Nuprl Lemma : csm-rev_fill_term
∀[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]]}]. ∀[Delta:j⊢]. ∀[s:Delta j⟶ Gamma].
((rev_fill_term(Gamma;cA;phi;u;a1))s+
= rev_fill_term(Delta;(cA)s+;(phi)s;(u)s+;(a1)s)
∈ {Delta.𝕀 ⊢ _:(A)s+[((phi)s)p |⟶ (u)s+]})
Proof
Definitions occuring in Statement :
rev_fill_term: rev_fill_term(Gamma;cA;phi;u;a1),
csm-comp-structure: (cA)tau,
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-1: 1(𝕀),
interval-type: 𝕀,
csm+: tau+,
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 ⊢ _},
cube_set_map: A ⟶ B,
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,
csm+: tau+,
csm-comp: G o F,
guard: {T},
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
squash: ↓T,
cubical-type: {X ⊢ _},
csm-ap-term: (t)s,
csm-adjoin: (s;u),
csm-ap: (s)x,
pi1: fst(t),
compose: f o g,
rev-type-line: (A)-,
all: ∀x:A. B[x],
implies: P ⇒ Q,
prop: ℙ,
true: True,
partial-term-0: u[0],
partial-term-1: u[1],
interval-1: 1(𝕀),
csm-id-adjoin: [u],
interval-rev: 1-(r),
interval-0: 0(𝕀),
csm-id: 1(X),
cubical-term-at: u(a),
pi2: snd(t),
rev_fill_term: rev_fill_term(Gamma;cA;phi;u;a1),
rev-type-comp: rev-type-comp(Gamma;cA),
csm-comp-structure: (cA)tau
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,
csm-constrained-cubical-term,
csm+_wf_interval,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
rev_fill_term_wf,
cube_set_map_wf,
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,
thin-context-subset,
composition-structure_wf,
cubical-type_wf,
cubical_set_wf,
csm_ap_term_fst_adjoin_lemma,
rev-type-line_wf,
csm-adjoin_wf,
context-subset-map,
csm_id_adjoin_fst_term_lemma,
squash_wf,
true_wf,
csm-ap-id-term,
cubical-type-cumulativity,
rev-type-line-0,
dma-neg-dM0,
csm-id-adjoin_wf,
interval-1_wf,
csm-fill_term,
rev-type-comp_wf,
rev-rev-type-line,
csm-context-subset-subtype2,
context-subset-term-subtype
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,
inhabitedIsType,
applyLambdaEquality,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
productElimination,
dependent_functionElimination,
lambdaEquality_alt,
cumulativity,
universeEquality,
lambdaFormation_alt,
hyp_replacement,
equalityIstype,
independent_functionElimination,
natural_numberEquality
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]]\}]. \mforall{}[Delta:j\mvdash{}].
\mforall{}[s:Delta j{}\mrightarrow{} Gamma].
((rev\_fill\_term(Gamma;cA;phi;u;a1))s+ = rev\_fill\_term(Delta;(cA)s+;(phi)s;(u)s+;(a1)s))
Date html generated:
2020_05_20-PM-04_52_21
Last ObjectModification:
2020_04_13-PM-09_45_25
Theory : cubical!type!theory
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