MDP.OptimalCost

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noncomputable def MDP.EC {State : Type u_1} {Act : Type u_2} (M : MDP State Act) (c : M.Costs) (𝒮 : 𝔖[M]) (n : ℕ) (s : State) :

ENNReal

Equations

M.EC c 𝒮 n s = ∑' (π : ↑Path[M,s,=n]), MDP.Path.ECost c 𝒮 ↑π

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@[reducible, inline]

noncomputable abbrev MDP.OEC {State : Type u_1} {Act : Type u_2} {M : MDP State Act} (c : M.Costs) (s : State) :

ENNReal

Equations

MDP.OEC c s = ⨅ (𝒮 : 𝔖[M]), ⨆ (n : ℕ), M.EC c 𝒮 n s

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@[simp]

theorem MDP.EC_zero {State : Type u_1} {Act : Type u_2} {M : MDP State Act} {c : M.Costs} {𝒮 : 𝔖[M]} :

M.EC c 𝒮 0 = 0

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@[simp]

theorem MDP.EC_one {State : Type u_1} {Act : Type u_2} {M : MDP State Act} {c : M.Costs} {𝒮 : 𝔖[M]} :

M.EC c 𝒮 1 = c

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@[simp]

theorem MDP.EC_one' {State : Type u_1} {Act : Type u_2} {M : MDP State Act} {c : M.Costs} {𝒮 : 𝔖[M]} {s : State} :

M.EC c 𝒮 1 s = c s

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theorem MDP.EC_le_succ {State : Type u_1} {Act : Type u_2} {M : MDP State Act} {c : M.Costs} {𝒮 : 𝔖[M]} {n : ℕ} {s : State} [DecidableEq State] :

M.EC c 𝒮 n s ≤ M.EC c 𝒮 (n + 1) s

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theorem MDP.EC_monotone {State : Type u_1} {Act : Type u_2} {M : MDP State Act} {c : M.Costs} {𝒮 : 𝔖[M]} {s : State} [DecidableEq State] :

Monotone fun (x : ℕ) => M.EC c 𝒮 x s

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theorem MDP.EC_succ {State : Type u_1} {Act : Type u_2} {M : MDP State Act} {c : M.Costs} {n : ℕ} [DecidableEq State] (𝒮 : 𝔖[M]) :

M.EC c 𝒮 (n + 1) = c + fun (s : State) => ∑' (s' : ↑(M.succs_univ s)), M.P s (𝒮 {s}) ↑s' * M.EC c (𝒮.specialize s s') n ↑s'

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theorem MDP.EC_eq {State : Type u_1} {Act : Type u_2} {M : MDP State Act} {n : ℕ} {s : State} {𝒮 𝒮' : 𝔖[M]} {c : M.Costs} (h : ∀ π ∈ Path[M,s,≤n], 𝒮 π = 𝒮' π) :

M.EC c 𝒮 n s = M.EC c 𝒮' n s

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theorem MDP.EC_le {State : Type u_1} {Act : Type u_2} {M : MDP State Act} {n : ℕ} {s : State} {𝒮 𝒮' : 𝔖[M]} {c : M.Costs} (h : ∀ π ∈ Path[M,s,≤n], 𝒮 π = 𝒮' π) :

M.EC c 𝒮 n s ≤ M.EC c 𝒮' n s

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@[simp]

theorem MDP.EC_markovian_scheduler_specialize {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] {c : M.Costs} {s₀ : State} {s : ↑(M.succs_univ s₀)} {n : ℕ} {𝒮 : 𝔖[M]} [𝒮.Markovian] :

M.EC c (𝒮.specialize s₀ s) n ↑s = M.EC c 𝒮 n ↑s

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theorem MDP.bound_EC_succ_eq_bound_EC {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] {n : ℕ} {c : M.Costs} (s : State) (s' : ↑(M.succs_univ s)) :

⨅ (ℬ : 𝔖[M,s,≤n + 1]), M.EC c (↑(ℬ.specialize s s')) n ↑s' = ⨅ (ℬ : 𝔖[M,↑s',≤n]), M.EC c (↑ℬ) n ↑s'

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theorem MDP.iInf_EC_specialized_eq_bounded {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] {c : M.Costs} {n : ℕ} (s : State) (s' : ↑(M.succs_univ s)) :

⨅ (𝒮 : 𝔖[M]), M.EC c (𝒮.specialize s s') n ↑s' = ⨅ (ℬ : 𝔖[M,s,≤n + 1]), M.EC c (↑(ℬ.specialize s s')) n ↑s'

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theorem MDP.iInf_scheduler_eq_iInf_act_iInf_scheduler {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] {s : State} {c : M.Costs} {n : ℕ} :

⨅ (𝒮 : 𝔖[M]), ∑' (s' : ↑(M.succs_univ s)), M.P s (𝒮 {s}) ↑s' * M.EC c (𝒮.specialize s s') n ↑s' = ⨅ (α : ↑(M.act s)), ⨅ (𝒮 : 𝔖[M]), ∑' (s' : ↑(M.succs_univ s)), M.P s ↑α ↑s' * M.EC c (𝒮.specialize s s') n ↑s'

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theorem MDP.tsum_iInf_bounded_comm {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] [M.FiniteBranching] {s : State} {n : ℕ} (f : (s' : ↑(M.succs_univ s)) → 𝔖[M,↑s',≤n] → ENNReal) :

∑' (s' : ↑(M.succs_univ s)), ⨅ (ℬ : 𝔖[M,↑s',≤n]), f s' ℬ = ⨅ (ℬ : 𝔖[M,s,≤n + 1]), ∑' (s' : ↑(M.succs_univ s)), f s' (ℬ.specialize s s')

