more latex for inf rules
This commit is contained in:
Francesco Mecca
parent
d19c25abbd
commit
6b528d921e
6 changed files with 735 additions and 39 deletions
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@ -3,7 +3,7 @@ import re
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from sys import argv
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from sys import argv
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allsymbols = json.load(open('./unicode-latex.json'))
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allsymbols = json.load(open('./unicode-latex.json'))
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mysymbols = ['≡', '≠', '≼', '→', '←', '⊀', '⋠', '≺', '∀', '∈', 'ε','₀', '₂', '₁', '₃', 'ₐ', 'ₖ', 'ₘ', 'ₙ', 'ᵢ', 'ⁱ', '⋮', 'ₛ', 'ₜ', '≃', '⇔', '∧', '∅', 'ℕ', 'ⱼ', 'ʲ', '⊥', 'π', 'α', 'β', '∞', 'σ', '≤', '⊈', '∧', '∨', '∃', '⇒' ]
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mysymbols = ['≡', '≠', '≼', '→', '←', '⊀', '⋠', '≺', '∀', '∈', 'ε','₀', '₂', '₁', '₃', 'ₐ', 'ₖ', 'ₘ', 'ₙ', 'ᵢ', 'ⁱ', '⋮', 'ₛ', 'ₜ', '≃', '⇔', '∧', '∅', 'ℕ', 'ⱼ', 'ʲ', '⊥', 'π', 'α', 'β', '∞', 'σ', '≤', '⊈', '∧', '∨', '∃', '⇒', '∩', '∉', '⋃' ]
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extrasymbols = {'〚': '\llbracket', '〛': '\rrbracket'}
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extrasymbols = {'〚': '\llbracket', '〛': '\rrbracket'}
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symbols = {s: allsymbols[s] for s in mysymbols}
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symbols = {s: allsymbols[s] for s in mysymbols}
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469
tesi/mathpartir.sty
Normal file
469
tesi/mathpartir.sty
Normal file
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@ -0,0 +1,469 @@
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%%
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%% This is file `mathpartir.sty',
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%% generated with the docstrip utility.
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%%
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%% The original source files were:
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%%
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%% mathpartir.dtx (with options: `package')
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%%
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%% This is a generated file.
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%%
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%% Copyright (C) 2015, 2020 by Didier Remy <didier.remy(at)inria(dot)fr>
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%%
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%% This file is part of mathpartir.
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%%
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%% mathpartir is free software: you can redistribute it and/or modify
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%% it under the terms of the GNU General Public License as published by
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%% the Free Software Foundation, either version 2 of the License, or
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%% (at your option) any later version.
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%%
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%% mathpartir is distributed in the hope that it will be useful,
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%% but WITHOUT ANY WARRANTY; without even the implied warranty of
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%% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
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%% GNU General Public License for more details.
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%%
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%% You should have received a copy of the GNU General Public License
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%% along with mathpartir. If not, see <http://www.gnu.org/licenses/>.
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%%
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\NeedsTeXFormat{LaTeX2e}
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\ProvidesPackage{mathpartir}
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[2020/02/15 version 1.4.0 Math Paragraph for Typesetting Inference Rules]
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%% Identification
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%% Preliminary declarations
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\RequirePackage {keyval}
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%% Options
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%% More declarations
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%% PART I: Typesetting maths in paragraphe mode
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%% \newdimen \mpr@tmpdim
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%% Dimens are a precious ressource. Uses seems to be local.
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\let \mpr@tmpdim \@tempdima
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\let \mpr@hva \empty
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%% normal paragraph parametters, should rather be taken dynamically
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\def \mpr@savepar {%
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\edef \MathparNormalpar
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{\noexpand \lineskiplimit \the\lineskiplimit
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\noexpand \lineskip \the\lineskip}%
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}
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\def \mpr@rulelineskip {\lineskiplimit=0.3em\lineskip=0.2em plus 0.1em}
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\def \mpr@lesslineskip {\lineskiplimit=0.6em\lineskip=0.5em plus 0.2em}
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\def \mpr@lineskip {\lineskiplimit=1.2em\lineskip=1.2em plus 0.2em}
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\let \MathparLineskip \mpr@lineskip
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\def \mpr@paroptions {\MathparLineskip}
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\let \mpr@prebindings \relax
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\newskip \mpr@andskip \mpr@andskip 2em plus 0.5fil minus 0.5em
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\def \mpr@goodbreakand
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{\hskip -\mpr@andskip \penalty -1000\hskip \mpr@andskip}
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\def \mpr@and {\hskip \mpr@andskip}
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\def \mpr@andcr {\penalty 50\mpr@and}
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\def \mpr@cr {\penalty -10000\mpr@and}
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\def \mpr@eqno #1{\mpr@andcr #1\hskip 0em plus -1fil \penalty 10}
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\def \mpr@bindings {%
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\let \and \mpr@andcr
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\let \par \mpr@andcr
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\let \\\mpr@cr
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\let \eqno \mpr@eqno
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\let \hva \mpr@hva
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}
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\let \MathparBindings \mpr@bindings
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\def \MathparBeginhook {}
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\def \MathparEndhook {}
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\let \MathparCentering \centering
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\newenvironment{mathpar}[1][]
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{$$
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\MathparBeginhook
