### Nuprl Lemma : non-forking-wellfounded-linorder

`∀[T:Type]. ∀[R:T ⟶ T ⟶ ℙ].`
`  (decidable-non-minimal(T;x,y.R[x;y])`
`  `` WellFnd{i}(T;x,y.R[x;y])`
`  `` (∀m:T. (unique-minimal(T;x,y.R[x;y];m) `` non-forking(T;x,y.R[x;y]) `` WeakLinorder(T;x,y.x (R^*) y))))`

Proof

Definitions occuring in Statement :  non-forking: `non-forking(T;x,y.R[x; y])` decidable-non-minimal: `decidable-non-minimal(T;x,y.R[x; y])` unique-minimal: `unique-minimal(T;x,y.R[x; y];m)` rel_star: `R^*` weak-linorder: `WeakLinorder(T;x,y.R[x; y])` wellfounded: `WellFnd{i}(A;x,y.R[x; y])` uall: `∀[x:A]. B[x]` prop: `ℙ` infix_ap: `x f y` so_apply: `x[s1;s2]` all: `∀x:A. B[x]` implies: `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` weak-linorder: `WeakLinorder(T;x,y.R[x; y])` and: `P ∧ Q` infix_ap: `x f y` so_apply: `x[s1;s2]` prop: `ℙ` so_lambda: `λ2x y.t[x; y]` weak-connex: `weak-connex(T; x,y.R[x; y])` squash: `↓T` true: `True` subtype_rel: `A ⊆r B` uimplies: `b supposing a` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` rel_star: `R^*` exists: `∃x:A. B[x]` nat: `ℕ` decidable: `Dec(P)` or: `P ∨ Q` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` not: `¬A` top: `Top` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtract: `n - m` non-forking: `non-forking(T;x,y.R[x; y])`
Lemmas referenced :  unique-minimal-wellfounded-implies rel_star_order non-forking_wf unique-minimal_wf wellfounded_wf decidable-non-minimal_wf rel_star_wf equal_wf squash_wf true_wf eta_conv iff_weakening_equal decidable__lt subtract_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf le_wf rel_exp_add_iff infix_ap_wf rel_exp_wf exists_wf nat_wf minus-one-mul add-swap add-mul-special zero-mul add-zero non-forking-rel_exp add-commutes add-associates zero-add
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesisEquality independent_functionElimination hypothesis dependent_functionElimination independent_pairFormation sqequalRule cumulativity lambdaEquality applyEquality functionExtensionality functionEquality universeEquality imageElimination imageMemberEquality baseClosed isect_memberEquality hyp_replacement equalitySymmetry instantiate equalityTransitivity natural_numberEquality independent_isectElimination productElimination setElimination rename unionElimination inlFormation dependent_pairFormation dependent_set_memberEquality int_eqEquality intEquality voidElimination voidEquality computeAll applyLambdaEquality inrFormation

Latex:
\mforall{}[T:Type].  \mforall{}[R:T  {}\mrightarrow{}  T  {}\mrightarrow{}  \mBbbP{}].
(decidable-non-minimal(T;x,y.R[x;y])
{}\mRightarrow{}  WellFnd\{i\}(T;x,y.R[x;y])
{}\mRightarrow{}  (\mforall{}m:T
(unique-minimal(T;x,y.R[x;y];m)
{}\mRightarrow{}  non-forking(T;x,y.R[x;y])
{}\mRightarrow{}  WeakLinorder(T;x,y.x  rel\_star(T;  R)  y))))

Date html generated: 2018_05_21-PM-09_05_34
Last ObjectModification: 2017_07_26-PM-06_28_24

Theory : general

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