Nuprl Lemma : fillpath_wf
∀[Gamma:j⊢]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ CompOp(A)]. ∀[x:{Gamma ⊢ _:(A)[1(𝕀)]}]. ∀[y:{Gamma ⊢ _:(A)[0(𝕀)]}].
∀[z:{Gamma.𝕀 ⊢ _:((A)[0(𝕀)])p}].
(fillpath(Gamma;A;cA;x;y;z) ∈ {t:{Gamma.𝕀 ⊢ _:((A)[1(𝕀)])p}|
((t)[0(𝕀)] = x ∈ {Gamma ⊢ _:(A)[1(𝕀)]})
∧ ((t)[1(𝕀)] = app(transport-fun(Gamma;A;cA); y) ∈ {Gamma ⊢ _:(A)[1(𝕀)]})} ) supposing
(((z)[0(𝕀)] = app(rev-transport-fun(Gamma;A;cA); x) ∈ {Gamma ⊢ _:(A)[0(𝕀)]}) and
((z)[1(𝕀)] = y ∈ {Gamma ⊢ _:(A)[0(𝕀)]}))
Proof
Definitions occuring in Statement :
fillpath: fillpath(Gamma;A;cA;x;y;z),
rev-transport-fun: rev-transport-fun(Gamma;A;cA),
transport-fun: transport-fun(Gamma;A;cA),
composition-op: Gamma ⊢ CompOp(A),
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
cubical-app: app(w; u),
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,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
cubical-type: {X ⊢ _},
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-ap-type: (AF)s,
csm-id: 1(X),
csm-adjoin: (s;u),
csm-ap: (s)x,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
fillpath: fillpath(Gamma;A;cA;x;y;z),
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
constant-cubical-type: (X),
csm+: tau+,
csm-comp: G o F,
interval-1: 1(𝕀),
pi2: snd(t),
compose: f o g,
pi1: fst(t),
guard: {T},
implies: P ⇒ Q,
csm-ap-term: (t)s,
squash: ↓T,
prop: ℙ,
true: True,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
same-cubical-term: X ⊢ u=v:A,
case-endpoints: [r=0 ⊢→ a; r=1 ⊢→ b],
respects-equality: respects-equality(S;T),
and: P ∧ Q,
cand: A c∧ B,
face-term-implies: Gamma ⊢ (phi ⇒ psi),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
face-one: (i=1),
cubical-term-at: u(a),
face-zero: (i=0),
or: P ∨ Q,
dm-neg: ¬(x),
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,
dM0: 0,
lattice-0: 0,
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],
ifthenelse: if b then t else f fi ,
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),
btrue: tt,
empty-fset: {},
nil: [],
it: ⋅,
opposite-lattice: opposite-lattice(L),
lattice-1: 1,
fset-singleton: {x},
cons: [a / b],
dM1: 1,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
so_apply: x[s],
cubical-type-at: A(a),
face-type: 𝔽,
I_cube: A(I),
functor-ob: ob(F),
face-presheaf: 𝔽,
lattice-point: Point(l),
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),
DeMorgan-algebra: DeMorganAlgebra,
cubical-app: app(w; u)
Lemmas referenced :
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-id-adjoin_wf,
interval-0_wf,
cc-fst_wf_interval,
interval-1_wf,
csm_id_adjoin_fst_type_lemma,
cubical-term-eqcd,
csm-ap-term_wf,
composition-term_wf,
cubical_set_cumulativity-i-j,
face-or_wf,
face-zero_wf,
cc-snd_wf,
face-one_wf,
csm+_wf_interval,
csm-composition_wf,
case-endpoints_wf,
cubical-app_wf_fun,
fill-type-down_wf,
fill-type-up_wf,
csm-interval-type,
rev-transport-fun_wf,
istype-cubical-term,
composition-op_wf,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf,
subset-cubical-term,
context-subset_wf,
face-type_wf,
csm-face-type,
context-adjoin-subset4,
csm-context-subset-subtype2,
cube_set_map_cumulativity-i-j,
csm-face-zero,
csm-face-one,
csm-face-or,
csm-case-endpoints,
csm_id_adjoin_fst_term_lemma,
term-p+0,
term-p+1,
squash_wf,
true_wf,
fill-type-down-0,
cube_set_map_wf,
fill-type-up-0,
constrained-cubical-term-eqcd,
fill-type-down-1,
fill-type-up-1,
context-subset-is-subset,
case-term-same2,
cubical-term-1-q1,
cubical-term-0-q0,
thin-context-subset,
respects-equality-context-subset-term,
context-subset-map,
sub_cubical_set_transitivity,
face-1_wf,
context-1-subset,
face-term-implies-subset,
face-or-eq-1,
equal_wf,
istype-universe,
lattice-point_wf,
face_lattice_wf,
dM-to-FL-dM1,
subtype_rel_self,
iff_weakening_equal,
dM-to-FL_wf,
dM0_wf,
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,
cubical-term-at_wf,
lattice-1_wf,
I_cube_wf,
fset_wf,
nat_wf,
case-endpoints-0,
csm-id_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-id-term,
dm-neg_wf,
names_wf,
names-deq_wf,
dM1_wf,
subtype_rel-equal,
dM_wf,
free-DeMorgan-lattice_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-axioms_wf,
case-endpoints-1,
transport-fun_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
instantiate,
hypothesis,
sqequalRule,
dependent_functionElimination,
Error :memTop,
setElimination,
rename,
productElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
applyEquality,
lambdaEquality_alt,
cumulativity,
universeIsType,
universeEquality,
inhabitedIsType,
hyp_replacement,
axiomEquality,
equalityIstype,
isect_memberEquality_alt,
isectIsTypeImplies,
independent_functionElimination,
imageElimination,
applyLambdaEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
dependent_set_memberEquality_alt,
independent_pairFormation,
lambdaFormation_alt,
inlFormation_alt,
productEquality,
isectEquality,
inrFormation_alt,
productIsType
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} CompOp(A)]. \mforall{}[x:\{Gamma \mvdash{} \_:(A)[1(\mBbbI{})]\}].
\mforall{}[y:\{Gamma \mvdash{} \_:(A)[0(\mBbbI{})]\}]. \mforall{}[z:\{Gamma.\mBbbI{} \mvdash{} \_:((A)[0(\mBbbI{})])p\}].
(fillpath(Gamma;A;cA;x;y;z) \mmember{} \{t:\{Gamma.\mBbbI{} \mvdash{} \_:((A)[1(\mBbbI{})])p\}|
((t)[0(\mBbbI{})] = x) \mwedge{} ((t)[1(\mBbbI{})] = app(transport-fun(Gamma;A;cA); y))\} \000C) supposing
(((z)[0(\mBbbI{})] = app(rev-transport-fun(Gamma;A;cA); x)) and
((z)[1(\mBbbI{})] = y))
Date html generated:
2020_05_20-PM-04_56_27
Last ObjectModification:
2020_05_02-PM-03_20_30
Theory : cubical!type!theory
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