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%% We save \lineskip parameters so that the user can restore them
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%% inside math \MathparNormalpar
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\mpr@savepar
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%% We define the shape of the paragrah
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\parskip 0em \hsize \linewidth \MathparCentering
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\vbox \bgroup \mpr@prebindings \mpr@paroptions #1\ifmmode $\else
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\noindent $\displaystyle\fi
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\MathparBindings}
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{\MathparEndhook \unskip \ifmmode $\fi\egroup $$\ignorespacesafterend}
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\newenvironment{mathparpagebreakable}[1][]
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{\begingroup
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\par
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\mpr@savepar \parskip 0em \hsize \linewidth \centering
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\mpr@prebindings \mpr@paroptions #1%
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\vskip \abovedisplayskip \vskip -\lineskip%
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\ifmmode \else $\displaystyle\fi
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\MathparBindings
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}
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{\unskip
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\ifmmode $\fi \par\endgroup
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\vskip \belowdisplayskip
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\noindent
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\ignorespacesafterend}
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%%% HOV BOXES
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\def \mathvbox@ #1{\hbox \bgroup \mpr@normallineskip
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\vbox \bgroup \tabskip 0em \let \\ \cr
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\halign \bgroup \hfil $##$\hfil\cr #1\crcr \egroup \egroup
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\egroup}
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\def \mathhvbox@ #1{\setbox0 \hbox {\let \\\qquad $#1$}\ifnum \wd0 < \hsize
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\box0\else \mathvbox {#1}\fi}
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%% Part II -- operations on lists
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\newtoks \mpr@lista
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\newtoks \mpr@listb
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\long \def\mpr@cons #1\mpr@to#2{\mpr@lista {\\{#1}}\mpr@listb \expandafter
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{#2}\edef #2{\the \mpr@lista \the \mpr@listb}}
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\long \def\mpr@snoc #1\mpr@to#2{\mpr@lista {\\{#1}}\mpr@listb \expandafter
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{#2}\edef #2{\the \mpr@listb\the\mpr@lista}}
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\long \def \mpr@concat#1=#2\mpr@to#3{\mpr@lista \expandafter {#2}\mpr@listb
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\expandafter {#3}\edef #1{\the \mpr@listb\the\mpr@lista}}
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\def \mpr@head #1\mpr@to #2{\expandafter \mpr@head@ #1\mpr@head@ #1#2}
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\long \def \mpr@head@ #1#2\mpr@head@ #3#4{\def #4{#1}\def#3{#2}}
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\def \mpr@flatten #1\mpr@to #2{\expandafter \mpr@flatten@ #1\mpr@flatten@ #1#2}
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\long \def \mpr@flatten@ \\#1\\#2\mpr@flatten@ #3#4{\def #4{#1}\def #3{\\#2}}
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\def \mpr@makelist #1\mpr@to #2{\def \mpr@all {#1}%
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\mpr@lista {\\}\mpr@listb \expandafter {\mpr@all}\edef \mpr@all {\the
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\mpr@lista \the \mpr@listb \the \mpr@lista}\let #2\empty
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\def \mpr@stripof ##1##2\mpr@stripend{\def \mpr@stripped{##2}}\loop
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\mpr@flatten \mpr@all \mpr@to \mpr@one
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\expandafter \mpr@snoc \mpr@one \mpr@to #2\expandafter \mpr@stripof
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\mpr@all \mpr@stripend
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\ifx \mpr@stripped \empty \let \mpr@isempty 0\else \let \mpr@isempty 1\fi
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\ifx 1\mpr@isempty
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\repeat
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}
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\def \mpr@rev #1\mpr@to #2{\let \mpr@tmp \empty
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\def \\##1{\mpr@cons ##1\mpr@to \mpr@tmp}#1\let #2\mpr@tmp}
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%% Part III -- Type inference rules
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\newif \if@premisse
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\newbox \mpr@hlist
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\newbox \mpr@vlist
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\newif \ifmpr@center \mpr@centertrue
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\def \mpr@vskip {}
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\let \mpr@vbox \vbox
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\def \mpr@htovlist {%
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\setbox \mpr@hlist
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\hbox {\strut
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\ifmpr@center \hskip -0.5\wd\mpr@hlist\fi
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\unhbox \mpr@hlist}%
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\setbox \mpr@vlist
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\mpr@vbox {\if@premisse
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\box \mpr@hlist
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\ifx \mpr@vskip \empty \else
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\hrule height 0em depth \mpr@vskip width 0em
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\fi
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\unvbox \mpr@vlist
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\else
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\unvbox \mpr@vlist
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\ifx \mpr@vskip \empty \else
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\hrule height 0em depth \mpr@vskip width 0em
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\fi
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\box \mpr@hlist
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\fi}%
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}
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\def \mpr@item #1{$\displaystyle #1$}
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\def \mpr@sep{2em}
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\def \mpr@blank { }
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\def \mpr@hovbox #1#2{\hbox
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\bgroup
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\ifx #1T\@premissetrue
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\else \ifx #1B\@premissefalse
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\else
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\PackageError{mathpartir}
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{Premisse orientation should either be T or B}
