### Nuprl Lemma : transitive-closure-minimal

`∀[A:Type]. ∀[R,Q:A ⟶ A ⟶ ℙ].  (R => Q `` Trans(A;x,y.x Q y) `` TC(R) => Q)`

Proof

Definitions occuring in Statement :  transitive-closure: `TC(R)` rel_implies: `R1 => R2` trans: `Trans(T;x,y.E[x; y])` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` implies: `P `` Q` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` rel_implies: `R1 => R2` all: `∀x:A. B[x]` trans: `Trans(T;x,y.E[x; y])` transitive-closure: `TC(R)` infix_ap: `x f y` member: `t ∈ T` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` and: `P ∧ Q` subtype_rel: `A ⊆r B` or: `P ∨ Q` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` false: `False` cons: `[a / b]` rel_path: `rel_path(A;L;x;y)` so_lambda: `so_lambda(x,y,z.t[x; y; z])` top: `Top` so_apply: `x[s1;s2;s3]` pi1: `fst(t)` pi2: `snd(t)` hd: `hd(l)` spreadn: spread3 sq_exists: `∃x:{A| B[x]}` nat: `ℕ` ge: `i ≥ j ` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` guard: `{T}` colength: `colength(L)` decidable: `Dec(P)` nil: `[]` it: `⋅` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)` cand: `A c∧ B`
Lemmas referenced :  transitive-closure_wf trans_wf rel_implies_wf list-cases length_of_nil_lemma product_subtype_list list_ind_cons_lemma reduce_tl_cons_lemma equal_wf nat_properties satisfiable-full-omega-tt intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf ge_wf less_than_wf rel_path_wf equal-wf-T-base nat_wf colength_wf_list less_than_transitivity1 less_than_irreflexivity list_wf list_ind_nil_lemma list_accum_nil_lemma spread_cons_lemma intformeq_wf itermAdd_wf int_formula_prop_eq_lemma int_term_value_add_lemma decidable__le intformnot_wf int_formula_prop_not_lemma le_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma subtype_base_sq set_subtype_base int_subtype_base decidable__equal_int list_accum_cons_lemma subtype_rel_self subtype_rel_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation sqequalHypSubstitution sqequalRule rename applyEquality cut introduction extract_by_obid isectElimination thin cumulativity hypothesisEquality functionExtensionality hypothesis lambdaEquality functionEquality universeEquality setElimination productElimination productEquality dependent_functionElimination unionElimination imageElimination voidElimination promote_hyp hypothesis_subsumption isect_memberEquality voidEquality hyp_replacement independent_functionElimination equalityTransitivity equalitySymmetry applyLambdaEquality instantiate intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation int_eqEquality intEquality independent_pairFormation computeAll axiomEquality because_Cache dependent_set_memberEquality addEquality baseClosed dependent_pairEquality

Latex:
\mforall{}[A:Type].  \mforall{}[R,Q:A  {}\mrightarrow{}  A  {}\mrightarrow{}  \mBbbP{}].    (R  =>  Q  {}\mRightarrow{}  Trans(A;x,y.x  Q  y)  {}\mRightarrow{}  TC(R)  =>  Q)

Date html generated: 2017_04_17-AM-09_25_53
Last ObjectModification: 2017_02_27-PM-05_26_53

Theory : relations2

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