Added decomposed overapproximate for lazy intersection between CPA an…

…d Polyhedron

Squashed commit of the following:

commit 89913f2
Merge: 39d80d6 a96a698
Author: Unknown <[email protected]>
Date:   Mon Jun 17 01:09:40 2019 +1000

    Merge remote-tracking branch 'origin/kostemkin/1041_new' into kostemkin/1041_new

commit 39d80d6
Author: Kostiantyn Potomkin <[email protected]>
Date:   Mon Jun 17 00:46:43 2019 +1000

    Added decomposed overapproximate for lazy intersection between CPA and Polyhedron

    Squashed commit of the following:

    commit ac50169
    Author: Unknown <[email protected]>
    Date:   Sat Jun 15 17:40:24 2019 +1000

        added one more test for old overapproximate of lazy intersection

    commit 509b7f8
    Author: Unknown <[email protected]>
    Date:   Sat Jun 15 04:08:49 2019 +1000

        added using project

    commit 795f593
    Author: Unknown <[email protected]>
    Date:   Sat Jun 15 03:46:50 2019 +1000

        fixed tests

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    Author: Unknown <[email protected]>
    Date:   Fri Jun 14 19:01:32 2019 +1000

        small refactoring in unit_overapproximate

    commit 89fe48f
    Merge: 8f6f735 a5545ae
    Author: Unknown <[email protected]>
    Date:   Fri Jun 14 16:29:31 2019 +1000

        Merge remote-tracking branch 'origin/master' into kostemkin/1041_new

    commit 8f6f735
    Author: Unknown <[email protected]>
    Date:   Fri Jun 14 16:28:19 2019 +1000

        fixed tests

    commit 8448622
    Author: Unknown <[email protected]>
    Date:   Fri Jun 14 14:38:23 2019 +1000

        try to push the test

    commit a5545ae
    Author: Marcelo Forets <[email protected]>
    Date:   Thu Jun 13 15:15:09 2019 -0300

        Update Project.toml (#1437)

    commit 9dd9cc9
    Author: Marcelo Forets <[email protected]>
    Date:   Thu Jun 13 13:41:53 2019 -0300

        Pr/1408 (#1433)

        * Add TaylorModels optional requirement
        * Linear case
        * Nonlinear case

    commit 413743e
    Author: Unknown <[email protected]>
    Date:   Fri Jun 14 01:39:07 2019 +1000

        fixed tests

    commit 47f9226
    Author: Unknown <[email protected]>
    Date:   Thu Jun 13 16:54:11 2019 +1000

        added new tests for decomposed overapproximate and minor refactoring

    commit a7e8305
    Author: Sebastián Guadalupe <[email protected]>
    Date:   Wed Jun 12 18:29:49 2019 -0300

        remove redundant vertices of LinearMap (#1428)

    commit c0fb6d6
    Author: Unknown <[email protected]>
    Date:   Thu Jun 13 02:58:12 2019 +1000

        some code clean ups according to Christian`s comments

    commit 6d58c75
    Merge: 02916e5 c52a2c4
    Author: Christian Schilling <[email protected]>
    Date:   Wed Jun 12 09:56:12 2019 +0200

        Merge pull request #1432 from JuliaReach/schillic/times

        Export 'times' symbol

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    Author: schillic <[email protected]>
    Date:   Wed Jun 12 08:42:55 2019 +0200

        export 'times' symbol

    commit d70ddef
    Merge: 6427a29 0aa9825
    Author: Unknown <[email protected]>
    Date:   Wed Jun 12 14:18:06 2019 +1000

        Merge remote-tracking branch 'origin/master' into kostemkin/1041_new

    commit 6427a29
    Author: Unknown <[email protected]>
    Date:   Wed Jun 12 04:03:30 2019 +1000

        fixed docs

    commit 02916e5
    Merge: 19b51ea 32a6cfa
    Author: Christian Schilling <[email protected]>
    Date:   Tue Jun 11 12:17:22 2019 +0200

        Merge pull request #1425 from JuliaReach/schillic/merge

        Simplifications in overapproximate

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    Author: Unknown <[email protected]>
    Date:   Tue Jun 11 20:10:58 2019 +1000

        further changes in documentation and removing redundant code and comments

    commit 19b51ea
    Author: Marcelo Forets <[email protected]>
    Date:   Tue Jun 11 06:34:19 2019 -0300

        Update README.md (#1423)

