### Nuprl Lemma : transitive-closure-minimal-uniform

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

Proof

Definitions occuring in Statement :  transitive-closure: `TC(R)` rel_implies: `R1 => R2` utrans: `UniformlyTrans(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]` utrans: `UniformlyTrans(T;x,y.E[x; y])` transitive-closure: `TC(R)` infix_ap: `x f y` member: `t ∈ T` subtype_rel: `A ⊆r B` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` and: `P ∧ Q` or: `P ∨ Q` less_than: `a < b` squash: `↓T` less_than': `less_than'(a;b)` length: `||as||` list_ind: list_ind nil: `[]` it: `⋅` 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 uimplies: `b supposing a` nat: `ℕ` ge: `i ≥ j ` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` decidable: `Dec(P)` colength: `colength(L)` guard: `{T}` so_lambda: `λ2x.t[x]` so_apply: `x[s]` sq_type: `SQType(T)`
Lemmas referenced :  transitive-closure_wf subtype_rel_self utrans_wf rel_implies_wf istype-universe list-cases product_subtype_list list_ind_cons_lemma istype-void reduce_tl_cons_lemma subtype_rel-equal subtype_rel_wf nat_properties full-omega-unsat intformand_wf intformle_wf itermConstant_wf itermVar_wf intformless_wf istype-int 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 istype-less_than list_ind_nil_lemma list_accum_nil_lemma colength-cons-not-zero colength_wf_list decidable__le intformnot_wf int_formula_prop_not_lemma istype-le list_wf subtract-1-ge-0 subtype_base_sq intformeq_wf int_formula_prop_eq_lemma set_subtype_base int_subtype_base spread_cons_lemma decidable__equal_int subtract_wf itermSubtract_wf itermAdd_wf int_term_value_subtract_lemma int_term_value_add_lemma le_wf list_accum_cons_lemma rel_path_wf istype-nat equal_wf subtype_rel_function
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  Error :lambdaFormation_alt,  sqequalHypSubstitution sqequalRule rename Error :universeIsType,  cut applyEquality introduction extract_by_obid isectElimination thin hypothesisEquality hypothesis instantiate because_Cache Error :lambdaEquality_alt,  Error :inhabitedIsType,  Error :functionIsType,  universeEquality setElimination productElimination productEquality dependent_functionElimination unionElimination imageElimination voidElimination promote_hyp hypothesis_subsumption Error :isect_memberEquality_alt,  independent_isectElimination equalitySymmetry Error :dependent_set_memberEquality_alt,  independent_pairFormation Error :productIsType,  Error :equalityIstype,  applyLambdaEquality equalityTransitivity hyp_replacement independent_functionElimination intWeakElimination natural_numberEquality approximateComputation Error :dependent_pairFormation_alt,  int_eqEquality axiomEquality Error :functionIsTypeImplies,  baseApply closedConclusion baseClosed intEquality sqequalBase Error :isectIsType,  functionExtensionality

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

Date html generated: 2019_06_20-PM-02_01_28
Last ObjectModification: 2018_12_07-AM-01_41_48

Theory : relations2

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