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11: Differential Equations & Proofs Logical Foundations of Cyber-Physical Systems Andr Platzer Logical Foundations of Cyber-Physical Systems Andr Platzer Andr Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11

  1. Differential Substitution Lemmas ❀ Proofs Lemma (Differential lemma) (Differential value vs. Time-derivative) = x ′ = f ( x ) ∧ Q for duration r > 0 , then for all 0 ≤ z ≤ r, FV ( e ) ⊆ { x } : If ϕ | ] = d ϕ ( t )[ [ e ] ] [( e ) ′ ] ϕ ( z )[ ( z ) d t Lemma (Differential assignment) (Effect on Differentials) DE [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ][ x ′ := f ( x )] P Lemma (Derivations) (Equations of Differentials) ( e + k ) ′ = ( e ) ′ +( k ) ′ + ′ ( e · k ) ′ = ( e ) ′ · k + e · ( k ) ′ · ′ ( c ()) ′ = 0 c ′ ( x ) ′ = x ′ x ′ André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 9 / 24

  2. Differential Weakening x ¬ Q ν Q ω t r 0 x ′ = f ( x )& Q [ x ′ = f ( x )& Q ] = x ′ = f ( x ) ∧ Q [ ] = { ( ϕ ( 0 ) | { x ′ } ∁ , ϕ ( r )) : ϕ | for some ϕ : [ 0 , r ] → S , some r ∈ R } ODE ϕ ( z )( x ′ ) = d ϕ ( t )( x ) ( z ) d t André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 10 / 24

  3. Differential Weakening x ¬ Q ν Q ω t r DW [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ]( Q → P ) 0 x ′ = f ( x )& Q [ x ′ = f ( x )& Q ] = x ′ = f ( x ) ∧ Q [ ] = { ( ϕ ( 0 ) | { x ′ } ∁ , ϕ ( r )) : ϕ | for some ϕ : [ 0 , r ] → S , some r ∈ R } ODE ϕ ( z )( x ′ ) = d ϕ ( t )( x ) ( z ) d t Differential equations cannot leave their domains. André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 10 / 24

  4. Differential Weakening x ¬ Q ν Q ω t r DW [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ]( Q → P ) 0 x ′ = f ( x )& Q Example (Bouncing ball) DW ⊢ [ x ′ = v , v ′ = − g & x ≥ 0 ] 0 ≤ x No need to solve any ODEs to prove that bouncing ball is above ground. André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 10 / 24

  5. Differential Weakening x ¬ Q ν Q ω t r DW [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ]( Q → P ) 0 x ′ = f ( x )& Q Example (Bouncing ball) G ⊢ [ x ′ = v , v ′ = − g & x ≥ 0 ]( x ≥ 0 → 0 ≤ x ) DW ⊢ [ x ′ = v , v ′ = − g & x ≥ 0 ] 0 ≤ x No need to solve any ODEs to prove that bouncing ball is above ground. André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 10 / 24

  6. Differential Weakening x ¬ Q ν Q ω t r DW [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ]( Q → P ) 0 x ′ = f ( x )& Q Example (Bouncing ball) R ⊢ x ≥ 0 → 0 ≤ x G ⊢ [ x ′ = v , v ′ = − g & x ≥ 0 ]( x ≥ 0 → 0 ≤ x ) DW ⊢ [ x ′ = v , v ′ = − g & x ≥ 0 ] 0 ≤ x No need to solve any ODEs to prove that bouncing ball is above ground. André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 10 / 24

  7. Differential Weakening x ¬ Q ν Q ω t r DW [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ]( Q → P ) 0 x ′ = f ( x )& Q Example (Bouncing ball) ∗ R ⊢ x ≥ 0 → 0 ≤ x G ⊢ [ x ′ = v , v ′ = − g & x ≥ 0 ]( x ≥ 0 → 0 ≤ x ) DW ⊢ [ x ′ = v , v ′ = − g & x ≥ 0 ] 0 ≤ x No need to solve any ODEs to prove that bouncing ball is above ground. André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 10 / 24

  8. Differential Weakening x Differential Weakening ¬ Q ν dW Γ ⊢ [ x ′ = f ( x )& Q ] P , ∆ Q ω t r DW [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ]( Q → P ) 0 x ′ = f ( x )& Q Example (Bouncing ball) ∗ R ⊢ x ≥ 0 → 0 ≤ x G ⊢ [ x ′ = v , v ′ = − g & x ≥ 0 ]( x ≥ 0 → 0 ≤ x ) DW ⊢ [ x ′ = v , v ′ = − g & x ≥ 0 ] 0 ≤ x No need to solve any ODEs to prove that bouncing ball is above ground. André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 10 / 24

  9. Differential Weakening x Differential Weakening ¬ Q ν Q ⊢ P dW Γ ⊢ [ x ′ = f ( x )& Q ] P , ∆ Q ω t r DW [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ]( Q → P ) 0 x ′ = f ( x )& Q Example (Bouncing ball) ∗ R ⊢ x ≥ 0 → 0 ≤ x G ⊢ [ x ′ = v , v ′ = − g & x ≥ 0 ]( x ≥ 0 → 0 ≤ x ) DW ⊢ [ x ′ = v , v ′ = − g & x ≥ 0 ] 0 ≤ x No need to solve any ODEs to prove that bouncing ball is above ground. André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 10 / 24

  10. Differential Invariant Terms for Differential Equations Differential Invariant ⊢ [ x ′ := f ( x )]( e ) ′ = 0 dI e = 0 ⊢ [ x ′ = f ( x )& Q ] e = 0 � [ x ′ = f ( x )] e = 0 ↔ e = 0 � ← [ x ′ = f ( x )]( e ) ′ = 0 DI DE [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ][ x ′ := f ( x )] P DW [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ]( Q → P ) André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 11 / 24

  11. Differential Invariant Terms for Differential Equations Differential Invariant Q ⊢ [ x ′ := f ( x )]( e ) ′ = 0 dI e = 0 ⊢ [ x ′ = f ( x )& Q ] e = 0 � [ x ′ = f ( x )& Q ] e = 0 ↔ [? Q ] e = 0 � ← [ x ′ = f ( x )& Q ]( e ) ′ = 0 DI DE [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ][ x ′ := f ( x )] P DW [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ]( Q → P ) André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 11 / 24