        * Update README.md

        * update README and About

    commit 5f17132
    Author: Unknown <[email protected]>
    Date:   Tue Jun 11 15:49:43 2019 +1000

        exporting common intersection preprocess part

    commit b965fb3
    Author: Unknown <[email protected]>
    Date:   Tue Jun 11 03:13:14 2019 +1000

        export common intersection preprocess part

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    Author: Unknown <[email protected]>
    Date:   Tue Jun 11 02:47:23 2019 +1000

        better functions definition in docs and minor refactoring

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    Author: Unknown <[email protected]>
    Date:   Mon Jun 10 19:27:27 2019 +1000

        add missing docs

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    Author: Unknown <[email protected]>
    Date:   Mon Jun 10 18:59:09 2019 +1000

        updated doc

    commit f67024f
    Author: Unknown <[email protected]>
    Date:   Mon Jun 10 18:54:06 2019 +1000

        correct import

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    Author: Unknown <[email protected]>
    Date:   Mon Jun 10 18:51:16 2019 +1000

        small refactoring and updated doc

    commit 0aa9825
    Merge: 847ffe8 a56ebc3
    Author: Christian Schilling <[email protected]>
    Date:   Fri Jun 7 13:19:45 2019 +0200

        Merge pull request #1426 from JuliaReach/schillic/aspectratio

        Set default aspect ratio in plot recipes to :none

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    Author: Unknown <[email protected]>
    Date:   Fri Jun 7 21:00:27 2019 +1000

        refactoring to use Vector{Tuple{Int, Int}} for block indices

    commit a56ebc3
    Author: schillic <[email protected]>
    Date:   Fri Jun 7 07:52:54 2019 +0200

        set default aspectratio in plot recipes to :none

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    Author: schillic <[email protected]>
    Date:   Thu Jun 6 17:01:58 2019 +0200

        simplifications in overapproximate

    commit 847ffe8
    Merge: 2770dd6 af32212
    Author: Christian Schilling <[email protected]>
    Date:   Thu Jun 6 13:20:05 2019 +0200

        Merge pull request #1414 from JuliaReach/schillic/empty_polytope_intersection

        Check for emptiness in concrete intersection using Polyhedra

    commit 2770dd6
    Merge: 875ecaf 03823d1
    Author: Christian Schilling <[email protected]>
    Date:   Thu Jun 6 13:02:08 2019 +0200

        Merge pull request #1419 from JuliaReach/kostemkin/1091

        Decomposed linear map of CartesianProductArray

    commit 875ecaf
    Merge: 9bc6422 fd2f305
    Author: Christian Schilling <[email protected]>
    Date:   Thu Jun 6 12:26:06 2019 +0200

        Merge pull request #1409 from JuliaReach/schillic/378

        #378 - Add isapprox between sets

    commit 03823d1
    Author: Unknown <[email protected]>
    Date:   Thu Jun 6 18:15:10 2019 +1000

        updated tests due to sparse

    commit 9b8f5be
    Author: Unknown <[email protected]>
    Date:   Thu Jun 6 18:11:07 2019 +1000

        Code clean up

    commit 16f4a71
    Author: Unknown <[email protected]>
    Date:   Thu Jun 6 16:34:15 2019 +1000

        fixed tests and function signature

    commit 2c48134
    Author: Unknown <[email protected]>
    Date:   Thu Jun 6 16:27:08 2019 +1000

        Moved intersection to overapproximation

    commit e20224a
    Author: Unknown <[email protected]>
    Date:   Thu Jun 6 15:50:32 2019 +1000

        added new tests and refactoring

        added functionality to keep the original structure and preallocate corresponding  set

    commit 9bc6422
    Author: Marcelo Forets <[email protected]>
    Date:   Wed Jun 5 19:16:55 2019 -0300

        Update README.md

    commit 617de10
    Author: Sebastián Guadalupe <[email protected]>
    Date:   Wed Jun 5 19:15:32 2019 -0300

        Add three point case for concrete convex hull (#1411)

    commit 4221528
    Merge: e517950 f1aa1ae
    Author: Christian Schilling <[email protected]>
    Date:   Wed Jun 5 19:05:48 2019 +0200

        Merge pull request #1415 from JuliaReach/schillic/571

        #571 - Restrict iterative refinement to AbstractFloat

    commit f1aa1ae
    Author: schillic <[email protected]>
    Date:   Fri May 31 22:13:59 2019 +0200

        restrict type for iterative refinement

    commit 14a5980
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 20:31:08 2019 +1000

        fixed is_intersection_empty to use correct projection

    commit 5f033d7
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 20:12:34 2019 +1000

        removed redundant function

    commit 6b62873
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 20:08:21 2019 +1000

        added correct projection

    commit 6160bee
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 20:06:44 2019 +1000

        Removed workarounds for LinearMap projection

    commit 9f0a8ce
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 18:40:25 2019 +1000

        Added tests

    commit a03e868
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 18:18:35 2019 +1000

        Added missing is_intersection_empty functions

    commit 13e7b9c
    Author: kostakoida <[email protected]>
    Date:   Wed Jun 5 18:00:12 2019 +1000

