hcju/isabelleps huggingface dataset:
tauInduct: fixes P :: "pi"   and P' :: "pi"   and F :: "'a \<Rightarrow> pi \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'a" assumes "P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'"   and "\<And>P C. F C (\<tau>.P) P"   and "\<And>P P' a C. \<lbrakk>P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'; \<And>C. F C P P'\<rbrakk> \<Longrightarrow> F C ([a\<frown>a]P) P'"   and "\<And>P P' a b C. \<lbrakk>P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'; \<And>C. F C P P'; a \<noteq> b\<rbrakk> \<Longrightarrow> F C ([a\<noteq>b]P) P'"   and "\<And>P P' Q C. \<lbrakk>P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'; \<And>C. F C P P'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) P'"   and "\<And>Q Q' P C. \<lbrakk>Q \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e Q'; \<And>C. F C Q Q'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) Q'"   and "\<And>P P' Q C. \<lbrakk>P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'; \<And>C. F C P P'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (P' \<parallel> Q)"   and "\<And>Q Q' P C. \<lbrakk>Q \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e Q'; \<And>C. F C Q Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (P \<parallel> Q')"   and "\<And>P a b P' Q Q' C. \<lbrakk>P \<longmapsto>\<^sub>e a<b> \<prec>\<^sub>e P'; Q \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (P' \<parallel> Q')"   and "\<And>P a b P' Q Q' C. \<lbrakk>P \<longmapsto>\<^sub>e a[b] \<prec>\<^sub>e P'; Q \<longmapsto>\<^sub>e a<b> \<prec>\<^sub>e Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (P' \<parallel> Q')"   and "\<And>P a x P' Q Q' C. \<lbrakk>P \<longmapsto>\<^sub>e a<x> \<prec>\<^sub>e P'; Q \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e Q'; x \<sharp> P; x \<sharp> Q; x \<noteq> a; x \<sharp> C\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (<\<nu>x>(P' \<parallel> Q'))"   and "\<And>P a x P' Q Q' C. \<lbrakk>P \<longmapsto>\<^sub>e a<\<nu>x> \<prec>\<^sub>e P'; Q \<longmapsto>\<^sub>e a<x> \<prec>\<^sub>e Q'; x \<sharp> P; x \<sharp> Q; x \<noteq> a; x \<sharp> C\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) (<\<nu>x>(P' \<parallel> Q'))"   and "\<And>P P' x C. \<lbrakk>P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'; x \<sharp> C; \<And>C. F C P P'\<rbrakk> \<Longrightarrow> F C (<\<nu>x>P) (<\<nu>x>P')"   and "\<And>P P' C. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>e \<tau> \<prec>\<^sub>e P'; \<And>C. F C (P \<parallel> !P) P'\<rbrakk> \<Longrightarrow> F C (!P) P'" shows "F C P P'"
conjugate_square_eq_0_vec: fixes v :: "'b vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "(v \<bullet>c v = (0::'b)) = (v = 0\<^sub>v n)"
has_laplace_add: fixes f :: "real \<Rightarrow> complex"   and F :: "complex"   and S :: "complex"   and g :: "real \<Rightarrow> complex"   and G :: "complex" assumes "(f has_laplace F) S"   and "(g has_laplace G) S" shows "((\<lambda>x. f x + g x) has_laplace F + G) S"
WT_elim_cases_6: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and F :: "char list"   and Cs :: "char list list"   and T :: "ty" assumes "P,E \<turnstile> e\<bullet>F{Cs} :: T" obtains C :: "char list" where "P,E \<turnstile> e :: Class C" and "P \<turnstile> C has least F:T via Cs"