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{Fatal error in Package}%
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\fi \fi
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\def \@test {#2}\ifx \@test \mpr@blank\else
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\setbox \mpr@hlist \hbox {}%
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\setbox \mpr@vlist \vbox {}%
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\if@premisse \let \snoc \mpr@cons \else \let \snoc \mpr@snoc \fi
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\let \@hvlist \empty \let \@rev \empty
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\mpr@tmpdim 0em
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\expandafter \mpr@makelist #2\mpr@to \mpr@flat
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\if@premisse \mpr@rev \mpr@flat \mpr@to \@rev \else \let \@rev \mpr@flat \fi
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\def \\##1{%
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\def \@test {##1}\ifx \@test \empty
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\mpr@htovlist
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\mpr@tmpdim 0em %%% last bug fix not extensively checked
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\else
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\setbox0 \hbox{\mpr@item {##1}}\relax
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\advance \mpr@tmpdim by \wd0
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%\mpr@tmpdim 1.02\mpr@tmpdim
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\ifnum \mpr@tmpdim < \hsize
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\ifnum \wd\mpr@hlist > 0
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\if@premisse
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\setbox \mpr@hlist
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\hbox {\unhbox0 \hskip \mpr@sep \unhbox \mpr@hlist}%
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\else
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\setbox \mpr@hlist
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\hbox {\unhbox \mpr@hlist \hskip \mpr@sep \unhbox0}%
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\fi
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\else
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\setbox \mpr@hlist \hbox {\unhbox0}%
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\fi
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\else
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\ifnum \wd \mpr@hlist > 0
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\mpr@htovlist
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\mpr@tmpdim \wd0
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\fi
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\setbox \mpr@hlist \hbox {\unhbox0}%
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\fi
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\advance \mpr@tmpdim by \mpr@sep
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\fi
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}%
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\@rev
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\mpr@htovlist
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\ifmpr@center \hskip \wd\mpr@vlist\fi \box \mpr@vlist
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\fi
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\egroup
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}
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%%% INFERENCE RULES
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\@ifundefined{@@over}{%
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\let\@@over\over % fallback if amsmath is not loaded
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\let\@@overwithdelims\overwithdelims
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\let\@@atop\atop \let\@@atopwithdelims\atopwithdelims
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\let\@@above\above \let\@@abovewithdelims\abovewithdelims
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}{}
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%% The default
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\def \mpr@@fraction #1#2{\hbox {\advance \hsize by -0.5em
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$\displaystyle {#1\mpr@over #2}$}}
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\def \mpr@@nofraction #1#2{\hbox {\advance \hsize by -0.5em
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$\displaystyle {#1\@@atop #2}$}}
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\let \mpr@fraction \mpr@@fraction
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%% A generic solution to arrow
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\def \mpr@@fractionaboveskip {0ex}
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\def \mpr@@fractionbelowskip {0.22ex}
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\def \mpr@make@fraction #1#2#3#4#5{\hbox {%
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\def \mpr@tail{#1}%
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\def \mpr@body{#2}%
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\def \mpr@head{#3}%
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\setbox1=\hbox{$#4$}\setbox2=\hbox{$#5$}%
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\setbox3=\hbox{$\mkern -3mu\mpr@body\mkern -3mu$}%
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\dimen0\ht3\advance\dimen0 by \dp3\relax
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\dimen0 0.5\dimen0\relax
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\advance \dimen0 by \mpr@@fractionaboveskip
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\setbox1=\hbox {\raise \dimen0 \box1}\relax
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\dimen0 -\dimen0\advance \dimen0 \mpr@@fractionaboveskip\dimen0 -\dimen0
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\advance \dimen0 by \mpr@@fractionbelowskip
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\setbox2=\hbox {\lower \dimen0 \box2}\relax
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\setbox0=\hbox {$\displaystyle {\box1 \atop \box2}$}%
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\dimen0=\wd0\box0
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\box0 \hskip -\dimen0\relax
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\hbox to \dimen0 {$%\color{blue}
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\mathrel{\mpr@tail}\joinrel
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\xleaders\hbox{\copy3}\hfil\joinrel\mathrel{\mpr@head}%
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$}}}
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%% Old stuff should be removed in next version
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\def \mpr@@nothing #1#2
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{$\lower 0.01pt \mpr@@nofraction {#1}{#2}$}
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\def \mpr@@reduce #1#2{\hbox
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{$\lower 0.01pt \mpr@@fraction {#1}{#2}\mkern -15mu\rightarrow$}}
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\def \mpr@@rewrite #1#2#3{\hbox
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{$\lower 0.01pt \mpr@@fraction {#2}{#3}\mkern -8mu#1$}}
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\def \mpr@infercenter #1{\vcenter {\mpr@hovbox{T}{#1}}}
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\def \mpr@empty {}
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\def \mpr@inferrule
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{\bgroup
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\ifnum \linewidth<\hsize \hsize \linewidth\fi
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\mpr@rulelineskip
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\let \and \qquad
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\let \hva \mpr@hva
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\let \@rulename \mpr@empty