        Update src/Approximations/overapproximate.jl

        Co-Authored-By: Christian Schilling <[email protected]>

    commit 80eed63
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 17:55:26 2019 +1000

        small refactoring according to Christian's recommendations

    commit a66586a
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 15:47:37 2019 +1000

        intersection works

    commit ece7d74
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 05:23:36 2019 +1000

        fixed tests and type instability

    commit 16521c3
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 05:06:09 2019 +1000

        small fixes and added intersection between Cartesian products

    commit 54d73fd
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 04:11:41 2019 +1000

        updated linear map to use correct rows and updated tests

    commit 8742462
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 01:23:36 2019 +1000

        added tests

    commit c7fc98d
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 01:10:10 2019 +1000

        cleaned up the code

    commit 80ab0cb
    Author: Unknown <[email protected]>
    Date:   Tue Jun 4 23:43:19 2019 +1000

        Correct minkowski sum addition

    commit e517950
    Merge: 873bbf1 cfae8fd
    Author: Christian Schilling <[email protected]>
    Date:   Tue Jun 4 14:11:15 2019 +0200

        Merge pull request #1420 from JuliaReach/schillic/README

        Update README

    commit cfae8fd
    Author: schillic <[email protected]>
    Date:   Tue Jun 4 10:42:22 2019 +0200

        update README

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    Author: Unknown <[email protected]>
    Date:   Tue Jun 4 18:57:12 2019 +1000

        small fixes

    commit cf01ae6
    Author: Unknown <[email protected]>
    Date:   Tue Jun 4 18:50:19 2019 +1000

        updated according to comments

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    Author: Unknown <[email protected]>
    Date:   Tue Jun 4 17:52:32 2019 +1000

        added getting block_structure for CPA and intersection by union of blocks

    commit a79b038
    Author: Unknown <[email protected]>
    Date:   Tue Jun 4 16:33:53 2019 +1000

        added block_indices and variable_indices

    commit b98b789
    Author: Unknown <[email protected]>
    Date:   Tue Jun 4 16:31:30 2019 +1000

        Added using @view

    commit ecabee5
    Author: Unknown <[email protected]>
    Date:   Tue Jun 4 14:45:14 2019 +1000

        updated overaproximation of  decomposed linear map according to Christian comments

    commit ef60e2e
    Author: Unknown <[email protected]>
    Date:   Sun Jun 2 01:46:13 2019 +1000

        Added overapproximation of Lazy linear map of cartesian product array

    commit af32212
    Author: schillic <[email protected]>
    Date:   Fri May 31 17:27:23 2019 +0200

        check for emptiness in concrete intersection using Polyhedra

    commit fd2f305
    Author: schillic <[email protected]>
    Date:   Tue May 28 20:17:35 2019 +0200

        add isapprox between 'LazySet's

commit a96a698
Author: Kostiantyn Potomkin <[email protected]>
Date:   Mon Jun 17 00:46:43 2019 +1000

    Added decomposed overapproximate for lazy intersection between CPA and Polyhedron

    Squashed commit of the following:

    commit ac50169
    Author: Unknown <[email protected]>
    Date:   Sat Jun 15 17:40:24 2019 +1000

        added one more test for old overapproximate of lazy intersection

    commit 509b7f8
    Author: Unknown <[email protected]>
    Date:   Sat Jun 15 04:08:49 2019 +1000

        added using project

    commit 795f593
    Author: Unknown <[email protected]>
    Date:   Sat Jun 15 03:46:50 2019 +1000

        fixed tests

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    Author: Unknown <[email protected]>
    Date:   Fri Jun 14 19:01:32 2019 +1000

        small refactoring in unit_overapproximate

    commit 89fe48f
    Merge: 8f6f735 a5545ae
    Author: Unknown <[email protected]>
    Date:   Fri Jun 14 16:29:31 2019 +1000

        Merge remote-tracking branch 'origin/master' into kostemkin/1041_new

    commit 8f6f735
    Author: Unknown <[email protected]>
    Date:   Fri Jun 14 16:28:19 2019 +1000

        fixed tests

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    Author: Unknown <[email protected]>
    Date:   Fri Jun 14 14:38:23 2019 +1000

        try to push the test

    commit a5545ae
    Author: Marcelo Forets <[email protected]>
    Date:   Thu Jun 13 15:15:09 2019 -0300

        Update Project.toml (#1437)

    commit 9dd9cc9
    Author: Marcelo Forets <[email protected]>
    Date:   Thu Jun 13 13:41:53 2019 -0300

        Pr/1408 (#1433)

        * Add TaylorModels optional requirement
        * Linear case
        * Nonlinear case

    commit 413743e
    Author: Unknown <[email protected]>
    Date:   Fri Jun 14 01:39:07 2019 +1000

        fixed tests

    commit 47f9226
    Author: Unknown <[email protected]>
    Date:   Thu Jun 13 16:54:11 2019 +1000

        added new tests for decomposed overapproximate and minor refactoring

    commit a7e8305
    Author: Sebastián Guadalupe <[email protected]>
    Date:   Wed Jun 12 18:29:49 2019 -0300

        remove redundant vertices of LinearMap (#1428)

    commit c0fb6d6
    Author: Unknown <[email protected]>
    Date:   Thu Jun 13 02:58:12 2019 +1000

        some code clean ups according to Christian`s comments

    commit 6d58c75
    Merge: 02916e5 c52a2c4
    Author: Christian Schilling <[email protected]>
    Date:   Wed Jun 12 09:56:12 2019 +0200

