Logical Foundations of Cyber-Physical Systems Andr Platzer Andr - - PowerPoint PPT Presentation

13: Differential Invariants & Proof Theory

Logical Foundations of Cyber-Physical Systems

Logical Foundations of Cyber-Physical Systems

André Platzer

André Platzer

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 1 / 23

Outline

1

Learning Objectives

2

Recap: Proofs for Differential Equations

3

Differential Equation Proof Theory Propositional Equivalences Differential Invariants & Arithmetic Differential Structure Differential Invariant Equations Equational Incompleteness Strict Differential Invariant Inequalities Differential Invariant Equations to Differential Invariant Inequalities Differential Invariant Atoms

4

Differential Cut Power & Differential Ghost Power

5

Curves Playing with Norms and Degrees

6

Summary

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 2 / 23

Outline

1

Learning Objectives

2

Recap: Proofs for Differential Equations

3

Differential Equation Proof Theory Propositional Equivalences Differential Invariants & Arithmetic Differential Structure Differential Invariant Equations Equational Incompleteness Strict Differential Invariant Inequalities Differential Invariant Equations to Differential Invariant Inequalities Differential Invariant Atoms

4

Differential Cut Power & Differential Ghost Power

5

Curves Playing with Norms and Degrees

6

Summary

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 2 / 23

Learning Objectives

Differential Invariants & Proof Theory

CT M&C CPS limits of computation proof theory for differential equations provability of differential equations nonprovability of differential equations proofs about proofs relativity theory of proofs inform differential invariant search intuition for differential equation proofs core argumentative principles tame analytic complexity improved analysis

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 3 / 23

Outline

1

Learning Objectives

2

Recap: Proofs for Differential Equations

3

Differential Equation Proof Theory Propositional Equivalences Differential Invariants & Arithmetic Differential Structure Differential Invariant Equations Equational Incompleteness Strict Differential Invariant Inequalities Differential Invariant Equations to Differential Invariant Inequalities Differential Invariant Atoms

4

Differential Cut Power & Differential Ghost Power

5

Curves Playing with Norms and Degrees

6

Summary

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 3 / 23

Differential Invariants for Differential Equations

Differential Weakening Q ⊢ F P ⊢ [x′ = f(x)&Q]F

t x Q w u r x′ = f(x) & Q ¬Q

Differential Invariant Q ⊢ [x′ := f(x)](F)′ F ⊢ [x′ = f(x)&Q]F Differential Cut F ⊢ [x′ = f(x)&Q]C F ⊢ [x′ = f(x)&Q ∧ C]F F ⊢ [x′ = f(x)&Q]F DW [x′ = f(x)&Q]F ↔ [x′ = f(x)&Q](Q → F) DI [x′ = f(x)&Q]F ←

• Q → F ∧[x′ = f(x)&Q](F)′

DC

• [x′ = f(x)&Q]F ↔ [x′ = f(x)&Q ∧ C]F
• ← [x′ = f(x)&Q]C

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 5 / 23

Differential Invariants for Differential Equations

Differential Weakening Q ⊢ F P ⊢ [x′ = f(x)&Q]F

t x Q w u r x′ = f(x) & Q ¬Q

Differential Invariant Q ⊢ [x′ := f(x)](F)′ F ⊢ [x′ = f(x)&Q]F Differential Cut F ⊢ [x′ = f(x)&Q]C F ⊢ [x′ = f(x)&Q ∧ C]F F ⊢ [x′ = f(x)&Q]F DW [x′ = f(x)&Q]F ↔ [x′ = f(x)&Q](Q → F) DI [x′ = f(x)&Q]F ←

• Q → F ∧[x′ = f(x)&Q](F)′

DC

• [x′ = f(x)&Q]F ↔ [x′ = f(x)&Q ∧ C]F
• ← [x′ = f(x)&Q]C

DE [x′ = f(x)&Q]F ↔ [x′ = f(x)&Q][x′ := f(x)]F

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 5 / 23

Outline

1

Learning Objectives

2

Recap: Proofs for Differential Equations

3

Differential Equation Proof Theory Propositional Equivalences Differential Invariants & Arithmetic Differential Structure Differential Invariant Equations Equational Incompleteness Strict Differential Invariant Inequalities Differential Invariant Equations to Differential Invariant Inequalities Differential Invariant Atoms

4

Differential Cut Power & Differential Ghost Power

5

Curves Playing with Norms and Degrees

6

Summary

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 5 / 23

Relativity Theory of Proofs

Differential Invariant Q ⊢ [x′ := f(x)](F)′ F ⊢ [x′ = f(x)&Q]F But generalizations are helpful to find the right F in the first place:

cut,MR

A ⊢ F F ⊢ [x′ = f(x)&Q]F F ⊢ B A ⊢ [x′ = f(x)&Q]B

Compare Provability with Classes Ω of Differential Invariants DIΩ : properties provable with differential invariants in Ω⊆{≥,>,=,∧,∨} A ≤ B iff all properties provable with A are also provable somehow with B A ≤ B otherwise, i.e., some property can be proved with A but not with B A ≡ B iff A ≤ B and B ≤ A so same deductive power A < B iff A ≤ B and B ≤ A so A has strictly less deductive power

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 6 / 23

Relativity Theory of Proofs

Differential Invariant Q ⊢ [x′ := f(x)](F)′ F ⊢ [x′ = f(x)&Q]F

DIe=k≡DIe=0 by considering (e − k) = 0

But generalizations are helpful to find the right F in the first place:

cut,MR

A ⊢ F F ⊢ [x′ = f(x)&Q]F F ⊢ B A ⊢ [x′ = f(x)&Q]B

Compare Provability with Classes Ω of Differential Invariants DIΩ : properties provable with differential invariants in Ω⊆{≥,>,=,∧,∨} A ≤ B iff all properties provable with A are also provable somehow with B A ≤ B otherwise, i.e., some property can be proved with A but not with B A ≡ B iff A ≤ B and B ≤ A so same deductive power A < B iff A ≤ B and B ≤ A so A has strictly less deductive power

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 6 / 23

Propositional Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is a propositional tautology then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof.

Can use any propositional normal form

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 7 / 23

Propositional Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is a propositional tautology then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof.

MR,cutF ⊢ [x′ = f(x)&Q]F

Can use any propositional normal form

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 7 / 23

Propositional Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is a propositional tautology then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof.

dI

G ⊢ [x′ = f(x)&Q]G

MR,cutF ⊢ [x′ = f(x)&Q]F

Can use any propositional normal form

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 7 / 23

Propositional Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is a propositional tautology then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof.

[:=] Q ⊢ [x′:=f(x)](G)′

dI

G ⊢ [x′ = f(x)&Q]G

MR,cutF ⊢ [x′ = f(x)&Q]F

Can use any propositional normal form

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 7 / 23

Propositional Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is a propositional tautology then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof. ∗

[:=] Q ⊢ [x′:=f(x)](F)′

dI

G ⊢ [x′ = f(x)&Q]G

MR,cutF ⊢ [x′ = f(x)&Q]F

Can use any propositional normal form

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 7 / 23

Propositional Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is a propositional tautology then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof. ∗

[:=] Q ⊢ [x′:=f(x)](F)′

dI

G ⊢ [x′ = f(x)&Q]G

MR,cutF ⊢ [x′ = f(x)&Q]F

F ↔ G propositionally equivalent, so

(F)′ ↔ (G)′ propositionally equivalent

Can use any propositional normal form

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 7 / 23

Propositional Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is a propositional tautology then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof. ∗

[:=] Q ⊢ [x′:=f(x)](F)′

dI

G ⊢ [x′ = f(x)&Q]G

MR,cutF ⊢ [x′ = f(x)&Q]F

F ↔ G propositionally equivalent, so

(F)′ ↔ (G)′ propositionally equivalent

since (F1 ∧ F2)′ ≡ (F1)′ ∧(F2)′ . . . Can use any propositional normal form

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 7 / 23

Arithmetic Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is real-arithmetic equivalence then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof.

