Nuprl Lemma : list-injection

`∀[T:Type]`
`  ((∀x,y:T.  Dec(x = y ∈ T))`
`  `` (∀L:T List. ∀f:{x:T| (x ∈ L)}  ⟶ {x:T| (x ∈ L)} .`
`        ∀x:{x:T| (x ∈ L)} . ∃m:{1..||L|| + 1-}. ((f^m x) = x ∈ {x:T| (x ∈ L)} ) supposing Inj({x:T| (x ∈ L)} ;{x:T| (x ∈\000C L)} ;f)))`

Proof

Definitions occuring in Statement :  l_member: `(x ∈ l)` length: `||as||` list: `T List` fun_exp: `f^n` inject: `Inj(A;B;f)` int_seg: `{i..j-}` decidable: `Dec(P)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` set: `{x:A| B[x]} ` apply: `f a` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` implies: `P `` Q` all: `∀x:A. B[x]` uimplies: `b supposing a` member: `t ∈ T` inject: `Inj(A;B;f)` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` less_than: `a < b` squash: `↓T` surject: `Surj(A;B;f)` sq_stable: `SqStable(P)` l_member: `(x ∈ l)` nat: `ℕ` le: `A ≤ B` cand: `A c∧ B` ge: `i ≥ j `
Lemmas referenced :  equal_wf l_member_wf finite-injection decidable__equal_set length_wf_nat select_wf list-subtype int_seg_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma int_seg_wf length_wf set_wf inject_wf list_wf all_wf decidable_wf sq_stable__l_member lelt_wf select_member nat_properties
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation lambdaFormation cut introduction sqequalRule sqequalHypSubstitution lambdaEquality dependent_functionElimination thin hypothesisEquality axiomEquality hypothesis extract_by_obid isectElimination setEquality cumulativity applyEquality functionExtensionality setElimination rename dependent_set_memberEquality because_Cache independent_functionElimination equalityTransitivity equalitySymmetry independent_isectElimination productElimination unionElimination natural_numberEquality dependent_pairFormation int_eqEquality intEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll imageElimination functionEquality universeEquality imageMemberEquality baseClosed

Latex:
\mforall{}[T:Type]
((\mforall{}x,y:T.    Dec(x  =  y))
{}\mRightarrow{}  (\mforall{}L:T  List.  \mforall{}f:\{x:T|  (x  \mmember{}  L)\}    {}\mrightarrow{}  \{x:T|  (x  \mmember{}  L)\}  .
\mforall{}x:\{x:T|  (x  \mmember{}  L)\}  .  \mexists{}m:\{1..||L||  +  1\msupminus{}\}.  ((f\^{}m  x)  =  x)  supposing  Inj(\{x:T|  (x  \mmember{}  L)\}  ;\{x:T|  (x\000C  \mmember{}  L)\}  ;f)))

Date html generated: 2017_04_17-AM-07_47_32
Last ObjectModification: 2017_02_27-PM-04_18_28

Theory : list_1

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