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\let \@rule@options \mpr@empty
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\let \mpr@over \@@over
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\mpr@inferrule@}
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\newcommand {\mpr@inferrule@}[3][]
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{\everymath={\displaystyle}%
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\def \@test {#2}\ifx \empty \@test
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\def \@test {#3}\ifx \empty \@test
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\PackageWarning {mathpartir}
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{\string\inferrule\space empty arguments substituted}{}%
|
||||||
|
\setbox0 \mpr@fraction {?}{?}%
|
||||||
|
\else
|
||||||
|
\setbox0 \hbox {$\vcenter {\mpr@hovbox{B}{#3}}$}%
|
||||||
|
\fi
|
||||||
|
\else
|
||||||
|
\def \@test {#3}\ifx \empty \@test
|
||||||
|
\setbox0 \hbox {$\vcenter {\mpr@hovbox{T}{#2}}$}%
|
||||||
|
\else
|
||||||
|
\setbox0 \mpr@fraction {\mpr@hovbox{T}{#2}}{\mpr@hovbox{B}{#3}}%
|
||||||
|
\fi \fi
|
||||||
|
\def \@test {#1}\ifx \@test\empty \box0
|
||||||
|
\else \vbox
|
||||||
|
%%% Suggestion de Francois pour les etiquettes longues
|
||||||
|
%%% {\hbox to \wd0 {\RefTirName {#1}\hfil}\box0}\fi
|
||||||
|
{\hbox {\DefTirName {#1}}\box0}\fi
|
||||||
|
\egroup}
|
||||||
|
|
||||||
|
\def \mpr@vdotfil #1{\vbox to #1{\leaders \hbox{$\cdot$} \vfil}}
|
||||||
|
|
||||||
|
|
||||||
|
%% Version for Hoare triples
|
||||||
|
|
||||||
|
\def \triple@hlinebox{\noalign{\setbox0\hbox {}\dp0 0.1ex\ht0 0.1ex\box0}}
|
||||||
|
\def \triple@hline {\triple@hlinebox\hline\triple@hlinebox}
|
||||||
|
|
||||||
|
\newcommand{\triplerule}[4][]
|
||||||
|
{\bgroup
|
||||||
|
\ifnum \linewidth<\hsize \hsize \linewidth\fi
|
||||||
|
\mpr@rulelineskip
|
||||||
|
\let \and \qquad
|
||||||
|
\let \hva \mpr@hva
|
||||||
|
\setbox0 \hbox {\begin{array}[b]{@{}c@{}}
|
||||||
|
\mpr@hovbox{T}{#2}\cr\triple@hline
|
||||||
|
\mpr@hovbox{T}{#3}\cr\triple@hline
|
||||||
|
\let \mpr@vbox \vtop \ht0 0em
|
||||||
|
\setbox0 \strut \@tempdima \ht0
|
||||||
|
\setbox0 \mpr@hovbox{B}{#4}\advance \@tempdima by -\ht0
|
||||||
|
\setbox0 \hbox{\raise \@tempdima \box0}\box0
|
||||||
|
\end{array}}%
|
||||||
|
\def \@test {#1}\ifx \@test\empty \box0
|
||||||
|
\else \vbox
|
||||||
|
%%% Suggestion de Francois pour les etiquettes longues
|
||||||
|
%%% {\hbox to \wd0 {\RefTirName {#1}\hfil}\box0}\fi
|
||||||
|
{\hbox {\DefTirName {#1}}\box0}\fi
|
||||||
|
\egroup}
|
||||||
|
|
||||||
|
|
||||||
|
%% Keys that make sence in all kinds of rules
|
||||||
|
\def \mprset #1{\setkeys{mprset}{#1}}
|
||||||
|
\define@key {mprset}{andskip}[]{\mpr@andskip=#1}
|
||||||
|
\define@key {mprset}{lineskip}[]{\lineskip=#1}
|
||||||
|
\define@key {mprset}{lessskip}[]{\lineskip=0.5\lineskip}
|
||||||
|
\define@key {mprset}{flushleft}[]{\mpr@centerfalse}
|
||||||
|
\define@key {mprset}{center}[]{\mpr@centertrue}
|
||||||
|
\define@key {mprset}{rewrite}[]{\let \mpr@fraction \mpr@@rewrite}
|
||||||
|
\define@key {mprset}{atop}[]{\let \mpr@fraction \mpr@@nofraction}
|
||||||
|
\define@key {mprset}{myfraction}[]{\let \mpr@fraction #1}
|
||||||
|
\define@key {mprset}{fraction}[]{\def \mpr@fraction {\mpr@make@fraction #1}}
|
||||||
|
\define@key {mprset}{defaultfraction}[]{\let \mpr@fraction \mpr@@fraction}
|
||||||
|
\define@key {mprset}{sep}{\def\mpr@sep{#1}}
|
||||||
|
\define@key {mprset}{fractionaboveskip}{\def\mpr@@fractionaboveskip{#1}}
|
||||||
|
\define@key {mprset}{fractionbelowskip}{\def\mpr@@fractionbelowskip{#1}}
|
||||||
|
\define@key {mprset}{style}[1]{\def\TirNameStyle{#1}def}
|
||||||
|
\define@key {mprset}{rightstyle}[1]{\def\RightTirNameStyle{#1}}
|
||||||
|
\define@key {mprset}{leftstyle}[1]{\def\LeftTirNameStyle{#1}}
|
||||||
|
\define@key {mprset}{vskip}[1]{\def \mpr@vskip{#1}}
|
||||||
|
|
||||||
|
\newbox \mpr@right
|
||||||
|
\define@key {mpr}{flushleft}[]{\mpr@centerfalse}
|
||||||
|
\define@key {mpr}{center}[]{\mpr@centertrue}
|
||||||
|
\define@key {mpr}{rewrite}[]{\let \mpr@fraction \mpr@@rewrite}
|
||||||
|
\define@key {mpr}{myfraction}[]{\let \mpr@fraction #1}
|
||||||
|
\define@key {mpr}{fraction}[]{\def \mpr@fraction {\mpr@make@fraction #1}}
|
||||||
|
\define@key {mpr}{width}{\hsize #1}
|
||||||
|
\define@key {mpr}{sep}{\def\mpr@sep{#1}}
|
||||||
|
\define@key {mpr}{before}{#1}
|
||||||
|
\define@key {mpr}{lab}{\let \DefTirName \LabTirName \def \mpr@rulename {#1}}
|
||||||
|
\define@key {mpr}{Lab}{\let \DefTirName \LabTirName \def \mpr@rulename {#1}}
|
||||||
|
\define@key {mpr}{style}[1]{\def\TirNameStyle{#1}def}
|
||||||
|
\define@key {mpr}{rightstyle}[1]{\def\RightTirNameStyle{#1}}
|
||||||
|
\define@key {mpr}{leftstyle}[1]{\def\LeftTirNameStyle{#1}}
|
||||||
|
\define@key {mpr}{vskip}[1]{\def \mpr@vskip{#1}}
|
||||||
|
\define@key {mpr}{narrower}{\hsize #1\hsize}
|
||||||
|
\define@key {mpr}{leftskip}{\hskip -#1}
|
||||||
|
\define@key {mpr}{reduce}[]{\let \mpr@fraction \mpr@@reduce}
|
||||||
|
\define@key {mpr}{rightskip}
|
||||||
|
{\setbox \mpr@right \hbox {\unhbox \mpr@right \hskip -#1}}
|
||||||
|
\define@key {mpr}{LEFT}{\setbox0 \hbox {$#1$}\relax
|
||||||
|
\advance \hsize by -\wd0\box0}
|
||||||
|
|
||||||
|
\define@key {mpr}{left}{\setbox0 \hbox {$\LeftTirName {#1}\;$}\relax
|
||||||
|
\advance \hsize by -\wd0\box0}
|
||||||
|
\define@key {mpr}{Left}{\llap{$\LeftTirName {#1}\;$}}
|
||||||
|
\define@key {mpr}{right}
|
||||||
|
{\setbox0 \hbox {$\;\RightTirName {#1}$}\relax \advance \hsize by -\wd0
|
||||||
|
\setbox \mpr@right \hbox {\unhbox \mpr@right \unhbox0}}
|
||||||
|
\define@key {mpr}{RIGHT}
|
||||||
|
{\setbox0 \hbox {$#1$}\relax \advance \hsize by -\wd0
|
||||||
|
\setbox \mpr@right \hbox {\unhbox \mpr@right \unhbox0}}
|
||||||
|
\define@key {mpr}{Right}
|
||||||
|
{\setbox \mpr@right \hbox {\unhbox \mpr@right \rlap {$\;\RightTirName {#1}$}}}
|
||||||
|
\define@key {mpr}{vdots}{\def \mpr@vdots {\@@atop \mpr@vdotfil{#1}}}
|
||||||
|
\define@key {mpr}{after}{\edef \mpr@after {\mpr@after #1}}
|
||||||
|
\define@key {mpr}{vcenter}[]{\mpr@vcentertrue}
|
||||||
|
|
||||||
|
\newif \ifmpr@vcenter \mpr@vcenterfalse
|
||||||
|
\newcommand \mpr@inferstar@ [3][]{\begingroup
|
||||||
|
\setbox0 \hbox
|
||||||
|
{\let \mpr@rulename \mpr@empty \let \mpr@vdots \relax
|
||||||
|
\setbox \mpr@right \hbox{}%
|
||||||
|
\setkeys{mpr}{#1}%
|
||||||
|
$\ifx \mpr@rulename \mpr@empty \mpr@inferrule {#2}{#3}\else
|
||||||
|
\mpr@inferrule [{\mpr@rulename}]{#2}{#3}\fi
|
||||||
|
\box \mpr@right \mpr@vdots$
|
||||||
|
\ifmpr@vcenter \aftergroup \mpr@vcentertrue \fi}
|
||||||
|
\ifmpr@vcenter
|
||||||
|
\box0
|
||||||
|
\else
|
||||||
|
\setbox1 \hbox {\strut}
|
||||||
|
\@tempdima \dp0 \advance \@tempdima by -\dp1
|
||||||
|
\raise \@tempdima \box0
|
||||||
|
\fi
|
||||||
|
\endgroup}
|
||||||
|
|
||||||
|
\def \mpr@infer {\@ifnextchar *{\mpr@inferstar}{\mpr@inferrule}}
|
||||||
|
\newcommand \mpr@err@skipargs[3][]{}
|
||||||
|
\def \mpr@inferstar*{\ifmmode
|
||||||
|
\let \@do \mpr@inferstar@
|
||||||
|
\else
|
||||||
|
\let \@do \mpr@err@skipargs
|
||||||
|
\PackageError {mathpartir}
|
||||||
|
{\string\inferrule* can only be used in math mode}{}%
|
||||||
|
\fi \@do}
|
||||||
|
|
||||||
|
%%% Exports
|
||||||
|
|
||||||
|
|
||||||
|
\let \inferrule \mpr@infer
|
||||||
|
|
||||||
|
\@ifundefined {infer}{\let \infer \mpr@infer}{}
|
||||||
|
|
||||||
|
\def \TirNameStyle #1{\small \textsc{#1}}
|
||||||
|
\def \LeftTirNameStyle #1{\TirNameStyle {#1}}
|
||||||
|
\def \RightTirNameStyle #1{\TirNameStyle {#1}}
|
||||||
|
|
||||||
|
\def \lefttir@name #1{\hbox {\small \LeftTirNameStyle{#1}}}
|
||||||
|
\def \righttir@name #1{\hbox {\small \RightTirNameStyle{#1}}}
|
||||||
|
\let \TirName \lefttir@name
|
||||||
|
\let \LeftTirName \lefttir@name
|
||||||
|
\let \DefTirName \lefttir@name
|
||||||
|
\let \LabTirName \lefttir@name
|
||||||
|
\let \RightTirName \righttir@name
|
||||||
|
|
||||||
|
%%% Other Exports
|
||||||
|
|
||||||
|
\endinput
|
||||||
|
%%
|
||||||
|
%% End of file `mathpartir.sty'.