  12. Differential Invariant Terms for Differential Equations Differential Invariant Q ⊢ [ x ′ := f ( x )]( e ) ′ = 0 dI e = 0 ⊢ [ x ′ = f ( x )& Q ] e = 0 � [ x ′ = f ( x )& Q ] e = 0 ↔ [? Q ] e = 0 � ← [ x ′ = f ( x )& Q ]( e ) ′ = 0 DI DE [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ][ x ′ := f ( x )] P DW [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ]( Q → P ) Proof (dI is a derived rule). DI e = 0 ⊢ [ x ′ = f ( x )& Q ] e = 0 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 11 / 24

  13. Differential Invariant Terms for Differential Equations Differential Invariant Q ⊢ [ x ′ := f ( x )]( e ) ′ = 0 dI e = 0 ⊢ [ x ′ = f ( x )& Q ] e = 0 � [ x ′ = f ( x )& Q ] e = 0 ↔ [? Q ] e = 0 � ← [ x ′ = f ( x )& Q ]( e ) ′ = 0 DI DE [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ][ x ′ := f ( x )] P DW [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ]( Q → P ) Proof (dI is a derived rule). ⊢ [ x ′ = f ( x )& Q ]( e ) ′ = 0 DE DI e = 0 ⊢ [ x ′ = f ( x )& Q ] e = 0 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 11 / 24

  14. Differential Invariant Terms for Differential Equations Differential Invariant Q ⊢ [ x ′ := f ( x )]( e ) ′ = 0 dI e = 0 ⊢ [ x ′ = f ( x )& Q ] e = 0 � [ x ′ = f ( x )& Q ] e = 0 ↔ [? Q ] e = 0 � ← [ x ′ = f ( x )& Q ]( e ) ′ = 0 DI DE [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ][ x ′ := f ( x )] P DW [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ]( Q → P ) Proof (dI is a derived rule). ⊢ [ x ′ = f ( x )& Q ][ x ′ := f ( x )]( e ) ′ = 0 DW ⊢ [ x ′ = f ( x )& Q ]( e ) ′ = 0 DE DI e = 0 ⊢ [ x ′ = f ( x )& Q ] e = 0 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 11 / 24

  15. Differential Invariant Terms for Differential Equations Differential Invariant Q ⊢ [ x ′ := f ( x )]( e ) ′ = 0 dI e = 0 ⊢ [ x ′ = f ( x )& Q ] e = 0 � [ x ′ = f ( x )& Q ] e = 0 ↔ [? Q ] e = 0 � ← [ x ′ = f ( x )& Q ]( e ) ′ = 0 DI DE [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ][ x ′ := f ( x )] P DW [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ]( Q → P ) Proof (dI is a derived rule). ⊢ [ x ′ = f ( x )& Q ]( Q → [ x ′ := f ( x )]( e ) ′ = 0 ) G, → R ⊢ [ x ′ = f ( x )& Q ][ x ′ := f ( x )]( e ) ′ = 0 DW ⊢ [ x ′ = f ( x )& Q ]( e ) ′ = 0 DE DI e = 0 ⊢ [ x ′ = f ( x )& Q ] e = 0 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 11 / 24

  16. Differential Invariant Terms for Differential Equations Differential Invariant Q ⊢ [ x ′ := f ( x )]( e ) ′ = 0 dI e = 0 ⊢ [ x ′ = f ( x )& Q ] e = 0 � [ x ′ = f ( x )& Q ] e = 0 ↔ [? Q ] e = 0 � ← [ x ′ = f ( x )& Q ]( e ) ′ = 0 DI DE [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ][ x ′ := f ( x )] P DW [ x ′ = f ( x )& Q ] P ↔ [ x ′ = f ( x )& Q ]( Q → P ) Proof (dI is a derived rule). Q ⊢ [ x ′ := f ( x )]( e ) ′ = 0 ⊢ [ x ′ = f ( x )& Q ]( Q → [ x ′ := f ( x )]( e ) ′ = 0 ) G, → R P ⊢ [ x ′ = f ( x )& Q ][ x ′ := f ( x )]( e ) ′ = 0 DW G [ α ] P ⊢ [ x ′ = f ( x )& Q ]( e ) ′ = 0 DE DI e = 0 ⊢ [ x ′ = f ( x )& Q ] e = 0 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 11 / 24

  17. Differential Invariant Equations Lemma (Differential lemma) (Differential value vs. Time-derivative) ] = d ϕ ( t )[ [ e ] ] = x ′ = f ( x ) ∧ Q for r > 0 ⇒ ∀ [( e ) ′ ] ϕ | 0 ≤ z ≤ r ϕ ( z )[ ( z ) d t Differential Invariant dI e = k ⊢ [ x ′ = f ( x )] e = k André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 12 / 24

  18. Differential Invariant Equations Lemma (Differential lemma) (Differential value vs. Time-derivative) ] = d ϕ ( t )[ [ e ] ] = x ′ = f ( x ) ∧ Q for r > 0 ⇒ ∀ [( e ) ′ ] ϕ | 0 ≤ z ≤ r ϕ ( z )[ ( z ) d t Differential Invariant ⊢ [ x ′ := f ( x )]( e ) ′ = ( k ) ′ dI e = k ⊢ [ x ′ = f ( x )] e = k � [ x ′ = f ( x )] e = k ↔ e = k � ← [ x ′ = f ( x )]( e ) ′ = ( k ) ′ DI André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 12 / 24

  19. Differential Invariant Equations Lemma (Differential lemma) (Differential value vs. Time-derivative) ] = d ϕ ( t )[ [ e ] ] = x ′ = f ( x ) ∧ Q for r > 0 ⇒ ∀ [( e ) ′ ] ϕ | 0 ≤ z ≤ r ϕ ( z )[ ( z ) d t Differential Invariant e ⊢ [ x ′ := f ( x )]( e ) ′ = ( k ) ′ k dI e = k ⊢ [ x ′ = f ( x )] e = k 0 t � [ x ′ = f ( x )] e = k ↔ e = k � ← [ x ′ = f ( x )]( e ) ′ = ( k ) ′ DI Proof ( = rate of change from = initial value. Mean-value theorem). d ϕ ( t )[ [ e ] ] ] = d ϕ ( t )[ [ k ] ] [( e ) ′ ] [( k ) ′ ] ( z ) = ϕ ( z )[ ] = ϕ ( z )[ ( z ) d t d t André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 12 / 24

