diff -r 6a19d9f6021d -r ff4ba1ed4527 src/FOL/ex/NewLocaleTest.thy --- a/src/FOL/ex/NewLocaleTest.thy Tue Jan 06 09:03:37 2009 -0800 +++ /dev/null Thu Jan 01 00:00:00 1970 +0000 @@ -1,486 +0,0 @@ -(* Title: FOL/ex/NewLocaleTest.thy - Author: Clemens Ballarin, TU Muenchen - -Testing environment for locale expressions --- experimental. -*) - -theory NewLocaleTest -imports NewLocaleSetup -begin - -ML_val {* set Toplevel.debug *} - - -typedecl int arities int :: "term" -consts plus :: "int => int => int" (infixl "+" 60) - zero :: int ("0") - minus :: "int => int" ("- _") - -axioms - int_assoc: "(x + y::int) + z = x + (y + z)" - int_zero: "0 + x = x" - int_minus: "(-x) + x = 0" - int_minus2: "-(-x) = x" - -section {* Inference of parameter types *} - -locale param1 = fixes p -print_locale! param1 - -locale param2 = fixes p :: 'b -print_locale! param2 - -(* -locale param_top = param2 r for r :: "'b :: {}" - Fails, cannot generalise parameter. -*) - -locale param3 = fixes p (infix ".." 50) -print_locale! param3 - -locale param4 = fixes p :: "'a => 'a => 'a" (infix ".." 50) -print_locale! param4 - - -subsection {* Incremental type constraints *} - -locale constraint1 = - fixes prod (infixl "**" 65) - assumes l_id: "x ** y = x" - assumes assoc: "(x ** y) ** z = x ** (y ** z)" -print_locale! constraint1 - -locale constraint2 = - fixes p and q - assumes "p = q" -print_locale! constraint2 - - -section {* Inheritance *} - -locale semi = - fixes prod (infixl "**" 65) - assumes assoc: "(x ** y) ** z = x ** (y ** z)" -print_locale! semi thm semi_def - -locale lgrp = semi + - fixes one and inv - assumes lone: "one ** x = x" - and linv: "inv(x) ** x = one" -print_locale! lgrp thm lgrp_def lgrp_axioms_def - -locale add_lgrp = semi "op ++" for sum (infixl "++" 60) + - fixes zero and neg - assumes lzero: "zero ++ x = x" - and lneg: "neg(x) ++ x = zero" -print_locale! add_lgrp thm add_lgrp_def add_lgrp_axioms_def - -locale rev_lgrp = semi "%x y. y ++ x" for sum (infixl "++" 60) -print_locale! rev_lgrp thm rev_lgrp_def - -locale hom = f: semi f + g: semi g for f and g -print_locale! hom thm hom_def - -locale perturbation = semi + d: semi "%x y. delta(x) ** delta(y)" for delta -print_locale! perturbation thm perturbation_def - -locale pert_hom = d1: perturbation f d1 + d2: perturbation f d2 for f d1 d2 -print_locale! pert_hom thm pert_hom_def - -text {* Alternative expression, obtaining nicer names in @{text "semi f"}. *} -locale pert_hom' = semi f + d1: perturbation f d1 + d2: perturbation f d2 for f d1 d2 -print_locale! pert_hom' thm pert_hom'_def - - -section {* Syntax declarations *} - -locale logic = - fixes land (infixl "&&" 55) - and lnot ("-- _" [60] 60) - assumes assoc: "(x && y) && z = x && (y && z)" - and notnot: "-- (-- x) = x" -begin - -definition lor (infixl "||" 50) where - "x || y = --(-- x && -- y)" - -end -print_locale! logic - -locale use_decl = logic + semi "op ||" -print_locale! use_decl thm use_decl_def - -locale extra_type = - fixes a :: 'a - and P :: "'a => 'b => o" -begin - -definition test :: "'a => o" where - "test(x) <-> (ALL b. P(x, b))" - -end - -term extra_type.test thm extra_type.test_def - -interpretation var: extra_type "0" "%x y. x = 0" . - -thm var.test_def - - -section {* Foundational versions of theorems *} - -thm logic.assoc -thm logic.lor_def - - -section {* Defines *} - -locale logic_def = - fixes land (infixl "&&" 55) - and lor (infixl "||" 50) - and lnot ("-- _" [60] 60) - assumes assoc: "(x && y) && z = x && (y && z)" - and notnot: "-- (-- x) = x" - defines "x || y == --(-- x && -- y)" -begin - -thm lor_def -(* Can we get rid the the additional hypothesis, caused by LocalTheory.notes? *) - -lemma "x || y = --(-- x && --y)" - by (unfold lor_def) (rule