Formalized Theory $\mathsf{R_0}$ #
@[simp]
TODO: move
@[simp]
@[simp]
@[simp]
@[simp]
class
LO.Arith.Formalized.R₀Theory
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
:
Type u_1
- refl : T ⊢ ((LO.Arith.bv 0 ⋯).equals (LO.Arith.bv 0 ⋯)).all
- replace : (p : ⌜ℒₒᵣ⌝.Semiformula (0 + 1)) → T ⊢ ((LO.Arith.bv 1 ⋯).equals (LO.Arith.bv 0 ⋯) ➝ p^/[(LO.Arith.bv 1 ⋯).sing] ➝ p^/[(LO.Arith.bv 0 ⋯).sing]).all.all
- add : (n m : V) → T ⊢ (LO.Arith.Formalized.typedNumeral 0 n + LO.Arith.Formalized.typedNumeral 0 m).equals (LO.Arith.Formalized.typedNumeral 0 (n + m))
- mul : (n m : V) → T ⊢ (LO.Arith.Formalized.typedNumeral 0 n * LO.Arith.Formalized.typedNumeral 0 m).equals (LO.Arith.Formalized.typedNumeral 0 (n * m))
- ne : {n m : V} → n ≠ m → T ⊢ (LO.Arith.Formalized.typedNumeral 0 n).notEquals (LO.Arith.Formalized.typedNumeral 0 m)
- ltNumeral : (n : V) → T ⊢ ((LO.Arith.bv 0 ⋯).lessThan (LO.Arith.Formalized.typedNumeral (0 + 1) n) ⭤ (LO.Arith.Formalized.tSubstItr (LO.Arith.bv 0 ⋯).sing ((LO.Arith.bv 1 ⋯).equals (LO.Arith.bv 0 ⋯)) n).disj).all
Instances
@[reducible, inline]
abbrev
LO.Arith.Formalized.oneAbbrev
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
{n : V}
:
⌜ℒₒᵣ⌝.Semiterm n
Equations
- LO.Arith.Formalized.oneAbbrev = LO.Arith.Formalized.typedNumeral n 1
Instances For
Equations
- LO.Arith.Formalized.«term^1» = Lean.ParserDescr.node `LO.Arith.Formalized.term^1 1024 (Lean.ParserDescr.symbol "^1")
Instances For
def
LO.Arith.Formalized.TProof.eqRefl
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t : ⌜ℒₒᵣ⌝.Term)
:
Equations
- LO.Arith.Formalized.TProof.eqRefl T t = let_fun this := LO.Arith.Formalized.R₀Theory.refl; ⋯.mp (LO.Arith.Language.Theory.TProof.specialize this t)
Instances For
theorem
LO.Arith.Formalized.TProof.eq_refl!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t : ⌜ℒₒᵣ⌝.Term)
:
noncomputable def
LO.Arith.Formalized.TProof.replace
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(p : ⌜ℒₒᵣ⌝.Semiformula (0 + 1))
(t : ⌜ℒₒᵣ⌝.Term)
(u : ⌜ℒₒᵣ⌝.Term)
:
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
LO.Arith.Formalized.TProof.replace!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(p : ⌜ℒₒᵣ⌝.Semiformula (0 + 1))
(t : ⌜ℒₒᵣ⌝.Term)
(u : ⌜ℒₒᵣ⌝.Term)
:
def
LO.Arith.Formalized.TProof.eqSymm
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t₁ : ⌜ℒₒᵣ⌝.Term)
(t₂ : ⌜ℒₒᵣ⌝.Term)
:
T ⊢ LO.Arith.Language.Semiterm.equals t₁ t₂ ➝ LO.Arith.Language.Semiterm.equals t₂ t₁
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
LO.Arith.Formalized.TProof.eq_symm!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t₁ : ⌜ℒₒᵣ⌝.Term)
(t₂ : ⌜ℒₒᵣ⌝.Term)
:
theorem
LO.Arith.Formalized.TProof.eq_symm'!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
{t₁ : ⌜ℒₒᵣ⌝.Term}
{t₂ : ⌜ℒₒᵣ⌝.Term}
(h : T ⊢! LO.Arith.Language.Semiterm.equals t₁ t₂)
:
T ⊢! LO.Arith.Language.Semiterm.equals t₂ t₁
def
LO.Arith.Formalized.TProof.eqTrans
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t₁ : ⌜ℒₒᵣ⌝.Term)
(t₂ : ⌜ℒₒᵣ⌝.Term)
(t₃ : ⌜ℒₒᵣ⌝.Term)
:
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