        Merge pull request #1432 from JuliaReach/schillic/times

        Export 'times' symbol

    commit c52a2c4
    Author: schillic <[email protected]>
    Date:   Wed Jun 12 08:42:55 2019 +0200

        export 'times' symbol

    commit d70ddef
    Merge: 6427a29 0aa9825
    Author: Unknown <[email protected]>
    Date:   Wed Jun 12 14:18:06 2019 +1000

        Merge remote-tracking branch 'origin/master' into kostemkin/1041_new

    commit 6427a29
    Author: Unknown <[email protected]>
    Date:   Wed Jun 12 04:03:30 2019 +1000

        fixed docs

    commit 02916e5
    Merge: 19b51ea 32a6cfa
    Author: Christian Schilling <[email protected]>
    Date:   Tue Jun 11 12:17:22 2019 +0200

        Merge pull request #1425 from JuliaReach/schillic/merge

        Simplifications in overapproximate

    commit b33cab8
    Author: Unknown <[email protected]>
    Date:   Tue Jun 11 20:10:58 2019 +1000

        further changes in documentation and removing redundant code and comments

    commit 19b51ea
    Author: Marcelo Forets <[email protected]>
    Date:   Tue Jun 11 06:34:19 2019 -0300

        Update README.md (#1423)

        * Update README.md

        * update README and About

    commit 5f17132
    Author: Unknown <[email protected]>
    Date:   Tue Jun 11 15:49:43 2019 +1000

        exporting common intersection preprocess part

    commit b965fb3
    Author: Unknown <[email protected]>
    Date:   Tue Jun 11 03:13:14 2019 +1000

        export common intersection preprocess part

    commit 1d55f57
    Author: Unknown <[email protected]>
    Date:   Tue Jun 11 02:47:23 2019 +1000

        better functions definition in docs and minor refactoring

    commit 77640db
    Author: Unknown <[email protected]>
    Date:   Mon Jun 10 19:27:27 2019 +1000

        add missing docs

    commit 133e962
    Author: Unknown <[email protected]>
    Date:   Mon Jun 10 18:59:09 2019 +1000

        updated doc

    commit f67024f
    Author: Unknown <[email protected]>
    Date:   Mon Jun 10 18:54:06 2019 +1000

        correct import

    commit 8b14a4a
    Author: Unknown <[email protected]>
    Date:   Mon Jun 10 18:51:16 2019 +1000

        small refactoring and updated doc

    commit 0aa9825
    Merge: 847ffe8 a56ebc3
    Author: Christian Schilling <[email protected]>
    Date:   Fri Jun 7 13:19:45 2019 +0200

        Merge pull request #1426 from JuliaReach/schillic/aspectratio

        Set default aspect ratio in plot recipes to :none

    commit 06bc249
    Author: Unknown <[email protected]>
    Date:   Fri Jun 7 21:00:27 2019 +1000

        refactoring to use Vector{Tuple{Int, Int}} for block indices

    commit a56ebc3
    Author: schillic <[email protected]>
    Date:   Fri Jun 7 07:52:54 2019 +0200

        set default aspectratio in plot recipes to :none

    commit 32a6cfa
    Author: schillic <[email protected]>
    Date:   Thu Jun 6 17:01:58 2019 +0200

        simplifications in overapproximate

    commit 847ffe8
    Merge: 2770dd6 af32212
    Author: Christian Schilling <[email protected]>
    Date:   Thu Jun 6 13:20:05 2019 +0200

        Merge pull request #1414 from JuliaReach/schillic/empty_polytope_intersection

        Check for emptiness in concrete intersection using Polyhedra

    commit 2770dd6
    Merge: 875ecaf 03823d1
    Author: Christian Schilling <[email protected]>
    Date:   Thu Jun 6 13:02:08 2019 +0200

        Merge pull request #1419 from JuliaReach/kostemkin/1091

        Decomposed linear map of CartesianProductArray

    commit 875ecaf
    Merge: 9bc6422 fd2f305
    Author: Christian Schilling <[email protected]>
    Date:   Thu Jun 6 12:26:06 2019 +0200

        Merge pull request #1409 from JuliaReach/schillic/378

        #378 - Add isapprox between sets

    commit 03823d1
    Author: Unknown <[email protected]>
    Date:   Thu Jun 6 18:15:10 2019 +1000

        updated tests due to sparse

    commit 9b8f5be
    Author: Unknown <[email protected]>
    Date:   Thu Jun 6 18:11:07 2019 +1000

        Code clean up

    commit 16f4a71
    Author: Unknown <[email protected]>
    Date:   Thu Jun 6 16:34:15 2019 +1000

        fixed tests and function signature

    commit 2c48134
    Author: Unknown <[email protected]>
    Date:   Thu Jun 6 16:27:08 2019 +1000

