Nuprl Lemma : test-example4
∀[Dom:Type]. ∀[A,B,C,D,P,Q,G:Dom ⟶ ℙ]. ∀[R1,R2:Dom ⟶ Dom ⟶ ℙ].
((∀x:Dom. (A[x] ∨ B[x]))
⇒ (∀x:Dom. (C[x] ∨ D[x]))
⇒ (∀x:Dom. ((A[x] ∧ C[x]) ⇒ (∃y:Dom. (R1[x;y] ∧ P[y]))))
⇒ (∀x:Dom. ((A[x] ∧ D[x]) ⇒ (∃y:Dom. (R1[x;y] ∧ Q[y]))))
⇒ (∀x:Dom. ((B[x] ∧ C[x]) ⇒ (∃y:Dom. (R1[x;y] ∧ P[y]))))
⇒ (∀x:Dom. ((B[x] ∧ D[x]) ⇒ (∃y:Dom. (R1[x;y] ∧ Q[y]))))
⇒ (∀x:Dom. ((A[x] ∧ C[x]) ⇒ (∃y:Dom. (R2[x;y] ∧ Q[y]))))
⇒ (∀x:Dom. ((A[x] ∧ D[x]) ⇒ (∃y:Dom. (R2[x;y] ∧ P[y]))))
⇒ (∀x:Dom. ((B[x] ∧ C[x]) ⇒ (∃y:Dom. (R2[x;y] ∧ Q[y]))))
⇒ (∀x:Dom. ((B[x] ∧ D[x]) ⇒ (∃y:Dom. (R2[x;y] ∧ P[y]))))
⇒ (∀x,y,z:Dom. (((R1[x;y] ∧ R2[x;z]) ∧ ((P[y] ∧ Q[z]) ∨ (Q[y] ∧ P[z]))) ⇒ G[x]))
⇒ (∀x:Dom. G[x]))
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
or: P ∨ Q,
and: P ∧ Q,
exists: ∃x:A. B[x],
prop: ℙ,
so_apply: x[s],
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
subtype_rel: A ⊆r B
Lemmas referenced :
and_wf,
all_wf,
or_wf,
exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
rename,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
unionElimination,
because_Cache,
independent_functionElimination,
independent_pairFormation,
productElimination,
inlFormation,
lemma_by_obid,
isectElimination,
applyEquality,
sqequalRule,
inrFormation,
cumulativity,
lambdaEquality,
functionEquality,
productEquality,
universeEquality
Latex:
\mforall{}[Dom:Type]. \mforall{}[A,B,C,D,P,Q,G:Dom {}\mrightarrow{} \mBbbP{}]. \mforall{}[R1,R2:Dom {}\mrightarrow{} Dom {}\mrightarrow{} \mBbbP{}].
((\mforall{}x:Dom. (A[x] \mvee{} B[x]))
{}\mRightarrow{} (\mforall{}x:Dom. (C[x] \mvee{} D[x]))
{}\mRightarrow{} (\mforall{}x:Dom. ((A[x] \mwedge{} C[x]) {}\mRightarrow{} (\mexists{}y:Dom. (R1[x;y] \mwedge{} P[y]))))
{}\mRightarrow{} (\mforall{}x:Dom. ((A[x] \mwedge{} D[x]) {}\mRightarrow{} (\mexists{}y:Dom. (R1[x;y] \mwedge{} Q[y]))))
{}\mRightarrow{} (\mforall{}x:Dom. ((B[x] \mwedge{} C[x]) {}\mRightarrow{} (\mexists{}y:Dom. (R1[x;y] \mwedge{} P[y]))))
{}\mRightarrow{} (\mforall{}x:Dom. ((B[x] \mwedge{} D[x]) {}\mRightarrow{} (\mexists{}y:Dom. (R1[x;y] \mwedge{} Q[y]))))
{}\mRightarrow{} (\mforall{}x:Dom. ((A[x] \mwedge{} C[x]) {}\mRightarrow{} (\mexists{}y:Dom. (R2[x;y] \mwedge{} Q[y]))))
{}\mRightarrow{} (\mforall{}x:Dom. ((A[x] \mwedge{} D[x]) {}\mRightarrow{} (\mexists{}y:Dom. (R2[x;y] \mwedge{} P[y]))))
{}\mRightarrow{} (\mforall{}x:Dom. ((B[x] \mwedge{} C[x]) {}\mRightarrow{} (\mexists{}y:Dom. (R2[x;y] \mwedge{} Q[y]))))
{}\mRightarrow{} (\mforall{}x:Dom. ((B[x] \mwedge{} D[x]) {}\mRightarrow{} (\mexists{}y:Dom. (R2[x;y] \mwedge{} P[y]))))
{}\mRightarrow{} (\mforall{}x,y,z:Dom. (((R1[x;y] \mwedge{} R2[x;z]) \mwedge{} ((P[y] \mwedge{} Q[z]) \mvee{} (Q[y] \mwedge{} P[z]))) {}\mRightarrow{} G[x]))
{}\mRightarrow{} (\mforall{}x:Dom. G[x]))
Date html generated:
2016_05_15-PM-03_19_23
Last ObjectModification:
2015_12_27-PM-01_04_03
Theory : general
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