|
||||||
|
|
@ -12,7 +12,7 @@ Test(S,true) = Yes
|
||||||
|
|
||||||
|
|
||||||
-------------------------------
|
-------------------------------
|
||||||
equiv \emptyset Cs Ct gs ~> Yes
|
equiv \emptyset Cs Ct gs \to Yes
|
||||||
|
|
||||||
-----------------------------------------
|
-----------------------------------------
|
||||||
equiv S Failure (Leaf BBt) gs ~>No(vs, vt)
|
equiv S Failure (Leaf BBt) gs ~>No(vs, vt)
|
||||||
|
|
|
||||||
36
tesi/notations.sty
Normal file
36
tesi/notations.sty
Normal file
|
|
@ -0,0 +1,36 @@
|
||||||
|
\newcommand{\bnfeq}{\mathrel{::=}\;}
|
||||||
|
\newcommand{\bnfor}{\mathrel{\vert}}
|
||||||
|
|
||||||
|
\newcommand{\paramrel}[3]{#2 \mathrel{\approx_{#1}} #3}
|
||||||
|
\newcommand{\vrel}[2]{\paramrel{\mathsf{val}}{#1}{#2}}
|
||||||
|
\newcommand{\trel}[2]{\paramrel{\mathsf{term}}{#1}{#2}}
|
||||||
|
|
||||||
|
\newcommand{\sem}[1]{{\llbracket{#1}\rrbracket}}
|
||||||
|
|
||||||
|
\newcommand{\Fam}[2]{{(#2)}^{#1}}
|
||||||
|
|
||||||
|
\newcommand{\match}[2]{\mathtt{match}(#1,#2)}
|
||||||
|
|
||||||
|
\newcommand{\var}[1]{\mathtt{#1}}
|
||||||
|
\newcommand{\pK}{\mathtt{K}}
|
||||||
|
\newcommand{\any}{\mathtt{\_}}
|
||||||
|
|
||||||
|
% \equiv is already taken
|
||||||
|
\newcommand{\Equiv}[4]{\mathsf{equiv}(#1, #2, #3, #4)}
|
||||||
|
\newcommand{\Equivrel}[4]{#1 \vdash_{#4} #2 \approx #3}
|
||||||
|
|
||||||
|
\newcommand{\covers}[2]{#1 \mathrel{\mathsf{covers}} #2}
|
||||||
|
|
||||||
|
\newcommand{\Yes}{\mathsf{Yes}}
|
||||||
|
\newcommand{\No}[2]{\mathsf{No}(#1, #2)}
|
||||||
|
|
||||||
|
\newcommand{\Leaf}[1]{\mathsf{Leaf}(#1)}
|
||||||
|
\newcommand{\Failure}{\mathsf{Failure}}
|
||||||
|
\newcommand{\Switch}[3]{\mathsf{Switch}(#1, #2, #3)}
|
||||||
|
\newcommand{\Guard}[3]{\mathsf{Guard}(#1, #2, #3)}
|
||||||
|
|
||||||
|
\newcommand{\emptyqueue}{\mathord{[]}}
|
||||||
|
|
||||||
|
\newcommand{\Cfb}{C_{\mathsf{fb}}}
|
||||||
|
|
||||||
|
\newcommand{\trim}[2]{\mathsf{trim}(#1, #2)}
|
||||||
BIN
tesi/tesi.pdf
BIN
tesi/tesi.pdf
Binary file not shown.
|
|
@ -1,11 +1,11 @@
|
||||||
\begin{comment}
|
\begin{comment}
|
||||||
* TODO Scaletta [1/6]
|
* TODO Scaletta [1/6]
|
||||||
- [X] Introduction
|
- [X] Introduction
|
||||||
- [-] Background [60%]
|
- [-] Background [80%]
|
||||||
- [X] Low level representation
|
- [X] Low level representation
|
||||||
- [X] Lambda code [0%]
|
- [X] Lambda code [0%]
|
||||||
- [X] Pattern matching
|
- [X] Pattern matching
|
||||||
- [ ] Symbolic execution
|
- [X] Symbolic execution
|
||||||
- [ ] Translation Validation
|
- [ ] Translation Validation
|
||||||
- [ ] Translation validation of the Pattern Matching Compiler
|
- [ ] Translation validation of the Pattern Matching Compiler
|
||||||
- [ ] Source translation
|
- [ ] Source translation
|
||||||
|
|
@ -46,6 +46,27 @@ Magari prima pattern matching poi compilatore?
|
||||||
#+LaTeX_HEADER: \newcommand{\covers}[2]{#1 \mathrel{\mathsf{covers}} #2}
|
#+LaTeX_HEADER: \newcommand{\covers}[2]{#1 \mathrel{\mathsf{covers}} #2}
|
||||||
#+LaTeX_HEADER: \newcommand{\Yes}{\mathsf{Yes}}
|
#+LaTeX_HEADER: \newcommand{\Yes}{\mathsf{Yes}}
|
||||||
#+LaTeX_HEADER: \newcommand{\No}[2]{\mathsf{No}(#1, #2)}
|
#+LaTeX_HEADER: \newcommand{\No}[2]{\mathsf{No}(#1, #2)}
|
||||||
|
#+LaTeX_HEADER: \usepackage{comment}
|
||||||
|
#+LaTeX_HEADER: \usepackage{mathpartir}
|
||||||
|
#+LaTeX_HEADER: \usepackage{stmaryrd} % llbracket, rrbracket
|
||||||
|
#+LaTeX_HEADER: \usepackage{listings}
|
||||||
|
#+LaTeX_HEADER: \usepackage{notations}
|
||||||
|
#+LaTeX_HEADER: \lstset{
|
||||||
|
#+LaTeX_HEADER: mathescape=true,
|
||||||
|
#+LaTeX_HEADER: language=[Objective]{Caml},
|
||||||
|
#+LaTeX_HEADER: basicstyle=\ttfamily,
|
||||||
|
#+LaTeX_HEADER: extendedchars=true,
|
||||||
|
#+LaTeX_HEADER: showstringspaces=false,
|
||||||
|
#+LaTeX_HEADER: aboveskip=\smallskipamount,
|
||||||
|
#+LaTeX_HEADER: % belowskip=\smallskipamount,
|
||||||
|
#+LaTeX_HEADER: columns=fullflexible,
|
||||||
|
#+LaTeX_HEADER: moredelim=**[is][\color{blue}]{/*}{*/},
|
||||||
|
#+LaTeX_HEADER: moredelim=**[is][\color{green!60!black}]{/!}{!/},
|
||||||
|
#+LaTeX_HEADER: moredelim=**[is][\color{orange}]{/(}{)/},
|
||||||
|
#+LaTeX_HEADER: moredelim=[is][\color{red}]{/[}{]/},
|
||||||
|
#+LaTeX_HEADER: xleftmargin=1em,
|
||||||
|
#+LaTeX_HEADER: }
|
||||||
|
#+LaTeX_HEADER: \lstset{aboveskip=0.4ex,belowskip=0.4ex}
|
||||||
|
|
||||||
#+EXPORT_SELECT_TAGS: export
|
#+EXPORT_SELECT_TAGS: export
|
||||||
#+EXPORT_EXCLUDE_TAGS: noexport
|
#+EXPORT_EXCLUDE_TAGS: noexport
|
||||||
|
|
@ -257,22 +278,22 @@ The algorithm respects the following correctness statement:
|
||||||
|
|
||||||
The algorithm proceeds by case analysis. Inference rules are shown.