        Moved intersection to overapproximation

    commit e20224a
    Author: Unknown <[email protected]>
    Date:   Thu Jun 6 15:50:32 2019 +1000

        added new tests and refactoring

        added functionality to keep the original structure and preallocate corresponding  set

    commit 9bc6422
    Author: Marcelo Forets <[email protected]>
    Date:   Wed Jun 5 19:16:55 2019 -0300

        Update README.md

    commit 617de10
    Author: Sebastián Guadalupe <[email protected]>
    Date:   Wed Jun 5 19:15:32 2019 -0300

        Add three point case for concrete convex hull (#1411)

    commit 4221528
    Merge: e517950 f1aa1ae
    Author: Christian Schilling <[email protected]>
    Date:   Wed Jun 5 19:05:48 2019 +0200

        Merge pull request #1415 from JuliaReach/schillic/571

        #571 - Restrict iterative refinement to AbstractFloat

    commit f1aa1ae
    Author: schillic <[email protected]>
    Date:   Fri May 31 22:13:59 2019 +0200

        restrict type for iterative refinement

    commit 14a5980
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 20:31:08 2019 +1000

        fixed is_intersection_empty to use correct projection

    commit 5f033d7
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 20:12:34 2019 +1000

        removed redundant function

    commit 6b62873
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 20:08:21 2019 +1000

        added correct projection

    commit 6160bee
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 20:06:44 2019 +1000

        Removed workarounds for LinearMap projection

    commit 9f0a8ce
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 18:40:25 2019 +1000

        Added tests

    commit a03e868
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 18:18:35 2019 +1000

        Added missing is_intersection_empty functions

    commit 13e7b9c
    Author: kostakoida <[email protected]>
    Date:   Wed Jun 5 18:00:12 2019 +1000

        Update src/Approximations/overapproximate.jl

        Co-Authored-By: Christian Schilling <[email protected]>

    commit 80eed63
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 17:55:26 2019 +1000

        small refactoring according to Christian's recommendations

    commit a66586a
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 15:47:37 2019 +1000

        intersection works

    commit ece7d74
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 05:23:36 2019 +1000

        fixed tests and type instability

    commit 16521c3
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 05:06:09 2019 +1000

        small fixes and added intersection between Cartesian products

    commit 54d73fd
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 04:11:41 2019 +1000

        updated linear map to use correct rows and updated tests

    commit 8742462
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 01:23:36 2019 +1000

        added tests

    commit c7fc98d
    Author: Unknown <[email protected]>
    Date:   Wed Jun 5 01:10:10 2019 +1000

        cleaned up the code

    commit 80ab0cb
    Author: Unknown <[email protected]>
    Date:   Tue Jun 4 23:43:19 2019 +1000

        Correct minkowski sum addition

    commit e517950
    Merge: 873bbf1 cfae8fd
    Author: Christian Schilling <[email protected]>
    Date:   Tue Jun 4 14:11:15 2019 +0200

        Merge pull request #1420 from JuliaReach/schillic/README

        Update README

    commit cfae8fd
    Author: schillic <[email protected]>
    Date:   Tue Jun 4 10:42:22 2019 +0200

        update README

    commit 5b97aec
    Author: Unknown <[email protected]>
    Date:   Tue Jun 4 18:57:12 2019 +1000

        small fixes

    commit cf01ae6
    Author: Unknown <[email protected]>
    Date:   Tue Jun 4 18:50:19 2019 +1000

        updated according to comments

    commit 7e2891f
    Author: Unknown <[email protected]>
    Date:   Tue Jun 4 17:52:32 2019 +1000

        added getting block_structure for CPA and intersection by union of blocks

    commit a79b038
    Author: Unknown <[email protected]>
    Date:   Tue Jun 4 16:33:53 2019 +1000

        added block_indices and variable_indices

    commit b98b789
    Author: Unknown <[email protected]>
    Date:   Tue Jun 4 16:31:30 2019 +1000

        Added using @view

    commit ecabee5
    Author: Unknown <[email protected]>
    Date:   Tue Jun 4 14:45:14 2019 +1000

        updated overaproximation of  decomposed linear map according to Christian comments

    commit ef60e2e
    Author: Unknown <[email protected]>
    Date:   Sun Jun 2 01:46:13 2019 +1000

        Added overapproximation of Lazy linear map of cartesian product array

    commit af32212
    Author: schillic <[email protected]>
    Date:   Fri May 31 17:27:23 2019 +0200

        check for emptiness in concrete intersection using Polyhedra

    commit fd2f305
    Author: schillic <[email protected]>
    Date:   Tue May 28 20:17:35 2019 +0200

        add isapprox between 'LazySet's

commit 894fda3
Merge: d9da4d4 0fd6c1e
Author: Christian Schilling <[email protected]>
Date:   Fri Jun 14 22:27:55 2019 +0200

