Nuprl Lemma : context-map-cube+-csm+
∀[Gamma:j⊢]. ∀[I,J:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[j:{j:ℕ| ¬j ∈ J} ]. ∀[g:J ⟶ I]. ∀[rho:Gamma(I+i)].
(<g,i=j(rho)> o cube+(J;j) = <rho> o cube+(I;i) o <g>+ ∈ formal-cube(J).𝕀 j⟶ Gamma)
Proof
Definitions occuring in Statement :
cube+: cube+(I;i),
interval-type: 𝕀,
csm+: tau+,
cube-context-adjoin: X.A,
csm-comp: G o F,
context-map: <rho>,
cube_set_map: A ⟶ B,
formal-cube: formal-cube(I),
cube-set-restriction: f(s),
I_cube: A(I),
cubical_set: CubicalSet,
nc-e': g,i=j,
add-name: I+i,
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,
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,
names-hom: I ⟶ J,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
formal-cube: formal-cube(I),
so_lambda: λ2x.t[x],
so_apply: x[s],
cube-context-adjoin: X.A,
context-map: <rho>,
cube+: cube+(I;i),
csm-comp: G o F,
csm+: tau+,
compose: f o g,
csm-adjoin: (s;u),
cc-fst: p,
csm-ap: (s)x,
cc-snd: q,
functor-arrow: arrow(F),
pi2: snd(t),
cube-set-restriction: f(s),
squash: ↓T,
names: names(I),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
cubical-type-at: A(a),
interval-type: 𝕀,
constant-cubical-type: (X),
interval-presheaf: 𝕀,
lattice-point: Point(l),
record-select: r.x,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
record-update: r[x := v],
eq_atom: x =a y,
bfalse: ff,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
DeMorgan-algebra: DeMorganAlgebra,
guard: {T},
uiff: uiff(P;Q),
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nc-e': g,i=j,
nequal: a ≠ b ∈ T ,
dM-lift: dM-lift(I;J;f),
free-dma-lift: free-dma-lift(T;eq;dm;eq2;f),
free-DeMorgan-algebra-property,
free-dist-lattice-property,
union-deq: union-deq(A;B;a;b)
Lemmas referenced :
csm-equal,
cube-context-adjoin_wf,
formal-cube_wf1,
interval-type_wf,
csm-comp_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,
cube+_wf,
context-map_wf,
cube-set-restriction_wf,
nc-e'_wf,
cubical_set_cumulativity-i-j,
csm+_wf_interval,
subtype_rel_self,
I_cube_wf,
cube-set-map-subtype,
names-hom_wf,
fset-member_wf,
nat_wf,
int-deq_wf,
istype-void,
istype-nat,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
fset_wf,
cubical_set_wf,
I_cube_pair_redex_lemma,
arrow_pair_lemma,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cube-set-restriction-comp,
eq_int_wf,
lattice-point_wf,
dM_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
not-added-name,
names_wf,
nh-comp_wf,
iff_weakening_equal,
interval-type-at,
eqtt_to_assert,
assert_of_eq_int,
nh-comp-sq,
dM-lift-inc,
trivial-member-add-name1,
intformeq_wf,
int_formula_prop_eq_lemma,
dM-lift_wf2,
int_subtype_base,
free-DeMorgan-algebra-property,
free-dist-lattice-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
instantiate,
hypothesisEquality,
dependent_set_memberEquality_alt,
setElimination,
rename,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
sqequalRule,
independent_pairFormation,
universeIsType,
voidElimination,
because_Cache,
applyEquality,
setIsType,
functionIsType,
intEquality,
functionExtensionality,
productElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
inhabitedIsType,
lambdaFormation_alt,
equalityElimination,
productEquality,
cumulativity,
isectEquality,
equalityIstype,
promote_hyp,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[I,J:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ]. \mforall{}[j:\{j:\mBbbN{}| \mneg{}j \mmember{} J\} ]. \mforall{}[g:J {}\mrightarrow{} I]. \mforall{}[rho:Gamma(I+i)\000C].
(<g,i=j(rho)> o cube+(J;j) = <rho> o cube+(I;i) o <g>+)
Date html generated:
2020_05_20-PM-02_39_39
Last ObjectModification:
2020_04_04-PM-06_58_53
Theory : cubical!type!theory
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