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(* Title: FOL/ex/NewLocaleTest.thy 
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ID: $Id$ 

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Author: Clemens Ballarin, TU Muenchen 

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Testing environment for locale expressions  experimental. 

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*) 

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theory NewLocaleTest 

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imports NewLocaleSetup 

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begin 

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ML_val {* set new_locales *} 

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ML_val {* set Toplevel.debug *} 

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ML_val {* set show_hyps *} 

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typedecl int arities int :: "term" 

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consts plus :: "int => int => int" (infixl "+" 60) 

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zero :: int ("0") 

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minus :: "int => int" (" _") 

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text {* Inference of parameter types *} 

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locale param1 = fixes p 

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print_locale! param1 

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locale param2 = fixes p :: 'b 
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print_locale! param2 
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(* 

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locale param_top = param2 r for r :: "'b :: {}" 

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print_locale! param_top 

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*) 

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locale param3 = fixes p (infix ".." 50) 

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print_locale! param3 

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locale param4 = fixes p :: "'a => 'a => 'a" (infix ".." 50) 

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print_locale! param4 

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text {* Incremental type constraints *} 

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locale constraint1 = 

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fixes prod (infixl "**" 65) 

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assumes l_id: "x ** y = x" 

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assumes assoc: "(x ** y) ** z = x ** (y ** z)" 

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print_locale! constraint1 

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locale constraint2 = 

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fixes p and q 

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assumes "p = q" 

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print_locale! constraint2 

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text {* Inheritance *} 
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locale semi = 

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fixes prod (infixl "**" 65) 

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assumes assoc: "(x ** y) ** z = x ** (y ** z)" 

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print_locale! semi thm semi_def 

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locale lgrp = semi + 

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fixes one and inv 

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assumes lone: "one ** x = x" 

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and linv: "inv(x) ** x = one" 

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print_locale! lgrp thm lgrp_def lgrp_axioms_def 

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locale add_lgrp = semi "op ++" for sum (infixl "++" 60) + 

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fixes zero and neg 

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assumes lzero: "zero ++ x = x" 

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and lneg: "neg(x) ++ x = zero" 

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print_locale! add_lgrp thm add_lgrp_def add_lgrp_axioms_def 

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locale rev_lgrp = semi "%x y. y ++ x" for sum (infixl "++" 60) 

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print_locale! rev_lgrp thm rev_lgrp_def 

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locale hom = f: semi f + g: semi g for f and g 

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print_locale! hom thm hom_def 

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locale perturbation = semi + d: semi "%x y. delta(x) ** delta(y)" for delta 

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print_locale! perturbation thm perturbation_def 

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locale pert_hom = d1: perturbation f d1 + d2: perturbation f d2 for f d1 d2 

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print_locale! pert_hom thm pert_hom_def 

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text {* Alternative expression, obtaining nicer names in @{text "semi f"}. *} 

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locale pert_hom' = semi f + d1: perturbation f d1 + d2: perturbation f d2 for f d1 d2 

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print_locale! pert_hom' thm pert_hom'_def 

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text {* Syntax declarations *} 
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locale logic = 

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fixes land (infixl "&&" 55) 

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and lnot (" _" [60] 60) 

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assumes assoc: "(x && y) && z = x && (y && z)" 

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and notnot: " ( x) = x" 

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begin 

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definition lor (infixl "" 50) where 

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"x  y = ( x &&  y)" 

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end 

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print_locale! logic 

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locale use_decl = logic + semi "op " 

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print_locale! use_decl thm use_decl_def 

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text {* Theorem statements *} 

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lemma (in lgrp) lcancel: 
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"x ** y = x ** z <> y = z" 

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proof 

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assume "x ** y = x ** z" 

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then have "inv(x) ** x ** y = inv(x) ** x ** z" by (simp add: assoc) 

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then show "y = z" by (simp add: lone linv) 

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qed simp 

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print_locale! lgrp 
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locale rgrp = semi + 

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fixes one and inv 

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assumes rone: "x ** one = x" 

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and rinv: "x ** inv(x) = one" 

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begin 
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lemma rcancel: 
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"y ** x = z ** x <> y = z" 
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proof 

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assume "y ** x = z ** x" 

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then have "y ** (x ** inv(x)) = z ** (x ** inv(x))" 

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by (simp add: assoc [symmetric]) 

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then show "y = z" by (simp add: rone rinv) 

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qed simp 

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end 

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print_locale! rgrp 
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text {* Patterns *} 