    Merge pull request #1442 from JuliaReach/schillic/1364

    #1364 - Concrete linear map of a translation

commit 0fd6c1e
Author: schillic <[email protected]>
Date:   Fri Jun 14 16:19:22 2019 +0200

    concrete linear_map of lazy Translation

docs/src/lib/binary_functions.md

@@ -37,6 +37,8 @@ is_intersection_empty(::Universe{N}, ::LazySet{N}, ::Bool=false) where {N<:Real}
 is_intersection_empty(::Complement{N}, ::LazySet{N}, ::Bool=false) where {N<:Real}
 is_intersection_empty(::Zonotope{N}, ::Zonotope{N}, ::Bool=false) where {N<:Real}
 is_intersection_empty(::Interval{N}, ::Interval{N}, ::Bool=false) where {N<:Real}
+is_intersection_empty(X::CartesianProductArray{N}, Y::HPolyhedron{N}) where {N<:Real}
+is_intersection_empty(X::CartesianProductArray{N}, Y::CartesianProductArray{N}) where {N}
 ```
 
 ## Convex hull
@@ -64,6 +66,7 @@ intersection(::UnionSet{N}, ::LazySet{N}) where {N<:Real}
 intersection(::UnionSetArray{N}, ::LazySet{N}) where {N<:Real}
 intersection(::Universe{N}, ::LazySet{N}) where {N<:Real}
 intersection(::AbstractPolyhedron{N}, ::ResetMap{N}) where {N<:Real}
+intersection(X::CartesianProductArray{N}, Y::CartesianProductArray{N}) where {N<:Real}
 ```
 
 ## Subset check

docs/src/lib/operations.md

@@ -52,6 +52,10 @@ isempty(::CartesianProductArray)
 constraints_list(::CartesianProductArray{N}) where {N<:Real}
 vertices_list(::CartesianProductArray{N}) where {N<:Real}
 array(::CartesianProductArray{N, S}) where {N<:Real, S<:LazySet{N}}
+block_structure(cpa::CartesianProductArray{N}) where {N}
+block_to_dimension_indices(cpa::CartesianProductArray{N}, vars::Vector{Int}) where {N}
+substitute_blocks(low_dim_cpa::CartesianProductArray{N}, orig_cpa::CartesianProductArray{N},
+blocks::Vector{Tuple{Int,Int}}) where {N}
 ```
 Inherited from [`LazySet`](@ref):
 * [`norm`](@ref norm(::LazySet, ::Real))
@@ -349,6 +353,7 @@ an_element(::Translation)
 isempty(::Translation)
 constraints_list(::Translation{N}, ::Val{true}) where {N<:Real}
 LinearMap(::AbstractMatrix{N}, ::Translation{N}) where {N<:Real}
+linear_map(M::AbstractMatrix{N}, tr::Translation{N}) where {N<:Real}
 ∈(::AbstractVector{N}, ::Translation{N}) where {N<:Real}
 ```
 

docs/src/lib/utils.md

@@ -35,6 +35,7 @@ same_block_structure
 substitute
 substitute!
 σ_helper
+get_constrained_lowdimset
 @neutral
 @absorbing
 @neutral_absorbing

src/Approximations/overapproximate.jl

@@ -1,3 +1,4 @@
+using LazySets: block_to_dimension_indices, substitute_blocks, get_constrained_lowdimset
 """
     overapproximate(X::S, ::Type{S}) where {S<:LazySet}
 
@@ -911,3 +912,63 @@ function _overapproximate_lm_cpa!(arr, M, cpa, overapprox_option)
 
     return CartesianProductArray(arr)
 end
+
+"""
+    overapproximate(cap::Intersection{N,
+                            <:CartesianProductArray{N},
+                            <:AbstractPolyhedron{N}},
+                       ::Type{CartesianProductArray}, oa) where {N}
+
+Return the intersection of the cartesian product of a finite number of convex sets and a polyhedron.
+
+### Input
+
+ - `cap` -- Lazy intersection of cartesian product array and polyhedron
+ - `CartesianProductArray` -- type for dispatch
+ - `oa`  -- template directions
+
+### Output
+
+The intersection between `cpa` and `P` with overapproximation in each constrained block.
+
+### Algorithm
+
+The intersection is only needed to be taken in the elements of the cartesian product array (subsets of
+variables, or "blocks") which are constrained in `P`.
+Hence we first search for constrained blocks and then take the intersection of
+a lower-dimensional Cartesian product of these blocks with the projection of `Y`
+onto the variables of these blocks. (This projection is syntactic and exact.)
+The result is a `CartesianProductArray` with the same block structure as in `X`.
+However, when we decompose back our set we overapproximate during projection operation,
+therefore we need to specify overapproximation strategy (it is Hyperrectangle by default)
+"""
+function overapproximate(cap::Intersection{N,
+                                            <:CartesianProductArray{N},
+                                            <:AbstractPolyhedron{N}},
+                            ::Type{CartesianProductArray}, oa) where {N}
+
+    cpa, P = cap.X, cap.Y
+
+    cpa_low_dim, vars, block_structure, blocks = get_constrained_lowdimset(cpa, P)
+
+    low_intersection = intersection(cpa_low_dim, project(P, vars))
+
+    if isempty(low_intersection)
+        return EmptySet{N}()
+    end
+
+    decomposed_low_set = decompose(low_intersection, block_structure, oa)
+
+    return substitute_blocks(decomposed_low_set, cpa, blocks)
+end
+
+# symmetric method
+function overapproximate(cap::Intersection{N,
+                                            <:AbstractPolyhedron{N},
+                                            <:CartesianProductArray{N}},
+                            ::Type{CartesianProductArray}, oa) where {N}
+    overapproximate(Intersection(cap.Y, cap.X), oa)
+end
+
+
+end

