Nuprl Lemma : term-to-path_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀a:{X.𝕀 ⊢ _:(A)p}. (<>(a) ∈ {X ⊢ _:(Path_A (a)[0(𝕀)] (a)[1(𝕀)])})
Proof
Definitions occuring in Statement :
term-to-path: <>(a),
path-type: (Path_A a b),
interval-1: 1(𝕀),
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],
all: ∀x:A. B[x],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
term-to-path: <>(a),
path-type: (Path_A a b),
member: t ∈ T,
so_lambda: so_lambda3,
prop: ℙ,
and: P ∧ Q,
pathtype: Path(A),
cubical-fun: (A ⟶ B),
cubical-fun-family: cubical-fun-family(X; A; B; I; a),
subtype_rel: A ⊆r B,
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],
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,
cubical-type-at: A(a),
pi1: fst(t),
interval-type: 𝕀,
constant-cubical-type: (X),
I_cube: A(I),
functor-ob: ob(F),
interval-presheaf: 𝕀,
uimplies: b supposing a,
squash: ↓T,
true: True,
so_apply: x[s1;s2;s3],
cand: A c∧ B,
csm-id-adjoin: [u],
csm-ap-term: (t)s,
cubical-term-at: u(a),
cubical-lambda: (λb),
csm-ap: (s)x,
interval-0: 0(𝕀),
csm-id: 1(X),
csm-adjoin: (s;u),
cc-adjoin-cube: (v;u),
cc-fst: p,
interval-1: 1(𝕀)
Lemmas referenced :
cubical-subset-term,
pathtype_wf,
equal_wf,
cubical-type-at_wf,
cubical_type_at_pair_lemma,
nh-id_wf,
dM0_wf,
subtype_rel_self,
interval-type_wf,
cube-set-restriction_wf,
cubical-term-at_wf,
csm-ap-term_wf,
cubical_set_cumulativity-i-j,
cube-context-adjoin_wf,
csm-ap-type_wf,
cc-fst_wf,
csm-id-adjoin_wf-interval-0,
subset-cubical-term2,
sub_cubical_set_self,
csm_id_adjoin_fst_type_lemma,
csm-ap-id-type,
dM1_wf,
subtype_rel-equal,
cube-set-restriction-id,
csm-id-adjoin_wf-interval-1,
istype-cubical-type-at,
I_cube_wf,
path-restriction,
cubical-term_wf,
cubical-type-cumulativity2,
cubical-type_wf,
cubical_set_wf,
cubical-lambda_wf,
cubical-fun-as-cubical-pi,
cc-adjoin-cube_wf,
squash_wf,
true_wf,
fset_wf,
nat_wf,
csm-ap-type-at
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality_alt,
productEquality,
applyEquality,
dependent_functionElimination,
Error :memTop,
setElimination,
rename,
because_Cache,
instantiate,
independent_isectElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeIsType,
independent_pairFormation,
dependent_set_memberEquality_alt,
equalityTransitivity,
equalitySymmetry,
productIsType,
equalityIstype,
inhabitedIsType,
applyLambdaEquality,
productElimination,
hyp_replacement
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}a:\{X.\mBbbI{} \mvdash{} \_:(A)p\}. (<>(a) \mmember{} \{X \mvdash{} \_:(Path\_A (a)[0(\mBbbI{})] (a)[1(\mBbbI{})])\})
Date html generated:
2020_05_20-PM-03_18_17
Last ObjectModification:
2020_04_06-PM-06_34_50
Theory : cubical!type!theory
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