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 8 / 23

Arithmetic Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is real-arithmetic equivalence then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof.

dI −5≤x∧x≤5 ⊢ [x′ = −x](−5≤x∧x≤5)

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 8 / 23

Arithmetic Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is real-arithmetic equivalence then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof.

[:=]

⊢ [x′:=−x](0≤x′∧x′≤0)

dI −5≤x∧x≤5 ⊢ [x′ = −x](−5≤x∧x≤5)

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 8 / 23

Arithmetic Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is real-arithmetic equivalence then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof. ⊢ 0 ≤ −x ∧−x ≤ 0

[:=]

⊢ [x′:=−x](0≤x′∧x′≤0)

dI −5≤x∧x≤5 ⊢ [x′ = −x](−5≤x∧x≤5)

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 8 / 23

Arithmetic Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is real-arithmetic equivalence then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof.

not valid

⊢ 0 ≤ −x ∧−x ≤ 0

[:=]

⊢ [x′:=−x](0≤x′∧x′≤0)

dI −5≤x∧x≤5 ⊢ [x′ = −x](−5≤x∧x≤5)

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 8 / 23

Arithmetic Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is real-arithmetic equivalence then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof.

not valid

⊢ 0 ≤ −x ∧−x ≤ 0

[:=]

⊢ [x′:=−x](0≤x′∧x′≤0)

dI −5≤x∧x≤5 ⊢ [x′ = −x](−5≤x∧x≤5) dI x2≤52 ⊢ [x′ = −x]x2≤52

arithmetic equivalence −5≤x∧x≤5 ↔ x2≤52

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 8 / 23

Arithmetic Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is real-arithmetic equivalence then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof.

not valid

⊢ 0 ≤ −x ∧−x ≤ 0

[:=]

⊢ [x′:=−x](0≤x′∧x′≤0)

dI −5≤x∧x≤5 ⊢ [x′ = −x](−5≤x∧x≤5)

[:=]

⊢ [x′:=−x]2xx′≤0

dI x2≤52 ⊢ [x′ = −x]x2≤52

arithmetic equivalence −5≤x∧x≤5 ↔ x2≤52

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 8 / 23

Arithmetic Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is real-arithmetic equivalence then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof.

not valid

⊢ 0 ≤ −x ∧−x ≤ 0

[:=]

⊢ [x′:=−x](0≤x′∧x′≤0)

dI −5≤x∧x≤5 ⊢ [x′ = −x](−5≤x∧x≤5)

R

⊢ −x2x≤0

[:=]

⊢ [x′:=−x]2xx′≤0

dI x2≤52 ⊢ [x′ = −x]x2≤52

arithmetic equivalence −5≤x∧x≤5 ↔ x2≤52

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 8 / 23

Arithmetic Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is real-arithmetic equivalence then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof.

not valid

⊢ 0 ≤ −x ∧−x ≤ 0

[:=]

⊢ [x′:=−x](0≤x′∧x′≤0)

dI −5≤x∧x≤5 ⊢ [x′ = −x](−5≤x∧x≤5)

∗

R

⊢ −x2x≤0

[:=]

⊢ [x′:=−x]2xx′≤0

dI x2≤52 ⊢ [x′ = −x]x2≤52

arithmetic equivalence −5≤x∧x≤5 ↔ x2≤52

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 8 / 23

Arithmetic Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is real-arithmetic equivalence then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof.

not valid

⊢ 0 ≤ −x ∧−x ≤ 0

[:=]

⊢ [x′:=−x](0≤x′∧x′≤0)

dI −5≤x∧x≤5 ⊢ [x′ = −x](−5≤x∧x≤5)

∗

R

⊢ −x2x≤0

[:=]

⊢ [x′:=−x]2xx′≤0

dI x2≤52 ⊢ [x′ = −x]x2≤52

Despite arithmetic equivalence −5≤x∧x≤5 ↔ x2≤52 Differential structure matters! Higher degree helps here

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 8 / 23

Different Differential Structure for Equivalent Solutions ≥0

3 2 1 1 2 3 4 x 5 10 15 p 3 2 1 1 2 3 4 x 6 4 2 2 4 6 8 p 3 2 1 1 2 3 4 x 1000 2000 3000 4000 p 3 2 1 1 2 3 4 x 2000 1000 1000 2000 3000 p 2 2 4 6 x 5 10 15 20 25 30 p 2 2 4 6 x 20 10 10 20 p

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 9 / 23

Different Differential Structure for Equivalent Solutions ≥0

3 2 1 1 2 3 4 x 5 10 15 p 3 2 1 1 2 3 4 x 6 4 2 2 4 6 8 p 3 2 1 1 2 3 4 x 1000 2000 3000 4000 p 3 2 1 1 2 3 4 x 2000 1000 1000 2000 3000 p 2 2 4 6 x 5 10 15 20 25 30 p 2 2 4 6 x 20 10 10 20 p

Same p ≥ 0. But different p′ ≥ 0.

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 9 / 23

Different Differential Structure for Equivalent Solutions ≥0

3 2 1 1 2 3 4 x 5 10 15 p 3 2 1 1 2 3 4 x 6 4 2 2 4 6 8 p 3 2 1 1 2 3 4 x 1000 2000 3000 4000 p 3 2 1 1 2 3 4 x 2000 1000 1000 2000 3000 p 2 2 4 6 x 5 10 15 20 25 30 p 2 2 4 6 x 20 10 10 20 p

Same p ≥ 0. But different p′ ≥ 0. Can still normalize atomic formulas to e = 0,e ≥ 0,e > 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 9 / 23

Differential Invariant Equations

Proposition (Equational deductive power [6, 2]) DI= DI=,∧,∨ Proof core. Full: [6, 2].

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 10 / 23

Differential Invariant Equations

Proposition (Equational deductive power [6, 2])

atomic equations are enough:

DI= ≡ DI=,∧,∨ Proof core. Full: [6, 2].

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 10 / 23

Differential Invariant Equations

Proposition (Equational deductive power [6, 2])

atomic equations are enough:

DI= ≡ DI=,∧,∨ Proof core. Full: [6, 2].

e1 = e2 ∨ k1 = k2 e1 = e2 ∧ k1 = k2

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 10 / 23

Differential Invariant Equations

Proposition (Equational deductive power [6, 2])

atomic equations are enough:

DI= ≡ DI=,∧,∨ Proof core. Full: [6, 2].

e1 = e2 ∨ k1 = k2 ↔ (e1 − e2)(k1 − k2) = 0 e1 = e2 ∧ k1 = k2 ↔ (e1 − e2)2 +(k1 − k2)2 = 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 10 / 23

Differential Invariant Equations

Proposition (Equational deductive power [6, 2])

atomic equations are enough:

DI= ≡ DI=,∧,∨ Proof core. Full: [6, 2].

e1 = e2 ∨ k1 = k2 ↔ (e1 − e2)(k1 − k2) = 0

[x′:=f(x)]((e1)′ = (e2)′ ∧(k1)′ = (k2)′)

e1 = e2 ∧ k1 = k2 ↔ (e1 − e2)2 +(k1 − k2)2 = 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 10 / 23

Differential Invariant Equations

Proposition (Equational deductive power [6, 2])

atomic equations are enough:

DI= ≡ DI=,∧,∨ Proof core. Full: [6, 2].

e1 = e2 ∨ k1 = k2 ↔ (e1 − e2)(k1 − k2) = 0

[x′:=f(x)]((e1)′ = (e2)′ ∧(k1)′ = (k2)′)

So [x′:=f(x)]((e1 − e2)(k1 − k2))′ = 0

≡[x′:=f(x)]