src/CartesianProduct.jl

@@ -3,7 +3,8 @@ import Base: *, ∈, isempty
 export CartesianProduct,
        CartesianProductArray,
        CartesianProduct!,
-       array
+       array,
+       same_block_structure
 
 """
     CartesianProduct{N<:Real, S1<:LazySet{N}, S2<:LazySet{N}} <: LazySet{N}
@@ -562,3 +563,143 @@ function same_block_structure(x::AbstractVector{S1}, y::AbstractVector{S2}
     end
     return true
 end
+
+"""
+    block_structure(cpa::CartesianProductArray{N}) where {N}
+
+Returns an array containing the dimension ranges of each block in a `CartesianProductArray`.
+
+### Input
+
+- `cpa` -- Cartesian product array
+
+### Output
+
+A vector of ranges
+
+### Example
+
+```jldoctest
+julia> using LazySets: block_structure
+
+julia> cpa = CartesianProductArray([BallInf(zeros(n), 1.0) for n in [3, 1, 2]]);
+
+julia> block_structure(cpa)
+3-element Array{UnitRange{Int64},1}:
+ 1:3
+ 4:4
+ 5:6
+```
+"""
+function block_structure(cpa::CartesianProductArray{N}) where {N}
+    result = Vector{UnitRange{Int}}(undef, length(array(cpa)))
+    start_index = 1
+    @inbounds for (i, bi) in enumerate(array(cpa))
+        end_index = start_index + dim(bi) - 1
+        result[i] = start_index:end_index
+        start_index = end_index + 1
+    end
+    return result
+end
+
+
+"""
+    block_to_dimension_indices(cpa::CartesianProductArray{N}, vars::Vector{Int}) where {N}
+
+Returns a vector mapping block index `i`
+to tuple `(f, l)` such that either `f = l = -1` or `f` is the first dimension index and `l` is the last dimension
+index of the `i`-th block, depending on whether one of the block's dimension indices is specified in `vars`.
+
+### Input
+
+- `cpa` -- Cartesian product array
+- `vars` -- list containing the variables of interest, sorted in ascending order
+
+### Output
+
+A vector of tuples, where values in tuple relate to range of dimensions in the i-th block.
+
+### Example
+
+```jldoctest
+julia> using LazySets: block_to_dimension_indices
+
+julia> cpa = CartesianProductArray([BallInf(zeros(n), 1.0) for n in [1, 3, 2, 3]]);
+
+julia> block_to_dimension_indices(cpa, [2, 4, 8])
+(Tuple{Int64,Int64}[(-1, -1), (2, 4), (-1, -1), (7, 9)], 2)
+```
+This vector represents the mapping "second block from dimension 2 to dimension 4,
+fourth block from dimension 7 to dimension 9."
+These blocks contain the dimensions specified in `[2, 4, 8]`.
+"""
+function block_to_dimension_indices(cpa::CartesianProductArray{N}, vars::Vector{Int}) where {N}
+    result = fill((-1, -1), length(array(cpa)))
+    non_empty_length = 0
+    start_index, end_index = 1, 0
+    v_i = 1
+    for i in 1:length(cpa.array)
+        end_index += dim(cpa.array[i])
+        if v_i <= length(vars) && vars[v_i] <= end_index
+            result[i] = (start_index, end_index)
+            non_empty_length += 1
+            while v_i <= length(vars) && vars[v_i] <= end_index
+                v_i += 1
+            end
+        end
+        if v_i > length(vars)
+            break
+        end
+        start_index = end_index + 1
+    end
+    return result, non_empty_length
+end
+
+#same but for all variables
+function block_to_dimension_indices(cpa::CartesianProductArray{N}) where {N}
+    result = Vector{Tuple{Int, Int}}(undef, length(cpa.array))
+
+    start_index, end_index = 1, 0
+    for i in 1:length(cpa.array)
+        end_index += dim(cpa.array[i])
+        result[i] = (start_index, end_index)
+        start_index = end_index + 1
+    end
+    return result, length(cpa.array)
+end
+
+"""
+    substitute_blocks(low_dim_cpa::CartesianProductArray{N},
+                         orig_cpa::CartesianProductArray{N},
+                           blocks::Vector{Tuple{Int,Int}}) where {N}
+
+Return merged Cartesian Product Array between original CPA and some low-dimensional CPA,
+which represents updated subset of variables in specified blocks.
+
+### Input
+
+- `low_dim_cpa` -- low-dimensional cartesian product array
+- `orig_cpa` -- original high-dimensional Cartesian product array
+- `blocks` -- index of the first variable in each block of `orig_cpa`
+
+### Output
+
+Merged cartesian product array
+"""
+function substitute_blocks(low_dim_cpa::CartesianProductArray{N},
+                           orig_cpa::CartesianProductArray{N},
+                           blocks::Vector{Tuple{Int,Int}}) where {N}
+
+    array = Vector{LazySet{N}}(undef, length(orig_cpa.array))
+    index = 1
+    for bi in 1:length(orig_cpa.array)
+        start_ind, end_index = blocks[bi]
+        if start_ind == -1
+            array[bi] = orig_cpa.array[bi]
+        else
+            array[bi] = low_dim_cpa.array[index]
+            index += 1
+        end
+    end
+    return CartesianProductArray(array)
+end