LO.Arith.Formalized.TProof.eq_trans!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t₁ : ⌜ℒₒᵣ⌝.Term)
(t₂ : ⌜ℒₒᵣ⌝.Term)
(t₃ : ⌜ℒₒᵣ⌝.Term)
:
noncomputable def
LO.Arith.Formalized.TProof.addExt
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t₁ : ⌜ℒₒᵣ⌝.Term)
(t₂ : ⌜ℒₒᵣ⌝.Term)
(u₁ : ⌜ℒₒᵣ⌝.Term)
(u₂ : ⌜ℒₒᵣ⌝.Term)
:
T ⊢ LO.Arith.Language.Semiterm.equals t₁ t₂ ➝ LO.Arith.Language.Semiterm.equals u₁ u₂ ➝ LO.Arith.Language.Semiterm.equals (t₁ + u₁) (t₂ + u₂)
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
LO.Arith.Formalized.TProof.add_ext!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t₁ : ⌜ℒₒᵣ⌝.Term)
(t₂ : ⌜ℒₒᵣ⌝.Term)
(u₁ : ⌜ℒₒᵣ⌝.Term)
(u₂ : ⌜ℒₒᵣ⌝.Term)
:
T ⊢! LO.Arith.Language.Semiterm.equals t₁ t₂ ➝ LO.Arith.Language.Semiterm.equals u₁ u₂ ➝ LO.Arith.Language.Semiterm.equals (t₁ + u₁) (t₂ + u₂)
noncomputable def
LO.Arith.Formalized.TProof.mulExt
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t₁ : ⌜ℒₒᵣ⌝.Term)
(t₂ : ⌜ℒₒᵣ⌝.Term)
(u₁ : ⌜ℒₒᵣ⌝.Term)
(u₂ : ⌜ℒₒᵣ⌝.Term)
:
T ⊢ LO.Arith.Language.Semiterm.equals t₁ t₂ ➝ LO.Arith.Language.Semiterm.equals u₁ u₂ ➝ LO.Arith.Language.Semiterm.equals (t₁ * u₁) (t₂ * u₂)
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
LO.Arith.Formalized.TProof.mul_ext!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t₁ : ⌜ℒₒᵣ⌝.Term)
(t₂ : ⌜ℒₒᵣ⌝.Term)
(u₁ : ⌜ℒₒᵣ⌝.Term)
(u₂ : ⌜ℒₒᵣ⌝.Term)
:
T ⊢! LO.Arith.Language.Semiterm.equals t₁ t₂ ➝ LO.Arith.Language.Semiterm.equals u₁ u₂ ➝ LO.Arith.Language.Semiterm.equals (t₁ * u₁) (t₂ * u₂)
noncomputable def
LO.Arith.Formalized.TProof.eqExt
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t₁ : ⌜ℒₒᵣ⌝.Term)
(t₂ : ⌜ℒₒᵣ⌝.Term)
(u₁ : ⌜ℒₒᵣ⌝.Term)
(u₂ : ⌜ℒₒᵣ⌝.Term)
:
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
LO.Arith.Formalized.TProof.eq_ext
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t₁ : ⌜ℒₒᵣ⌝.Term)
(t₂ : ⌜ℒₒᵣ⌝.Term)
(u₁ : ⌜ℒₒᵣ⌝.Term)
(u₂ : ⌜ℒₒᵣ⌝.Term)
:
noncomputable def
LO.Arith.Formalized.TProof.neExt
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t₁ : ⌜ℒₒᵣ⌝.Term)
(t₂ : ⌜ℒₒᵣ⌝.Term)
(u₁ : ⌜ℒₒᵣ⌝.Term)
(u₂ : ⌜ℒₒᵣ⌝.Term)
:
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
LO.Arith.Formalized.TProof.ne_ext
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t₁ : ⌜ℒₒᵣ⌝.Term)
(t₂ : ⌜ℒₒᵣ⌝.Term)
(u₁ : ⌜ℒₒᵣ⌝.Term)
(u₂ : ⌜ℒₒᵣ⌝.Term)
:
noncomputable def
LO.Arith.Formalized.TProof.ltExt
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t₁ : ⌜ℒₒᵣ⌝.Term)
(t₂ : ⌜ℒₒᵣ⌝.Term)
(u₁ : ⌜ℒₒᵣ⌝.Term)
(u₂ : ⌜ℒₒᵣ⌝.Term)
:
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
LO.Arith.Formalized.TProof.lt_ext!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t₁ : ⌜ℒₒᵣ⌝.Term)
(t₂ : ⌜ℒₒᵣ⌝.Term)
(u₁ : ⌜ℒₒᵣ⌝.Term)
(u₂ : ⌜ℒₒᵣ⌝.Term)
:
noncomputable def
LO.Arith.Formalized.TProof.nltExt
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t₁ : ⌜ℒₒᵣ⌝.Term)
(t₂ : ⌜ℒₒᵣ⌝.Term)
(u₁ : ⌜ℒₒᵣ⌝.Term)
(u₂ : ⌜ℒₒᵣ⌝.Term)
:
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
LO.Arith.Formalized.TProof.nlt_ext
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t₁ : ⌜ℒₒᵣ⌝.Term)