• ((e1)′ −(e2)′)(k1 − k2)+(e1 − e2)((k1)′ −(k2)′) = 0
• e1 = e2 ∧ k1 = k2 ↔ (e1 − e2)2 +(k1 − k2)2 = 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 10 / 23

Differential Invariant Equations

Proposition (Equational deductive power [6, 2])

atomic equations are enough:

DI= ≡ DI=,∧,∨ Proof core. Full: [6, 2].

e1 = e2 ∨ k1 = k2 ↔ (e1 − e2)(k1 − k2) = 0

[x′:=f(x)]((e1)′ = (e2)′ ∧(k1)′ = (k2)′)

So [x′:=f(x)]((e1 − e2)(k1 − k2))′ = 0

≡[x′:=f(x)]

• ((e1)′ −(e2)′)(k1 − k2)+(e1 − e2)((k1)′ −(k2)′) = 0
• e1 = e2 ∧ k1 = k2 ↔ (e1 − e2)2 +(k1 − k2)2 = 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 10 / 23

Differential Invariant Equations

Proposition (Equational deductive power [6, 2])

atomic equations are enough:

DI= ≡ DI=,∧,∨ Proof core. Full: [6, 2].

e1 = e2 ∨ k1 = k2 ↔ (e1 − e2)(k1 − k2) = 0

[x′:=f(x)]((e1)′ = (e2)′ ∧(k1)′ = (k2)′)

So [x′:=f(x)]((e1 − e2)(k1 − k2))′ = 0

≡[x′:=f(x)]

• ((e1)′ −(e2)′)(k1 − k2)+(e1 − e2)((k1)′ −(k2)′) = 0
• e1 = e2 ∧ k1 = k2 ↔ (e1 − e2)2 +(k1 − k2)2 = 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 10 / 23

Differential Invariant Equations

Proposition (Equational deductive power [6, 2])

atomic equations are enough:

DI= ≡ DI=,∧,∨ Proof core. Full: [6, 2].

e1 = e2 ∨ k1 = k2 ↔ (e1 − e2)(k1 − k2) = 0

[x′:=f(x)]((e1)′ = (e2)′ ∧(k1)′ = (k2)′)

So [x′:=f(x)]((e1 − e2)(k1 − k2))′ = 0

≡[x′:=f(x)]

• ((e1)′ −(e2)′)(k1 − k2)+(e1 − e2)((k1)′ −(k2)′) = 0
• e1 = e2 ∧ k1 = k2 ↔ (e1 − e2)2 +(k1 − k2)2 = 0

[x′:=f(x)]

• (e1)′ = (e2)′ ∧(k1)′ = (k2)′

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 10 / 23

Differential Invariant Equations

Proposition (Equational deductive power [6, 2])

atomic equations are enough:

DI= ≡ DI=,∧,∨ Proof core. Full: [6, 2].

e1 = e2 ∨ k1 = k2 ↔ (e1 − e2)(k1 − k2) = 0

[x′:=f(x)]((e1)′ = (e2)′ ∧(k1)′ = (k2)′)

So [x′:=f(x)]((e1 − e2)(k1 − k2))′ = 0

≡[x′:=f(x)]

• ((e1)′ −(e2)′)(k1 − k2)+(e1 − e2)((k1)′ −(k2)′) = 0
• e1 = e2 ∧ k1 = k2 ↔ (e1 − e2)2 +(k1 − k2)2 = 0

[x′:=f(x)]

• (e1)′ = (e2)′ ∧(k1)′ = (k2)′

So [x′:=f(x)]

• ((e1 − e2)2 +(k1 − k2)2)′=0
• ≡ [x′:=f(x)]
• 2(e1−e2)((e1)′−(e2)′)+ 2(k1−k2)((k1)′−(k2)′)=0
• André Platzer (CMU)

LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 10 / 23

Differential Invariant Equations

Proposition (Equational deductive power [6, 2])

atomic equations are enough:

DI= ≡ DI=,∧,∨ Proof core. Full: [6, 2].

e1 = e2 ∨ k1 = k2 ↔ (e1 − e2)(k1 − k2) = 0

[x′:=f(x)]((e1)′ = (e2)′ ∧(k1)′ = (k2)′)

So [x′:=f(x)]((e1 − e2)(k1 − k2))′ = 0

≡[x′:=f(x)]

• ((e1)′ −(e2)′)(k1 − k2)+(e1 − e2)((k1)′ −(k2)′) = 0
• e1 = e2 ∧ k1 = k2 ↔ (e1 − e2)2 +(k1 − k2)2 = 0

[x′:=f(x)]

• (e1)′ = (e2)′ ∧(k1)′ = (k2)′

So [x′:=f(x)]

• ((e1 − e2)2 +(k1 − k2)2)′=0
• ≡ [x′:=f(x)]
• 2(e1−e2)((e1)′−(e2)′)+ 2(k1−k2)((k1)′−(k2)′)=0
• André Platzer (CMU)

LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 10 / 23

Differential Invariant Equations

Proposition (Equational deductive power [6, 2])

atomic equations are enough:

DI= ≡ DI=,∧,∨ Proof core. Full: [6, 2].

e1 = e2 ∨ k1 = k2 ↔ (e1 − e2)(k1 − k2) = 0

[x′:=f(x)]((e1)′ = (e2)′ ∧(k1)′ = (k2)′)

So [x′:=f(x)]((e1 − e2)(k1 − k2))′ = 0

≡[x′:=f(x)]

• ((e1)′ −(e2)′)(k1 − k2)+(e1 − e2)((k1)′ −(k2)′) = 0
• e1 = e2 ∧ k1 = k2 ↔ (e1 − e2)2 +(k1 − k2)2 = 0

[x′:=f(x)]

• (e1)′ = (e2)′ ∧(k1)′ = (k2)′

So [x′:=f(x)]

• ((e1 − e2)2 +(k1 − k2)2)′=0
• ≡ [x′:=f(x)]
• 2(e1−e2)((e1)′−(e2)′)+ 2(k1−k2)((k1)′−(k2)′)=0
• André Platzer (CMU)

LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 10 / 23

Equational

Proposition (Equational [2]) DI=≡DI=,∧,∨ DI DI≥ DI= Proof core.

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 11 / 23

Equational Incompleteness

Proposition (Equational incompleteness [2])

Equations are not enough:

DI=≡DI=,∧,∨ < DI since DI≥ ≤ DI= Proof core.

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 11 / 23

Equational Incompleteness

Proposition (Equational incompleteness [2])

Equations are not enough:

DI=≡DI=,∧,∨ < DI since DI≥ ≤ DI= Proof core.

Provable with DI≥ Unprovable with DI=

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 11 / 23

Equational Incompleteness

Proposition (Equational incompleteness [2])

Equations are not enough:

DI=≡DI=,∧,∨ < DI since DI≥ ≤ DI= Proof core.

Provable with DI≥

dI x ≥ 0 ⊢ [x′ = 5]x ≥ 0

Unprovable with DI=

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 11 / 23

Equational Incompleteness

Proposition (Equational incompleteness [2])

Equations are not enough:

DI=≡DI=,∧,∨ < DI since DI≥ ≤ DI= Proof core.

Provable with DI≥

[:=]

⊢ [x′:=5]x′ ≥ 0

dI x ≥ 0 ⊢ [x′ = 5]x ≥ 0

Unprovable with DI=

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 11 / 23

Equational Incompleteness

Proposition (Equational incompleteness [2])

Equations are not enough:

DI=≡DI=,∧,∨ < DI since DI≥ ≤ DI= Proof core.

Provable with DI≥

R

⊢ 5 ≥ 0

[:=]

⊢ [x′:=5]x′ ≥ 0

dI x ≥ 0 ⊢ [x′ = 5]x ≥ 0

Unprovable with DI=

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 11 / 23

Equational Incompleteness

Proposition (Equational incompleteness [2])

Equations are not enough:

DI=≡DI=,∧,∨ < DI since DI≥ ≤ DI= Proof core.