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theorem MDP.tsum_iInf_EC_comm {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] [M.FiniteBranching] {s : State} {α : Act} {c : M.Costs} {n : ℕ} :

∑' (s' : ↑(M.succs_univ s)), ⨅ (𝒮 : 𝔖[M]), M.P s α ↑s' * M.EC c (𝒮.specialize s s') n ↑s' = ⨅ (𝒮 : 𝔖[M]), ∑' (s' : ↑(M.succs_univ s)), M.P s α ↑s' * M.EC c (𝒮.specialize s s') n ↑s'

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theorem MDP.iInf_EC_eq_specialized {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] {c : M.Costs} {n : ℕ} (s : State) (s' : ↑(M.succs_univ s)) :

⨅ (𝒮 : 𝔖[M]), M.EC c 𝒮 n ↑s' = ⨅ (𝒮 : 𝔖[M]), M.EC c (𝒮.specialize s s') n ↑s'

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theorem MDP.iInf_EC_succ_eq_Φ𝒟 {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] {c : M.Costs} {n : ℕ} [M.FiniteBranching] :

⨅ (𝒮 : 𝔖[M]), M.EC c 𝒮 (n + 1) = (Φ Optimization.Demonic c) (⨅ (𝒮 : 𝔖[M]), M.EC c 𝒮 n)

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theorem MDP.iInf_EC_eq_Φ𝒟 {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] {c : M.Costs} {n : ℕ} [M.FiniteBranching] :

⨅ (𝒮 : 𝔖[M]), M.EC c 𝒮 n = (⇑(Φ Optimization.Demonic c))^[n] ⊥

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theorem MDP.iSup_iInf_EC_eq_iSup_Φ𝒟 {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] {c : M.Costs} [M.FiniteBranching] :

⨆ (n : ℕ), ⨅ (𝒮 : 𝔖[M]), M.EC c 𝒮 n = ⨆ (n : ℕ), (⇑(Φ Optimization.Demonic c))^[n] ⊥

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theorem MDP.iSup_iInf_EC_eq_lfp_Φ𝒟 {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] {c : M.Costs} [M.FiniteBranching] :

⨆ (n : ℕ), ⨅ (𝒮 : 𝔖[M]), M.EC c 𝒮 n = OrderHom.lfp (Φ Optimization.Demonic c)

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theorem MDP.Φℒ_step_ECℒ {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] {n : ℕ} (c : M.Costs) (ℒ : 𝔏[M]) :

M.EC c (↑ℒ) (n + 1) = (Φℒ ℒ c) (M.EC c (↑ℒ) n)

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theorem MDP.iSup_ECℒ_eq_lfp_Φℒ {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] {c : M.Costs} (ℒ : 𝔏[M]) [M.FiniteBranching] :

⨆ (n : ℕ), M.EC c (↑ℒ) n = lfp_Φℒ ℒ c

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noncomputable def MDP.optℒ {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [M.FiniteBranching] (c : M.Costs) :

𝔏[M]

Equations

One or more equations did not get rendered due to their size.

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noncomputable def MDP.optℒ_spec {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [M.FiniteBranching] (c : M.Costs) (s : State) :

⨅ (α : ↑(M.act s)), (Φf s ↑α) (OrderHom.lfp (Φ Optimization.Demonic c)) = (fun (x : Act) => (Φf s x) (OrderHom.lfp (Φ Optimization.Demonic c))) ((optℒ c) {s})

Equations

⋯ = ⋯

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theorem MDP.lfp_Φℒ_eq_lfp_Φ {State : Type u_1} {Act : Type u_2} {M : MDP State Act} {c : M.Costs} [M.FiniteBranching] :

lfp_Φℒ (optℒ c) c = OrderHom.lfp (Φ Optimization.Demonic c)

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theorem MDP.iSup_iInf_EC_eq_iInf_iSup_EC {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] {c : M.Costs} [M.FiniteBranching] :

⨆ (n : ℕ), ⨅ (𝒮 : 𝔖[M]), M.EC c 𝒮 n = ⨅ (𝒮 : 𝔖[M]), ⨆ (n : ℕ), M.EC c 𝒮 n

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theorem MDP.iInf_iSup_EC_eq_iInf_iSup_ECℒ {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] {c : M.Costs} [M.FiniteBranching] :

⨅ (𝒮 : 𝔖[M]), ⨆ (n : ℕ), M.EC c 𝒮 n = ⨅ (ℒ : 𝔏[M]), ⨆ (n : ℕ), M.EC c (↑ℒ) n

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theorem MDP.iSup_iInf_EC_le_iSup_iInf_ECℒ {State : Type u_1} {Act : Type u_2} {M : MDP State Act} {c : M.Costs} :

⨆ (n : ℕ), ⨅ (𝒮 : 𝔖[M]), M.EC c 𝒮 n ≤ ⨆ (n : ℕ), ⨅ (ℒ : 𝔏[M]), M.EC c (↑ℒ) n

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theorem MDP.iSup_iInf_ECℒ_eq_iInf_iSup_ECℒ {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] {c : M.Costs} [M.FiniteBranching] :

⨆ (n : ℕ), ⨅ (ℒ : 𝔏[M]), M.EC c (↑ℒ) n = ⨅ (ℒ : 𝔏[M]), ⨆ (n : ℕ), M.EC c (↑ℒ) n

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theorem MDP.iInf_iSup_EC_eq_lfp_Φ𝒟 {State : Type u_1} {Act : Type u_2} {M : MDP State Act} [DecidableEq State] {c : M.Costs} [M.FiniteBranching] :

⨅ (𝒮 : 𝔖[M]), ⨆ (n : ℕ), M.EC c 𝒮 n = OrderHom.lfp (Φ Optimization.Demonic c)