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lemma (in rgrp) 

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assumes "y ** x = z ** x" (is ?a) 

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shows "y = z" (is ?t) 

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proof  

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txt {* Weird proof involving patterns from context element and conclusion. *} 

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{ 

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assume ?a 

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then have "y ** (x ** inv(x)) = z ** (x ** inv(x))" 

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by (simp add: assoc [symmetric]) 

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then have ?t by (simp add: rone rinv) 

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} 

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note x = this 

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show ?t by (rule x [OF `?a`]) 

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qed 

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text {* Interpretation between locales: sublocales *} 

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sublocale lgrp < right: rgrp 

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print_facts 

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proof unfold_locales 
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{ 
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fix x 

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have "inv(x) ** x ** one = inv(x) ** x" by (simp add: linv lone) 

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then show "x ** one = x" by (simp add: assoc lcancel) 

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} 

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note rone = this 

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{ 

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fix x 

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have "inv(x) ** x ** inv(x) = inv(x) ** one" 

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by (simp add: linv lone rone) 

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then show "x ** inv(x) = one" by (simp add: assoc lcancel) 

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} 

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qed 
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(* effect on printed locale *) 

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print_locale! lgrp 

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(* use of derived theorem *) 

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lemma (in lgrp) 

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"y ** x = z ** x <> y = z" 

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apply (rule rcancel) 

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done 

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(* circular interpretation *) 

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sublocale rgrp < left: lgrp 

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proof unfold_locales 
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{ 
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fix x 
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have "one ** (x ** inv(x)) = x ** inv(x)" by (simp add: rinv rone) 
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then show "one ** x = x" by (simp add: assoc [symmetric] rcancel) 
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} 
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note lone = this 
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{ 
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fix x 
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have "inv(x) ** (x ** inv(x)) = one ** inv(x)" 
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by (simp add: rinv lone rone) 
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then show "inv(x) ** x = one" by (simp add: assoc [symmetric] rcancel) 
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} 
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qed 
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(* effect on printed locale *) 

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print_locale! rgrp 

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print_locale! lgrp 

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(* Duality *) 
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locale order = 
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fixes less :: "'a => 'a => o" (infix "<<" 50) 
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assumes refl: "x << x" 
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and trans: "[ x << y; y << z ] ==> x << z" 
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sublocale order < dual: order "%x y. y << x" 
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apply unfold_locales apply (rule refl) apply (blast intro: trans) 
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done 
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print_locale! order (* Only two instances of order. *) 
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locale order' = 
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fixes less :: "'a => 'a => o" (infix "<<" 50) 
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assumes refl: "x << x" 
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and trans: "[ x << y; y << z ] ==> x << z" 
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locale order_with_def = order' 
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begin 
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definition greater :: "'a => 'a => o" (infix ">>" 50) where 
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"x >> y <> y << x" 
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end 
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sublocale order_with_def < dual: order' "op >>" 
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apply unfold_locales 
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unfolding greater_def 
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apply (rule refl) apply (blast intro: trans) 
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done 
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print_locale! order_with_def 
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(* Note that decls come after theorems that make use of them. 
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Appears to be harmless at least in this example. *) 
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(* locale with many parameters  
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interpretations generate alternating group A5 *) 
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locale A5 = 
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fixes A and B and C and D and E 
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assumes eq: "A <> B <> C <> D <> E" 
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sublocale A5 < 1: A5 _ _ D E C 
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print_facts 
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using eq apply (blast intro: A5.intro) done 
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sublocale A5 < 2: A5 C _ E _ A 
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print_facts 
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using eq apply (blast intro: A5.intro) done 
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sublocale A5 < 3: A5 B C A _ _ 
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print_facts 
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using eq apply (blast intro: A5.intro) done 
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(* Any even permutation of parameters is subsumed by the above. *) 
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print_locale! A5 
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(* Free arguments of instance *) 
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locale trivial = 
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fixes P and Q :: o 
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assumes Q: "P <> P <> Q" 
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begin 
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lemma Q_triv: "Q" using Q by fast 
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end 
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sublocale trivial < x: trivial x _ 
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apply unfold_locales using Q by fast 
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print_locale! trivial 
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context trivial begin thm x.Q [where ?x = True] end 
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sublocale trivial < y: trivial Q Q 
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by unfold_locales 
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(* Succeeds since previous interpretation is more general. *) 
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print_locale! trivial (* No instance for y created (subsumed). *) 
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end 