src/LazySets.jl

@@ -89,8 +89,14 @@ include("Rectification.jl")
 # =============================
 # Conversions between set types
 # =============================
+include("intersection_helper.jl")
 include("convert.jl")
 
+# =====================
+# Approximations module
+# =====================
+include("Approximations/Approximations.jl")
+
 # ===========================
 # Concrete operations on sets
 # ===========================
@@ -104,11 +110,6 @@ include("difference.jl")
 # =======
 include("aliases.jl")
 
-# =====================
-# Approximations module
-# =====================
-include("Approximations/Approximations.jl")
-
 # ==========================
 # Parallel algorithms module
 # ==========================

src/Translation.jl

@@ -2,7 +2,8 @@ import Base: isempty
 
 export Translation,
        an_element,
-       constraints_list
+       constraints_list,
+       linear_map
 
 """
     Translation{N<:Real, VN<:AbstractVector{N}, S<:LazySet{N}} <: LazySet{N}
@@ -364,3 +365,29 @@ This implementation relies on the set membership function for the wrapped set
 function ∈(x::AbstractVector{N}, tr::Translation{N})::Bool where {N<:Real}
     return ∈(x - tr.v, tr.X)
 end
+
+"""
+    linear_map(M::AbstractMatrix{N}, tr::Translation{N}) where {N<:Real}
+
+Concrete linear map of a polyhedron in constraint representation.
+
+### Input
+
+- `M`  -- matrix
+- `tr` -- translation of a convex set
+
+### Output
+
+A concrete set corresponding to the linear map.
+The type of the result depends on the type of the set wrapped by `tr`.
+
+### Algorithm
+
+We compute `translate(linear_map(M, tr.X), M * tr.v)`.
+"""
+function linear_map(M::AbstractMatrix{N}, tr::Translation{N}) where {N<:Real}
+    @assert dim(tr) == size(M, 2) "a linear map of size $(size(M)) cannot be " *
+        "applied to a set of dimension $(dim(tr))"
+
+    return translate(linear_map(M, tr.X), M * tr.v)
+end

src/concrete_intersection.jl

@@ -678,7 +678,7 @@ Return the intersection of a lazy linear map and a convex set.
 
 - `L` -- linear map
 - `S` -- convex set
-  
+
 ### Output
 
 The polytope obtained by the intersection of `l.M * L.X` and `S`.
@@ -782,3 +782,36 @@ function intersection(rm::ResetMap{N, <:AbstractPolytope},
                       P::AbstractPolyhedron{N}) where {N<:Real}
     return intersection(P, rm)
 end
+
+function intersection(U::Universe{N}, X::CartesianProductArray{N}) where {N<:Real}
+    return X
+end
+
+# symmetric method
+function intersection(X::CartesianProductArray{N}, U::Universe{N}) where {N<:Real}
+    return intersection(U, X)
+end
+
+"""
+        intersection(X::CartesianProductArray{N}, Y::CartesianProductArray{N})
+
+Return the intersection between cartesian products of a finite number of convex sets.
+
+### Input
+
+ - `X` -- cartesian product of a finite number of convex sets
+ - `Y` -- cartesian product of a finite number of convex sets
+
+### Output
+
+The decomposed set which represents concrete intersection between `X` and `Y`
+
+### Algorithm
+
+This algorithm intersect corresponding blocks between sets.
+"""
+function intersection(X::CartesianProductArray{N}, Y::CartesianProductArray{N}) where {N<:Real}
+    @assert same_block_structure(array(X), array(Y)) "block structure has to be the same"
+
+    return CartesianProductArray([intersection(array(X)[i], array(Y)[i]) for i in eachindex(X.array)])
+end

src/helper_functions.jl

@@ -542,9 +542,9 @@ julia> subtypes(AbstractPolytope, true)
 function subtypes(interface, concrete::Bool)
 
     subtypes_to_test = subtypes(interface)
-    
+
     # do not seek the concrete subtypes further
-    if !concrete 
+    if !concrete
         return sort(subtypes_to_test, by=string)
     end