Nuprl Lemma : funinv-funinv

`∀[n:ℕ]. ∀[f:{f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} ].  (inv(inv(f)) = f ∈ {f:ℕn ⟶ ℕn| Inj(ℕn;ℕn;f)} )`

Proof

Definitions occuring in Statement :  funinv: `inv(f)` inject: `Inj(A;B;f)` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` set: `{x:A| B[x]} ` function: `x:A ⟶ B[x]` natural_number: `\$n` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` squash: `↓T` nat: `ℕ` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` inject: `Inj(A;B;f)` all: `∀x:A. B[x]` subtype_rel: `A ⊆r B` implies: `P `` Q` and: `P ∧ Q` guard: `{T}` int_seg: `{i..j-}` lelt: `i ≤ j < k` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top`
Lemmas referenced :  int_formula_prop_less_lemma intformless_wf decidable__lt int_term_value_constant_lemma int_formula_prop_le_lemma itermConstant_wf intformle_wf decidable__le int_formula_prop_wf int_term_value_var_lemma int_formula_prop_eq_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermVar_wf intformeq_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__equal_int nat_properties lelt_wf int_seg_properties funinv-property nat_wf set_wf inject_wf int_seg_wf funinv_wf2
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis applyEquality lambdaEquality setElimination rename sqequalRule imageMemberEquality baseClosed setEquality functionEquality natural_numberEquality imageElimination dependent_set_memberEquality functionExtensionality isect_memberEquality axiomEquality because_Cache dependent_functionElimination independent_functionElimination equalityTransitivity equalitySymmetry productElimination intEquality unionElimination independent_isectElimination dependent_pairFormation int_eqEquality voidElimination voidEquality independent_pairFormation computeAll

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f:\{f:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}n|  Inj(\mBbbN{}n;\mBbbN{}n;f)\}  ].    (inv(inv(f))  =  f)

Date html generated: 2016_05_14-AM-07_30_51
Last ObjectModification: 2016_01_14-PM-10_01_36

Theory : int_2

Home Index