functor_axioms: shows "functor (\<odot>) S.comp HomC.map"
transrec3_0: fixes a :: "V"   and b :: "V \<Rightarrow> V \<Rightarrow> V"   and c :: "V \<Rightarrow> (V \<Rightarrow> V) \<Rightarrow> V" shows "transrec3 a b c 0 = a"
Tarskian_logic_axioms: shows "Tarskian_logic Cn"
in_eqvt: fixes pi :: "(atom \<times> atom) list"   and x :: "'a"   and X :: "'a set" shows "pi \<bullet> (x \<in> X) = (pi \<bullet> x \<in> pi \<bullet> X)"
zeros_len: fixes n :: "nat"   and v :: "nat" shows "length (zeros n v) = n"
borel_eq_sigmaI4: fixes G :: "'d \<Rightarrow> 'e \<Rightarrow> 'c set"   and A :: "('d \<times> 'e) set"   and F :: "'f \<Rightarrow> 'c set" assumes "borel = sigma UNIV ((\<lambda>(i, j). G i j) ` A)"   and "\<And>i j. (i, j) \<in> A \<Longrightarrow> G i j \<in> sets (sigma UNIV (range F))"   and "\<And>i. F i \<in> sets borel" shows "borel = sigma UNIV (range F)"
intro: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and partition :: "'b set multiset multiset" assumes "incidence_system.resolution point_set block_collection partition" shows "resolvable_design_axioms point_set block_collection partition"
twiddleStore_def: fixes \<beta> :: "(Location \<times> Location) set"   and s1 :: "Var \<Rightarrow> Val"   and s2 :: "Var \<Rightarrow> Val" shows "(s1  \<approx>\<^sub>\<beta> s2) = (\<forall>x. CONTEXT x = low \<longrightarrow> (\<beta>, s1 x, s2 x) \<in> twiddleVal)"
all_types_next_eqiv: fixes P :: "latin_type \<Rightarrow> bool" shows "(\<forall>t. P (next t)) = (\<forall>t. P t)"
left: fixes x :: "'a"   and y :: "'b" assumes "x \<in> carrier \<X>"   and "y \<in> carrier \<Y>"   and "x \<sqsubseteq>\<^bsub>\<X>\<^esub> \<pi>\<^sub>* y" shows "\<pi>\<^sup>* x \<sqsubseteq>\<^bsub>\<Y>\<^esub> y"
set_empty_eq: fixes xo :: "'b option" shows "(set_option xo = {}) = (xo = None)"
con_comp_ipurge_tr_aux: fixes U :: "'e option set"   and xs :: "'f list"   and p :: "'g \<Rightarrow> 'f"   and I :: "('e \<times> 'e) set"   and D :: "'g \<Rightarrow> 'e"   and E :: "'h \<Rightarrow> 'e"   and q :: "'h \<Rightarrow> 'f" assumes "U \<subseteq> range Some"   and "set xs \<subseteq> range p" shows "ipurge_tr_aux I D (the ` U) (map (inv p) xs) = map (inv p) (ipurge_tr_aux (con_comp_pol I) (con_comp_map D E p q) U xs)"
add_0_right: fixes a :: "'b" shows "a + (0::'b) = a"
apex_def: shows "\<nu>.apex \<equiv> C.dom \<nu>.leg0"
one_add_one: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b" assumes "class.numeral plus" shows "plus one one = numeral.numeral one plus (num.Bit0 num.One)"
gen_lset_def: fixes A :: "'c set"   and xs :: "'c llist" shows "gen_lset A xs = A \<union> lset xs"
odd_add: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "(\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (plus a b)) = ((\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a) \<noteq> (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) b))"
verit_minus_simplify_3: fixes b :: "'d" shows "(0::'d) - b = - b"
forest_mono: fixes E :: "'c set"   and V :: "'d set"   and vertices :: "'c \<Rightarrow> 'd set"   and joins :: "'d \<Rightarrow> 'd \<Rightarrow> 'c \<Rightarrow> bool"   and forest :: "'c set \<Rightarrow> bool"   and connected :: "'c set \<Rightarrow> ('d \<times> 'd) set"   and X :: "'c set"   and Y :: "'c set" assumes "Kruskal_interface E V vertices joins forest connected"   and "forest X"   and "Y \<subseteq> X" shows "forest Y"