Provable with DI≥

∗

R

⊢ 5 ≥ 0

[:=]

⊢ [x′:=5]x′ ≥ 0

dI x ≥ 0 ⊢ [x′ = 5]x ≥ 0

Unprovable with DI=

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 11 / 23

Proving Differences in Set Theory & Linear Algebra

Example (Sets Bijective or Not)

1 2 3 4 5 6 a b c d e f

Example (Vector Spaces Isomorphic or Not)

x y x′ y′

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 12 / 23

Proving Differences in Set Theory & Linear Algebra

Example (Sets Bijective or Not)

1 2 3 4 5 6 a b c d e f

Example (Vector Spaces Isomorphic or Not)

x y x′ y′

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 12 / 23

Proving Differences in Set Theory & Linear Algebra

Example (Sets Bijective or Not)

1 2 3 4 5 6 a b c d e f 1 2 3 4 5 6 a b c d e

Example (Vector Spaces Isomorphic or Not)

x y x′ y′

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 12 / 23

Proving Differences in Set Theory & Linear Algebra

Example (Sets Bijective or Not)

1 2 3 4 5 6 a b c d e f 1 2 3 4 5 6 a b c d e criterion: cardinality |{1,...,6}| = 6 = |{a,b,c,d,e}| = 5 Need an indirect criterion especially if these sets are infinite

Example (Vector Spaces Isomorphic or Not)

x y x′ y′

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 12 / 23

Proving Differences in Set Theory & Linear Algebra

Example (Sets Bijective or Not)

1 2 3 4 5 6 a b c d e f 1 2 3 4 5 6 a b c d e criterion: cardinality |{1,...,6}| = 6 = |{a,b,c,d,e}| = 5 Need an indirect criterion especially if these sets are infinite

Example (Vector Spaces Isomorphic or Not)

x y x′ y′

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 12 / 23

Proving Differences in Set Theory & Linear Algebra

Example (Sets Bijective or Not)

1 2 3 4 5 6 a b c d e f 1 2 3 4 5 6 a b c d e criterion: cardinality |{1,...,6}| = 6 = |{a,b,c,d,e}| = 5 Need an indirect criterion especially if these sets are infinite

Example (Vector Spaces Isomorphic or Not)

x y x′ y′ x y z x′ y′

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 12 / 23

Proving Differences in Set Theory & Linear Algebra

Example (Sets Bijective or Not)

1 2 3 4 5 6 a b c d e f 1 2 3 4 5 6 a b c d e criterion: cardinality |{1,...,6}| = 6 = |{a,b,c,d,e}| = 5 Need an indirect criterion especially if these sets are infinite

Example (Vector Spaces Isomorphic or Not)

x y x′ y′ x y z x′ y′ criterion: dimension 3 = 2

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 12 / 23

Equational Incompleteness

Proposition (Equational incompleteness [2])

Equations are not enough:

DI=≡DI=,∧,∨ < DI since DI≥ ≤ DI= Proof core.

Provable with DI≥

∗

R

⊢ 5 ≥ 0

[:=]

⊢ [x′:=5]x′ ≥ 0

dI x ≥ 0 ⊢ [x′ = 5]x ≥ 0

Unprovable with DI=

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 13 / 23

Equational Incompleteness

Proposition (Equational incompleteness [2])

Equations are not enough:

DI=≡DI=,∧,∨ < DI since DI≥ ≤ DI= Proof core.

Provable with DI≥

∗

R

⊢ 5 ≥ 0

[:=]

⊢ [x′:=5]x′ ≥ 0

dI x ≥ 0 ⊢ [x′ = 5]x ≥ 0

Unprovable with DI=

cut,MR

x ≥ 0 ⊢ [x′ = 5]x ≥ 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 13 / 23

Equational Incompleteness

Proposition (Equational incompleteness [2])

Equations are not enough:

DI=≡DI=,∧,∨ < DI since DI≥ ≤ DI= Proof core.

Provable with DI≥

∗

R

⊢ 5 ≥ 0

[:=]

⊢ [x′:=5]x′ ≥ 0

dI x ≥ 0 ⊢ [x′ = 5]x ≥ 0

Unprovable with DI=

dI

p(x) = 0 ⊢ [x′ = 5]p(x) = 0

cut,MR

x ≥ 0 ⊢ [x′ = 5]x ≥ 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 13 / 23

Equational Incompleteness

Proposition (Equational incompleteness [2])

Equations are not enough:

DI=≡DI=,∧,∨ < DI since DI≥ ≤ DI= Proof core.

Provable with DI≥

∗

R

⊢ 5 ≥ 0

[:=]

⊢ [x′:=5]x′ ≥ 0

dI x ≥ 0 ⊢ [x′ = 5]x ≥ 0

Unprovable with DI=

⊢ [x′:=5](p(x))′ = 0

dI

p(x) = 0 ⊢ [x′ = 5]p(x) = 0

cut,MR

x ≥ 0 ⊢ [x′ = 5]x ≥ 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 13 / 23

Equational Incompleteness

Proposition (Equational incompleteness [2])

Equations are not enough:

DI=≡DI=,∧,∨ < DI since DI≥ ≤ DI= Proof core.

Provable with DI≥

∗

R

⊢ 5 ≥ 0

[:=]

⊢ [x′:=5]x′ ≥ 0

dI x ≥ 0 ⊢ [x′ = 5]x ≥ 0

Unprovable with DI=

??? ⊢ [x′:=5](p(x))′ = 0

dI

p(x) = 0 ⊢ [x′ = 5]p(x) = 0

cut,MR

x ≥ 0 ⊢ [x′ = 5]x ≥ 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 13 / 23

Equational Incompleteness

Proposition (Equational incompleteness [2])

Equations are not enough:

DI=≡DI=,∧,∨ < DI since DI≥ ≤ DI= Proof core.

Provable with DI≥

∗

R

⊢ 5 ≥ 0

[:=]

⊢ [x′:=5]x′ ≥ 0

dI x ≥ 0 ⊢ [x′ = 5]x ≥ 0

Unprovable with DI=

??? ⊢ [x′:=5](p(x))′ = 0

dI

p(x) = 0 ⊢ [x′ = 5]p(x) = 0

cut,MR

x ≥ 0 ⊢ [x′ = 5]x ≥ 0 Univariate polynomial p(x) is 0 if 0 on all x≥0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 13 / 23

Strict Inequality

Proposition (Strict barrier ) DI> DI DI= DI> Proof core.

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 14 / 23

Strict Inequality Incompleteness

Proposition (Strict barrier incompleteness)

Strict inequalities are not enough:

DI> < DI because DI= ≤ DI> Proof core.

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 14 / 23

Strict Inequality Incompleteness

Proposition (Strict barrier incompleteness)

Strict inequalities are not enough:

DI> < DI because DI= ≤ DI> Proof core.

Provable with DI= Unprovable with DI>

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 14 / 23

Strict Inequality Incompleteness

Proposition (Strict barrier incompleteness)

Strict inequalities are not enough:

DI> < DI because DI= ≤ DI> Proof core.

Provable with DI=

dI v2+w2=c2 ⊢ [v′ = w,w′ = −v]v2+w2=c2

Unprovable with DI>

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 14 / 23

Strict Inequality Incompleteness

Proposition (Strict barrier incompleteness)

Strict inequalities are not enough:

DI> < DI because DI= ≤ DI> Proof core.

Provable with DI=

[:=]

⊢ [v′:=w][w′:=−v]2vv′ + 2ww′ = 0

dI v2+w2=c2 ⊢ [v′ = w,w′ = −v]v2+w2=c2

Unprovable with DI>

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 14 / 23

Strict Inequality Incompleteness

Proposition (Strict barrier incompleteness)

Strict inequalities are not enough:

DI> < DI because DI= ≤ DI> Proof core.

