,
pi-comp-nu-property,
nh-id-right,
nh-comp-assoc,
s-comp-nc-1,
cubical-subset-restriction,
cube-set-restriction-id,
subtype_rel_weakening,
ext-eq_weakening,
csm-ap-restriction,
csm-ap_wf,
csm-ap-context-map,
subtype_rel_wf,
csm-cubical-pi,
csm-adjoin_wf,
cc-fst_wf,
cc-snd_wf,
csm-ap-type-at,
csm-adjoin-ap,
cc_snd_adjoin_cube_lemma,
csm_comp_fst_adjoin_cube_lemma,
implies-nh-comp-satisfies,
uiff_transitivity2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
sqequalHypSubstitution,
cut,
introduction,
extract_by_obid,
isectElimination,
thin,
Error :memTop,
hypothesis,
setElimination,
rename,
hypothesisEquality,
dependent_set_memberEquality_alt,
because_Cache,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
applyEquality,
instantiate,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
functionIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
productElimination,
universeEquality,
setIsType,
intEquality,
functionExtensionality,
inhabitedIsType,
lambdaFormation_alt,
callbyvalueReduce,
setEquality,
productEquality,
cumulativity,
isectEquality,
hyp_replacement,
functionEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[B:\{Gamma.A \mvdash{} \_\}]. \mforall{}[cA:Gamma \mvdash{} CompOp(A)].
\mforall{}[cB:Gamma.A \mvdash{} CompOp(B)]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ]. \mforall{}[rho:Gamma(I+i)]. \mforall{}[phi:\mBbbF{}(I)].
\mforall{}[mu:\{I+i,s(phi) \mvdash{} \_:(\mPi{}A B)<rho> o iota\}]. \mforall{}[lambda:cubical-path-0(Gamma;\mPi{}A B;I;i;rho;phi;mu)].
\mforall{}[J:fset(\mBbbN{})]. \mforall{}[f:I,phi(J)].
((pi-comp(Gamma;A;B;cA;cB) I i rho phi mu lambda (i1)(rho) f) = mu((i1) \mcdot{} f))
Date html generated:
2020_05_20-PM-04_02_30
Last ObjectModification:
2020_04_09-PM-11_13_48
Theory : cubical!type!theory
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