assms: shows "WT_rest \<I>_adv \<I>_usr I_rest rest"
equal_size: shows "length cms\<^sub>1 = length cms\<^sub>2"
exhaust: fixes y :: "('b, 'c) fm" obtains x1 :: "'b" where "y = Pro x1"   | x2 :: "'c" where "y = Nom x2"   | x3 :: "('b, 'c) fm" where "y = (\<^bold>\<not> x3)"   | x41 :: "('b, 'c) fm" and x42 :: "('b, 'c) fm" where "y = (x41 \<^bold>\<or> x42)"   | x5 :: "('b, 'c) fm" where "y = (\<^bold>\<diamond> x5)"   | x61 :: "'c" and x62 :: "('b, 'c) fm" where "y = (\<^bold>@ x61 x62)"
frac_idempotent: fixes x :: "'d" shows "frac (frac x) = frac x"
unique_fun_value: fixes f :: "ZF"   and x :: "ZF" assumes "isFun f"   and "x |\<in>| HOLZF.Domain f" shows "\<exists>!y. Opair x y |\<in>| f"
induct: fixes P :: "v sem_env \<Rightarrow> 'a state \<Rightarrow> exp0 \<Rightarrow> bool"   and a0 :: "v sem_env"   and a1 :: "'a state"   and a2 :: "exp0" assumes "\<And>env s l. P env s (Lit l)"   and "\<And>env s e. P env s e \<Longrightarrow> P env s (Raise e)"   and "\<And>env s e pes. \<lbrakk>P env s e; \<And>x y x2 x1 x1a xa ya. \<lbrakk>(x, y) = Evaluate_Single.evaluate env s e; y = Rerr x2; x2 = Rraise x1; match_result env x x1 pes x1 = Rval x1a; (xa, ya) = x1a\<rbrakk> \<Longrightarrow> P (update_v (\<lambda>_. nsAppend (alist_to_ns ya) (sem_env.v env)) env) x xa\<rbrakk> \<Longrightarrow> P env s (Handle e pes)"   and "\<And>env s cn es. (\<And>e sa. \<lbrakk>do_con_check (c env) cn (length es); e \<in> set (rev es); clock sa \<le> clock s\<rbrakk> \<Longrightarrow> P env sa e) \<Longrightarrow> P env s (Con cn es)"   and "\<And>env s n. P env s (Var n)"   and "\<And>env s n e. P env s (Fun n e)"   and "\<And>env s op0 es. \<lbrakk>\<And>e sa. \<lbrakk>e \<in> set (rev es); clock sa \<le> clock s\<rbrakk> \<Longrightarrow> P env sa e; \<And>x y x1 x2 xa ya. \<lbrakk>(x, y) = Evaluate_Single.evaluate_list (Evaluate_Single.evaluate env) s (rev es); y = Rval x1; op0 = Opapp; do_opapp (rev x1) = Some x2; (xa, ya) = x2; clock x \<noteq> 0\<rbrakk> \<Longrightarrow> P xa (dec_clock x) ya\<rbrakk> \<Longrightarrow> P env s (App op0 es)"   and "\<And>env s op0 e1 e2. \<lbrakk>P env s e1; \<And>x y x1 x2 x1a. \<lbrakk>(x, y) = Evaluate_Single.evaluate env s e1; y = Rval x1; do_log op0 x1 e2 = Some x2; x2 = Exp x1a\<rbrakk> \<Longrightarrow> P env x x1a\<rbrakk> \<Longrightarrow> P env s (Log op0 e1 e2)"   and "\<And>env s e1 e2 e3. \<lbrakk>P env s e1; \<And>x y x1 x2. \<lbrakk>(x, y) = Evaluate_Single.evaluate env s e1; y = Rval x1; do_if x1 e2 e3 = Some x2\<rbrakk> \<Longrightarrow> P env x x2\<rbrakk> \<Longrightarrow> P env s (exp0.If e1 e2 e3)"   and "\<And>env s e pes. \<lbrakk>P env s e; \<And>x y x1 x1a xa ya. \<lbrakk>(x, y) = Evaluate_Single.evaluate env s e; y = Rval x1; match_result env x x1 pes Bindv = Rval x1a; (xa, ya) = x1a\<rbrakk> \<Longrightarrow> P (update_v (\<lambda>_. nsAppend (alist_to_ns ya) (sem_env.v env)) env) x xa\<rbrakk> \<Longrightarrow> P env s (Mat e pes)"   and "\<And>env s n e1 e2. \<lbrakk>P env s e1; \<And>x y x1. \<lbrakk>(x, y) = Evaluate_Single.evaluate env s e1; y = Rval x1\<rbrakk> \<Longrightarrow> P (update_v (\<lambda>_. nsOptBind n x1 (sem_env.v env)) env) x e2\<rbrakk> \<Longrightarrow> P env s (exp0.Let n e1 e2)"   and "\<And>env s funs e. (distinct (map (\<lambda>(x, y, z). x) funs) \<Longrightarrow> P (update_v (\<lambda>_. build_rec_env funs env (sem_env.v env)) env) s e) \<Longrightarrow> P env s (Letrec funs e)"   and "\<And>env s e t0. P env s e \<Longrightarrow> P env s (Tannot e t0)"   and "\<And>env s e l. P env s e \<Longrightarrow> P env s (Lannot e l)" shows "P a0 a1 a2"