  20. Differential Invariant Inequalities Lemma (Differential lemma) (Differential value vs. Time-derivative) ] = d ϕ ( t )[ [ e ] ] = x ′ = f ( x ) ∧ Q for r > 0 ⇒ ∀ [( e ) ′ ] ϕ | 0 ≤ z ≤ r ϕ ( z )[ ( z ) d t Differential Invariant e ⊢ [ x ′ := f ( x )]( e ) ′ ≥ ( k ) ′ k dI e ≥ k ⊢ [ x ′ = f ( x )] e ≥ k 0 t � [ x ′ = f ( x )] e ≥ k ↔ e ≥ k � ← [ x ′ = f ( x )]( e ) ′ ≥ ( k ) ′ DI Proof ( ≥ rate of change from ≥ initial value. Mean-value theorem). d ϕ ( t )[ [ e ] ] ] = d ϕ ( t )[ [ k ] ] [( e ) ′ ] [( k ) ′ ] ( z ) = ϕ ( z )[ ] ≥ ϕ ( z )[ ( z ) d t d t André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 13 / 24

  21. Differential Invariant Inequalities Lemma (Differential lemma) (Differential value vs. Time-derivative) ] = d ϕ ( t )[ [ e ] ] = x ′ = f ( x ) ∧ Q for r > 0 ⇒ ∀ [( e ) ′ ] ϕ | 0 ≤ z ≤ r ϕ ( z )[ ( z ) d t Differential Invariant k e ⊢ [ x ′ := f ( x )]( e ) ′ ≤ ( k ) ′ dI e ≤ k ⊢ [ x ′ = f ( x )] e ≤ k 0 t � [ x ′ = f ( x )] e ≤ k ↔ e ≤ k � ← [ x ′ = f ( x )]( e ) ′ ≤ ( k ) ′ DI Proof ( ≤ rate of change from ≤ initial value. Mean-value theorem). d ϕ ( t )[ [ e ] ] ] = d ϕ ( t )[ [ k ] ] [( e ) ′ ] [( k ) ′ ] ( z ) = ϕ ( z )[ ] ≤ ϕ ( z )[ ( z ) d t d t André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 13 / 24

  22. Differential Invariant Inequalities Lemma (Differential lemma) (Differential value vs. Time-derivative) ] = d ϕ ( t )[ [ e ] ] = x ′ = f ( x ) ∧ Q for r > 0 ⇒ ∀ [( e ) ′ ] ϕ | 0 ≤ z ≤ r ϕ ( z )[ ( z ) d t Differential Invariant e ⊢ [ x ′ := f ( x )]( e ) ′ > ( k ) ′ k dI e > k ⊢ [ x ′ = f ( x )] e > k 0 t � [ x ′ = f ( x )] e > k ↔ e > k � ← [ x ′ = f ( x )]( e ) ′ > ( k ) ′ DI Proof ( > rate of change from > initial value. Mean-value theorem). d ϕ ( t )[ [ e ] ] ] = d ϕ ( t )[ [ k ] ] [( e ) ′ ] [( k ) ′ ] ( z ) = ϕ ( z )[ ] > ϕ ( z )[ ( z ) d t d t André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 13 / 24

  23. Differential Invariant Inequalities Lemma (Differential lemma) (Differential value vs. Time-derivative) ] = d ϕ ( t )[ [ e ] ] = x ′ = f ( x ) ∧ Q for r > 0 ⇒ ∀ [( e ) ′ ] ϕ | 0 ≤ z ≤ r ϕ ( z )[ ( z ) d t Differential Invariant e ⊢ [ x ′ := f ( x )]( e ) ′ ≥ ( k ) ′ k dI e > k ⊢ [ x ′ = f ( x )] e > k 0 t � [ x ′ = f ( x )] e > k ↔ e > k � ← [ x ′ = f ( x )]( e ) ′ ≥ ( k ) ′ DI Proof ( ≥ rate of change from > initial value. Mean-value theorem). d ϕ ( t )[ [ e ] ] ] = d ϕ ( t )[ [ k ] ] [( e ) ′ ] [( k ) ′ ] ( z ) = ϕ ( z )[ ] ≥ ϕ ( z )[ ( z ) d t d t André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 13 / 24

  24. Differential Invariant Inequalities Lemma (Differential lemma) (Differential value vs. Time-derivative) ] = d ϕ ( t )[ [ e ] ] = x ′ = f ( x ) ∧ Q for r > 0 ⇒ ∀ [( e ) ′ ] ϕ | 0 ≤ z ≤ r ϕ ( z )[ ( z ) d t Differential Invariant e ⊢ [ x ′ := f ( x )]( e ) ′ � = ( k ) ′ k dI e � = k ⊢ [ x ′ = f ( x )] e � = k 0 t � [ x ′ = f ( x )] e � = k ↔ e � = k � ← [ x ′ = f ( x )]( e ) ′ � = ( k ) ′ DI Proof ( � = rate of change from � = initial value. Mean-value theorem). d ϕ ( t )[ [ e ] ] ] = d ϕ ( t )[ [ k ] ] [( e ) ′ ] [( k ) ′ ] ( z ) = ϕ ( z )[ ] � = ϕ ( z )[ ( z ) d t d t André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 14 / 24

  25. Differential Invariant Inequalities Lemma (Differential lemma) (Differential value vs. Time-derivative) ] = d ϕ ( t )[ [ e ] ] = x ′ = f ( x ) ∧ Q for r > 0 ⇒ ∀ [( e ) ′ ] ϕ | 0 ≤ z ≤ r ϕ ( z )[ ( z ) d t Differential Invariant ⊢ [ x ′ := f ( x )]( e ) ′ � = ( k ) ′ k dI e � = k ⊢ [ x ′ = f ( x )] e � = k e 0 t � [ x ′ = f ( x )] e � = k ↔ e � = k � ← [ x ′ = f ( x )]( e ) ′ � = ( k ) ′ DI Proof ( � = rate of change from � = initial value. Mean-value theorem). d ϕ ( t )[ [ e ] ] ] = d ϕ ( t )[ [ k ] ] [( e ) ′ ] [( k ) ′ ] ( z ) = ϕ ( z )[ ] � = ϕ ( z )[ ( z ) d t d t André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 14 / 24

