Nuprl Lemma : path_comp_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[a,b:{G ⊢ _:A}]. ∀[cA:G ⊢ Compositon(A)].
(path_comp(G;A;a;b;
cA) ∈ G ⊢ Compositon((Path_A a b)))
Proof
Definitions occuring in Statement :
path_comp: path_comp,
composition-structure: Gamma ⊢ Compositon(A),
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,
composition-structure: Gamma ⊢ Compositon(A),
prop: ℙ,
csm-ap-term: (t)s,
pi1: fst(t),
pi2: snd(t),
csm-id: 1(X),
csm-id-adjoin: [u],
interval-1: 1(𝕀),
cubical-type: {X ⊢ _},
csm-ap: (s)x,
csm-adjoin: (s;u),
compose: f o g,
csm-comp-structure: (cA)tau,
csm-comp: G o F,
csm+: tau+,
constant-cubical-type: (X),
csm-ap-type: (AF)s,
cc-fst: p,
interval-type: 𝕀,
cc-snd: q,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
squash: ↓T,
composition-function: composition-function{j:l,i:l}(Gamma;A),
path_comp: path_comp,
interval-0: 0(𝕀),
path-elim: path-elim(pth),
istype: istype(T),
cubical-term-at: u(a),
context-subset: Gamma, phi,
respects-equality: respects-equality(S;T),
term-to-pathtype: <>a,
cubical-subset: cubical-subset,
path-type: (Path_A a b),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
path_term: path_term(phi; w; a; b; r)
Lemmas referenced :
uniform-comp-function_wf,
path-type_wf,
composition-structure_wf,
istype-cubical-term,
cubical-type_wf,
cubical_set_wf,
cube_set_map_wf,
cubical-type-cumulativity2,
csm-id-adjoin_wf-interval-0,
constrained-cubical-term_wf,
context-adjoin-subset4,
csm-context-subset-subtype3,
context-adjoin-subset1,
sub_cubical_set_self,
sub_cubical_set_transitivity,
subset-cubical-term,
context-subset-map,
csm-context-subset-subtype2,
csm-comp-structure_wf,
csm+_wf_interval,
csm-comp-structure-composition-function,
face-one_wf,
cc-snd_wf,
face-zero_wf,
csm-face-type,
face-type_wf,
face-or_wf,
comp_term_wf,
csm-interval-type,
cubical_set_cumulativity-i-j,
cc-fst_wf_interval,
csm+_wf,
cubical-term-eqcd,
iff_weakening_equal,
subtype_rel_self,
csm-ap-term_wf,
csm-ap-type_wf,
context-subset-is-subset,
sub_cubical_set_functionality,
subset-cubical-type,
csm-path-type,
interval-type_wf,
context-subset_wf,
cube-context-adjoin_wf,
istype-universe,
true_wf,
squash_wf,
equal_wf,
path_term_wf,
thin-context-subset-adjoin,
csm-face-or,
path_term-0,
cubical-term_wf,
cc-fst_wf,
cubical-type-cumulativity,
cubical-path-app_wf,
path-term-equal,
csm-id-adjoin_wf-interval-1,
cubical-path-app-sq,
context-subset-term-subtype,
thin-context-subset,
path-type-sub-pathtype,
path-type-subset,
interval-0_wf,
csm-id-adjoin_wf,
path-elim_wf,
cubical-path-app-0,
subset-cubical-term2,
cubical-term-equal,
nat_wf,
fset_wf,
I_cube_wf,
face-zero-eq-1,
I_cube_pair_redex_lemma,
cubical-path-app-1,
interval-1_wf,
face-one-eq-1,
path-term_wf,
respects-equality-context-subset-term,
path_term-1,
csm-face-zero,
csm-face-one,
context-1-subset,
face-one-interval-1,
csm-id_wf,
csm_id_adjoin_fst_term_lemma,
sub_cubical_set_wf,
face-or-1,
csm-path-term,
cc_snd_csm_id_adjoin_lemma,
path-term-1,
face-zero-interval-0,
face-1-or,
path-term-0,
term-to-path_wf,
paths-equal,
pathtype_wf,
pathtype-subset,
term-to-pathtype_wf,
csm-id-adjoin-subset,
csm-context-subset-subtype,
path-term-case1,
face-term-implies-or1,
face-term-implies-subset,
context-adjoin-subset2,
term-to-pathtype-eta,
path-type-subtype,
term-to-path-subset,
cubical_type_at_pair_lemma,
constrained-cubical-term-eqcd,
csm-constrained-cubical-term,
csm-paths-equal,
sub_cubical_set-cumulativity1,
csm-comp_term,
csm-path_term,
csm-cubical-path-app,
subtype_rel-equal,
cube_set_map_cumulativity-i-j,
csm-pathtype,
csm-term-to-pathtype
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
dependent_set_memberEquality_alt,
sqequalHypSubstitution,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
instantiate,
independent_pairFormation,
Error :memTop,
hyp_replacement,
rename,
setElimination,
cumulativity,
independent_functionElimination,
productElimination,
baseClosed,
imageMemberEquality,
natural_numberEquality,
independent_isectElimination,
dependent_functionElimination,
universeEquality,
imageElimination,
lambdaEquality_alt,
applyEquality,
sqequalRule,
functionExtensionality,
inhabitedIsType,
applyLambdaEquality,
equalityIstype,
lambdaFormation_alt,
equalityElimination,
promote_hyp,
functionEquality,
setEquality
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A:\{G \mvdash{} \_\}]. \mforall{}[a,b:\{G \mvdash{} \_:A\}]. \mforall{}[cA:G \mvdash{} Compositon(A)].
(path\_comp(G;A;a;b;
cA) \mmember{} G \mvdash{} Compositon((Path\_A a b)))
Date html generated:
2020_05_20-PM-05_12_09
Last ObjectModification:
2020_05_01-PM-11_52_08
Theory : cubical!type!theory
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