Nuprl Lemma : pres-a0-constraint
∀[G:j⊢]. ∀[phi:{G ⊢ _:𝔽}]. ∀[A,T:{G.𝕀 ⊢ _}]. ∀[f:{G.𝕀 ⊢ _:(T ⟶ A)}]. ∀[t:{G.𝕀, (phi)p ⊢ _:T}].
∀[t0:{G ⊢ _:(T)[0(𝕀)][phi |⟶ t[0]]}]. ∀[cT:G.𝕀 +⊢ Compositon(T)].
((pres-a0(G;f;t0))p ∈ {G.𝕀 ⊢ _:((A)p+)[0(𝕀)][((phi)p ∨ (q=1)) |⟶ ((presw(G;phi;f;t;t0;cT))p+)[0(𝕀)]]})
Proof
Definitions occuring in Statement :
presw: presw(G;phi;f;t;t0;cT),
pres-a0: pres-a0(G;f;t0),
composition-structure: Gamma ⊢ Compositon(A),
partial-term-0: u[0],
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
context-subset: Gamma, phi,
face-one: (i=1),
face-or: (a ∨ b),
face-type: 𝔽,
interval-0: 0(𝕀),
interval-type: 𝕀,
cubical-fun: (A ⟶ B),
csm+: tau+,
csm-id-adjoin: [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],
member: t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
guard: {T},
cc-snd: q,
interval-type: 𝕀,
cc-fst: p,
csm-ap-type: (AF)s,
constant-cubical-type: (X),
uimplies: b supposing a,
composition-structure: Gamma ⊢ Compositon(A),
all: ∀x:A. B[x],
constrained-cubical-term: {Gamma ⊢ _:A[phi |⟶ t]},
true: True,
squash: ↓T,
prop: ℙ,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cubical-type: {X ⊢ _},
interval-0: 0(𝕀),
csm-id-adjoin: [u],
csm+: tau+,
csm-ap: (s)x,
csm-id: 1(X),
csm-adjoin: (s;u),
csm-comp: G o F,
pi2: snd(t),
compose: f o g,
pi1: fst(t),
csm-ap-term: (t)s,
same-cubical-term: X ⊢ u=v:A,
pres-a0: pres-a0(G;f;t0),
presw: presw(G;phi;f;t;t0;cT),
pres-v: pres-v(G;phi;t;t0;cT)
Lemmas referenced :
csm+_wf,
interval-type_wf,
cc-fst_wf_interval,
csm-interval-type,
context-subset-term-subtype,
cube-context-adjoin_wf,
cubical-fun_wf,
csm-ap-term_wf,
face-type_wf,
csm-face-type,
cubical-app_wf_fun,
thin-context-subset,
cubical-fun-subset,
subset-cubical-term,
context-subset_wf,
face-or_wf,
face-one_wf,
cc-snd_wf,
sub_cubical_set-cumulativity1,
sub_cubical_set_functionality,
context-subset-is-subset,
csm-ap-type_wf,
cubical_set_cumulativity-i-j,
csm-id-adjoin_wf,
interval-1_wf,
presw_wf,
composition-function-cumulativity,
constrained-cubical-term-eqcd,
composition-structure_wf,
interval-0_wf,
partial-term-0_wf,
istype-cubical-term,
cubical-type_wf,
cubical_set_wf,
cubical-term-eqcd,
csm-comp-type,
equal_wf,
squash_wf,
true_wf,
istype-universe,
cube_set_map_wf,
cc-fst+-comp-0,
subtype_rel_self,
iff_weakening_equal,
pres-a0_wf,
csm-same-cubical-term,
csm-cubical-app,
csm-cubical-fun,
fill_term_0,
csm-id-adjoin_wf-interval-0,
csm-context-subset-subtype2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
sqequalRule,
Error :memTop,
applyEquality,
instantiate,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
setElimination,
rename,
dependent_functionElimination,
lambdaEquality_alt,
cumulativity,
universeIsType,
universeEquality,
dependent_set_memberEquality_alt,
hyp_replacement,
natural_numberEquality,
imageElimination,
inhabitedIsType,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
applyLambdaEquality,
equalityIstype,
lambdaFormation_alt
Latex:
\mforall{}[G:j\mvdash{}]. \mforall{}[phi:\{G \mvdash{} \_:\mBbbF{}\}]. \mforall{}[A,T:\{G.\mBbbI{} \mvdash{} \_\}]. \mforall{}[f:\{G.\mBbbI{} \mvdash{} \_:(T {}\mrightarrow{} A)\}]. \mforall{}[t:\{G.\mBbbI{}, (phi)p \mvdash{} \_:T\}].
\mforall{}[t0:\{G \mvdash{} \_:(T)[0(\mBbbI{})][phi |{}\mrightarrow{} t[0]]\}]. \mforall{}[cT:G.\mBbbI{} +\mvdash{} Compositon(T)].
((pres-a0(G;f;t0))p \mmember{} \{G.\mBbbI{} \mvdash{} \_:((A)p+)[0(\mBbbI{})][((phi)p \mvee{} (q=1))
|{}\mrightarrow{} ((presw(G;phi;f;t;t0;cT))p+)[0(\mBbbI{})]]\})
Date html generated:
2020_05_20-PM-05_27_51
Last ObjectModification:
2020_05_02-PM-03_33_14
Theory : cubical!type!theory
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