  26. Differential Invariant Inequalities Lemma (Differential lemma) (Differential value vs. Time-derivative) ] = d ϕ ( t )[ [ e ] ] = x ′ = f ( x ) ∧ Q for r > 0 ⇒ ∀ [( e ) ′ ] ϕ | 0 ≤ z ≤ r ϕ ( z )[ ( z ) d t Differential Invariant ⊢ [ x ′ := f ( x )]( e ) ′ � = ( k ) ′ k dI e � = k ⊢ [ x ′ = f ( x )] e � = k e 0 t � [ x ′ = f ( x )] e � = k ↔ e � = k � ← [ x ′ = f ( x )]( e ) ′ � = ( k ) ′ DI Proof ( � = rate of change from � = initial value. Mean-value theorem). d ϕ ( t )[ [ e ] ] ] = d ϕ ( t )[ [ k ] ] [( e ) ′ ] [( k ) ′ ] ( z ) = ϕ ( z )[ ] � = ϕ ( z )[ ( z ) d t d t André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 14 / 24

  27. Differential Invariant Inequalities Lemma (Differential lemma) (Differential value vs. Time-derivative) ] = d ϕ ( t )[ [ e ] ] = x ′ = f ( x ) ∧ Q for r > 0 ⇒ ∀ [( e ) ′ ] ϕ | 0 ≤ z ≤ r ϕ ( z )[ ( z ) d t Differential Invariant e ⊢ [ x ′ := f ( x )]( e ) ′ = ( k ) ′ k dI e � = k ⊢ [ x ′ = f ( x )] e � = k 0 t � [ x ′ = f ( x )] e � = k ↔ e � = k � ← [ x ′ = f ( x )]( e ) ′ = ( k ) ′ DI Proof ( = rate of change from � = initial value. Mean-value theorem). d ϕ ( t )[ [ e ] ] ] = d ϕ ( t )[ [ k ] ] [( e ) ′ ] [( k ) ′ ] ( z ) = ϕ ( z )[ ] = ϕ ( z )[ ( z ) d t d t André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 14 / 24

  28. Example: Differential Invariant Inequalities ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y & ω ≥ 0 ∧ d ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 1.0 ��� x ��� 0.5 ��� 1 2 3 4 5 6 y - ��� � 0.5 - ��� � 1.0 - ��� � 1.5 - ��� - ��� - ��� ��� ��� ��� ��� André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 15 / 24

  29. Example: Differential Invariant Inequalities: Oscillator ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y & ω ≥ 0 ∧ d ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 1.0 ��� x ��� 0.5 ��� 1 2 3 4 5 6 y - ��� � 0.5 - ��� � 1.0 - ��� � 1.5 damped oscillator - ��� - ��� - ��� ��� ��� ��� ��� André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 15 / 24

  30. Example: Differential Invariant Inequalities: Oscillator ω ≥ 0 ∧ d ≥ 0 ⊢ [ x ′ := y ][ y ′ := − ω 2 x − 2 d ω y ] 2 ω 2 xx ′ + 2 yy ′ ≤ 0 ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y & ω ≥ 0 ∧ d ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 1.0 ��� x ��� 0.5 ��� 1 2 3 4 5 6 y - ��� � 0.5 - ��� � 1.0 - ��� � 1.5 damped oscillator - ��� - ��� - ��� ��� ��� ��� ��� André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 15 / 24

  31. Example: Differential Invariant Inequalities: Oscillator ω ≥ 0 ∧ d ≥ 0 ⊢ 2 ω 2 xy + 2 y ( − ω 2 x − 2 d ω y ) ≤ 0 ω ≥ 0 ∧ d ≥ 0 ⊢ [ x ′ := y ][ y ′ := − ω 2 x − 2 d ω y ] 2 ω 2 xx ′ + 2 yy ′ ≤ 0 ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y & ω ≥ 0 ∧ d ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 1.0 ��� x ��� 0.5 ��� 1 2 3 4 5 6 y - ��� � 0.5 - ��� � 1.0 - ��� � 1.5 damped oscillator - ��� - ��� - ��� ��� ��� ��� ��� André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 15 / 24

  32. Example: Differential Invariant Inequalities: Oscillator ∗ ω ≥ 0 ∧ d ≥ 0 ⊢ 2 ω 2 xy + 2 y ( − ω 2 x − 2 d ω y ) ≤ 0 ω ≥ 0 ∧ d ≥ 0 ⊢ [ x ′ := y ][ y ′ := − ω 2 x − 2 d ω y ] 2 ω 2 xx ′ + 2 yy ′ ≤ 0 ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y & ω ≥ 0 ∧ d ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 1.0 ��� x ��� 0.5 ��� 1 2 3 4 5 6 y - ��� � 0.5 - ��� � 1.0 - ��� � 1.5 damped oscillator - ��� - ��� - ��� ��� ��� ��� ��� André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 15 / 24

  33. Example: Differential Invariant Inequalities: Oscillator ∗ ω ≥ 0 ∧ d ≥ 0 ⊢ 2 ω 2 xy + 2 y ( − ω 2 x − 2 d ω y ) ≤ 0 ω ≥ 0 ∧ d ≥ 0 ⊢ [ x ′ := y ][ y ′ := − ω 2 x − 2 d ω y ] 2 ω 2 xx ′ + 2 yy ′ ≤ 0 ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y & ω ≥ 0 ∧ d ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 1.0 ��� x ��� 0.5 ��� 1 2 3 4 5 6 y - ��� � 0.5 - ��� � 1.0 - ��� � 1.5 damped oscillator - ��� - ��� - ��� ��� ��� ��� ��� André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 15 / 24

  34. Differential Invariant Conjunctions Differential Invariant dI A ∧ B ⊢ [ x ′ = f ( x )]( A ∧ B ) André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 16 / 24