Provable with DI=

R

⊢ 2vw + 2w(−v) = 0

[:=]

⊢ [v′:=w][w′:=−v]2vv′ + 2ww′ = 0

dI v2+w2=c2 ⊢ [v′ = w,w′ = −v]v2+w2=c2

Unprovable with DI>

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 14 / 23

Strict Inequality Incompleteness

Proposition (Strict barrier incompleteness)

Strict inequalities are not enough:

DI> < DI because DI= ≤ DI> Proof core.

Provable with DI=

∗

R

⊢ 2vw + 2w(−v) = 0

[:=]

⊢ [v′:=w][w′:=−v]2vv′ + 2ww′ = 0

dI v2+w2=c2 ⊢ [v′ = w,w′ = −v]v2+w2=c2

Unprovable with DI>

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 14 / 23

Strict Inequality Incompleteness

Proposition (Strict barrier incompleteness)

Strict inequalities are not enough:

DI> < DI because DI= ≤ DI> Proof core.

Provable with DI=

∗

R

⊢ 2vw + 2w(−v) = 0

[:=]

⊢ [v′:=w][w′:=−v]2vv′ + 2ww′ = 0

dI v2+w2=c2 ⊢ [v′ = w,w′ = −v]v2+w2=c2

v2+w2=c2 is a closed set Unprovable with DI> e > 0 is open set. closed v2+w2≤1 with full boundary

• pen v2+w2<1

without boundary

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 14 / 23

Strict Inequality Incompleteness

Proposition (Strict barrier incompleteness)

Strict inequalities are not enough:

DI> < DI because DI= ≤ DI> Proof core.

Provable with DI=

∗

R

⊢ 2vw + 2w(−v) = 0

[:=]

⊢ [v′:=w][w′:=−v]2vv′ + 2ww′ = 0

dI v2+w2=c2 ⊢ [v′ = w,w′ = −v]v2+w2=c2

v2+w2=c2 is a closed set Unprovable with DI> e > 0 is open set. Only true/false are both closed v2+w2≤1 with full boundary

• pen v2+w2<1

without boundary

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 14 / 23

Strict Inequality Incompleteness

Proposition (Strict barrier incompleteness)

Strict inequalities are not enough:

DI> < DI because DI= ≤ DI> Proof core.

Provable with DI=

∗

R

⊢ 2vw + 2w(−v) = 0

[:=]

⊢ [v′:=w][w′:=−v]2vv′ + 2ww′ = 0

dI v2+w2=c2 ⊢ [v′ = w,w′ = −v]v2+w2=c2

v2+w2=c2 is a closed set Unprovable with DI> e > 0 is open set. Only true/false are both but don’t help proof closed v2+w2≤1 with full boundary

• pen v2+w2<1

without boundary

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 14 / 23

Differential Invariant Equations to Inequalities

Proposition (Equational ) DI=,∧,∨ DI≥ Proof core.

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 15 / 23

Differential Invariant Equations to Inequalities

Proposition (Equational definability)

Equations are definable by weak inequalities:

DI=,∧,∨ ≤ DI≥ Proof core.

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 15 / 23

Differential Invariant Equations to Inequalities

Proposition (Equational definability)

Equations are definable by weak inequalities:

DI=,∧,∨ ≤ DI≥ Proof core.

Provable with DI= Provable with DI≥

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 15 / 23

Differential Invariant Equations to Inequalities

Proposition (Equational definability)

Equations are definable by weak inequalities:

DI=,∧,∨ ≤ DI≥ Proof core.

Provable with DI=

dIe = 0 ⊢ [x′ = f(x)&Q]e = 0

Provable with DI≥

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 15 / 23

Differential Invariant Equations to Inequalities

Proposition (Equational definability)

Equations are definable by weak inequalities:

DI=,∧,∨ ≤ DI≥ Proof core.

Provable with DI= Q ⊢ [x′:=f(x)](e)′ = 0

dIe = 0 ⊢ [x′ = f(x)&Q]e = 0

Provable with DI≥

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 15 / 23

Differential Invariant Equations to Inequalities

Proposition (Equational definability)

Equations are definable by weak inequalities:

DI=,∧,∨ ≤ DI≥ Proof core.

Provable with DI=

∗

Q ⊢ [x′:=f(x)](e)′ = 0

dIe = 0 ⊢ [x′ = f(x)&Q]e = 0

Provable with DI≥

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 15 / 23

Differential Invariant Equations to Inequalities

Proposition (Equational definability)

Equations are definable by weak inequalities:

DI=,∧,∨ ≤ DI≥ Proof core.

Provable with DI=

∗

Q ⊢ [x′:=f(x)](e)′ = 0

dIe = 0 ⊢ [x′ = f(x)&Q]e = 0

Provable with DI≥

dI−e2 ≥ 0 ⊢ [x′ = f(x)&Q](−e2 ≥ 0)

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 15 / 23

Differential Invariant Equations to Inequalities

Proposition (Equational definability)

Equations are definable by weak inequalities:

DI=,∧,∨ ≤ DI≥ Proof core.

Provable with DI=

∗

Q ⊢ [x′:=f(x)](e)′ = 0

dIe = 0 ⊢ [x′ = f(x)&Q]e = 0

Provable with DI≥ Q ⊢ [x′:=f(x)]− 2e(e)′ ≥ 0

dI−e2 ≥ 0 ⊢ [x′ = f(x)&Q](−e2 ≥ 0)

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 15 / 23

Differential Invariant Equations to Inequalities

Proposition (Equational definability)

Equations are definable by weak inequalities:

DI=,∧,∨ ≤ DI≥ Proof core.

Provable with DI=

∗

Q ⊢ [x′:=f(x)](e)′ = 0

dIe = 0 ⊢ [x′ = f(x)&Q]e = 0

Provable with DI≥

∗

Q ⊢ [x′:=f(x)]− 2e(e)′ ≥ 0

dI−e2 ≥ 0 ⊢ [x′ = f(x)&Q](−e2 ≥ 0)

Local view of logic on differentials is crucial for this proof. Degree increases

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 15 / 23

Differential Invariant Atoms

Theorem (Atomic ) DI≥ DI≥,∧,∨ and DI> DI>,∧,∨ Proof idea.

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 16 / 23

Differential Invariant Atoms

Theorem (Atomic incompleteness)

Atomic inequalities not enough:

DI≥ < DI≥,∧,∨ and DI> < DI>,∧,∨ Proof idea.

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 16 / 23

Differential Invariant Atoms

Theorem (Atomic incompleteness)

Atomic inequalities not enough:

DI≥ < DI≥,∧,∨ and DI> < DI>,∧,∨ Proof idea.

Provable with DI≥,∧,∨ Unprovable with DI≥

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 16 / 23

Differential Invariant Atoms

Theorem (Atomic incompleteness)

Atomic inequalities not enough:

DI≥ < DI≥,∧,∨ and DI> < DI>,∧,∨ Proof idea.

Provable with DI≥,∧,∨

∗

R

⊢ 5 ≥ 0∧ y2 ≥ 0

[:=]

⊢ [x′:=5][y′:=y2](x′≥0∧y′≥0)

dI x≥0∧y≥0 ⊢ [x′ = 5,y′ = y2](x≥0∧y≥0)

Unprovable with DI≥

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 16 / 23

Differential Invariant Atoms

Theorem (Atomic incompleteness)

Atomic inequalities not enough:

DI≥ < DI≥,∧,∨ and DI> < DI>,∧,∨ Proof idea.