finite_roots_unity: fixes n :: "nat" assumes "1 \<le> n" shows "finite {z. z ^ n = (1::'a)}"
mark_guards_simps_9: fixes f :: "'f"   and f' :: "'g"   and g :: "'d set"   and c :: "('d, 'e, 'g) com" shows "mark_guards f (Guard f' g c) = Guard f g (mark_guards f c)"
simps_235: fixes f :: "'c \<Rightarrow> 'b"   and x131 :: "'c ltlc"   and x132 :: "'c ltlc" shows "map_ltlc f (x131 W\<^sub>c x132) = map_ltlc f x131 W\<^sub>c map_ltlc f x132"
mat_adjoint_def: fixes A :: "'a mat" shows "mat_adjoint A \<equiv> mat_of_rows (dim_row A) (map conjugate (cols A))"
tm_comp_exec_after_first: fixes c :: "nat \<times> cell list \<times> cell list"   and A :: "(action \<times> nat) list"   and tp :: "cell list \<times> cell list"   and B :: "(action \<times> nat) list" assumes "\<not> is_final c"   and "step0 c A = (0, tp)"   and "fst c \<le> length A div 2" shows "step0 c (A |+| B) = (Suc (length A div 2), tp)"
xmy: shows "\<forall>w. P w (x \<ominus> y) = (P w x \<and> \<not> O w y)"
assms_8: fixes y :: "'b" assumes "y \<in> T" shows "f (g y) = y"
hyps: assumes "x < y in s1 = x < y in s2" shows "x < y in swaps ss s1 = x < y in (swap 0 ^^ ALG_P ss x y s1) s2"
intro: fixes G :: "('a, 'b) monoid_scheme"   and p :: "nat"   and a :: "nat"   and m :: "nat"   and M1 :: "'a set"   and M :: "'a set set" assumes "sylow G p a m"   and "sylow_central_axioms G p a m M1 M" shows "sylow_central G p a m M1 M"
lemma_4_5_b: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a \<le> c"   and "b \<le> c" shows "a \<squnion>2 b \<le> c"
isnpolyh_zero_iff: fixes p :: "poly"   and n0 :: "nat" assumes "isnpolyh p n0"   and "\<forall>bs. wf_bs bs p \<longrightarrow> \<lparr>p\<rparr>\<^sub>p\<^bsup>bs\<^esup> = (0::'a)" shows "p = 0\<^sub>p"
isomorphic_trans: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and T :: "'c \<Rightarrow> 'd"   and W :: "'d set"   and smult'' :: "('b, 'a) aezfun \<Rightarrow> 'e \<Rightarrow> 'e"   and X :: "'e set" assumes "FGModuleIso G smult V smult' T W"   and "\<exists>T. FGModuleIso G smult' W smult'' T X" shows "\<exists>T. FGModuleIso G smult V smult'' T X"
G'_3: shows "\<omega> = \<omega>\<^bsub>G'\<^esub>"
eval_def: shows "eval \<equiv> {((xa, x), xc, xb). evalp xa x xc xb}"
strand_sem_append'_2: fixes M :: "('fun, 'var) Term.term set"   and S :: "('fun, 'var) strand_step list"   and \<theta> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and S' :: "('fun, 'var) strand_step list" assumes "\<lbrakk>M; S\<rbrakk>\<^sub>d' \<theta>"   and "\<lbrakk>M \<union> ik\<^sub>s\<^sub>t S; S'\<rbrakk>\<^sub>d' \<theta>" shows "\<lbrakk>M; S @ S'\<rbrakk>\<^sub>d' \<theta>"
f_def: fixes x :: "real" shows "f x = a * cos x + b * sin x + c"
substitution_def: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr" shows "substitution subst \<equiv> (\<forall>\<alpha> \<beta> e x e'. \<R>\<^sub>E \<alpha> \<beta> (subst e x e') = subst (\<R>\<^sub>E \<alpha> \<beta> e) x (\<R>\<^sub>E \<alpha> \<beta> e')) \<and> (\<forall>e x e'. RID\<^sub>E (subst e x e') \<subseteq> RID\<^sub>E e \<union> RID\<^sub>E e') \<and> (\<forall>e x e'. LID\<^sub>E (subst e x e') \<subseteq> LID\<^sub>E e \<union> LID\<^sub>E e')"