  35. Differential Invariant Conjunctions v Differential Invariant dist( x , v ) ∧ slow( v ) ⊢ [ x ′ := f ( x )](( A ) ′ ∧ ( B ) ′ ) dI A ∧ B ⊢ [ x ′ = f ( x )]( A ∧ B ) x � [ x ′ = f ( x )]( A ∧ B ) ↔ ( A ∧ B ) � ← [ x ′ = f ( x ))](( A ) ′ ∧ ( B ) ′ ) DI André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 16 / 24

  36. Differential Invariant Conjunctions v Differential Invariant dist( x, v ) ∧ slow( v ) ⊢ [ x ′ := f ( x )](( A ) ′ ∧ ( B ) ′ ) dI A ∧ B ⊢ [ x ′ = f ( x )]( A ∧ B ) x � [ x ′ = f ( x )]( A ∧ B ) ↔ ( A ∧ B ) � ← [ x ′ = f ( x ))](( A ) ′ ∧ ( B ) ′ ) DI Proof (separately). ⊢ [ x ′ = f ( x )]( A ) ′ ⊢ [ x ′ = f ( x )]( B ) ′ DI A ⊢ [ x ′ = f ( x )] A DI B ⊢ [ x ′ = f ( x )] B A ∧ B ⊢ [ x ′ = f ( x )]( A ∧ B ) [] ∧ ,WL [] ∧ [ α ]( P ∧ Q ) ↔ [ α ] P ∧ [ α ] Q André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 16 / 24

  37. Quantum’s Back for a Differential Invariant Proof 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ]( 2 gx = 2 gH − v 2 ∧ x ≥ 0 ) No solutions but still a proof. Simple proof with simple arithmetic. Independent proofs for independent questions. André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 17 / 24

  38. Quantum’s Back for a Differential Invariant Proof [] ∧ [ α ]( P ∧ Q ) ↔ [ α ] P ∧ [ α ] Q [] ∧ 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ] 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ] x ≥ 0 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ]( 2 gx = 2 gH − v 2 ∧ x ≥ 0 ) No solutions but still a proof. Simple proof with simple arithmetic. Independent proofs for independent questions. André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 17 / 24

  39. Quantum’s Back for a Differential Invariant Proof x ≥ 0 ⊢ [ x ′ := v ][ v ′ := − g ] 2 gx ′ = − 2 vv ′ dI 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ] 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ] x ≥ 0 [] ∧ 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ]( 2 gx = 2 gH − v 2 ∧ x ≥ 0 ) No solutions but still a proof. Simple proof with simple arithmetic. Independent proofs for independent questions. André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 17 / 24

  40. Quantum’s Back for a Differential Invariant Proof x ≥ 0 ⊢ 2 gv = − 2 v ( − g ) [:=] x ≥ 0 ⊢ [ x ′ := v ][ v ′ := − g ] 2 gx ′ = − 2 vv ′ dI 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ] 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ] x ≥ 0 [] ∧ 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ]( 2 gx = 2 gH − v 2 ∧ x ≥ 0 ) No solutions but still a proof. Simple proof with simple arithmetic. Independent proofs for independent questions. André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 17 / 24

  41. Quantum’s Back for a Differential Invariant Proof ∗ R x ≥ 0 ⊢ 2 gv = − 2 v ( − g ) [:=] x ≥ 0 ⊢ [ x ′ := v ][ v ′ := − g ] 2 gx ′ = − 2 vv ′ dI 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ] 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ] x ≥ 0 [] ∧ 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ]( 2 gx = 2 gH − v 2 ∧ x ≥ 0 ) No solutions but still a proof. Simple proof with simple arithmetic. Independent proofs for independent questions. André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 17 / 24

  42. Quantum’s Back for a Differential Invariant Proof ∗ R x ≥ 0 ⊢ 2 gv = − 2 v ( − g ) [:=] x ≥ 0 ⊢ [ x ′ := v ][ v ′ := − g ] 2 gx ′ = − 2 vv ′ x ≥ 0 ⊢ x ≥ 0 dI 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ] 2 gx = 2 gH − v 2 dW ⊢ [ x ′′ = − g & x ≥ 0 ] x ≥ 0 [] ∧ 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ]( 2 gx = 2 gH − v 2 ∧ x ≥ 0 ) No solutions but still a proof. Simple proof with simple arithmetic. Independent proofs for independent questions. André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 17 / 24

  43. Quantum’s Back for a Differential Invariant Proof ∗ R ∗ x ≥ 0 ⊢ 2 gv = − 2 v ( − g ) [:=] id x ≥ 0 ⊢ [ x ′ := v ][ v ′ := − g ] 2 gx ′ = − 2 vv ′ x ≥ 0 ⊢ x ≥ 0 dI 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ] 2 gx = 2 gH − v 2 dW ⊢ [ x ′′ = − g & x ≥ 0 ] x ≥ 0 [] ∧ 2 gx = 2 gH − v 2 ⊢ [ x ′′ = − g & x ≥ 0 ]( 2 gx = 2 gH − v 2 ∧ x ≥ 0 ) No solutions but still a proof. Simple proof with simple arithmetic. Independent proofs for independent questions. André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 17 / 24

  44. Differential Invariant Conjunctions v Differential Invariant dist( x, v ) ∧ slow( v ) ⊢ [ x ′ := f ( x )](( A ) ′ ∧ ( B ) ′ ) dI A ∧ B ⊢ [ x ′ = f ( x )]( A ∧ B ) x � [ x ′ = f ( x )]( A ∧ B ) ↔ ( A ∧ B ) � ← [ x ′ = f ( x ))](( A ) ′ ∧ ( B ) ′ ) DI André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 18 / 24

  45. Differential Invariant Disjunctions v Differential Invariant dist( x, v ) ∨ slow( v ) ⊢ [ x ′ := f ( x )](( A ) ′ ∨ ( B ) ′ ) dI A ∨ B ⊢ [ x ′ = f ( x )]( A ∨ B ) x � [ x ′ = f ( x )]( A ∨ B ) ↔ ( A ∨ B ) � ← [ x ′ = f ( x ))](( A ) ′ ∨ ( B ) ′ ) DI André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 18 / 24

