A Coq Formalization of Lebesgue Integration of Nonnegative Functions - PowerPoint PPT Presentation

Formal Methods in Mathematics Carnegie Mellon University A Coq Formalization of Lebesgue Integration of Nonnegative Functions Sylvie Boldo Inria, Universit´ e Paris-Saclay January 7th, 2020

Disclaimers Disclaimer 1: this is joint work with Fran¸ cois Cl´ ement, Florian Faissole, Vincent Martin, Micaela Mayero. Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 2 / 39

Disclaimers Disclaimer 1: this is joint work with Fran¸ cois Cl´ ement, Florian Faissole, Vincent Martin, Micaela Mayero. Disclaimer 2: This formalization of mathematics is done in Coq. Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 2 / 39

Disclaimers Disclaimer 1: this is joint work with Fran¸ cois Cl´ ement, Florian Faissole, Vincent Martin, Micaela Mayero. Disclaimer 2: This formalization of mathematics is done in Coq. Disclaimer 3: There is (nearly) no computer arithmetic! Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 2 / 39

Outline Introduction 1 Towards the Finite Element Method 2 Lebesgue Integration 3 Measurability Measure Simple Functions and their Integral Lebesgue Integral of Nonnegative Functions Conclusion and Perspectives 4 Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 3 / 39

Introduction , ∂ 2 u � R , ▼❛t❤❡♠❛t✐❝s ∂ t 2 theorems Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 4 / 39

Introduction , ∂ 2 u � R , ▼❛t❤❡♠❛t✐❝s ∂ t 2 theorems numerical scheme, convergence Applied Mathematics algorithms + theorems Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 4 / 39

Introduction , ∂ 2 u � R , ▼❛t❤❡♠❛t✐❝s ∂ t 2 theorems numerical scheme, convergence Applied Mathematics algorithms + theorems floating-point numbers, implementation Computer programs + ? Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 4 / 39

Introduction , ∂ 2 u � R , PARANOIA ▼❛t❤❡♠❛t✐❝s ∂ t 2 theorems numerical scheme, convergence Applied Mathematics algorithms + theorems floating-point numbers, implementation Computer programs + ? Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 4 / 39

Motivations PDE (Partial Differential Equation) ⇒ weather forecast ⇒ nuclear simulation ⇒ optimal control ⇒ . . . Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 5 / 39

Motivations PDE (Partial Differential Equation) ⇒ weather forecast ⇒ nuclear simulation ⇒ optimal control ⇒ . . . Usually too complex to solve by an exact mathematical formula ⇒ approximated by numerical scheme over discrete grids/volumes ⇒ mathematical proofs of the convergence of the numerical scheme (we compute something close to the PDE solution if the size decreases) Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 5 / 39

Motivations PDE (Partial Differential Equation) ⇒ weather forecast ⇒ nuclear simulation ⇒ optimal control ⇒ . . . Usually too complex to solve by an exact mathematical formula ⇒ approximated by numerical scheme over discrete grids/volumes ⇒ mathematical proofs of the convergence of the numerical scheme (we compute something close to the PDE solution if the size decreases) ⇒ real program implementing the scheme/method Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 5 / 39

Motivations PDE (Partial Differential Equation) ⇒ weather forecast ⇒ nuclear simulation ⇒ optimal control ⇒ . . . Usually too complex to solve by an exact mathematical formula ⇒ approximated by numerical scheme over discrete grids/volumes ⇒ mathematical proofs of the convergence of the numerical scheme (we compute something close to the PDE solution if the size decreases) ⇒ real program implementing the scheme/method Let us machine-check this kind of programs! Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 5 / 39

Outline Introduction 1 Towards the Finite Element Method 2 Lebesgue Integration 3 Measurability Measure Simple Functions and their Integral Lebesgue Integral of Nonnegative Functions Conclusion and Perspectives 4 Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 6 / 39

Motivations http://www.ima.umn.edu/~arnold/disasters/sleipner.html Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 7 / 39

Motivations Real life applications need solving PDE (Partial Differential Equation) on complex 3D geometries. Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 8 / 39