hvzk_z_r: fixes r1 :: "nat"   and c :: "nat"   and x1 :: "nat" assumes "r1 < order \<G>" shows "r1 = ((r1 + c * x1) mod order \<G> + order \<G> * c * x1 - c * x1) mod order \<G>"
assms_2: fixes u :: "'a" assumes "u \<in> basis" shows "f *\<^sub>V u = g *\<^sub>V u"
sets_kernel: fixes K :: "'b \<Rightarrow> 'c measure"   and M :: "'b measure"   and N :: "'c measure"   and a :: "'b" assumes "K \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "a \<in> space M" shows "sets (K a) = sets N"
bl_to_bin_app_cat: fixes bsa :: "bool list"   and bs :: "bool list" shows "bl_to_bin (bsa @ bs) = concat_bit (length bs) (bl_to_bin bs) (bl_to_bin bsa)"
prob_space_distr: fixes f :: "'a \<Rightarrow> 'b"   and M' :: "'b measure" assumes "random_variable M' f" shows "prob_space (distr M M' f)"
qOp_alpha_iff: fixes delta :: "'h"   and inp :: "'d \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm option"   and binp :: "'e \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qAbs option"   and X' :: "('d, 'e, 'f, 'g, 'h) qTerm" shows "(qOp delta inp binp #= X') = (\<exists>inp' binp'. X' = qOp delta inp' binp' \<and> sameDom inp inp' \<and> sameDom binp binp' \<and> liftAll2 (#=) inp inp' \<and> liftAll2 ($=) binp binp')"
fold_deriv_FEx: fixes d0 :: "'x \<Rightarrow> 'a \<Rightarrow> ('a, 'k) aformula"   and k :: "'k"   and \<phi> :: "('a, 'k) aformula" shows "{\<langle>fold (deriv d0) xs (FEx k \<phi>)\<rangle> |xs. True} \<subseteq> {FEx k \<psi> |\<psi>. nf_ACI \<psi> \<and> disjuncts \<psi> \<subseteq> (\<Union>xs. disjuncts \<langle>fold (deriv d0) xs \<phi>\<rangle>)}"
Pn_def: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state"   and n :: "nat" shows "Pn cf n \<equiv> T.prob cf {bT \<in> space (T cf). N cf n bT}"
r\<^sub>\<psi>_def: shows "r\<^sub>\<psi> = run\<^sub>t \<MM>.\<delta>\<^sub>\<R> \<MM>.q\<^sub>\<R> w"
matches_add_match_MatchNot_no_matching_Goto_simp: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and rs :: "'c rule list" assumes "\<not> matches \<gamma> m p" shows "no_matching_Goto \<gamma> p (add_match m rs)"
mat_tensor_prod_2_prelim: fixes v :: "complex Matrix.mat"   and w :: "complex Matrix.mat" assumes "state 1 v"   and "state 1 w" shows "v \<Otimes> w = Tensor.mat_of_cols_list 4 [[v $$ (0, 0) * w $$ (0, 0), v $$ (0, 0) * w $$ (1, 0), v $$ (1, 0) * w $$ (0, 0), v $$ (1, 0) * w $$ (1, 0)]]"
lead_coeff_in_carrier: fixes K :: "'a set"   and a :: "'a"   and p :: "'a list" assumes "subring K R"   and "polynomial K (a # p)" shows "a \<in> carrier R - {\<zero>}"
numeral_ne_\<omega>: fixes n :: "num" shows "numeral n \<noteq> \<omega>"
split_sign_changes_correct: fixes x\<^sub>0 :: "real" assumes "poly (hd ps) x\<^sub>0 \<noteq> 0" shows "(\<Sum>ps'\<leftarrow>split_sign_changes ps x\<^sub>0. sign_changes ps' x\<^sub>0) = sign_changes ps x\<^sub>0"
mult_nonneg_nonneg: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) \<le> a"   and "(0::'a) \<le> b" shows "(0::'a) \<le> a * b"
val_eq_def: fixes v1 :: "val"   and v2 :: "val" shows "v1 \<sim> v2 \<equiv> v1 \<sqsubseteq> v2 \<and> v2 \<sqsubseteq> v1"