  46. Differential Invariant Disjunctions v Differential Invariant dist( x, v ) ∨ slow( v ) ⊢ [ x ′ := f ( x )](( A ) ′ ∨ ( B ) ′ ) dI A ∨ B ⊢ [ x ′ = f ( x )]( A ∨ B ) x � [ x ′ = f ( x )]( A ∨ B ) ↔ ( A ∨ B ) � ← [ x ′ = f ( x ))](( A ) ′ ∨ ( B ) ′ ) DI André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 18 / 24

  47. Differential Invariant Disjunctions v Differential Invariant dist( x, v ) ∨ slow( v ) ⊢ [ x ′ := f ( x )](( A ) ′ ∧ ( B ) ′ ) dI A ∨ B ⊢ [ x ′ = f ( x )]( A ∨ B ) x � [ x ′ = f ( x )]( A ∨ B ) ↔ ( A ∨ B ) � ← [ x ′ = f ( x ))](( A ) ′ ∧ ( B ) ′ ) DI André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 18 / 24

  48. Differential Invariant Disjunctions v Differential Invariant dist( x, v ) ∨ slow( v ) ⊢ [ x ′ := f ( x )](( A ) ′ ∧ ( B ) ′ ) dI A ∨ B ⊢ [ x ′ = f ( x )]( A ∨ B ) x � [ x ′ = f ( x )]( A ∨ B ) ↔ ( A ∨ B ) � ← [ x ′ = f ( x ))](( A ) ′ ∧ ( B ) ′ ) DI Proof (separately). ⊢ [ x ′ = f ( x )]( A ) ′ ⊢ [ x ′ = f ( x )]( B ) ′ ∗ ∗ DI A ⊢ [ x ′ = f ( x )] A DI B ⊢ [ x ′ = f ( x )] B A ⊢ A ∨ B B ⊢ A ∨ B A ⊢ [ x ′ = f ( x )]( A ∨ B ) B ⊢ [ x ′ = f ( x )]( A ∨ B ) MR MR A ∨ B ⊢ [ x ′ = f ( x )]( A ∨ B ) ∨ L André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 18 / 24

  49. Differential Invariant Disjunctions v Differential Invariant dist( x, v ) ∨ slow( v ) ⊢ [ x ′ := f ( x )](( A ) ′ ∧ ( B ) ′ ) dI A ∨ B ⊢ [ x ′ = f ( x )]( A ∨ B ) x � [ x ′ = f ( x )]( A ∨ B ) ↔ ( A ∨ B ) � ← [ x ′ = f ( x ))](( A ) ′ ∧ ( B ) ′ ) DI Proof (separately). ⊢ [ x ′ = f ( x )]( A ) ′ ⊢ [ x ′ = f ( x )]( B ) ′ ∗ ∗ DI A ⊢ [ x ′ = f ( x )] A DI B ⊢ [ x ′ = f ( x )] B A ⊢ A ∨ B B ⊢ A ∨ B A ⊢ [ x ′ = f ( x )]( A ∨ B ) B ⊢ [ x ′ = f ( x )]( A ∨ B ) MR MR A ∨ B ⊢ [ x ′ = f ( x )]( A ∨ B ) ∨ L [] ∧ [ α ]( P ∧ Q ) ↔ [ α ] P ∧ [ α ] Q André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 18 / 24

  50. Assuming Invariants F F ¬ ¬ F F ¬ F ¬ F Q → [ x ′ := f ( x )]( F ) ′ F ∧ Q → [ x ′ := f ( x )]( F ) ′ F ⊢ [ x ′ = f ( x )& Q ] F F ⊢ [ x ′ = f ( x )& Q ] F F ⊢ [ α ] F loop F ⊢ [ α ∗ ] F André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 19 / 24

  51. Assuming Invariants F F ¬ ¬ F F ¬ F ¬ F Q → [ x ′ := f ( x )]( F ) ′ F ∧ Q → [ x ′ := f ( x )]( F ) ′ F ⊢ [ x ′ = f ( x )& Q ] F F ⊢ [ x ′ = f ( x )& Q ] F Example (Restrictions) v 2 − 2 v + 1 = 0 ⊢ [ v ′ = w , w ′ = − v ] v 2 − 2 v + 1 = 0 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 19 / 24

  52. Assuming Invariants F F ¬ ¬ F F ¬ F ¬ F Q → [ x ′ := f ( x )]( F ) ′ F ∧ Q → [ x ′ := f ( x )]( F ) ′ F ⊢ [ x ′ = f ( x )& Q ] F F ⊢ [ x ′ = f ( x )& Q ] F Example (Restrictions) v 2 − 2 v + 1 = 0 ⊢ [ v ′ := w ][ w ′ := − v ] 2 vv ′ − 2 v ′ = 0 v 2 − 2 v + 1 = 0 ⊢ [ v ′ = w , w ′ = − v ] v 2 − 2 v + 1 = 0 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 19 / 24

  53. Assuming Invariants F F ¬ ¬ F F ¬ F ¬ F Q → [ x ′ := f ( x )]( F ) ′ F ∧ Q → [ x ′ := f ( x )]( F ) ′ F ⊢ [ x ′ = f ( x )& Q ] F F ⊢ [ x ′ = f ( x )& Q ] F Example (Restrictions) v 2 − 2 v + 1 = 0 ⊢ 2 vw − 2 w = 0 v 2 − 2 v + 1 = 0 ⊢ [ v ′ := w ][ w ′ := − v ] 2 vv ′ − 2 v ′ = 0 v 2 − 2 v + 1 = 0 ⊢ [ v ′ = w , w ′ = − v ] v 2 − 2 v + 1 = 0 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 19 / 24

  54. Assuming Invariants F F ¬ ¬ F F ¬ F ¬ F Q → [ x ′ := f ( x )]( F ) ′ F ∧ Q → [ x ′ := f ( x )]( F ) ′ F ⊢ [ x ′ = f ( x )& Q ] F F ⊢ [ x ′ = f ( x )& Q ] F Example (Restrictions) v v 2 − 2 v + 1 = 0 ⊢ 2 vw − 2 w = 0 w 0 v 2 − 2 v + 1 = 0 ⊢ [ v ′ := w ][ w ′ := − v ] 2 vv ′ − 2 v ′ = 0 v 2 − 2 v + 1 = 0 ⊢ [ v ′ = w , w ′ = − v ] v 2 − 2 v + 1 = 0 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 19 / 24

