hcju/isabelleps huggingface dataset:
fmdom'_notD: fixes x :: "'c"   and m :: "('c, 'd) fmap" assumes "x \<notin> fmdom' m" shows "fmlookup m x = None"
arr_dom: fixes f :: "'a" assumes "hom.arr f" shows "hom.arr (hom.dom f)"
is_span: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b span_data" assumes "span_in_category C S" shows "partial_magma.arr C (Leg0 S) \<and> partial_magma.arr C (Leg1 S) \<and> partial_magma.dom C (Leg0 S) = partial_magma.dom C (Leg1 S)"
fsubset_finsertI2: fixes A :: "'a fset"   and B :: "'a fset"   and b :: "'a" assumes "A |\<subseteq>| B" shows "A |\<subseteq>| finsert b B"
divide_eq_eq_numeral_1: fixes b :: "'a"   and c :: "'a"   and w :: "num" shows "(b / c = numeral w) = (if c \<noteq> (0::'a) then b = numeral w * c else numeral w = (0::'a))"
not_iszero_Numeral1: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b" assumes "class.ring_1 one times plus zero minus uminus" shows "\<not> ring_1.iszero zero (numeral.numeral one plus num.One)"
set_intros_16: fixes yn :: "'b"   and x131 :: "'b ltlc"   and x132 :: "'b ltlc" assumes "yn \<in> atoms_ltlc x131" shows "yn \<in> atoms_ltlc (x131 W\<^sub>c x132)"
assms_2: shows "B.arr (snd \<mu>\<nu>)"
M1_inj_H: shows "\<exists>f\<in>H \<rightarrow> M1. inj_on f H"
below_in_singleton: fixes x :: "'a"   and P :: "'a set"   and y :: "'a" assumes "x \<in> P"   and "order.greater_eq {y} (P .\<^bold>\<le> x)" shows "y = x"
T_append: fixes qs :: "'c list"   and as :: "'d list"   and step :: "'e \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> 'e"   and t :: "'e \<Rightarrow> 'c \<Rightarrow> 'd \<Rightarrow> nat"   and s :: "'e"   and q :: "'c"   and a :: "'d" assumes "length qs = length as" shows "On_Off.T step t s (qs @ [q]) (as @ [a]) = On_Off.T step t s qs as + t (On_Off.steps step s qs as) q a"
Int_iff: fixes c :: "'g"   and A :: "'g set"   and B :: "'g set" shows "(c \<in> A \<inter> B) = (c \<in> A \<and> c \<in> B)"
simps_3: fixes S1 :: "com"   and S2 :: "com"   and \<rho> :: "complex mat" shows "denote (S1;; S2) \<rho> = denote S2 (denote S1 \<rho>)"
Ln_Reals_eq: fixes x :: "complex" assumes "x \<in> \<real>"   and "0 < Re x" shows "Ln x = complex_of_real (ln (Re x))"
of_fun_inverse: fixes y :: "'c \<Rightarrow> 'c \<Rightarrow> 'd" assumes "y \<in> UNIV" shows "to_fun (of_fun y) = y"
Domain_insert: fixes a :: "'b"   and b :: "'c"   and r :: "('b \<times> 'c) set" shows "Domain (insert (a, b) r) = insert a (Domain r)"
correct_game_def: fixes m :: "'plain" shows "correct_game m = key_gen \<bind> (\<lambda>(ck, vk). commit ck m \<bind> (\<lambda>(c, d). return_spmf (verify vk m c d)))"
conv_isotone: fixes x :: "'a"   and y :: "'a" assumes "x \<le> y" shows "x\<^sup>T \<le> y\<^sup>T"
P1: shows "Bet B A A'' \<and> Cong A A'' E D \<and> Bet B C C'' \<and> Cong C C'' E F \<and> Bet E D D'' \<and> Cong D D'' B A \<and> Bet E F F'' \<and> Cong F F'' B C \<and> Cong A'' C'' D'' F''"
lin_indpt_list_length_eq_n_1: fixes xs :: "'a vec list" assumes "lin_indpt_list xs"   and "length xs = n" shows "local.span (set xs) = carrier_vec n"
RESUMPTION_corec: fixes f :: "'j \<Rightarrow> ('h, 'g, 'f, 'i, 'j) resumption_corec"   and e :: "'j" shows "RESUMPTION_corec f e = (case f e of Terminal_corec a \<Rightarrow> TERMINAL a | Linear_corec b e' \<Rightarrow> LINEAR b (RESUMPTION_corec f e') | Branch_corec c es \<Rightarrow> BRANCH c (\<lambda>d. RESUMPTION_corec f (es d)) | Resumption_corec r \<Rightarrow> Rep_resumption r)"
wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t_induct: fixes \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and P :: "('a \<Rightarrow> ('b, 'a) Term.term) \<Rightarrow> bool" assumes "wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta>"   and "P Var"   and "\<And>\<theta> v t. \<lbrakk>wf\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<theta>; P \<theta>; v \<notin> subst_domain \<theta>; v \<notin> range_vars \<theta>; fv t \<inter> insert v (subst_domain \<theta>) = {}\<rbrakk> \<Longrightarrow> P (\<theta>(v := t))" shows "P \<delta>"
cases: fixes a :: "Syntax.term" assumes "value a" obtains "a = Unit"   | uu_ :: "Syntax.var" where "a = Var uu_"   | uv_ :: "Syntax.var" and uw_ :: "Syntax.term" where "a = Lam uv_ uw_"   | ux_ :: "Syntax.var" and uy_ :: "Syntax.term" where "a = Rec ux_ uy_"   | v :: "Syntax.term" where "a = Inj1 va__" and "value va__"   | v :: "Syntax.term" where "a = Inj2 vb__" and "value vb__"   | v\<^sub>1 :: "Syntax.term" and v\<^sub>2 :: "Syntax.term" where "a = Syntax.Pair v\<^sub>1 v\<^sub>2" and "value v\<^sub>1" and "value v\<^sub>2"   | v :: "Syntax.term" where "a = Roll vc__" and "value vc__"   | uz_ :: "hash" where "a = Hash uz_"   | v :: "Syntax.term" and va_ :: "hash" where "a = Hashed va_ v" and "value v"
ovalidNF_def: fixes P :: "'e \<Rightarrow> bool"   and f :: "'e \<Rightarrow> 'f option"   and Q :: "'f \<Rightarrow> 'e \<Rightarrow> bool" shows "ovalidNF P f Q \<equiv> \<forall>s. P s \<longrightarrow> f s \<noteq> None \<and> (\<forall>r. f s = Some r \<longrightarrow> Q r s)"
ssorted_shift: fixes xs :: "'b list"   and s :: "'b stream" shows "ssorted (xs @- s) = (sorted xs \<and> ssorted s \<and> (\<forall>x\<in>set xs. \<forall>y\<in>sset s. x \<le> y))"
Un_absorb1: fixes A :: "'d set"   and B :: "'d set" assumes "A \<subseteq> B" shows "A \<union> B = B"
hoare\<^sub>3_def: fixes P :: "'b \<Rightarrow> bool"   and M :: "'b \<Rightarrow> ('c \<times> 'b) option"   and Q :: "'c \<Rightarrow> 'b \<Rightarrow> bool" shows "\<lbrace>P\<rbrace> M \<lbrace>Q\<rbrace> \<equiv> \<forall>\<sigma>. P \<sigma> \<longrightarrow> (case M \<sigma> of None \<Rightarrow> False | Some (x, \<sigma>') \<Rightarrow> Q x \<sigma>')"
Cons_3: shows "\<forall>c. c \<in> set (clause # Wl') \<longrightarrow> 0 \<le> c \<and> c < length (getF stateA)"
ideals_set_def: fixes R :: "('c, 'd) ring_scheme" shows "ideals_set R = \<lparr>carrier = {I. ideal I R}, mult = (\<cdot>\<^bsub>R\<^esub>), one = carrier R, zero = {\<zero>\<^bsub>R\<^esub>}, add = (<+>\<^bsub>R\<^esub>)\<rparr>"
cases: fixes a1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and a2 :: "'Var \<Rightarrow> 'Val"   and a3 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and a4 :: "'Var \<Rightarrow> 'Val" assumes "(a1, a2) \<rightarrow> (a3, a4)" obtains cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list" and n :: "nat" and mem :: "'Var \<Rightarrow> 'Val" and cm' :: "'Com \<times> (Mode \<Rightarrow> 'Var set)" and mem' :: "'Var \<Rightarrow> 'Val" where "a1 = cms" and "a2 = mem" and "a3 = cms[n := cm']" and "a4 = mem'" and "(cms ! n, mem) \<leadsto> (cm', mem')" and "n \<sqsubset> length cms"
atom_HPairE: fixes H :: "fm set"   and x :: "tm"   and y :: "tm"   and v :: "tm"   and A :: "fm" assumes "H \<turnstile> x EQ HPair (HPair Zero (HPair Zero Zero)) y" shows "insert (IndP x OR x NEQ v) H \<turnstile> A"
