Nuprl Lemma : rev-path_wf
∀[X:j⊢]. ∀[A:{X ⊢ _}]. ∀[b,a:{X ⊢ _:A}]. ∀[pth:{X ⊢ _:(Path_A a b)}]. (rev-path(X;pth) ∈ {X ⊢ _:(Path_A b a)})
Proof
Definitions occuring in Statement :
rev-path: rev-path(G;pth),
path-type: (Path_A a b),
cubical-term: {X ⊢ _: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,
ext-eq: A ≡ B,
and: P ∧ Q,
subtype_rel: A ⊆r B,
rev-path: rev-path(G;pth),
cand: A c∧ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
term-to-pathtype: <>a,
squash: ↓T,
true: True,
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
prop: ℙ,
uimplies: b supposing a,
cubicalpath-app: pth @ r,
cubical-path-app: pth @ r,
cubical-app: app(w; u),
csm-ap-term: (t)s,
term-to-path: <>(a),
cubical-lambda: (λb),
cc-adjoin-cube: (v;u),
interval-rev: 1-(r),
cubical-term-at: u(a),
pi2: snd(t),
cubical-term: {X ⊢ _:A},
interval-0: 0(𝕀),
interval-1: 1(𝕀),
dM1: 1,
lattice-1: 1,
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,
fset-singleton: {x},
cons: [a / b],
dma-neg: ¬(x),
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,
empty-fset: {},
nil: [],
it: ⋅,
opposite-lattice: opposite-lattice(L),
DeMorgan-algebra: DeMorganAlgebra
Lemmas referenced :
path-type-ext-eq,
cubical-term_wf,
path-type_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
cubical-type_wf,
cubical_set_wf,
cubicalpath-app_wf,
interval-0_wf,
interval-1_wf,
path-type-subtype,
term-to-pathtype_wf,
cube-context-adjoin_wf,
interval-type_wf,
csm-ap-type_wf,
cc-fst_wf,
csm-ap-term_wf,
pathtype_wf,
csm-pathtype,
interval-rev_wf,
cc-snd_wf,
cubical-path-app-1,
equal_wf,
squash_wf,
true_wf,
istype-universe,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term-equal,
cc_fst_adjoin_cube_lemma,
path-type-at,
subtype_rel-equal,
cubical-type-at_wf,
cube-set-restriction_wf,
nh-id_wf,
cube-set-restriction-id,
dma-neg_wf,
dM_wf,
dM0_wf,
cubical-path-app-0,
neg-dM1
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
sqequalRule,
universeIsType,
instantiate,
inhabitedIsType,
dependent_set_memberEquality_alt,
independent_pairFormation,
productIsType,
equalityIstype,
because_Cache,
lambdaFormation_alt,
dependent_functionElimination,
independent_functionElimination,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
universeEquality,
functionExtensionality,
independent_isectElimination,
Error :memTop,
setElimination,
rename
Latex:
\mforall{}[X:j\mvdash{}]. \mforall{}[A:\{X \mvdash{} \_\}]. \mforall{}[b,a:\{X \mvdash{} \_:A\}]. \mforall{}[pth:\{X \mvdash{} \_:(Path\_A a b)\}].
(rev-path(X;pth) \mmember{} \{X \mvdash{} \_:(Path\_A b a)\})
Date html generated:
2020_05_20-PM-03_22_24
Last ObjectModification:
2020_04_07-PM-03_32_58
Theory : cubical!type!theory
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