|
The algorithm proceeds by case analysis. Inference rules are shown.
|
||||||
If $S$ is empty the results is $\Yes$.
|
If $S$ is empty the results is $\Yes$.
|
||||||
|
\begin{mathpar}
|
||||||
\begin{verbatim}
|
\infer{ }
|
||||||
------------------------
|
{\Equivrel \emptyset {C_S} {C_T} G}
|
||||||
equiv \emptyset Cs Ct gs
|
\end{mathpar}
|
||||||
\end{verbatim}
|
|
||||||
|
|
||||||
If the two decision trees are both terminal nodes the algorithm checks
|
If the two decision trees are both terminal nodes the algorithm checks
|
||||||
for content equality.
|
for content equality.
|
||||||
\begin{verbatim}
|
\begin{mathpar}
|
||||||
--------------------------
|
\infer{ }
|
||||||
equiv S Failure Failure []
|
{\Equivrel S \Failure \Failure \emptyqueue}
|
||||||
|
\\
|
||||||
|
\infer
|
||||||
|
{\trel {t_S} {t_T}}
|
||||||
|
{\Equivrel S {\Leaf {t_S}} {\Leaf {t_T}} \emptyqueue}
|
||||||
|
|
||||||
equiv_BB BBs BBt
|
\end{mathpar}
|
||||||
-------------------------------
|
|
||||||
equiv S (Leaf BBs) (Leaf BBt) []
|
|
||||||
\end{verbatim}
|
|
||||||
|
|
||||||
If the source decision tree (left hand side) is a terminal while the
|
If the source decision tree (left hand side) is a terminal while the
|
||||||
target decistion tree (right hand side) is not, the algorithm proceeds
|
target decistion tree (right hand side) is not, the algorithm proceeds
|
||||||
|
|
@ -280,13 +301,16 @@ by \emph{explosion} of the right hand side. Explosion means that every
|
||||||
child of the right hand side is tested for equality against the left
|
child of the right hand side is tested for equality against the left
|
||||||
hand side.
|
hand side.
|
||||||
|
|
||||||
\begin{verbatim}
|
\begin{mathpar}
|
||||||
(equiv S Cs Ci gs)^i
|
\infer
|
||||||
equiv S Cs Cf gs
|
{C_S \in {\Leaf t, \Failure}
|
||||||
-----------------------------------------
|
\\
|
||||||
equiv S Cs (Node(a, (Domi,Ci)^i, Cf)) gs
|
\forall i,\; \Equivrel {(S \land a \in D_i)} {C_S} {C_i} G
|
||||||
\end{verbatim}
|
\\
|
||||||
|
\Equivrel {(S \land a \notin \Fam i {D_i})} {C_S} \Cfb G}
|
||||||
|
{\Equivrel S
|
||||||
|
{C_S} {\Switch a {\Fam i {D_i} {C_i}} \Cfb} G}
|
||||||
|
\end{mathpar}
|
||||||
|
|
||||||
\begin{comment}
|
\begin{comment}
|
||||||
% TODO: [Gabriel] in practice the $dom_S$ are constructors and the
|
% TODO: [Gabriel] in practice the $dom_S$ are constructors and the
|
||||||
|
|
@ -304,12 +328,20 @@ the accessors on both side are equal, and removing the branches that
|
||||||
result in an empty intersection. If the accessors are different,
|
result in an empty intersection. If the accessors are different,
|
||||||
\emph{$dom_T$} is left unchanged.
|
\emph{$dom_T$} is left unchanged.
|
||||||
|
|
||||||
\begin{verbatim}
|
\begin{mathpar}
|
||||||
equiv S Ci (trim Ct a=Ki) gs
|
\infer
|
||||||
equiv S Cf (trim Ct (a\notin(K_i)^i) gs
|
{\forall i,\;
|
||||||
-------------------------------------
|
\Equivrel
|
||||||
equiv S (Node(a, (Ki,Ci)^i, Cf) Ct gs
|
{(S \land a = K_i)}
|
||||||
\end{verbatim}
|
{C_i} {\trim {C_T} {a = K_i}} G
|
||||||
|
\\
|
||||||
|
\Equivrel
|
||||||
|
{(S \land a \notin \Fam i {K_i})}
|
||||||
|
\Cfb {\trim {C_T} {a \notin \Fam i {K_i}}} G
|
||||||
|
}
|
||||||
|
{\Equivrel S
|
||||||
|
{\Switch a {\Fam i {K_i, C_i}} \Cfb} {C_T} G}
|
||||||
|
\end{mathpar}
|
||||||
|
|
||||||
The equivalence checking algorithm deals with guards by storing a
|
The equivalence checking algorithm deals with guards by storing a
|
||||||
queue. A guard blackbox is pushed to the queue whenever the algorithm
|
queue. A guard blackbox is pushed to the queue whenever the algorithm
|
||||||
|
|
@ -321,17 +353,21 @@ continues by exploding to two subtrees, one in which the guard
|
||||||
condition evaluates to true, the other when it evaluates to false.
|
condition evaluates to true, the other when it evaluates to false.
|
||||||
Termination of the algorithm is successful only when the guards queue
|
Termination of the algorithm is successful only when the guards queue
|
||||||
is empty.
|
is empty.
|
||||||
\begin{verbatim}
|
\begin{mathpar}
|
||||||
equiv S Ctrue Ct (gs++[condition])
|
\infer
|
||||||
equiv S Cfalse Ct (gs++[condition])
|
{\Equivrel S {C_0} {C_T} {G, (t_S = 0)}
|
||||||
--------------------------------------------
|
\\
|
||||||
equiv S (Guard condition Ctrue Cfalse) Ct gs
|
\Equivrel S {C_1} {C_T} {G, (t_S = 1)}}
|
||||||
|
{\Equivrel S
|
||||||
|
{\Guard {t_S} {C_0} {C_1}} {C_T} G}
|
||||||
|
|
||||||
equiv S Cs Ctrue gs
|
\infer
|
||||||
equiv S Cs Cfalse gs
|
{\trel {t_S} {t_T}
|
||||||
--------------------------------------------
|
\\
|
||||||
equiv S Cs (Guard condition Ctrue Cfalse) ([condition]++gs)
|
\Equivrel S {C_S} {C_b} G}
|
||||||
\end{verbatim}
|
{\Equivrel S
|
||||||
|
{C_S} {\Guard {t_T} {C_0} {C_1}} {(t_S = b), G}}
|
||||||
|
\end{mathpar}
|
||||||
|
|
||||||
\begin{comment}
|
\begin{comment}
|
||||||
TODO: replace inference rules with good latex
|
TODO: replace inference rules with good latex
|
||||||
|
|
@ -1129,6 +1165,106 @@ following four rules:
|
||||||
apply, and P₂ → L₂ containing the remaining rows. The algorithm is
|
apply, and P₂ → L₂ containing the remaining rows. The algorithm is
|
||||||
applied to both matrices.