Lim_within_union: fixes f :: "'c \<Rightarrow> 'd"   and l :: "'d"   and x :: "'c"   and s :: "'c set"   and t :: "'c set" shows "(f \<longlongrightarrow> l) (at x within s \<union> t) = ((f \<longlongrightarrow> l) (at x within s) \<and> (f \<longlongrightarrow> l) (at x within t))"
psubstT_Var_not: fixes x :: "'var"   and txs :: "('trm \<times> 'var) list" assumes "x \<in> var"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "x \<notin> snd ` set txs" shows "psubstT (Var x) txs = Var x"
wf_domain_def: shows "wf_domain \<equiv> wf_types \<and> distinct (map pred (predicates D)) \<and> Ball (set (predicates D)) wf_predicate_decl \<and> distinct (map fst (consts D)) \<and> (\<forall>(n, y)\<in>set (consts D). wf_type y) \<and> distinct (map ast_action_schema.name (actions D)) \<and> Ball (set (actions D)) wf_action_schema"
inject_1: fixes x2 :: "vrs"   and x1 :: "ty"   and y2 :: "vrs"   and y1 :: "ty" shows "(VarB x2 x1 = VarB y2 y1) = (x2 = y2 \<and> x1 = y1)"
dist_prop: shows "\<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length (t # ts) \<longrightarrow> (t # ts) ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> (all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<union> all_unshared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) \<union> all_acquired (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)) \<inter> (all_shared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb') \<union> all_unshared (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb') \<union> all_acquired (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb')) = {}"
coreflexive_comp_top_inf_one: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_idempotent_semiring.coreflexive x" shows "x \<odot> mtop \<otimes> mone = x"
zhmset_of_times: fixes A :: "hmultiset"   and B :: "hmultiset" shows "zhmset_of (A * B) = zhmset_of A * zhmset_of B"
Q: shows "Q \<in> sets (M1 \<Otimes>\<^sub>M M2)"
arg_min_nat_lemma: fixes P :: "'a \<Rightarrow> bool"   and k :: "'a"   and m :: "'a \<Rightarrow> nat" assumes "P k" shows "P (arg_min m P) \<and> (\<forall>y. P y \<longrightarrow> m (arg_min m P) \<le> m y)"
preserves_ide: fixes a :: "'a arrow_of_spans_data" assumes "ide a" shows "ide (R a)"
hhom_is_subcategory: fixes a :: "'b"   and b :: "'b" shows "subcategory (\<cdot>\<^sub>D) (\<lambda>\<mu>. \<guillemotleft>\<mu> : a \<rightarrow>\<^sub>D b\<guillemotright>)"
clocked_min_counter: fixes env :: "v sem_env"   and s :: "'b state"   and e :: "exp0"   and s' :: "'b state"   and r' :: "(v, v) result" assumes "evaluate True env s e (s', r')" shows "evaluate True env (update_clock (\<lambda>_. clock s - clock s') s) e (update_clock (\<lambda>_. 0) s', r')"
assms_1: shows "\<not> hb (i, Append i e1) (r, Expunge e2 mo r)"
preserves_arr: fixes f :: "'b" assumes "B.arr f" shows "B.arr (trg\<^sub>B f)"
Red_F_of_subset: fixes N :: "'f set"   and N' :: "'f set" assumes "N \<subseteq> N'" shows "Red_F N \<subseteq> Red_F N'"
eq_vector_fraction_iff: fixes x :: "'a"   and u :: "real"   and v :: "real"   and a :: "'a" shows "(x = (u / v) *\<^sub>R a) = (if v = 0 then x = (0::'a) else v *\<^sub>R x = u *\<^sub>R a)"
pickQFreshEnv_card_of: fixes V :: "'var set"   and XS :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Rho :: "('varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm option) set"   and xs :: "'varSort" assumes "|V| <o |UNIV|"   and "|XS| <o |UNIV|"   and "Ball XS qGood"   and "|Rho| <o |UNIV|"   and "Ball Rho qGoodEnv" shows "pickQFreshEnv xs V XS Rho \<notin> V \<and> (\<forall>X\<in>XS. qFresh xs (pickQFreshEnv xs V XS Rho) X) \<and> (\<forall>rho\<in>Rho. qFreshEnv xs (pickQFreshEnv xs V XS Rho) rho)"