  55. Assuming Invariants F F ¬ ¬ F F ¬ F ¬ F Q → [ x ′ := f ( x )]( F ) ′ F ∧ Q → [ x ′ := f ( x )]( F ) ′ F ⊢ [ x ′ = f ( x )& Q ] F F ⊢ [ x ′ = f ( x )& Q ] F Example (Restrictions are unsound!) v (unsound) v 2 − 2 v + 1 = 0 ⊢ 2 vw − 2 w = 0 w 0 v 2 − 2 v + 1 = 0 ⊢ [ v ′ := w ][ w ′ := − v ] 2 vv ′ − 2 v ′ = 0 v 2 − 2 v + 1 = 0 ⊢ [ v ′ = w , w ′ = − v ] v 2 − 2 v + 1 = 0 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 19 / 24

  56. Outline Learning Objectives 1 Differential Invariants 2 Recap: Ingredients for Differential Equation Proofs Soundness: Derivations Lemma Differential Weakening Equational Differential Invariants Differential Invariant Inequalities Disequational Differential Invariants Example Proof: Damped Oscillator Conjunctive Differential Invariants Disjunctive Differential Invariants Assuming Invariants Differential Cuts 3 Soundness 4 Summary 5 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 19 / 24

  57. Differential Cuts Differential Cut F ⊢ [ x ′ = f ( x )] F Differential Cut André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 20 / 24

  58. Differential Cuts Differential Cut F ⊢ [ x ′ = f ( x )] C F ⊢ [ x ′ = f ( x )] F Differential Cut André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 20 / 24

  59. Differential Cuts Differential Cut F ⊢ [ x ′ = f ( x )] C F ⊢ [ x ′ = f ( x )& C ] F F ⊢ [ x ′ = f ( x )] F Differential Cut André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 20 / 24

  60. Differential Cuts Differential Cut F ⊢ [ x ′ = f ( x )& Q ] C F ⊢ [ x ′ = f ( x )& Q ∧ C ] F F ⊢ [ x ′ = f ( x )& Q ] F Differential Cut André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 20 / 24

  61. Differential Cuts Differential Cut F ⊢ [ x ′ = f ( x )& Q ] C F ⊢ [ x ′ = f ( x )& Q ∧ C ] F F ⊢ [ x ′ = f ( x )& Q ] F Differential Cut André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 20 / 24

  62. Differential Cuts Differential Cut F ⊢ [ x ′ = f ( x )& Q ] C F ⊢ [ x ′ = f ( x )& Q ∧ C ] F F ⊢ [ x ′ = f ( x )& Q ] F Differential Cut André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 20 / 24

  63. Differential Cuts Differential Cut F ⊢ [ x ′ = f ( x )& Q ] C F ⊢ [ x ′ = f ( x )& Q ∧ C ] F F ⊢ [ x ′ = f ( x )& Q ] F Differential Cut André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 20 / 24

  64. Differential Cuts Differential Cut F ⊢ [ x ′ = f ( x )& Q ] C F ⊢ [ x ′ = f ( x )& Q ∧ C ] F F ⊢ [ x ′ = f ( x )& Q ] F Differential Cut André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 20 / 24

  65. Differential Cuts Differential Cut F ⊢ [ x ′ = f ( x )& Q ] C F ⊢ [ x ′ = f ( x )& Q ∧ C ] F F ⊢ [ x ′ = f ( x )& Q ] F Proof (Soundness). Differential Cut = x ′ = f ( x ) ∧ Q starting in ω ∈ [ Let ϕ | [ F ] ] . [[ x ′ = f ( x )& Q ] C ] ω ∈ [ ] by left premise. = x ′ = f ( x ) ∧ Q ∧ C . Thus, ϕ | Thus, ϕ ( r ) ∈ [ [ F ] ] by second premise. André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 20 / 24

  66. Differential Cut Example: Increasingly Damped Oscillator dC ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 21 / 24

  67. Differential Cut Example: Increasingly Damped Oscillator dC ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 1.0 ��� x ��� 0.5 ��� 0.0 1 2 3 4 5 6 y - ��� - 0.5 - ��� - 1.0 - ��� increasingly damped oscillator - 1.5 - ��� - ��� - ��� ��� ��� ��� ��� André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 21 / 24

  68. Differential Cut Example: Increasingly Damped Oscillator dI ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ∧ d ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 dC ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 increasingly damped oscillator André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 21 / 24

  69. Differential Cut Example: Increasingly Damped Oscillator dI ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ∧ d ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 dC ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 dI d ≥ 0 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] d ≥ 0 increasingly damped oscillator André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 21 / 24

  70. Differential Cut Example: Increasingly Damped Oscillator dI ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ∧ d ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 dC ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 [:=] ω ≥ 0 ⊢ [ d ′ := 7 ] d ′ ≥ 0 dI d ≥ 0 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] d ≥ 0 increasingly damped oscillator André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 21 / 24

  71. Differential Cut Example: Increasingly Damped Oscillator dI ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ∧ d ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 dC ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 R ω ≥ 0 ⊢ 7 ≥ 0 [:=] ω ≥ 0 ⊢ [ d ′ := 7 ] d ′ ≥ 0 dI d ≥ 0 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] d ≥ 0 increasingly damped oscillator André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 21 / 24

  72. Differential Cut Example: Increasingly Damped Oscillator dI ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ∧ d ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 dC ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 ∗ R ω ≥ 0 ⊢ 7 ≥ 0 ask [:=] ω ≥ 0 ⊢ [ d ′ := 7 ] d ′ ≥ 0 dI d ≥ 0 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] d ≥ 0 increasingly damped oscillator André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 21 / 24

  73. Differential Cut Example: Increasingly Damped Oscillator ω ≥ 0 ∧ d ≥ 0 ⊢ [ x ′ := y ][ y ′ := − ω 2 x − 2 d ω y ] 2 ω 2 xx ′ + 2 yy ′ ≤ 0 [:=] dI ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ∧ d ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 dC ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 ∗ R ω ≥ 0 ⊢ 7 ≥ 0 [:=] ω ≥ 0 ⊢ [ d ′ := 7 ] d ′ ≥ 0 dI d ≥ 0 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] d ≥ 0 increasingly damped oscillator André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 21 / 24

