### Nuprl Lemma : urec-is-least-fixedpoint

`∀[F:Type ⟶ Type]. ∀T:Type. urec(F) ⊆r T supposing F T ≡ T supposing Monotone(T.F T)`

Proof

Definitions occuring in Statement :  urec: `urec(F)` type-monotone: `Monotone(T.F[T])` ext-eq: `A ≡ B` uimplies: `b supposing a` subtype_rel: `A ⊆r B` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` apply: `f a` function: `x:A ⟶ B[x]` universe: `Type`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` subtype_rel: `A ⊆r B` urec: `urec(F)` tunion: `⋃x:A.B[x]` pi2: `snd(t)` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` decidable: `Dec(P)` or: `P ∨ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` squash: `↓T` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` type-monotone: `Monotone(T.F[T])`
Lemmas referenced :  fun_exp_wf subtype_rel_weakening subtype_rel_transitivity iff_weakening_equal true_wf squash_wf subtype_rel_wf type-monotone_wf ext-eq_wf urec_wf subtract-add-cancel le_wf fun_exp_apply_add1 int_term_value_subtract_lemma int_formula_prop_not_lemma itermSubtract_wf intformnot_wf subtract_wf decidable__le fun_exp0_lemma less_than_wf ge_wf int_formula_prop_wf int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_and_lemma intformless_wf itermVar_wf itermConstant_wf intformle_wf intformand_wf satisfiable-full-omega-tt nat_properties
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lambdaFormation lambdaEquality sqequalHypSubstitution imageElimination productElimination thin sqequalRule hypothesisEquality applyEquality lemma_by_obid isectElimination hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination axiomEquality unionElimination instantiate because_Cache dependent_set_memberEquality universeEquality equalityTransitivity equalitySymmetry functionEquality imageMemberEquality baseClosed

Latex:
\mforall{}[F:Type  {}\mrightarrow{}  Type].  \mforall{}T:Type.  urec(F)  \msubseteq{}r  T  supposing  F  T  \mequiv{}  T  supposing  Monotone(T.F  T)

Date html generated: 2016_05_15-PM-06_54_53
Last ObjectModification: 2016_01_16-AM-09_48_47

Theory : general

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