Provable with DI≥,∧,∨

∗

R

⊢ 5 ≥ 0∧ y2 ≥ 0

[:=]

⊢ [x′:=5][y′:=y2](x′≥0∧y′≥0)

dI x≥0∧y≥0 ⊢ [x′ = 5,y′ = y2](x≥0∧y≥0)

Unprovable with DI≥ p(x,y)≥0 ↔ x≥0∧y≥0 impossible since this implies p(x,0)≥0 ↔ x≥0 so p(x,0) is 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 16 / 23

Differential Invariant Atoms

Theorem (Atomic incompleteness)

Atomic inequalities not enough:

DI≥ < DI≥,∧,∨ and DI> < DI>,∧,∨ Proof idea.

Provable with DI≥,∧,∨

∗

R

⊢ 5 ≥ 0∧ y2 ≥ 0

[:=]

⊢ [x′:=5][y′:=y2](x′≥0∧y′≥0)

dI x≥0∧y≥0 ⊢ [x′ = 5,y′ = y2](x≥0∧y≥0)

Unprovable with DI≥ p(x,y)≥0 ↔ x≥0∧y≥0 impossible since this implies p(x,0)≥0 ↔ x≥0 so p(x,0) is 0 Substantial remaining parts of the proof shown elsewhere [2].

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 16 / 23

Differential Invariant Atoms

Theorem (Atomic incompleteness)

Atomic inequalities not enough:

DI≥ < DI≥,∧,∨ and DI> < DI>,∧,∨ Proof idea.

Provable with DI≥,∧,∨

∗

R

⊢ 5 ≥ 0∧ y2 ≥ 0

[:=]

⊢ [x′:=5][y′:=y2](x′≥0∧y′≥0)

dI x≥0∧y≥0 ⊢ [x′ = 5,y′ = y2](x≥0∧y≥0)

Unprovable with DI≥ p(x,y)≥0 ↔ x≥0∧y≥0 impossible since this implies p(x,0)≥0 ↔ x≥0 so p(x,0) is 0 Substantial remaining parts of the proof shown elsewhere [2]. dC still possible here but more involved argument separates.

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 16 / 23

Outline

1

Learning Objectives

2

Recap: Proofs for Differential Equations

3

Differential Equation Proof Theory Propositional Equivalences Differential Invariants & Arithmetic Differential Structure Differential Invariant Equations Equational Incompleteness Strict Differential Invariant Inequalities Differential Invariant Equations to Differential Invariant Inequalities Differential Invariant Atoms

4

Differential Cut Power & Differential Ghost Power

5

Curves Playing with Norms and Degrees

6

Summary

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 16 / 23

Deductive Power of Differential Cuts & Differential Ghosts

Theorem (Gentzen’s Cut Elimination) (1935)

A ⊢ B ∨ C A∧ C ⊢ B A ⊢ B cut can be eliminated

Theorem (No Differential Cut Elimination) (LMCS 2012)

Deductive power with differential cuts exceeds deductive power without.

DI + DC > DI Theorem (Auxiliary Differential Variables) (LMCS 2012)

Deductive power with differential ghosts exceeds power without.

DI + DC+ DG > DI + DC

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 17 / 23

Ex: The Need for Differential Cuts

dI x3 ≥ −1∧ y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]x3 ≥ −1

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 18 / 23

Ex: The Need for Differential Cuts

[:=]

⊢ [x′:=(x − 2)4 + y5][y′:=y2]3x2x′ ≥ 0

dI x3 ≥ −1∧ y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]x3 ≥ −1

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 18 / 23

Ex: The Need for Differential Cuts

⊢ 3x2((x − 2)4 + y5) ≥ 0

[:=]

⊢ [x′:=(x − 2)4 + y5][y′:=y2]3x2x′ ≥ 0

dI x3 ≥ −1∧ y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]x3 ≥ −1

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 18 / 23

Ex: The Need for Differential Cuts

not valid

⊢ 3x2((x − 2)4 + y5) ≥ 0

[:=]

⊢ [x′:=(x − 2)4 + y5][y′:=y2]3x2x′ ≥ 0

dI x3 ≥ −1∧ y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]x3 ≥ −1

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 18 / 23

Ex: The Need for Differential Cuts

not valid

⊢ 3x2((x − 2)4 + y5) ≥ 0

[:=]

⊢ [x′:=(x − 2)4 + y5][y′:=y2]3x2x′ ≥ 0

dI x3 ≥ −1∧ y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]x3 ≥ −1

Have to know something about y5

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 18 / 23

Ex: Differential Cuts

dC x3 ≥ −1∧ y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]x3 ≥ −1

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 19 / 23

Ex: Differential Cuts

dC x3 ≥ −1∧ y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]x3 ≥ −1 dI y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]y5 ≥ 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 19 / 23

Ex: Differential Cuts

dC x3 ≥ −1∧ y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]x3 ≥ −1

[:=]

⊢ [x′:=(x − 2)4 + y5][y′:=y2]5y4y′ ≥ 0

dI y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]y5 ≥ 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 19 / 23

Ex: Differential Cuts

dC x3 ≥ −1∧ y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]x3 ≥ −1

R

⊢ 5y4y2 ≥ 0

[:=]

⊢ [x′:=(x − 2)4 + y5][y′:=y2]5y4y′ ≥ 0

dI y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]y5 ≥ 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 19 / 23

Ex: Differential Cuts

dC x3 ≥ −1∧ y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]x3 ≥ −1

∗

R

⊢ 5y4y2 ≥ 0

[:=]

⊢ [x′:=(x − 2)4 + y5][y′:=y2]5y4y′ ≥ 0

dI y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]y5 ≥ 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 19 / 23

Ex: Differential Cuts

dI

x3 ≥ −1 ⊢ [x′ = (x − 2)4 + y5,y′ = y2 &y5 ≥ 0]x3 ≥ −1 ⊲

dC x3 ≥ −1∧ y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]x3 ≥ −1

∗

R

⊢ 5y4y2 ≥ 0

[:=]

⊢ [x′:=(x − 2)4 + y5][y′:=y2]5y4y′ ≥ 0

dI y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]y5 ≥ 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 19 / 23

Ex: Differential Cuts

[:=]

y5 ≥ 0 ⊢ [x′:=(x − 2)4 + y5][y′:=y2]3x2x′ ≥ 0

dI

x3 ≥ −1 ⊢ [x′ = (x − 2)4 + y5,y′ = y2 &y5 ≥ 0]x3 ≥ −1 ⊲

dC x3 ≥ −1∧ y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]x3 ≥ −1

∗

R

⊢ 5y4y2 ≥ 0

[:=]

⊢ [x′:=(x − 2)4 + y5][y′:=y2]5y4y′ ≥ 0

dI y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]y5 ≥ 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 19 / 23

Ex: Differential Cuts

R

y5 ≥ 0 ⊢ 3x2((x − 2)4 + y5) ≥ 0

[:=]

y5 ≥ 0 ⊢ [x′:=(x − 2)4 + y5][y′:=y2]3x2x′ ≥ 0

dI

x3 ≥ −1 ⊢ [x′ = (x − 2)4 + y5,y′ = y2 &y5 ≥ 0]x3 ≥ −1 ⊲

dC x3 ≥ −1∧ y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]x3 ≥ −1

∗

R

⊢ 5y4y2 ≥ 0

[:=]

⊢ [x′:=(x − 2)4 + y5][y′:=y2]5y4y′ ≥ 0

dI y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]y5 ≥ 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 19 / 23

Ex: Differential Cuts

∗

R

y5 ≥ 0 ⊢ 3x2((x − 2)4 + y5) ≥ 0

[:=]

y5 ≥ 0 ⊢ [x′:=(x − 2)4 + y5][y′:=y2]3x2x′ ≥ 0

dI

x3 ≥ −1 ⊢ [x′ = (x − 2)4 + y5,y′ = y2 &y5 ≥ 0]x3 ≥ −1 ⊲

dC x3 ≥ −1∧ y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]x3 ≥ −1

∗

R

⊢ 5y4y2 ≥ 0

[:=]