(t₂ : ⌜ℒₒᵣ⌝.Term)
(u₁ : ⌜ℒₒᵣ⌝.Term)
(u₂ : ⌜ℒₒᵣ⌝.Term)
:
noncomputable def
LO.Arith.Formalized.TProof.ballReplace
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(p : ⌜ℒₒᵣ⌝.Semiformula (0 + 1))
(t : ⌜ℒₒᵣ⌝.Term)
(u : ⌜ℒₒᵣ⌝.Term)
:
Equations
- LO.Arith.Formalized.TProof.ballReplace T p t u = ⋯.mp (LO.Arith.Formalized.TProof.replace T (LO.Arith.Language.Semiformula.ball (LO.Arith.bv 0 ⋯) (p^/[(LO.Arith.bv 0 ⋯).sing])) t u)
Instances For
theorem
LO.Arith.Formalized.TProof.ball_replace!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(p : ⌜ℒₒᵣ⌝.Semiformula (0 + 1))
(t : ⌜ℒₒᵣ⌝.Term)
(u : ⌜ℒₒᵣ⌝.Term)
:
noncomputable def
LO.Arith.Formalized.TProof.bexReplace
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(p : ⌜ℒₒᵣ⌝.Semiformula (0 + 1))
(t : ⌜ℒₒᵣ⌝.Term)
(u : ⌜ℒₒᵣ⌝.Term)
:
Equations
- LO.Arith.Formalized.TProof.bexReplace T p t u = ⋯.mp (LO.Arith.Formalized.TProof.replace T (LO.Arith.Language.Semiformula.bex (LO.Arith.bv 0 ⋯) (p^/[(LO.Arith.bv 0 ⋯).sing])) t u)
Instances For
theorem
LO.Arith.Formalized.TProof.bex_replace!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(p : ⌜ℒₒᵣ⌝.Semiformula (0 + 1))
(t : ⌜ℒₒᵣ⌝.Term)
(u : ⌜ℒₒᵣ⌝.Term)
:
def
LO.Arith.Formalized.TProof.eqComplete
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
{n : V}
{m : V}
(h : n = m)
:
T ⊢ (LO.Arith.Formalized.typedNumeral 0 n).equals (LO.Arith.Formalized.typedNumeral 0 m)
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
LO.Arith.Formalized.TProof.eq_complete!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
{n : V}
{m : V}
(h : n = m)
:
T ⊢! (LO.Arith.Formalized.typedNumeral 0 n).equals (LO.Arith.Formalized.typedNumeral 0 m)
def
LO.Arith.Formalized.TProof.addComplete
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(n : V)
(m : V)
:
T ⊢ (LO.Arith.Formalized.typedNumeral 0 n + LO.Arith.Formalized.typedNumeral 0 m).equals
(LO.Arith.Formalized.typedNumeral 0 (n + m))
Equations
Instances For
theorem
LO.Arith.Formalized.TProof.add_complete!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(n : V)
(m : V)
:
T ⊢! (LO.Arith.Formalized.typedNumeral 0 n + LO.Arith.Formalized.typedNumeral 0 m).equals
(LO.Arith.Formalized.typedNumeral 0 (n + m))
def
LO.Arith.Formalized.TProof.mulComplete
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(n : V)
(m : V)
:
T ⊢ (LO.Arith.Formalized.typedNumeral 0 n * LO.Arith.Formalized.typedNumeral 0 m).equals
(LO.Arith.Formalized.typedNumeral 0 (n * m))
Equations
Instances For
theorem
LO.Arith.Formalized.TProof.mul_complete!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(n : V)
(m : V)
:
T ⊢! (LO.Arith.Formalized.typedNumeral 0 n * LO.Arith.Formalized.typedNumeral 0 m).equals
(LO.Arith.Formalized.typedNumeral 0 (n * m))
def
LO.Arith.Formalized.TProof.neComplete
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
{n : V}
{m : V}
(h : n ≠ m)
:
T ⊢ (LO.Arith.Formalized.typedNumeral 0 n).notEquals (LO.Arith.Formalized.typedNumeral 0 m)
Instances For
theorem
LO.Arith.Formalized.TProof.ne_complete!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
