Nuprl Lemma : rev_fill_term_0
∀[Gamma:j⊢]. ∀[phi:{Gamma ⊢ _:𝔽}]. ∀[A:{Gamma.𝕀 ⊢ _}]. ∀[cA:Gamma.𝕀 ⊢ Compositon(A)]. ∀[u:{Gamma.𝕀, (phi)p ⊢ _:A}].
∀[a1:{Gamma ⊢ _:(A)[1(𝕀)][phi |⟶ u[1]]}].
((rev_fill_term(Gamma;cA;phi;u;a1))[0(𝕀)]
= comp rev-type-comp(Gamma;cA) [phi ⊢→ (u)(p;1-(q))] a1
∈ {Gamma ⊢ _:(A)[0(𝕀)]})
Proof
Definitions occuring in Statement :
rev_fill_term: rev_fill_term(Gamma;cA;phi;u;a1),
comp_term: comp cA [phi ⊢→ u] a0,
rev-type-comp: rev-type-comp(Gamma;cA),
composition-structure: Gamma ⊢ Compositon(A),
partial-term-1: u[1],
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-type: 𝔽,
interval-rev: 1-(r),
interval-1: 1(𝕀),
interval-0: 0(𝕀),
interval-type: 𝕀,
csm-id-adjoin: [u],
csm-adjoin: (s;u),
cc-snd: q,
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],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rev_fill_term: rev_fill_term(Gamma;cA;phi;u;a1),
rev-type-line: (A)-,
guard: {T},
csm-ap-term: (t)s,
interval-rev: 1-(r),
csm-adjoin: (s;u),
cubical-term-at: u(a),
csm-ap: (s)x,
pi1: fst(t),
squash: ↓T,
prop: ℙ,
true: True,
partial-term-0: u[0],
partial-term-1: u[1],
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm-id: 1(X),
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
interval-1: 1(𝕀),
cube_set_map: A ⟶ B,
psc_map: A ⟶ B,
nat-trans: nat-trans(C;D;F;G),
cat-ob: cat-ob(C),
op-cat: op-cat(C),
spreadn: spread4,
cube-cat: CubeCat,
fset: fset(T),
quotient: x,y:A//B[x; y],
cat-arrow: cat-arrow(C),
type-cat: TypeCat,
names-hom: I ⟶ J,
cat-comp: cat-comp(C),
compose: f o g,
composition-structure: Gamma ⊢ Compositon(A),
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]}
Lemmas referenced :
interval-rev_wf,
cube-context-adjoin_wf,
interval-type_wf,
cc-snd_wf,
subset-cubical-term2,
sub_cubical_set_self,
csm-ap-type_wf,
cc-fst_wf,
csm-interval-type,
constrained-cubical-term_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf-interval-1,
cubical-type-cumulativity2,
partial-term-1_wf,
cubical-term_wf,
context-subset_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
thin-context-subset,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
context-subset-map,
csm-adjoin_wf,
squash_wf,
true_wf,
rev-type-line-0,
cubical-type-cumulativity,
context-adjoin-subset2,
cube_set_map_wf,
context-subset-adjoin-subtype,
interval-rev-0,
csm-id-adjoin_wf,
subtype_rel_self,
fill_term_1,
rev-type-line_wf,
rev-type-comp_wf,
rev-type-line-1,
csm-id-adjoin_wf-interval-0,
equal_wf,
istype-universe,
fill_term_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
instantiate,
hypothesis,
hypothesisEquality,
sqequalRule,
equalityTransitivity,
equalitySymmetry,
applyEquality,
because_Cache,
independent_isectElimination,
Error :memTop,
universeIsType,
lambdaEquality_alt,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
inhabitedIsType,
lambdaFormation_alt,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
applyLambdaEquality,
universeEquality,
setElimination,
rename
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A:\{Gamma.\mBbbI{} \mvdash{} \_\}]. \mforall{}[cA:Gamma.\mBbbI{} \mvdash{} Compositon(A)].
\mforall{}[u:\{Gamma.\mBbbI{}, (phi)p \mvdash{} \_:A\}]. \mforall{}[a1:\{Gamma \mvdash{} \_:(A)[1(\mBbbI{})][phi |{}\mrightarrow{} u[1]]\}].
((rev\_fill\_term(Gamma;cA;phi;u;a1))[0(\mBbbI{})] = comp rev-type-comp(Gamma;cA) [phi \mvdash{}\mrightarrow{} (u)(p;1-(q))] a1)
Date html generated:
2020_05_20-PM-04_52_00
Last ObjectModification:
2020_04_14-AM-11_54_05
Theory : cubical!type!theory
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