Nuprl Lemma : csm-fill_term
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[u:{Gamma.𝕀, (phi)p ⊢ _:A}].
∀[a0:{Gamma ⊢ _:(A)[0(𝕀)][phi |⟶ u[0]]}]. ∀[Delta:j⊢]. ∀[s:Delta j⟶ Gamma].
((fill cA [phi ⊢→ u] a0)s+ = fill (cA)s+ [(phi)s ⊢→ (u)s+] (a0)s ∈ {Delta.𝕀 ⊢ _:(A)s+[((phi)s)p |⟶ (u)s+]})
Proof
Definitions occuring in Statement :
fill_term: fill cA [phi ⊢→ u] a0,
csm-comp-structure: (cA)tau,
composition-structure: Gamma ⊢ Compositon(A),
partial-term-0: u[0],
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
interval-0: 0(𝕀),
interval-type: 𝕀,
csm+: tau+,
csm-id-adjoin: [u],
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
cubical-term: {X ⊢ _:A},
csm-ap-type: (AF)s,
cubical-type: {X ⊢ _},
cube_set_map: A ⟶ B,
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
face-term-implies: Gamma ⊢ (phi ⇒ psi),
all: ∀x:A. B[x],
implies: P ⇒ Q,
csm-ap-term: (t)s,
cc-fst: p,
cubical-term-at: u(a),
interval-type: 𝕀,
csm+: tau+,
csm-ap: (s)x,
cc-snd: q,
constant-cubical-type: (X),
csm-ap-type: (AF)s,
csm-comp: G o F,
csm-adjoin: (s;u),
pi1: fst(t),
compose: f o g,
member: t ∈ T,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
cubical-type-at: A(a),
face-type: 𝔽,
I_cube: A(I),
functor-ob: ob(F),
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,
same-cubical-type: Gamma ⊢ A = B,
guard: {T},
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
squash: ↓T,
cubical-type: {X ⊢ _},
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-id: 1(X),
pi2: snd(t),
true: True,
partial-term-0: u[0],
csm-comp-structure: (cA)tau,
composition-function: composition-function{j:l,i:l}(Gamma;A),
cubical_set: CubicalSet,
ps_context: __⊢,
cat-functor: Functor(C1;C2),
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
type-cat: TypeCat,
cube-context-adjoin: X.A,
interval-presheaf: 𝕀,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
functor-arrow: arrow(F),
cube-set-restriction: f(s),
cubical-type-ap-morph: (u a f),
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,
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,
fill_term: fill cA [phi ⊢→ u] a0,
filling-structure: Gamma ⊢ Filling(A),
uniform-filling-function: uniform-filling-function{j:l, i:l}(Gamma;A;fill),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
comp-to-fill: comp-to-fill(Gamma;cA),
csm-m: m,
cc-adjoin-cube: (v;u)
Lemmas referenced :
csm-ap-term_wf,
cube-context-adjoin_wf,
interval-type_wf,
face-type_wf,
csm-face-type,
cc-fst_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,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
cubical-term-at_wf,
subtype_rel_self,
lattice-1_wf,
I_cube_wf,
fset_wf,
nat_wf,
csm-ap-term-wf-subset,
csm-ap-type_wf,
csm+_wf_interval,
context-subset_wf,
csm-context-subset-subtype2,
cube_set_map_wf,
constrained-cubical-term_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
partial-term-0_wf,
cubical-term_wf,
thin-context-subset,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
squash_wf,
true_wf,
cubical-type-cumulativity,
csm-id-adjoin_wf,
interval-0_wf,
subset-cubical-term2,
sub_cubical_set_self,
subset-cubical-term,
context-subset-is-subset,
fill_term_wf,
csm-comp-structure-composition-function,
csm+_wf,
composition-function_wf,
comp-to-fill_wf2,
csm-id_wf,
istype-universe,
csm-ap-id-type,
subset-cubical-type,
iff_weakening_equal,
free-DeMorgan-algebra-property,
free-dist-lattice-property
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
lambdaFormation_alt,
sqequalRule,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
instantiate,
hypothesisEquality,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
inhabitedIsType,
equalityIstype,
universeIsType,
applyEquality,
lambdaEquality_alt,
productEquality,
cumulativity,
isectEquality,
because_Cache,
independent_isectElimination,
setElimination,
rename,
dependent_functionElimination,
independent_functionElimination,
dependent_set_memberEquality_alt,
hyp_replacement,
imageElimination,
productElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
applyLambdaEquality,
universeEquality
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} Compositon(A)].
\mforall{}[u:\{Gamma.\mBbbI{}, (phi)p \mvdash{} \_:A\}]. \mforall{}[a0:\{Gamma \mvdash{} \_:(A)[0(\mBbbI{})][phi |{}\mrightarrow{} u[0]]\}]. \mforall{}[Delta:j\mvdash{}].
\mforall{}[s:Delta j{}\mrightarrow{} Gamma].
((fill cA [phi \mvdash{}\mrightarrow{} u] a0)s+ = fill (cA)s+ [(phi)s \mvdash{}\mrightarrow{} (u)s+] (a0)s)
Date html generated:
2020_05_20-PM-04_48_20
Last ObjectModification:
2020_04_13-PM-09_43_47
Theory : cubical!type!theory
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