### Nuprl Lemma : stable-union-decomp

`∀[X,T:Type]. ∀[P:T ⟶ X ⟶ ℙ]. ∀[S:T ⟶ Type]. ∀[Q:i:T ⟶ S[i] ⟶ X ⟶ ℙ].`
`  stable-union(X;T;i,x.P[i;x]) ≡ stable-union(X;i:T × S[i];p,x.Q[fst(p);snd(p);x]) `
`  supposing (∀x:X. ∀i:T. ∀s:S[i].  (Q[i;s;x] `` (¬¬P[i;x]))) ∧ (∀x:X. ∀i:T.  (P[i;x] `` (¬¬(∃s:S[i]. Q[i;s;x]))))`

Proof

Definitions occuring in Statement :  stable-union: `stable-union(X;T;i,x.P[i; x])` ext-eq: `A ≡ B` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` prop: `ℙ` so_apply: `x[s1;s2;s3]` so_apply: `x[s1;s2]` so_apply: `x[s]` pi1: `fst(t)` pi2: `snd(t)` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` not: `¬A` implies: `P `` Q` and: `P ∧ Q` function: `x:A ⟶ B[x]` product: `x:A × B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` and: `P ∧ Q` ext-eq: `A ≡ B` subtype_rel: `A ⊆r B` stable-union: `stable-union(X;T;i,x.P[i; x])` prop: `ℙ` exists: `∃x:A. B[x]` so_apply: `x[s1;s2]` so_apply: `x[s]` so_apply: `x[s1;s2;s3]` so_lambda: `λ2x.t[x]` implies: `P `` Q` not: `¬A` false: `False` so_lambda: `λ2x y.t[x; y]` pi1: `fst(t)` pi2: `snd(t)` all: `∀x:A. B[x]`
Lemmas referenced :  double-negation-hyp-elim not_wf pi1_wf pi2_wf istype-void stable-union_wf subtype_rel_self istype-universe
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation_alt introduction cut sqequalHypSubstitution productElimination thin independent_pairFormation lambdaEquality_alt setElimination rename dependent_set_memberEquality_alt hypothesisEquality extract_by_obid isectElimination sqequalRule productEquality applyEquality universeIsType hypothesis independent_functionElimination lambdaFormation_alt productIsType because_Cache functionIsType dependent_functionElimination dependent_pairFormation_alt voidElimination independent_pairEquality axiomEquality instantiate universeEquality isect_memberEquality_alt isectIsTypeImplies inhabitedIsType dependent_pairEquality_alt

Latex:
\mforall{}[X,T:Type].  \mforall{}[P:T  {}\mrightarrow{}  X  {}\mrightarrow{}  \mBbbP{}].  \mforall{}[S:T  {}\mrightarrow{}  Type].  \mforall{}[Q:i:T  {}\mrightarrow{}  S[i]  {}\mrightarrow{}  X  {}\mrightarrow{}  \mBbbP{}].
stable-union(X;T;i,x.P[i;x])  \mequiv{}  stable-union(X;i:T  \mtimes{}  S[i];p,x.Q[fst(p);snd(p);x])
supposing  (\mforall{}x:X.  \mforall{}i:T.  \mforall{}s:S[i].    (Q[i;s;x]  {}\mRightarrow{}  (\mneg{}\mneg{}P[i;x])))
\mwedge{}  (\mforall{}x:X.  \mforall{}i:T.    (P[i;x]  {}\mRightarrow{}  (\mneg{}\mneg{}(\mexists{}s:S[i].  Q[i;s;x]))))

Date html generated: 2020_05_19-PM-09_36_17
Last ObjectModification: 2019_10_24-AM-10_17_20

Theory : int_1

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