Nuprl Lemma : csm-equiv_comp-sq
∀[H,K,tau,A,E,cA,cE:Top].
((equiv_comp(H;A;E;cA;cE))tau ~ sigma_comp(pi_comp(K;(A)tau;(cA)tau;((cE)tau)p);
pi_comp(K.((A)tau ⟶ (E)tau);((E)tau)p;((cE)tau)p;
contractible_comp(K.((A)tau ⟶ (E)tau).((E)tau)p;
(Fiber((q)p;q))tau++;
fiber-comp(K.((A)tau ⟶ (E)tau).((E)tau)p;
(((A)tau)p)p;(((E)tau)p)p;(q)p;q;
(((cA)tau)p)p;(((cE)tau)p)p)))))
Proof
Definitions occuring in Statement :
equiv_comp: equiv_comp(H;A;E;cA;cE),
contractible_comp: contractible_comp(X;A;cA),
fiber-comp: fiber-comp(X;T;A;w;a;cT;cA),
pi_comp: pi_comp(X;A;cA;cB),
sigma_comp: sigma_comp(cA;cB),
csm-comp-structure: (cA)tau,
cubical-fiber: Fiber(w;a),
cubical-fun: (A ⟶ B),
csm+: tau+,
cc-snd: q,
cc-fst: p,
cube-context-adjoin: X.A,
csm-ap-term: (t)s,
csm-ap-type: (AF)s,
uall: ∀[x:A]. B[x],
top: Top,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
equiv_comp: equiv_comp(H;A;E;cA;cE),
csm-comp-structure: (cA)tau,
csm-comp: G o F,
compose: f o g,
cc-fst: p,
pi1: fst(t),
csm+: tau+,
csm-adjoin: (s;u),
csm-ap: (s)x,
csm-ap-type: (AF)s,
interval-type: 𝕀,
cc-snd: q,
csm-ap-term: (t)s,
so_lambda: so_lambda4,
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
all: ∀x:A. B[x],
implies: P ⇒ Q,
pi2: snd(t)
Lemmas referenced :
csm-sigma_comp3,
istype-top,
csm-pi_comp,
lifting-strict-spread,
strict4-spread,
csm-contractible_comp,
csm-fiber-comp-sq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
Error :memTop,
sqequalRule,
hypothesis,
axiomSqEquality,
inhabitedIsType,
isect_memberEquality_alt,
isectIsTypeImplies,
baseClosed,
independent_isectElimination,
lambdaFormation_alt,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination
Latex:
\mforall{}[H,K,tau,A,E,cA,cE:Top].
((equiv\_comp(H;A;E;cA;cE))tau
\msim{} sigma\_comp(pi\_comp(K;(A)tau;(cA)tau;((cE)tau)p);
pi\_comp(K.((A)tau {}\mrightarrow{} (E)tau);((E)tau)p;((cE)tau)p;
contractible\_comp(K.((A)tau {}\mrightarrow{} (E)tau).((E)tau)p;(Fiber((q)p;q))tau++;
fiber-comp(K.((A)tau {}\mrightarrow{} (E)tau).((E)tau)p;(((A)tau)p)p;
(((E)tau)p)p;(q)p;q;(((cA)tau)p)p;
(((cE)tau)p)p)))))
Date html generated:
2020_05_20-PM-07_19_44
Last ObjectModification:
2020_04_27-PM-02_00_52
Theory : cubical!type!theory
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