Nuprl Lemma : fill_from_comp_wf
∀[Gamma:j⊢]. ∀[A:{Gamma ⊢ _}]. ∀[comp:Gamma ⊢ CompOp(A)]. (fill_from_comp(Gamma;A;comp) ∈ filling-op(Gamma;A))
Proof
Definitions occuring in Statement :
fill_from_comp: fill_from_comp(Gamma;A;comp),
filling-op: filling-op(Gamma;A),
composition-op: Gamma ⊢ CompOp(A),
cubical-type: {X ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
filling-op: filling-op(Gamma;A),
subtype_rel: A ⊆r B,
filling-uniformity: filling-uniformity(Gamma;A;fill),
all: ∀x:A. B[x],
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],
fill_from_comp: fill_from_comp(Gamma;A;comp),
has-value: (a)↓,
face-presheaf: 𝔽,
I_cube: A(I),
names: names(I),
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
composition-op: Gamma ⊢ CompOp(A),
cubical-path-1: cubical-path-1(Gamma;A;I;i;rho;phi;u),
cube-set-restriction: f(s),
pi2: snd(t),
fl-join: fl-join(I;x;y),
bdd-distributive-lattice: BoundedDistributiveLattice,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
functor-ob: ob(F),
pi1: fst(t),
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,
nc-e': g,i=j,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
DeMorgan-algebra: DeMorganAlgebra,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
nc-s: s,
compose: f o g,
sq_stable: SqStable(P),
label: ...$L... t,
context-map: <rho>,
subset-iota: iota,
csm-comp: G o F,
subset-trans: subset-trans(I;J;f;x),
functor-arrow: arrow(F),
cubical-term: {X ⊢ _:A},
csm-ap-term: (t)s,
fillterm: fillterm(Gamma;A;I;i;j;rho;a0;u),
csm-ap: (s)x,
cubical-term-at: u(a),
name-morph-satisfies: (psi f) = 1,
cubical-path-0: cubical-path-0(Gamma;A;I;i;rho;phi;u),
rev_uimplies: rev_uimplies(P;Q),
f-subset: xs ⊆ ys,
cubical-type: {X ⊢ _},
csm-ap-type: (AF)s,
cubical-path-condition: cubical-path-condition(Gamma;A;I;i;rho;phi;u;a0),
nc-0: (i0),
names-hom: I ⟶ J,
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
Lemmas referenced :
composition-op_wf,
cubical_set_cumulativity-i-j,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf,
fill_from_comp_wf2,
cubical-path-0_wf,
istype-cubical-term,
cubical-subset_wf,
add-name_wf,
cube-set-restriction_wf,
face-presheaf_wf2,
nc-s_wf,
f-subset-add-name,
csm-ap-type_wf,
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,
names-hom_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,
new-name_wf,
value-type-has-value,
not_wf,
set-value-type,
int-value-type,
ob_pair_lemma,
fl0_wf,
trivial-member-add-name1,
cubical-path-0-fillterm,
composition-op-uniformity,
nc-e'_wf,
nc-m_wf,
fl-join_wf,
fillterm_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-type-at_wf,
cube-set-restriction-comp,
nc-1_wf,
subtype_rel_self,
iff_weakening_equal,
nh-comp_wf,
nh-comp-assoc,
nc-m-nc-1,
nh-id-right,
cubical-type-ap-morph_wf,
cube-set-restriction-when-id,
subtype_rel_set,
subtype_rel_universe1,
cubical-path-condition'_wf,
istype-cubical-type-at,
subtype_rel-equal,
nc-e'-comp-m,
I_cube_pair_redex_lemma,
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,
fl-morph_wf,
fl-morph-join,
fl-morph-comp2,
nc-e'-lemma3,
fl-morph-fl0,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
dM-to-FL_wf,
neg-dM_inc,
dM-to-FL-opp,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
cube_set_restriction_pair_lemma,
dM_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-axioms_wf,
dm-neg_wf,
names_wf,
names-deq_wf,
free-DeMorgan-lattice_wf,
nh-comp-sq,
dM-lift-inc,
trivial-member-add-name2,
dM_inc_wf,
sq_stable__fset-member,
decidable__equal_int,
sq_stable__not,
cubical-term-equal,
cubical-term_wf,
subset-trans_wf,
csm-ap-comp-type,
cube_set_map_wf,
csm-equal,
cubical-subset-I_cube-member,
csm-ap-term_wf,
cubical-path-0-ap-morph,
fl-morph-restriction,
subset-trans-iota-lemma,
csm-ap-type-at,
iff_transitivity,
or_wf,
face_lattice-1-join-irreducible,
fset-member-add-name,
isdM0_wf,
assert-isdM0,
f-subset-add-name1,
cubical-type-ap-morph-comp-eq,
nc-0_wf,
nc-e'-lemma2,
nh-comp-nc-m-eq2,
nc-e'-lemma4,
nc-e'-lemma5,
csm-ap_wf,
cubical-subset-I_cube,
name-morph-satisfies_wf,
nc-e'-comp-m-2,
fset-member_witness,
dM0_wf,
uiff_transitivity3,
fl-morph-fl0-is-1,
cubical-term-at_wf,
nh-comp-nc-m-eq3,
set_subtype_base,
int_subtype_base,
s-comp-nc-0',
nc-m-nc-0,
nc-0-s-commute,
member-cubical-subset-I_cube,
s-comp-nc-0,
cubical-path-condition_wf,
cubical-path-1_wf,
cubical-subset-term-trans,
filling-uniformity_wf,
cubical-type-cumulativity,
face-lattice-property,
free-dist-lattice-with-constraints-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeIsType,
thin,
instantiate,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
isect_memberEquality_alt,
isectIsTypeImplies,
inhabitedIsType,
lambdaFormation_alt,
setElimination,
rename,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
Error :memTop,
independent_pairFormation,
voidElimination,
setIsType,
functionIsType,
intEquality,
callbyvalueReduce,
setEquality,
equalityIstype,
hyp_replacement,
imageElimination,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
cumulativity,
productEquality,
isectEquality,
equalityElimination,
promote_hyp,
functionEquality,
functionExtensionality,
unionIsType,
inlFormation_alt,
productIsType,
applyLambdaEquality,
sqequalBase,
inrFormation_alt
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma \mvdash{} \_\}]. \mforall{}[comp:Gamma \mvdash{} CompOp(A)].
(fill\_from\_comp(Gamma;A;comp) \mmember{} filling-op(Gamma;A))
Date html generated:
2020_05_20-PM-03_57_24
Last ObjectModification:
2020_04_20-PM-07_36_01
Theory : cubical!type!theory
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