refl) - -end - -(* Inheritance of defines *) - -locale logic_def2 = logic_def -begin - -lemma "x || y = --(-- x && --y)" - by (unfold lor_def) (rule refl) - -end - - -section {* Notes *} - -(* A somewhat arcane homomorphism example *) - -definition semi_hom where - "semi_hom(prod, sum, h) <-> (ALL x y. h(prod(x, y)) = sum(h(x), h(y)))" - -lemma semi_hom_mult: - "semi_hom(prod, sum, h) ==> h(prod(x, y)) = sum(h(x), h(y))" - by (simp add: semi_hom_def) - -locale semi_hom_loc = prod: semi prod + sum: semi sum - for prod and sum and h + - assumes semi_homh: "semi_hom(prod, sum, h)" - notes semi_hom_mult = semi_hom_mult [OF semi_homh] - -thm semi_hom_loc.semi_hom_mult -(* unspecified, attribute not applied in backgroud theory !!! *) - -lemma (in semi_hom_loc) "h(prod(x, y)) = sum(h(x), h(y))" - by (rule semi_hom_mult) - -(* Referring to facts from within a context specification *) - -lemma - assumes x: "P <-> P" - notes y = x - shows True .. - - -section {* Theorem statements *} - -lemma (in lgrp) lcancel: - "x ** y = x ** z <-> y = z" -proof - assume "x ** y = x ** z" - then have "inv(x) ** x ** y = inv(x) ** x ** z" by (simp add: assoc) - then show "y = z" by (simp add: lone linv) -qed simp -print_locale! lgrp - - -locale rgrp = semi + - fixes one and inv - assumes rone: "x ** one = x" - and rinv: "x ** inv(x) = one" -begin - -lemma rcancel: - "y ** x = z ** x <-> y = z" -proof - assume "y ** x = z ** x" - then have "y ** (x ** inv(x)) = z ** (x ** inv(x))" - by (simp add: assoc [symmetric]) - then show "y = z" by (simp add: rone rinv) -qed simp - -end -print_locale! rgrp - - -subsection {* Patterns *} - -lemma (in rgrp) - assumes "y ** x = z ** x" (is ?a) - shows "y = z" (is ?t) -proof - - txt {* Weird proof involving patterns from context element and conclusion. *} - { - assume ?a - then have "y ** (x ** inv(x)) = z ** (x ** inv(x))" - by (simp add: assoc [symmetric]) - then have ?t by (simp add: rone rinv) - } - note x = this - show ?t by (rule x [OF `?a`]) -qed - - -section {* Interpretation between locales: sublocales *} - -sublocale lgrp < right: rgrp -print_facts -proof unfold_locales - { - fix x - have "inv(x) ** x ** one = inv(x) ** x" by (simp add: linv lone) - then show "x ** one = x" by (simp add: assoc lcancel) - } - note rone = this - { - fix x - have "inv(x) ** x ** inv(x) = inv(x) ** one" - by (simp add: linv lone rone) - then show "x ** inv(x) = one" by (simp add: assoc lcancel) - } -qed - -(* effect on printed locale *) - -print_locale! lgrp - -(* use of derived theorem *) - -lemma (in lgrp) - "y ** x = z ** x <-> y = z" - apply (rule rcancel) - done - -(* circular interpretation *) - -sublocale rgrp < left: lgrp -proof unfold_locales - { - fix x - have "one ** (x ** inv(x)) = x ** inv(x)" by (simp add: rinv rone) - then show "one ** x = x" by (simp add: assoc [symmetric] rcancel) - } - note lone = this - { - fix x - have "inv(x) ** (x ** inv(x)) = one ** inv(x)" - by (simp add: rinv lone rone) - then show "inv(x) ** x = one" by (simp add: assoc [symmetric] rcancel) - } -qed - -(* effect on printed locale *) - -print_locale! rgrp -print_locale! lgrp - - -(* Duality *) - -locale order = - fixes less :: "'a => 'a => o" (infix "<<" 50) - assumes refl: "x << x" - and trans: "[| x << y; y << z |] ==> x << z" - -sublocale order < dual: order "%x y. y << x" - apply unfold_locales apply (rule refl) apply (blast intro: trans) - done - -print_locale! order (* Only two instances of order. *) - -locale order' = - fixes less :: "'a => 'a => o" (infix "<<" 50) - assumes refl: "x << x" - and trans: "[| x << y; y << z |] ==> x << z" - -locale order_with_def = order' -begin - -definition greater :: "'a => 'a => o" (infix ">>" 50) where - "x >> y <-> y << x" - -end - -sublocale order_with_def < dual: order' "op >>" - apply unfold_locales - unfolding greater_def - apply (rule refl) apply (blast intro: trans) - done - -print_locale! order_with_def -(* Note that decls come after theorems that make use of them. *) - - -(* locale with many parameters --- - interpretations generate alternating group A5 *) - - -locale A5 = - fixes A and B and C and D and E - assumes eq: "A <-> B <-> C <-> D <-> E" - -sublocale A5 < 1: A5 _ _ D E C -print_facts - using eq apply (blast intro: A5.intro) done - -sublocale A5 < 2: A5 C _ E _ A -print_facts - using eq apply (blast intro: A5.intro) done - -sublocale A5 < 3: A5 B C A _ _ -print_facts - using eq apply (blast intro: A5.intro) done - -(* Any even permutation of parameters is subsumed by the above. *) - -print_locale! A5 - - -(* Free arguments of instance *) - -locale trivial = - fixes P and Q :: o - assumes Q: "P <-> P <-> Q" -begin - -lemma Q_triv: "Q" using Q by fast - -end - -sublocale trivial < x: trivial x _ - apply unfold_locales using Q by fast - -print_locale! trivial - -context trivial begin thm x.Q [where ?x = True] end - -sublocale trivial < y: trivial Q Q - by unfold_locales - (* Succeeds since previous interpretation is more general. *) - -print_locale! trivial (* No instance for y created (subsumed). *) - - -subsection {* Sublocale, then interpretation in theory *} - -interpretation int: lgrp "op +" "0" "minus" -proof unfold_locales -qed (rule int_assoc int_zero int_minus)+ - -thm int.assoc int.semi_axioms - -interpretation int2: semi "op +" - by unfold_locales (* subsumed, thm int2.assoc not generated *) - -thm int.lone int.right.rone - (* the latter comes through the sublocale relation *) - - -subsection {* Interpretation in theory, then sublocale *} - -interpretation (* fol: *) logic "op +" "minus" -(* FIXME declaration of qualified names *) - by unfold_locales (rule int_assoc int_minus2)+ - -locale logic2 = - fixes land (infixl "&&" 55) - and lnot ("-- _" [60] 60) - assumes assoc: "(x && y) && z = x && (y && z)" - and notnot: "-- (-- x) = x" -begin - -(* FIXME interpretation below fails -definition lor (infixl "||" 50) where - "x || y = --(-- x && -- y)" -*) - -end - -sublocale logic < two: logic2 - by unfold_locales (rule assoc notnot)+ - -thm two.assoc - - -subsection {* Declarations and sublocale *} - -locale logic_a = logic -locale logic_b = logic - -sublocale logic_a < logic_b - by unfold_locales - - -subsection {* Equations *} - -locale logic_o = - fixes land (infixl "&&" 55) - and lnot ("-- _" [60] 60) - assumes assoc_o: "(x && y) && z <-> x && (y && z)" - and notnot_o: "-- (-- x) <-> x" -begin - -definition lor_o (infixl "||" 50) where - "x || y <-> --(-- x && -- y)" - -end - -interpretation x!: logic_o "op &" "Not" - where bool_logic_o: "logic_o.lor_o(op &, Not, x, y) <-> x | y" -proof - - show bool_logic_o: "PROP logic_o(op &, Not)" by unfold_locales fast+ - show "logic_o.lor_o(op &, Not, x, y) <-> x | y" - by (unfold logic_o.lor_o_def [OF bool_logic_o]) fast -qed - -thm x.lor_o_def bool_logic_o - -lemma lor_triv: "z <-> z" .. - -lemma (in logic_o) lor_triv: "x || y <-> x || y" by fast - -thm lor_triv [where z = True] (* Check strict prefix. *) - x.lor_triv - - -subsection {* Interpretation in proofs *} - -lemma True -proof - interpret "local": lgrp "op +" "0" "minus" - by unfold_locales (* subsumed *) - { - fix zero :: int - assume "!!x. zero + x = x" "!!x. (-x) + x = zero" - then interpret local_fixed: lgrp "op +" zero "minus" - by unfold_locales - thm local_fixed.lone - } - assume "!!x zero. zero + x = x" "!!x zero. (-x) + x = zero" - then interpret local_free: lgrp "op +" zero "minus" for zero - by unfold_locales - thm local_free.lone [where ?zero = 0] -qed - -end