  74. Differential Cut Example: Increasingly Damped Oscillator R ω ≥ 0 ∧ d ≥ 0 ⊢ 2 ω 2 xy + 2 y ( − ω 2 x − 2 d ω y ) ≤ 0 ω ≥ 0 ∧ d ≥ 0 ⊢ [ x ′ := y ][ y ′ := − ω 2 x − 2 d ω y ] 2 ω 2 xx ′ + 2 yy ′ ≤ 0 [:=] dI ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ∧ d ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 dC ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 ∗ R ω ≥ 0 ⊢ 7 ≥ 0 [:=] ω ≥ 0 ⊢ [ d ′ := 7 ] d ′ ≥ 0 dI d ≥ 0 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] d ≥ 0 increasingly damped oscillator André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 21 / 24

  75. Differential Cut Example: Increasingly Damped Oscillator ∗ R ω ≥ 0 ∧ d ≥ 0 ⊢ 2 ω 2 xy + 2 y ( − ω 2 x − 2 d ω y ) ≤ 0 ω ≥ 0 ∧ d ≥ 0 ⊢ [ x ′ := y ][ y ′ := − ω 2 x − 2 d ω y ] 2 ω 2 xx ′ + 2 yy ′ ≤ 0 [:=] dI ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ∧ d ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 dC ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 DC ∗ R ω ≥ 0 ⊢ 7 ≥ 0 [:=] ω ≥ 0 ⊢ [ d ′ := 7 ] d ′ ≥ 0 dI d ≥ 0 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] d ≥ 0 increasingly damped oscillator André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 21 / 24

  76. Differential Cut Example: Increasingly Damped Oscillator ∗ R ω ≥ 0 ∧ d ≥ 0 ⊢ 2 ω 2 xy + 2 y ( − ω 2 x − 2 d ω y ) ≤ 0 ω ≥ 0 ∧ d ≥ 0 ⊢ [ x ′ := y ][ y ′ := − ω 2 x − 2 d ω y ] 2 ω 2 xx ′ + 2 yy ′ ≤ 0 [:=] dI ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ∧ d ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 dC ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 ∗ init R ω ≥ 0 ⊢ 7 ≥ 0 [:=] ω ≥ 0 ⊢ [ d ′ := 7 ] d ′ ≥ 0 dI d ≥ 0 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] d ≥ 0 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 21 / 24

  77. Differential Cut Example: Increasingly Damped Oscillator ∗ R ω ≥ 0 ∧ d ≥ 0 ⊢ 2 ω 2 xy + 2 y ( − ω 2 x − 2 d ω y ) ≤ 0 ω ≥ 0 ∧ d ≥ 0 ⊢ [ x ′ := y ][ y ′ := − ω 2 x − 2 d ω y ] 2 ω 2 xx ′ + 2 yy ′ ≤ 0 [:=] dI ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ∧ d ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 dC ω 2 x 2 + y 2 ≤ c 2 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] ω 2 x 2 + y 2 ≤ c 2 ∗ init R ω ≥ 0 ⊢ 7 ≥ 0 [:=] ω ≥ 0 ⊢ [ d ′ := 7 ] d ′ ≥ 0 dI d ≥ 0 ⊢ [ x ′ = y , y ′ = − ω 2 x − 2 d ω y , d ′ = 7 & ω ≥ 0 ] d ≥ 0 Could repeatedly diffcut in formulas to help the proof André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 21 / 24

  78. Ex: Differential Cuts dC x 3 ≥ − 1 ∧ y 5 ≥ 0 ⊢ [ x ′ = ( x − 2 ) 4 + y 5 , y ′ = y 2 ] x 3 ≥ − 1 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 22 / 24

  79. Ex: Differential Cuts dC x 3 ≥ − 1 ∧ y 5 ≥ 0 ⊢ [ x ′ = ( x − 2 ) 4 + y 5 , y ′ = y 2 ] x 3 ≥ − 1 dI y 5 ≥ 0 ⊢ [ x ′ = ( x − 2 ) 4 + y 5 , y ′ = y 2 ] y 5 ≥ 0 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 22 / 24

  80. Ex: Differential Cuts dC x 3 ≥ − 1 ∧ y 5 ≥ 0 ⊢ [ x ′ = ( x − 2 ) 4 + y 5 , y ′ = y 2 ] x 3 ≥ − 1 ⊢ [ x ′ :=( x − 2 ) 4 + y 5 ][ y ′ := y 2 ] 5 y 4 y ′ ≥ 0 [:=] dI y 5 ≥ 0 ⊢ [ x ′ = ( x − 2 ) 4 + y 5 , y ′ = y 2 ] y 5 ≥ 0 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 22 / 24

  81. Ex: Differential Cuts dC x 3 ≥ − 1 ∧ y 5 ≥ 0 ⊢ [ x ′ = ( x − 2 ) 4 + y 5 , y ′ = y 2 ] x 3 ≥ − 1 ⊢ 5 y 4 y 2 ≥ 0 R ⊢ [ x ′ :=( x − 2 ) 4 + y 5 ][ y ′ := y 2 ] 5 y 4 y ′ ≥ 0 [:=] dI y 5 ≥ 0 ⊢ [ x ′ = ( x − 2 ) 4 + y 5 , y ′ = y 2 ] y 5 ≥ 0 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 22 / 24

  82. Ex: Differential Cuts dC x 3 ≥ − 1 ∧ y 5 ≥ 0 ⊢ [ x ′ = ( x − 2 ) 4 + y 5 , y ′ = y 2 ] x 3 ≥ − 1 ∗ ⊢ 5 y 4 y 2 ≥ 0 R ⊢ [ x ′ :=( x − 2 ) 4 + y 5 ][ y ′ := y 2 ] 5 y 4 y ′ ≥ 0 [:=] dI y 5 ≥ 0 ⊢ [ x ′ = ( x − 2 ) 4 + y 5 , y ′ = y 2 ] y 5 ≥ 0 André Platzer (CMU) LFCPS/11: Differential Equations & Proofs LFCPS/11 22 / 24

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