Nuprl Lemma : pi-comp-nu-uniformity
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[cA:Gamma ⊢ CompOp(A)]. ∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[rho:Gamma(I+i)].
∀[J,K:fset(ℕ)]. ∀[f:J ⟶ I]. ∀[g:K ⟶ J]. ∀[u:A(f((i1)(rho)))]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ∀[k:{k:ℕ| ¬k ∈ K} ].
((pi-comp-nu(Gamma;A;cA;I;i;rho;J;f;u;j) f,i=j(rho) g,j=k)
= pi-comp-nu(Gamma;A;cA;I;i;rho;K;f ⋅ g;(u f((i1)(rho)) g);k)
∈ A(f ⋅ g,i=k(rho)))
Proof
Definitions occuring in Statement :
pi-comp-nu: pi-comp-nu(Gamma;A;cA;I;i;rho;J;f;u1;j),
composition-op: Gamma ⊢ CompOp(A),
cubical-type-ap-morph: (u a f),
cubical-type-at: A(a),
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nc-e': g,i=j,
nc-1: (i1),
add-name: I+i,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
not: ¬A,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
pi-comp-nu: pi-comp-nu(Gamma;A;cA;I;i;rho;J;f;u1;j),
all: ∀x:A. B[x],
implies: P ⇒ Q,
filling-op: filling-op(Gamma;A),
let: let,
nat: ℕ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
cube-set-restriction: f(s),
pi2: snd(t),
face-presheaf: 𝔽,
fl-morph: <f>,
fl-lift: fl-lift(T;eq;L;eqL;f0;f1),
face-lattice-property,
free-dist-lattice-with-constraints-property,
lattice-extend-wc: lattice-extend-wc(L;eq;eqL;f;ac),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
fset-image: f"(s),
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
lattice-0: 0,
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,
empty-fset: {},
nil: [],
it: ⋅,
bdd-distributive-lattice: BoundedDistributiveLattice,
filling-uniformity: filling-uniformity(Gamma;A;fill),
lattice-point: Point(l),
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
int-deq: IntDeq,
nat-deq: NatDeq,
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u)
Lemmas referenced :
fill_from_comp_wf,
member-cubical-path-0-0,
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-r'_wf,
subtype_rel-equal,
cubical-type-at_wf,
nc-1_wf,
nc-0_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
I_cube_wf,
cube-set-restriction-comp,
subtype_rel_self,
iff_weakening_equal,
nh-comp_wf,
nc-r'-nc-0,
lattice-0_wf,
face_lattice_wf,
trivial-section_wf,
face-presheaf_wf2,
nc-s_wf,
f-subset-add-name,
istype-nat,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
istype-void,
istype-cubical-type-at,
names-hom_wf,
fset_wf,
composition-op_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf,
nc-r_wf,
trivial-member-add-name1,
nc-e'_wf,
cubical-type-ap-morph_wf,
nc-r'-r,
nat-deq_wf,
nc-r'-to-e'2,
nh-comp-assoc,
nc-e'-lemma6,
cubical-type-ap-morph-comp,
nc-r'-to-e',
nc-r-e'-r,
nc-e'-r,
cubical-path-condition-0,
cubical-path-condition_wf,
empty-cubical-subset-term,
face-lattice-property,
free-dist-lattice-with-constraints-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
inhabitedIsType,
lambdaFormation_alt,
rename,
setElimination,
sqequalRule,
because_Cache,
dependent_set_memberEquality_alt,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
universeIsType,
voidElimination,
applyEquality,
imageElimination,
instantiate,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
equalityIstype,
setIsType,
functionIsType,
intEquality,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
applyLambdaEquality,
hyp_replacement
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[cA:Gamma \mvdash{} CompOp(A)]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ].
\mforall{}[rho:Gamma(I+i)]. \mforall{}[J,K:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I]. \mforall{}[g:K {}\mrightarrow{} J]. \mforall{}[u:A(f((i1)(rho)))].
\mforall{}[j:\{j:\mBbbN{}| \mneg{}j \mmember{} J\} ]. \mforall{}[k:\{k:\mBbbN{}| \mneg{}k \mmember{} K\} ].
((pi-comp-nu(Gamma;A;cA;I;i;rho;J;f;u;j) f,i=j(rho) g,j=k)
= pi-comp-nu(Gamma;A;cA;I;i;rho;K;f \mcdot{} g;(u f((i1)(rho)) g);k))
Date html generated:
2020_05_20-PM-03_58_36
Last ObjectModification:
2020_04_09-PM-07_08_13
Theory : cubical!type!theory
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