{n : V}
{m : V}
(h : n ≠ m)
:
T ⊢! (LO.Arith.Formalized.typedNumeral 0 n).notEquals (LO.Arith.Formalized.typedNumeral 0 m)
def
LO.Arith.Formalized.TProof.ltNumeral
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t : ⌜ℒₒᵣ⌝.Term)
(n : V)
:
T ⊢ LO.Arith.Language.Semiterm.lessThan t (LO.Arith.Formalized.typedNumeral 0 n) ⭤ (LO.Arith.Formalized.tSubstItr (LO.Arith.Language.Semiterm.sing t) ((LO.Arith.bv 1 ⋯).equals (LO.Arith.bv 0 ⋯))
n).disj
Equations
- LO.Arith.Formalized.TProof.ltNumeral T t n = let_fun this := LO.Arith.Formalized.R₀Theory.ltNumeral n; ⋯.mp (LO.Arith.Language.Theory.TProof.specialize this t)
Instances For
noncomputable def
LO.Arith.Formalized.TProof.nltNumeral
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(t : ⌜ℒₒᵣ⌝.Term)
(n : V)
:
T ⊢ LO.Arith.Language.Semiterm.notLessThan t (LO.Arith.Formalized.typedNumeral 0 n) ⭤ (LO.Arith.Formalized.tSubstItr (LO.Arith.Language.Semiterm.sing t) ((LO.Arith.bv 1 ⋯).notEquals (LO.Arith.bv 0 ⋯))
n).conj
Equations
Instances For
def
LO.Arith.Formalized.TProof.ltComplete
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
{n : V}
{m : V}
(h : n < m)
:
T ⊢ (LO.Arith.Formalized.typedNumeral 0 n).lessThan (LO.Arith.Formalized.typedNumeral 0 m)
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
LO.Arith.Formalized.TProof.lt_complete!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
{n : V}
{m : V}
(h : n < m)
:
T ⊢! (LO.Arith.Formalized.typedNumeral 0 n).lessThan (LO.Arith.Formalized.typedNumeral 0 m)
noncomputable def
LO.Arith.Formalized.TProof.nltComplete
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
{n : V}
{m : V}
(h : m ≤ n)
:
T ⊢ (LO.Arith.Formalized.typedNumeral 0 n).notLessThan (LO.Arith.Formalized.typedNumeral 0 m)
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
LO.Arith.Formalized.TProof.nlt_complete
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
{n : V}
{m : V}
(h : m ≤ n)
:
T ⊢! (LO.Arith.Formalized.typedNumeral 0 n).notLessThan (LO.Arith.Formalized.typedNumeral 0 m)
noncomputable def
LO.Arith.Formalized.TProof.ballIntro
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(p : ⌜ℒₒᵣ⌝.Semiformula (0 + 1))
(n : V)
(bs : (i : V) → i < n → T ⊢ p^/[(LO.Arith.Formalized.typedNumeral 0 i).sing])
:
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
LO.Arith.Formalized.TProof.ball_intro!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(p : ⌜ℒₒᵣ⌝.Semiformula (0 + 1))
(n : V)
(bs : ∀ i < n, T ⊢! p^/[(LO.Arith.Formalized.typedNumeral 0 i).sing])
:
noncomputable def
LO.Arith.Formalized.TProof.bexIntro
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(p : ⌜ℒₒᵣ⌝.Semiformula (0 + 1))
(n : V)
{i : V}
(hi : i < n)
(b : T ⊢ p^/[(LO.Arith.Formalized.typedNumeral 0 i).sing])
:
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
LO.Arith.Formalized.TProof.bex_intro!
{V : Type u_1}
[LO.ORingStruc V]
[V ⊧ₘ* 𝐈𝚺₁]
(T : ⌜ℒₒᵣ⌝.TTheory)
[LO.Arith.Formalized.R₀Theory T]
(p : ⌜ℒₒᵣ⌝.Semiformula (0 + 1))
(n : V)
{i : V}
(hi : i < n)
(b : T ⊢! p^/[(LO.Arith.Formalized.typedNumeral 0 i).sing])
: