### Nuprl Lemma : un-half-squash-test

`∀[P,Q,R:ℙ].  (((P ∧ R) `` Q) `` ⇃(R) `` half-squash-stable(Q) `` ⇃(P) `` True `` {Q ∧ ⇃(P) ∧ (∀n:ℕ. Q)})`

Proof

Definitions occuring in Statement :  half-squash-stable: `half-squash-stable(P)` quotient: `x,y:A//B[x; y]` nat: `ℕ` uall: `∀[x:A]. B[x]` prop: `ℙ` guard: `{T}` all: `∀x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` true: `True`
Definitions unfolded in proof :  cand: `A c∧ B` half-squash-stable: `half-squash-stable(P)` all: `∀x:A. B[x]` so_apply: `x[s]` so_lambda: `λ2x.t[x]` uimplies: `b supposing a` so_apply: `x[s1;s2]` so_lambda: `λ2x y.t[x; y]` and: `P ∧ Q` prop: `ℙ` member: `t ∈ T` guard: `{T}` implies: `P `` Q` uall: `∀[x:A]. B[x]`
Lemmas referenced :  trivial-quotient-true half-squash-stable_wf implies-quotient-true half-squash-stable__all half-squash-stable__half-squash nat_wf all_wf equiv_rel_true true_wf quotient_wf half-squash-stable__and
Rules used in proof :  independent_pairFormation universeEquality functionEquality promote_hyp cumulativity independent_functionElimination independent_isectElimination hypothesis lambdaEquality because_Cache productEquality hypothesisEquality thin isectElimination sqequalHypSubstitution extract_by_obid introduction sqequalRule cut lambdaFormation isect_memberFormation sqequalReflexivity computationStep sqequalTransitivity sqequalSubstitution

Latex:
\mforall{}[P,Q,R:\mBbbP{}].
(((P  \mwedge{}  R)  {}\mRightarrow{}  Q)  {}\mRightarrow{}  \00D9(R)  {}\mRightarrow{}  half-squash-stable(Q)  {}\mRightarrow{}  \00D9(P)  {}\mRightarrow{}  True  {}\mRightarrow{}  \{Q  \mwedge{}  \00D9(P)  \mwedge{}  (\mforall{}n:\mBbbN{}.  Q)\})

Date html generated: 2017_09_29-PM-05_48_13
Last ObjectModification: 2017_08_30-AM-11_48_12

Theory : quot_1

Home Index