|
applied to both matrices.
|
||||||
|
|
||||||
|
** Target translation
|
||||||
|
|
||||||
|
TODO
|
||||||
|
|
||||||
|
** Equivalence checking
|
||||||
|
|
||||||
|
The equivalence checking algorithm takes as input a domain of
|
||||||
|
possible values \emph{S} and a pair of source and target decision trees and
|
||||||
|
in case the two trees are not equivalent it returns a counter example.
|
||||||
|
The algorithm respects the following correctness statement:
|
||||||
|
|
||||||
|
% TODO: we have to define what \covers mean for readers to understand the specifications
|
||||||
|
% (or we use a simplifying lie by hiding \covers in the statements).
|
||||||
|
|
||||||
|
\begin{align*}
|
||||||
|
\Equiv S {C_S} {C_T} \emptyqueue = \Yes \;\land\; \covers {C_T} S
|
||||||
|
& \implies
|
||||||
|
\forall v_S \approx v_T \in S,\; C_S(v_S) = C_T(v_T)
|
||||||
|
\\
|
||||||
|
\Equiv S {C_S} {C_T} \emptyqueue = \No {v_S} {v_T} \;\land\; \covers {C_T} S
|
||||||
|
& \implies
|
||||||
|
v_S \approx v_T \in S \;\land\; C_S(v_S) \neq C_T(v_T)
|
||||||
|
\end{align*}
|
||||||
|
|
||||||
|
Our equivalence-checking algorithm $\Equiv S {C_S} {C_T} G$ is
|
||||||
|
a exactly decision procedure for the provability of the judgment
|
||||||
|
$(\Equivrel S {C_S} {C_T} G)$, defined below.
|
||||||
|
\begin{mathpar}
|
||||||
|
\begin{array}{l@{~}r@{~}l}
|
||||||
|
& & \text{\emph{constraint trees}} \\
|
||||||
|
C & \bnfeq & \Leaf t \\
|
||||||
|
& \bnfor & \Failure \\
|
||||||
|
& \bnfor & \Switch a {\Fam i {\pi_i, C_i}} \Cfb \\
|
||||||
|
& \bnfor & \Guard t {C_0} {C_1} \\
|
||||||
|
\end{array}
|
||||||
|
|
||||||
|
\begin{array}{l@{~}r@{~}l}
|
||||||
|
& & \text{\emph{boolean result}} \\
|
||||||
|
b & \in & \{ 0, 1 \} \\[0.5em]
|
||||||
|
& & \text{\emph{guard queues}} \\
|
||||||
|
G & \bnfeq & (t_1 = b_1), \dots, (t_n = b_n) \\
|
||||||
|
\end{array}
|
||||||
|
|
||||||
|
\begin{array}{l@{~}r@{~}l}
|
||||||
|
& & \text{\emph{input space}} \\
|
||||||
|
S & \subseteq & \{ (v_S, v_T) \mid \vrel {v_S} {v_T} \} \\
|
||||||
|
\end{array}
|
||||||
|
\\
|
||||||
|
\infer{ }
|
||||||
|
{\Equivrel \emptyset {C_S} {C_T} G}
|
||||||
|
|
||||||
|
\infer{ }
|
||||||
|
{\Equivrel S \Failure \Failure \emptyqueue}
|
||||||
|
|
||||||
|
\infer
|
||||||
|
{\trel {t_S} {t_T}}
|
||||||
|
{\Equivrel S {\Leaf {t_S}} {\Leaf {t_T}} \emptyqueue}
|
||||||
|
|
||||||
|
\infer
|
||||||
|
{\forall i,\;
|
||||||
|
\Equivrel
|
||||||
|
{(S \land a = K_i)}
|
||||||
|
{C_i} {\trim {C_T} {a = K_i}} G
|
||||||
|
\\
|
||||||
|
\Equivrel
|
||||||
|
{(S \land a \notin \Fam i {K_i})}
|
||||||
|
\Cfb {\trim {C_T} {a \notin \Fam i {K_i}}} G
|
||||||
|
}
|
||||||
|
{\Equivrel S
|
||||||
|
{\Switch a {\Fam i {K_i, C_i}} \Cfb} {C_T} G}
|
||||||
|
|
||||||
|
\begin{comment}
|
||||||
|
% TODO explain somewhere why the judgment is not symmetric:
|
||||||
|
% we avoid defining trimming on source trees, which would
|
||||||
|
% require more expressive conditions than just constructors
|
||||||
|
\end{comment}
|
||||||
|
\infer
|
||||||
|
{C_S \in {\Leaf t, \Failure}
|
||||||
|
\\
|
||||||
|
\forall i,\; \Equivrel {(S \land a \in D_i)} {C_S} {C_i} G
|
||||||
|
\\
|
||||||
|
\Equivrel {(S \land a \notin \Fam i {D_i})} {C_S} \Cfb G}
|
||||||
|
{\Equivrel S
|
||||||
|
{C_S} {\Switch a {\Fam i {D_i} {C_i}} \Cfb} G}
|
||||||
|
|
||||||
|
\infer
|
||||||
|
{\Equivrel S {C_0} {C_T} {G, (t_S = 0)}
|
||||||
|
\\
|
||||||
|
\Equivrel S {C_1} {C_T} {G, (t_S = 1)}}
|
||||||
|
{\Equivrel S
|
||||||
|
{\Guard {t_S} {C_0} {C_1}} {C_T} G}
|
||||||
|
|
||||||
|
\infer
|
||||||
|
{\trel {t_S} {t_T}
|
||||||
|
\\
|
||||||
|
\Equivrel S {C_S} {C_b} G}
|
||||||
|
{\Equivrel S
|
||||||
|
{C_S} {\Guard {t_T} {C_0} {C_1}} {(t_S = b), G}}
|
||||||
|
\end{mathpar}
|
||||||
|
|
||||||
* Correctness of the algorithm
|
* Correctness of the algorithm
|
||||||
Running a program tₛ or its translation 〚tₛ〛 against an input vₛ
|
Running a program tₛ or its translation 〚tₛ〛 against an input vₛ
|
||||||
produces as a result a result /r/ in the following way:
|
produces as a result a result /r/ in the following way:
|
||||||
|
|
@ -1258,7 +1394,41 @@ We define
|
||||||
|
|
||||||
|
|
||||||
** Proof of equivalence checking
|
** Proof of equivalence checking
|
||||||
|
\begin{comment}
|
||||||
|
TODO: put ^i∈I where needed
|
||||||
|
\end{comment}
|
||||||
|
\subsubsection{The trimming lemma}
|
||||||
|
The trimming lemma allows to reduce the size of a decision tree given
|
||||||
|
an accessor → π relation (In other words???)