Motivations Real life applications need solving PDE (Partial Differential Equation) on complex 3D geometries. @ V. Martin Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 8 / 39

Motivations Real life applications need solving PDE (Partial Differential Equation) on complex 3D geometries. @ V. Martin Instead of regular 2D/3D grids, we consider meshes made of triangles/tetrahedra. Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 8 / 39

Motivations The Finite Element Method (FEM) is the most used method to solve PDEs over meshes. FEM encompasses methods for connecting many simple element equations over many small subdomains, named finite elements, to approximate a more complex equation over a larger domain. ( https://en.wikipedia.org/wiki/Finite_element_method ) Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 9 / 39

Motivations The Finite Element Method (FEM) is the most used method to solve PDEs over meshes. FEM encompasses methods for connecting many simple element equations over many small subdomains, named finite elements, to approximate a more complex equation over a larger domain. ( https://en.wikipedia.org/wiki/Finite_element_method ) ⇒ mathematical proofs of the FEM ⇒ C++ library (Felisce) implementing the FEM Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 9 / 39

Motivations The Finite Element Method (FEM) is the most used method to solve PDEs over meshes. FEM encompasses methods for connecting many simple element equations over many small subdomains, named finite elements, to approximate a more complex equation over a larger domain. ( https://en.wikipedia.org/wiki/Finite_element_method ) ⇒ mathematical proofs of the FEM ⇒ C++ library (Felisce) implementing the FEM Let us machine-check this program! Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 9 / 39

Motivations The Finite Element Method (FEM) is the most used method to solve PDEs over meshes. FEM encompasses methods for connecting many simple element equations over many small subdomains, named finite elements, to approximate a more complex equation over a larger domain. ( https://en.wikipedia.org/wiki/Finite_element_method ) ⇒ mathematical proofs of the FEM ⇒ C++ library (Felisce) implementing the FEM Let us machine-check this program! First, let us understand/formally prove the mathematics. Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 9 / 39

Mathematicians at work for Lax-Milgram theorem more 50 pages of mathematical proofs Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 10 / 39

Mathematicians at work for Lax-Milgram theorem more 50 pages of mathematical proofs very detailed! Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 10 / 39

Mathematicians at work for Lax-Milgram theorem more 50 pages of mathematical proofs very detailed! more than 7,000 lines and 220,000 characters Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 10 / 39

Mathematicians at work for Lax-Milgram theorem more 50 pages of mathematical proofs very detailed! more than 7,000 lines and 220,000 characters with dependencies! Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 10 / 39

Mathematicians at work for Lax-Milgram theorem more 50 pages of mathematical proofs very detailed! more than 7,000 lines and 220,000 characters with dependencies! Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 10 / 39

Mathematicians at work for Lax-Milgram theorem more 50 pages of mathematical proofs very detailed! more than 7,000 lines and 220,000 characters with dependencies! 175 174 173 171 169 128 139 170 141 168 166 88 133 114 127 56 137 58 59 132 138 140 160 72 167 113 86 136 92 130 135 159 154 131 85 134 129 161 156 153 126 4 103 144 158 157 11 148 152 125 74 102 143 155 10 165 163 147 164 149 13 151 9 87 123 112 124 77 90 172 12 3 162 146 145 8 150 91 6 79 84 44 121 122 119 120 71 73 76 69 83 7 142 57 89 5 2 78 55 81 38 39 1 118 68 65 37 75 67 101 108 53 80 117 116 111 51 70 29 61 66 34 35 64 82 96 115 105 36 107 33 46 50 49 48 43 28 27 60 23 32 26 54 98 40 104 100 19 106 110 16 31 42 45 47 22 24 18 63 25 62 97 15 99 95 109 14 41 30 21 20 52 94 93 Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 10 / 39

Proof engineering Let us build upon the Coquelicot library (Boldo, Lelay, Melquiond) + general spaces Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 11 / 39

Proof engineering Let us build upon the Coquelicot library (Boldo, Lelay, Melquiond) + general spaces + many existing theorems Sylvie Boldo (Inria) Lebesgue Integration January 7th, 2020 11 / 39