⊢ [x′:=(x − 2)4 + y5][y′:=y2]5y4y′ ≥ 0

dI y5 ≥ 0 ⊢ [x′ = (x − 2)4 + y5,y′ = y2]y5 ≥ 0

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 19 / 23

Outline

1

Learning Objectives

2

Recap: Proofs for Differential Equations

3

Differential Equation Proof Theory Propositional Equivalences Differential Invariants & Arithmetic Differential Structure Differential Invariant Equations Equational Incompleteness Strict Differential Invariant Inequalities Differential Invariant Equations to Differential Invariant Inequalities Differential Invariant Atoms

4

Differential Cut Power & Differential Ghost Power

5

Curves Playing with Norms and Degrees

6

Summary

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 19 / 23

Arithmetic Equivalences of Differential Invariants

Lemma (Differential invariants and propositional logic)

If F ↔ G is real-arithmetic equivalence then F differential invariant of x′ = f(x)&Q iff G differential invariant of x′ = f(x)&Q

Proof.

not valid

⊢ 0 ≤ −x ∧−x ≤ 0

[:=]

⊢ [x′:=−x](0≤x′∧x′≤0)

dI −5≤x∧x≤5 ⊢ [x′ = −x](−5≤x∧x≤5)

∗

R

⊢ −x2x≤0

[:=]

⊢ [x′:=−x]2xx′≤0

dI x2≤52 ⊢ [x′ = −x]x2≤52

Despite arithmetic equivalence −5≤x∧x≤5 ↔ x2≤52 Differential structure matters! Higher degree helps here

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 20 / 23

Curves Playing with Norms and Degrees

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)∞ ≤ t A

def

≡ v2+w2≤1∧ x=y=t=0 (x,y)∞ ≤ t

def

≡ −t ≤ x ≤ t ∧−t ≤ y ≤ t

Supremum norm

(x,y)2 ≤ t

def

≡ x2 + y2 ≤ t2

Euclidean norm

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 21 / 23

Curves Playing with Norms and Degrees

dI ⊳

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1&v2+w2≤1](x,y)∞ ≤ t

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)∞ ≤ t A

def

≡ v2+w2≤1∧ x=y=t=0 (x,y)∞ ≤ t

def

≡ −t ≤ x ≤ t ∧−t ≤ y ≤ t

Supremum norm

(x,y)2 ≤ t

def

≡ x2 + y2 ≤ t2

Euclidean norm

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 21 / 23

Curves Playing with Norms and Degrees

[:=]v2+w2≤1 ⊢ [x′:=v][y′:=w][v′:=ωw][w′:=−ωv][t′:=1](−t′≤x′≤t′∧−t′≤y′≤t′)

dI ⊳

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1&v2+w2≤1](x,y)∞ ≤ t

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)∞ ≤ t A

def

≡ v2+w2≤1∧ x=y=t=0 (x,y)∞ ≤ t

def

≡ −t ≤ x ≤ t ∧−t ≤ y ≤ t

Supremum norm

(x,y)2 ≤ t

def

≡ x2 + y2 ≤ t2

Euclidean norm

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 21 / 23

Curves Playing with Norms and Degrees

R v2+w2≤1 ⊢ −1 ≤ v ≤ 1∧−1 ≤ w ≤ 1

[:=]v2+w2≤1 ⊢ [x′:=v][y′:=w][v′:=ωw][w′:=−ωv][t′:=1](−t′≤x′≤t′∧−t′≤y′≤t′)

dI ⊳

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1&v2+w2≤1](x,y)∞ ≤ t

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)∞ ≤ t A

def

≡ v2+w2≤1∧ x=y=t=0 (x,y)∞ ≤ t

def

≡ −t ≤ x ≤ t ∧−t ≤ y ≤ t

Supremum norm

(x,y)2 ≤ t

def

≡ x2 + y2 ≤ t2

Euclidean norm

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 21 / 23

Curves Playing with Norms and Degrees

∗

R v2+w2≤1 ⊢ −1 ≤ v ≤ 1∧−1 ≤ w ≤ 1

[:=]v2+w2≤1 ⊢ [x′:=v][y′:=w][v′:=ωw][w′:=−ωv][t′:=1](−t′≤x′≤t′∧−t′≤y′≤t′)

dI ⊳

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1&v2+w2≤1](x,y)∞ ≤ t

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)∞ ≤ t A

def

≡ v2+w2≤1∧ x=y=t=0 (x,y)∞ ≤ t

def

≡ −t ≤ x ≤ t ∧−t ≤ y ≤ t

Supremum norm

(x,y)2 ≤ t

def

≡ x2 + y2 ≤ t2

Euclidean norm

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 21 / 23

Curves Playing with Norms and Degrees

∗

R v2+w2≤1 ⊢ −1 ≤ v ≤ 1∧−1 ≤ w ≤ 1

[:=]v2+w2≤1 ⊢ [x′:=v][y′:=w][v′:=ωw][w′:=−ωv][t′:=1](−t′≤x′≤t′∧−t′≤y′≤t′)

dI ⊳

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1&v2+w2≤1](x,y)∞ ≤ t

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)∞ ≤ t

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)2 ≤ t A

def

≡ v2+w2≤1∧ x=y=t=0 (x,y)∞ ≤ t

def

≡ −t ≤ x ≤ t ∧−t ≤ y ≤ t

Supremum norm

(x,y)2 ≤ t

def

≡ x2 + y2 ≤ t2

Euclidean norm

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 21 / 23

Curves Playing with Norms and Degrees

∗

R v2+w2≤1 ⊢ −1 ≤ v ≤ 1∧−1 ≤ w ≤ 1

[:=]v2+w2≤1 ⊢ [x′:=v][y′:=w][v′:=ωw][w′:=−ωv][t′:=1](−t′≤x′≤t′∧−t′≤y′≤t′)

dI ⊳

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1&v2+w2≤1](x,y)∞ ≤ t

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)∞ ≤ t

dI ⊳

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1&v2+w2≤1](x,y)2 ≤ t

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)2 ≤ t A

def

≡ v2+w2≤1∧ x=y=t=0 (x,y)∞ ≤ t

def

≡ −t ≤ x ≤ t ∧−t ≤ y ≤ t

Supremum norm

(x,y)2 ≤ t

def

≡ x2 + y2 ≤ t2

Euclidean norm

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 21 / 23

Curves Playing with Norms and Degrees

∗

R v2+w2≤1 ⊢ −1 ≤ v ≤ 1∧−1 ≤ w ≤ 1

[:=]v2+w2≤1 ⊢ [x′:=v][y′:=w][v′:=ωw][w′:=−ωv][t′:=1](−t′≤x′≤t′∧−t′≤y′≤t′)

dI ⊳

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1&v2+w2≤1](x,y)∞ ≤ t

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)∞ ≤ t

[:=]v2+w2≤1 ⊢ [x′:=v][y′:=w][v′:=ωw][w′:=−ωv][t′:=1](2xx′ + 2yy′ ≤ 2tt′)

dI ⊳

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1&v2+w2≤1](x,y)2 ≤ t

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)2 ≤ t A

def

≡ v2+w2≤1∧ x=y=t=0 (x,y)∞ ≤ t

def

≡ −t ≤ x ≤ t ∧−t ≤ y ≤ t

Supremum norm

(x,y)2 ≤ t

def

≡ x2 + y2 ≤ t2

Euclidean norm

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 21 / 23

Curves Playing with Norms and Degrees

∗

R v2+w2≤1 ⊢ −1 ≤ v ≤ 1∧−1 ≤ w ≤ 1

[:=]v2+w2≤1 ⊢ [x′:=v][y′:=w][v′:=ωw][w′:=−ωv][t′:=1](−t′≤x′≤t′∧−t′≤y′≤t′)

dI ⊳

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1&v2+w2≤1](x,y)∞ ≤ t

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)∞ ≤ t v2+w2≤1 ⊢ 2xv + 2yw ≤ 2t1

