Nuprl Lemma : fill_term_0
∀[H:j⊢]. ∀[phi:{H ⊢ _:𝔽}]. ∀[T:{H.𝕀 ⊢ _}]. ∀[u:{H.𝕀, (phi)p ⊢ _:T}]. ∀[a0:{H ⊢ _:(T)[0(𝕀)][phi |⟶ u[0]]}].
∀[cT:H.𝕀 ⊢ Compositon(T)].
((fill cT [phi ⊢→ u] a0)[0(𝕀)] = a0 ∈ {H ⊢ _:(T)[0(𝕀)]})
Proof
Definitions occuring in Statement :
fill_term: fill cA [phi ⊢→ u] a0,
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-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 ⊢ _},
cubical_set: CubicalSet,
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
composition-structure: Gamma ⊢ Compositon(A),
fill_term: fill cA [phi ⊢→ u] a0,
comp-to-fill: comp-to-fill(Gamma;cA),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
all: ∀x:A. B[x],
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
subtype_rel: A ⊆r B,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
uimplies: b supposing a,
implies: P ⇒ Q,
squash: ↓T,
prop: ℙ,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
same-cubical-type: Gamma ⊢ A = B,
partial-term-0: u[0],
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-m: m,
csm-comp: G o F,
cc-adjoin-cube: (v;u),
compose: f o g,
csm-id: 1(X),
csm-adjoin: (s;u),
csm-ap: (s)x,
same-cubical-term: X ⊢ u=v:A,
cubical-type: {X ⊢ _},
face-term-implies: Gamma ⊢ (phi ⇒ psi),
cubical-type-at: A(a),
pi1: fst(t),
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,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
so_apply: x[s],
context-subset: Gamma, phi,
cube-context-adjoin: X.A,
case-term: (u ∨ v),
cubical-term-at: u(a),
csm-ap-term: (t)s,
face-zero: (i=0),
pi2: snd(t),
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
dM0: 0,
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),
DeMorgan-algebra: DeMorganAlgebra,
interval-1: 1(𝕀),
lattice-meet: a ∧ b,
fset-ac-glb: fset-ac-glb(eq;ac1;ac2),
fset-minimals: fset-minimals(x,y.less[x; y]; s),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
lattice-0: 0,
empty-fset: {},
nil: [],
face-or: (a ∨ b),
face-1: 1(𝔽),
rev_uimplies: rev_uimplies(P;Q),
dm-neg: ¬(x),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
fset-image: f"(s),
opposite-lattice: opposite-lattice(L),
lattice-1: 1,
fset-singleton: {x},
cons: [a / b]
Lemmas referenced :
cube-context-adjoin_wf,
interval-type_wf,
csm-id-adjoin_wf,
interval-0_wf,
csm-comp_wf,
csm-m_wf,
csm-id_wf,
face-or_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
cc-fst_wf_interval,
face-zero_wf,
cc-snd_wf,
context-subset_wf,
thin-context-subset,
context-subset-map,
composition-structure_wf,
constrained-cubical-term_wf,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-0,
cubical-type-cumulativity2,
partial-term-0_wf,
istype-cubical-term,
cubical-type_wf,
cubical_set_wf,
cc-fst_wf,
subset-cubical-term,
sub_cubical_set_self,
context-subset-is-subset,
cubical-term_wf,
squash_wf,
true_wf,
csm-comp-type,
cube_set_map_wf,
csm-context-subset-subtype2,
equal_wf,
istype-universe,
0-comp-cc-fst-comp-m,
subtype_rel_self,
iff_weakening_equal,
interval-1_wf,
csm-m-comp-1,
csm-ap-id-type,
csm-comp-term,
csm-ap-term-wf-subset,
face-and_wf,
face-term-and-implies1,
csm-subset-domain,
sub_cubical_set-cumulativity1,
face-term-implies-subset,
face-term-and-implies2,
cubical-term-eqcd,
context-iterated-subset,
cube_set_map_cumulativity-i-j,
case-term_wf,
subtype_rel_transitivity,
thin-context-subset-adjoin,
lattice-point_wf,
face_lattice_wf,
cubical-term-at_wf,
I_cube_wf,
fset_wf,
nat_wf,
cc-fst-comp-csm-m-term,
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,
csm-face-or,
lattice-1_wf,
context-adjoin-subset3,
csm-m-comp-0,
I_cube_pair_redex_lemma,
face-or-eq-1,
fl-eq_wf,
eqtt_to_assert,
assert-fl-eq,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
cubical_type_at_pair_lemma,
cubical-type-at_wf,
cubical-type-cumulativity,
interval-type-at-is-point,
lattice-0-meet,
dM_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
DeMorgan-algebra-subtype,
DeMorgan-algebra_wf,
bdd-distributive-lattice_wf,
bdd-lattice_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
DeMorgan-algebra-axioms_wf,
istype-cubical-type-at,
csm-ap-term-at,
dM0_wf,
interval-type-at,
cubical-term-equal,
subset-cubical-type,
csm-comp-assoc,
csm-ap-id-term,
subset-cubical-term2,
csm-face-zero,
face-1_wf,
csm_id_adjoin_fst_term_lemma,
cc_snd_csm_id_adjoin_lemma,
face-type-at,
lattice-1-join,
dM-to-FL-eq-1,
dm-neg_wf,
names_wf,
names-deq_wf,
subtype_rel-equal,
free-DeMorgan-lattice_wf,
context-1-subset,
subtype_rel_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
thin,
instantiate,
introduction,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
promote_hyp,
setElimination,
rename,
sqequalRule,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
Error :memTop,
universeIsType,
applyEquality,
independent_isectElimination,
inhabitedIsType,
lambdaFormation_alt,
equalityIstype,
independent_functionElimination,
lambdaEquality_alt,
hyp_replacement,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
productElimination,
applyLambdaEquality,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
sqequalBase,
cumulativity,
productEquality,
isectEquality,
functionExtensionality,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
voidElimination,
dependent_pairEquality_alt
Latex:
\mforall{}[H:j\mvdash{}]. \mforall{}[phi:\{H \mvdash{} \_:\mBbbF{}\}]. \mforall{}[T:\{H.\mBbbI{} \mvdash{} \_\}]. \mforall{}[u:\{H.\mBbbI{}, (phi)p \mvdash{} \_:T\}].
\mforall{}[a0:\{H \mvdash{} \_:(T)[0(\mBbbI{})][phi |{}\mrightarrow{} u[0]]\}]. \mforall{}[cT:H.\mBbbI{} \mvdash{} Compositon(T)].
((fill cT [phi \mvdash{}\mrightarrow{} u] a0)[0(\mBbbI{})] = a0)
Date html generated:
2020_05_20-PM-04_50_55
Last ObjectModification:
2020_04_19-PM-02_11_24
Theory : cubical!type!theory
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