Nuprl Lemma : trans-const-path_wf
∀[G:j⊢]. ∀[A:{G ⊢ _}]. ∀[cA:G +⊢ Compositon(A)]. ∀[a:{G ⊢ _:A}].
(trans-const-path(G;cA;a) ∈ {G ⊢ _:(Path_A transprt-const(G;cA;a) a)})
Proof
Definitions occuring in Statement :
trans-const-path: trans-const-path(G;cA;a),
transprt-const: transprt-const(G;cA;a),
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,
trans-const-path: trans-const-path(G;cA;a),
subtype_rel: A ⊆r B,
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
squash: ↓T,
prop: ℙ,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
respects-equality: respects-equality(S;T),
composition-structure: Gamma ⊢ Compositon(A),
composition-function: composition-function{j:l,i:l}(Gamma;A),
uniform-comp-function: uniform-comp-function{j:l, i:l}(Gamma; A; comp),
same-cubical-term: X ⊢ u=v:A,
rev-type-comp: rev-type-comp(Gamma;cA),
csm-comp-structure: (cA)tau,
cc-fst: p,
interval-type: 𝕀,
csm-comp: G o F,
cc-snd: q,
interval-rev: 1-(r),
csm-adjoin: (s;u),
compose: f o g,
cubical-term-at: u(a),
csm-ap: (s)x,
pi1: fst(t),
transprt-const: transprt-const(G;cA;a)
Lemmas referenced :
rev_fill_term_wf,
face-0_wf,
csm-ap-type_wf,
cube-context-adjoin_wf,
interval-type_wf,
cc-fst_wf,
csm-comp-structure_wf,
composition-structure-cumulativity,
csm-face-0,
empty-context-subset-lemma3,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cubical-term_wf,
cubical-type-cumulativity2,
cubical_set_cumulativity-i-j,
csm-ap-type-fst-id-adjoin,
subtype_rel_self,
iff_weakening_equal,
csm_id_adjoin_fst_type_lemma,
istype-cubical-term,
context-subset_wf,
csm-id-adjoin_wf,
interval-1_wf,
csm-context-subset-subtype2,
respects-equality-context-subset-term,
csm-id-adjoin_wf-interval-1,
term-to-path-wf,
transprt-const_wf,
composition-structure_wf,
same-cubical-term_wf,
rev_fill_term_1,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-id-type,
rev_fill_term_0,
csm-id-adjoin_wf-interval-0,
cubical-type_wf,
cubical_set_wf,
equals-transprt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
instantiate,
applyEquality,
because_Cache,
Error :memTop,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality_alt,
lambdaEquality_alt,
imageElimination,
universeIsType,
universeEquality,
closedConclusion,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
hyp_replacement,
dependent_functionElimination,
equalityIstype,
setElimination,
rename,
inhabitedIsType,
axiomEquality,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[A:\{G \mvdash{} \_\}]. \mforall{}[cA:G +\mvdash{} Compositon(A)]. \mforall{}[a:\{G \mvdash{} \_:A\}].
(trans-const-path(G;cA;a) \mmember{} \{G \mvdash{} \_:(Path\_A transprt-const(G;cA;a) a)\})
Date html generated:
2020_05_20-PM-04_57_13
Last ObjectModification:
2020_04_14-PM-10_36_35
Theory : cubical!type!theory
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