[:=]v2+w2≤1 ⊢ [x′:=v][y′:=w][v′:=ωw][w′:=−ωv][t′:=1](2xx′ + 2yy′ ≤ 2tt′)

dI ⊳

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1&v2+w2≤1](x,y)2 ≤ t

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)2 ≤ t A

def

≡ v2+w2≤1∧ x=y=t=0 (x,y)∞ ≤ t

def

≡ −t ≤ x ≤ t ∧−t ≤ y ≤ t

Supremum norm

(x,y)2 ≤ t

def

≡ x2 + y2 ≤ t2

Euclidean norm

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 21 / 23

Curves Playing with Norms and Degrees

∗

R v2+w2≤1 ⊢ −1 ≤ v ≤ 1∧−1 ≤ w ≤ 1

[:=]v2+w2≤1 ⊢ [x′:=v][y′:=w][v′:=ωw][w′:=−ωv][t′:=1](−t′≤x′≤t′∧−t′≤y′≤t′)

dI ⊳

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1&v2+w2≤1](x,y)∞ ≤ t

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)∞ ≤ t not valid v2+w2≤1 ⊢ 2xv + 2yw ≤ 2t1

[:=]v2+w2≤1 ⊢ [x′:=v][y′:=w][v′:=ωw][w′:=−ωv][t′:=1](2xx′ + 2yy′ ≤ 2tt′)

dI ⊳

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1&v2+w2≤1](x,y)2 ≤ t

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)2 ≤ t A

def

≡ v2+w2≤1∧ x=y=t=0 (x,y)∞ ≤ t

def

≡ −t ≤ x ≤ t ∧−t ≤ y ≤ t

Supremum norm

(x,y)2 ≤ t

def

≡ x2 + y2 ≤ t2

Euclidean norm

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 21 / 23

Curves Playing with Norms and Degrees

∗

R v2+w2≤1 ⊢ −1 ≤ v ≤ 1∧−1 ≤ w ≤ 1

[:=]v2+w2≤1 ⊢ [x′:=v][y′:=w][v′:=ωw][w′:=−ωv][t′:=1](−t′≤x′≤t′∧−t′≤y′≤t′)

dI ⊳

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1&v2+w2≤1](x,y)∞ ≤ t

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)∞ ≤ t

Lower degree helps here

not valid v2+w2≤1 ⊢ 2xv + 2yw ≤ 2t1

[:=]v2+w2≤1 ⊢ [x′:=v][y′:=w][v′:=ωw][w′:=−ωv][t′:=1](2xx′ + 2yy′ ≤ 2tt′)

dI ⊳

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1&v2+w2≤1](x,y)2 ≤ t

dC

A ⊢ [x′ = v,y′ = w,v′ = ωw,w′ = −ωv,t′ = 1](x,y)2 ≤ t A

def

≡ v2+w2≤1∧ x=y=t=0 (x,y)∞ ≤ t

def

≡ −t ≤ x ≤ t ∧−t ≤ y ≤ t

Supremum norm

(x,y)2 ≤ t

def

≡ x2 + y2 ≤ t2

Euclidean norm

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 21 / 23

Interreducing Norms in Dimension n

∀x ∀y ((x,y)∞ ≤ (x,y)2 ≤ √

n(x,y)∞)

∀x ∀y ( 1 √

n(x,y)2 ≤ (x,y)∞ ≤ (x,y)2)

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 22 / 23

Interreducing Norms in Dimension n

∀x ∀y ((x,y)∞ ≤ (x,y)2 ≤ √

n(x,y)∞)

∀x ∀y ( 1 √

n(x,y)2 ≤ (x,y)∞ ≤ (x,y)2)

·2 ≤ 1

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 22 / 23

Interreducing Norms in Dimension n

∀x ∀y ((x,y)∞ ≤ (x,y)2 ≤ √

n(x,y)∞)

∀x ∀y ( 1 √

n(x,y)2 ≤ (x,y)∞ ≤ (x,y)2)

·2 ≤ 1 ·∞ ≤ 1

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 22 / 23

Interreducing Norms in Dimension n

∀x ∀y ((x,y)∞ ≤ (x,y)2 ≤ √

n(x,y)∞)

∀x ∀y ( 1 √

n(x,y)2 ≤ (x,y)∞ ≤ (x,y)2)

·∞ ≤

1

√

2

·2 ≤ 1 ·∞ ≤ 1

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 22 / 23

Interreducing Norms in Dimension n

∀x ∀y ((x,y)∞ ≤ (x,y)2 ≤ √

n(x,y)∞)

∀x ∀y ( 1 √

n(x,y)2 ≤ (x,y)∞ ≤ (x,y)2)

·∞ ≤

1

√

2

·2 ≤ 1 ·∞ ≤ 1 ·2 ≤ √

2

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 22 / 23

Interreducing Norms in Dimension n

∀x ∀y ((x,y)∞ ≤ (x,y)2 ≤ √

n(x,y)∞)

∀x ∀y ( 1 √

n(x,y)2 ≤ (x,y)∞ ≤ (x,y)2)

·∞ ≤

1

√

2

·2 ≤ 1 ·∞ ≤ 1 ·2 ≤ √

2 Benefit from norm relations but be mindful of approximation error factors

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 22 / 23

Outline

1

Learning Objectives

2

Recap: Proofs for Differential Equations

3

Differential Equation Proof Theory Propositional Equivalences Differential Invariants & Arithmetic Differential Structure Differential Invariant Equations Equational Incompleteness Strict Differential Invariant Inequalities Differential Invariant Equations to Differential Invariant Inequalities Differential Invariant Atoms

4

Differential Cut Power & Differential Ghost Power

5

Curves Playing with Norms and Degrees

6

Summary

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 22 / 23

Summary: Differential Invariance Chart

Theorem (Differential Invariance Chart) DI= DI=,∧,∨ DI> DI>,∧,∨ DI≥ DI≥,∧,∨ DI DI≥,=,∧,∨ DI>,=,∧,∨

Rich theory and structure behind differential invariants Scrutinize what property can be proved with what invariant Use provability sanity checks like open/closed/univariate Real differential semialgebraic geometry Exploit differential cuts to obtain more knowledge

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 23 / 23

André Platzer. Logical Foundations of Cyber-Physical Systems. Springer, Switzerland, 2018. URL: http://www.springer.com/978-3-319-63587-3,

doi:10.1007/978-3-319-63588-0.

André Platzer. The structure of differential invariants and differential cut elimination.

• Log. Meth. Comput. Sci., 8(4:16):1–38, 2012.

doi:10.2168/LMCS-8(4:16)2012.

André Platzer. Foundations of cyber-physical systems. Lecture Notes 15-424/624/824, Carnegie Mellon University, 2017. URL: http://lfcps.org/course/fcps17.html. André Platzer. A complete uniform substitution calculus for differential dynamic logic.

• J. Autom. Reas., 59(2):219–265, 2017.

doi:10.1007/s10817-016-9385-1.

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 23 / 23

André Platzer. A differential operator approach to equational differential invariants. In Lennart Beringer and Amy Felty, editors, ITP, volume 7406 of LNCS, pages 28–48, Berlin, 2012. Springer.

doi:10.1007/978-3-642-32347-8_3.

André Platzer. Differential-algebraic dynamic logic for differential-algebraic programs.

• J. Log. Comput., 20(1):309–352, 2010.

doi:10.1093/logcom/exn070.

André Platzer (CMU) LFCPS/13: Differential Invariants & Proof Theory LFCPS/13 23 / 23