map_of: shows "map_of (zip xs ys)(x \<mapsto> y) = map_of (zip xs zs)(x \<mapsto> z)"
not_le: fixes ko :: "'a key_order"   and x :: "'a"   and y :: "'a" shows "(\<not> le_of_key_order ko x y) = lt_of_key_order ko y x"
default_def: fixes k :: "'b"   and v :: "'c"   and m :: "('b, 'c) mapping" shows "Mapping.default k v m = (if k \<in> Mapping.keys m then m else Mapping.update k v m)"
assms: shows "(f \<circ> vec has_integral y) ((\<lambda>x. x $ 1) ` S)"
Z: shows "0 = emeasure M (\<Inter> (range K))"
has: shows "P \<turnstile> C has F,b:T in D"
assms_2: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> n" shows "P a"
functor_axioms: shows "functor (\<cdot>) (\<cdot>) L"
assms_7: shows "a \<cdot> l \<le> l \<cdot> a"
mult_1_left: fixes a :: "'a" shows "a \<cdot> (1::'a) = a"
natural_transformation_axioms: shows "natural_transformation \<F>\<^sub>SC.comp \<F>\<^sub>SC.comp DoS.map \<F>\<^sub>SC.map \<mu>.map"
comp_red_monic_basis_def: fixes xs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "comp_red_monic_basis xs = map monic (comp_red_basis xs)"
S_Max_Le: shows "S_Max_Le = {d - u c |c d. 0 < v c \<and> v c \<le> n \<and> M (v c) 0 = Le d}"
coeffM_zero_def: fixes A :: "(('c poly, 'd) vec, 'd) vec" shows "coeffM_zero A = (\<chi>i j. coeff (A $ i $ j) 0)"
simps_52: fixes x5 :: "'a OCL_Types.type"   and x4 :: "'a OCL_Types.type" shows "Set x5 \<noteq> Collection x4"
eq: fixes w :: "'a" assumes "w \<in> poly_vars p" shows "f w = g w"
assms_2: shows "RETURN x \<le> \<Down> R m"
bij_pointE: fixes f :: "'a \<Rightarrow> 'b"   and y :: "'b" assumes "bij f" obtains x :: "'a" where "y = f x" and "\<And>x'. y = f x' \<Longrightarrow> x' = x"
simps_3: shows "X5d X5.Xd2 = D2"
inv_mult_group: fixes x :: "'a \<Rightarrow> complex"   and y :: "'a \<Rightarrow> complex" assumes "x \<in> carrier C"   and "y \<in> carrier C" shows "inv\<^bsub>C\<^esub> (x \<otimes>\<^bsub>C\<^esub> y) = inv\<^bsub>C\<^esub> y \<otimes>\<^bsub>C\<^esub> inv\<^bsub>C\<^esub> x"
llist_all2_lhdD2: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'c llist"   and ys :: "'d llist" assumes "llist_all2 P xs ys"   and "\<not> lnull ys" shows "P (lhd xs) (lhd ys)"
left_neutral: fixes a :: "'b \<times> 'c" shows "times_prod' one_prod a = a"
modifies_lhsv: fixes \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi>: (c, s) \<Rightarrow> t" shows "modifies (lhsv \<pi> c) t s"
catchET_def: shows "catchET = (\<Lambda> m h. ErrorT\<cdot>(runErrorT\<cdot>m \<bind> (\<Lambda> n. case n of Err\<cdot>e \<Rightarrow> runErrorT\<cdot>(h\<cdot>e) | Ok\<cdot>x \<Rightarrow> return\<cdot>(Ok\<cdot>x))))"
rcos_j: shows "I +> j = \<one>\<^bsub>R Quot I\<^esub>"
dvd_add_times_triv_left_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and a :: "'c"   and c :: "'c"   and b :: "'c" assumes "class.comm_semiring_1_cancel plus minus zero times one" shows "dvd.dvd times a (plus (times c a) b) = dvd.dvd times a b"
Eq\<^sub>2E: fixes v :: "i"   and F :: "\<Pi>\<^sub>2"   and G :: "\<Pi>\<^sub>2" assumes "[F \<^bold>=\<^sub>2 G in v]" shows "F = G"
edge_succ_wu: shows "edge_succ HM (rev_H uw) = rev_H vw"