|
||||||
|
| ∀vₜ ∈ (a→π), Cₜ(vₜ) = C_{t/a→π(kᵢ)}(vₜ)
|
||||||
|
We prove this by induction on Cₜ:
|
||||||
|
a. Case where Cₜ = Leaf_{bb}:
|
||||||
|
Leaf_{bb/a→π}(v) = Leaf_{bb}(v)
|
||||||
|
That means that the result of trimming on a Leaf is
|
||||||
|
the Leaf itself
|
||||||
|
b. Same for failure terminal
|
||||||
|
e. Unreachabe → ⊥
|
||||||
|
c. When Cₜ = Node(b, (π→Cᵢ)ⁱ)_{/a→π} then
|
||||||
|
we define πᵢ' = πᵢ if a≠b else πᵢ∩π ;
|
||||||
|
Node(b, (π→Cᵢ)ⁱ)_{/a→π} := Node(b, (π'ᵢ→C_{i/a→π})ⁱ)
|
||||||
|
|
||||||
|
Goal: prove that Cₜ(vₜ) = Node(b, (πᵢ'→C_{i/a→π})ⁱ)(vₜ)
|
||||||
|
Either vₜ∉(b→πᵢ)ⁱ and that means that the node is a Failure node
|
||||||
|
Or vₜ∈(b→πₖ) for some k, then
|
||||||
|
C_{k/a→π}(vₜ) = Node(b, (πᵢ'→C_{i/a→π})ⁱ)(vₜ)
|
||||||
|
because when a ≠ b then πₖ'= πₖ => vₜ∈πₖ'
|
||||||
|
while when a = b then πₖ'=(πₖ∩π)
|
||||||
|
- vₜ∈π because of hypothesis
|
||||||
|
- we already know that vₜ∈πₖ
|
||||||
|
So vₜ ∈ πₖ' and by induction Cₖ(vₜ) = C_{k/a→π}(vₜ)
|
||||||
|
and Cₜ(vₜ) = Cₖ(vₜ) when vₜ∈(b→πₖ)
|
||||||
|
This proves that Node(b, (πᵢ'→C_{i/a→π})ⁱ)(vₜ) = Cₜ(vₜ)
|
||||||
|
|
||||||
|
Covering lemma:
|
||||||
|
∀a,π covers(Cₜ,S) => covers(C_{t/a→π}, (S∩a→π))
|
||||||
|
Uᵢπⁱ ≈ Uᵢπ'∩(a→π) ≈ (Uᵢπ')∩(a→π) %% # TODO swap π and π'
|
||||||
|
|
||||||
|
\subsubsection Proof of equivalence checking
|
||||||
The equivalence checking algorithm takes as parameters an input space
|
The equivalence checking algorithm takes as parameters an input space
|
||||||
/S/, a source decision tree /Cₛ/ and a target decision tree /Cₜ/:
|
/S/, a source decision tree /Cₛ/ and a target decision tree /Cₜ/:
|
||||||
| equiv(S, Cₛ, Cₜ) → Yes | No(vₛ, vₜ)
|
| equiv(S, Cₛ, Cₜ) → Yes | No(vₛ, vₜ)
|
||||||
|
|
@ -1297,6 +1467,7 @@ We proceed by case analysis:
|
||||||
Devo spiegarlo?
|
Devo spiegarlo?
|
||||||
\end{comment}
|
\end{comment}
|
||||||
In the other subcases S is always non-empty.
|
In the other subcases S is always non-empty.
|
||||||
|
0. in case of unreachable: Cₛ(vₛ) = Absurd(Unreachable) ≠ Cₜ(vₜ) ∀vₛ,vₜ
|
||||||
2. equiv(S, Failure, Failure) := Yes
|
2. equiv(S, Failure, Failure) := Yes
|
||||||
the statement holds because of equality between Failure nodes in
|
the statement holds because of equality between Failure nodes in
|
||||||
the case of every possible value /v/.
|
the case of every possible value /v/.
|
||||||
|
|
@ -1318,3 +1489,23 @@ In the other subcases S is always non-empty.
|
||||||
| vₛ≃vₜ∈S ∧ Cₛ(vₛ)≠Cₜ(vₜ)
|
| vₛ≃vₜ∈S ∧ Cₛ(vₛ)≠Cₜ(vₜ)
|
||||||
and the result of the algorithm is
|
and the result of the algorithm is
|
||||||
| equiv(Sᵢ, Cₛ, Cₜᵢ) = No(vₛ, vₜ) for some minimal k∈I
|
| equiv(Sᵢ, Cₛ, Cₜᵢ) = No(vₛ, vₜ) for some minimal k∈I
|
||||||
|
4. equiv(S, Node(a, (kᵢ → Cₛᵢ)ⁱ, C_{sf}), Cₜ) :=
|
||||||
|
let π' = ⋃π(kᵢ) ∀i in
|
||||||
|
Forall(equiv( S∩(a→π(kᵢ)ⁱ), Cₛᵢ, C_{t/a→π(kᵢ)})ⁱ +++ equiv(S∩(a→π(kᵢ)), Cₛ, C_{/a¬̸π'}))
|
||||||
|
The statement holds because:
|
||||||
|
a. Forall(equiv( S∩(a→π(kᵢ)ⁱ), Cₛᵢ, C_{t/a→π(kᵢ)})ⁱ = Yes
|
||||||
|
In the yes case let's reason by case analysis:
|
||||||
|
i. When k∈(kᵢ)ⁱ
|
||||||
|
there is a k=kₖ for some k and this means that Cₛ(vₛ) = Cₛᵢ(vₛ)
|
||||||
|
By induction we know that Cₛᵢ(vₛ) = c_{t/a→πᵢ}(vₜ)
|
||||||
|
and because of the trimming lemma:
|
||||||
|
Cₜ(vₜ) = C_{t/a→πᵢ}(vₜ)
|
||||||
|
Putting all together:
|
||||||
|
Cₛ(vₛ) = Cₛᵢ(vₛ) = C_{t/a→πᵢ}(vₜ) = Cₜ(vₜ)
|
||||||
|
|
||||||
|
ii. or k∉(kᵢ)ⁱ ???
|
||||||
|
|
||||||
|
b. Forall(...) = No(vₛ, vₜ)
|
||||||
|
for a minimum k, equiv(Sₖ, Cₛₖ, C_{t/a→πₖ} = No(vₛ, vₜ)
|
||||||
|
then Cₛₖ(vₛ) ≠ C_{t/a→πₖ}(vₜ) and C_{t/a→πₖ}(vₜ) = Cₜ(vt)
|
||||||
|
=> (Cₛₖ(vₛ) = Cₛ(vₛ)) ≠ Cₜ(vₜ) # Same for fallback?
|
||||||
|
|
|
||||||
Loading…
Reference in a new issue