CFGExit_axioms: shows "CFGExit src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry (lift_get_proc get_proc Main) (lift_get_return_edges get_return_edges valid_edge sourcenode targetnode kind) procs Main NewExit"
invert_opposite_sides_of_square: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d"   and g :: "'d"   and h :: "'d"   and k :: "'d" assumes "category C"   and "partial_magma.arr C (C f g)"   and "C f g = C h k"   and "category.iso C f"   and "category.iso C k" shows "partial_magma.arr C (C g (category.inv C k)) \<and> partial_magma.arr C (C (category.inv C f) h) \<and> C g (category.inv C k) = C (category.inv C f) h"
flat_ord_def: fixes b :: "'c"   and x :: "'c"   and y :: "'c" shows "flat_ord b x y = (x = b \<or> x = y)"
simps_2: fixes e :: "'c \<times> 'd \<times> bool"   and i :: "'c" shows "insert [] e (Some i) = None"
not_in_extended_FE: fixes S :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula set" assumes "fin_sat S"   and "\<not> sat (\<^bold>\<not> F \<triangleright> G)"   and "F \<notin> extended S"   and "G \<subseteq> extended S"   and "finite G" shows "False"
frameImpI: fixes F :: "'b frame"   and A\<^sub>F :: "name list"   and \<Psi>\<^sub>F :: "'b"   and \<phi> :: "'c" assumes "F = \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>"   and "A\<^sub>F \<sharp>* \<phi>"   and "\<Psi>\<^sub>F \<turnstile> \<phi>" shows "F \<turnstile>\<^sub>F \<phi>"
Conseq: fixes P :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and c :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set" assumes "\<forall>s\<in>P. \<exists>P' Q' A'. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P' c Q',A' \<and> s \<in> P' \<and> Q' \<subseteq> Q \<and> A' \<subseteq> A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c Q,A"
scale_right_diff_distrib: fixes a :: "real"   and x :: "'b"   and y :: "'b" shows "a *\<^sub>R (x - y) = a *\<^sub>R x - a *\<^sub>R y"
dsumM_carr: fixes I :: "'f set"   and M :: "'f \<Rightarrow> ('g, 'a, 'h) Module_scheme" shows "carrier (\<^bsub>R\<^esub>\<Sigma>\<^sub>d\<^bsub>I\<^esub> M) = carr_dsumag I M"
extended_simps_5: fixes x :: "'b extended" shows "\<not> \<stileturn> < x"
embed_def: fixes r :: "('c \<times> 'c) set"   and r' :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" shows "embed r r' f \<equiv> \<forall>a\<in>Field r. bij_betw f (under r a) (under r' (f a))"
tranclp_trans: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "r\<^sup>+\<^sup>+ x y"   and "r\<^sup>+\<^sup>+ y z" shows "r\<^sup>+\<^sup>+ x z"
ex: shows "\<exists>q. q \<in> fst ` set todo \<union> fst ` set old \<and> poly (real_of_int_poly q) x = 0"
of_int: fixes f :: "'b \<Rightarrow> 'c"   and n :: "int" assumes "periodic_fun_simple' f" shows "f (of_int n) = f (0::'b)"
is_coext_def: fixes x' :: "'d"   and f :: "'c"   and g :: "'d" shows "x\<epsilon>.is_coext x' f g \<equiv> \<guillemotleft>g : x' \<rightarrow>\<^sub>D G x\<guillemotright> \<and> f = \<epsilon>.map x \<cdot>\<^sub>C F g"
pos_le_minus_divideR_eq: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and c :: "real"   and a :: "'a"   and b :: "'a" assumes "class.ordered_real_vector plus zero minus uminus less_eq less scaleR"   and "0 < c" shows "less_eq a (uminus (scaleR (inverse c) b)) = less_eq (scaleR c a) (uminus b)"
HJ_fcs_eq_for_a': fixes b :: "'b" assumes "b \<in> arcs H \<inter> G.face_cycle_set a'" shows "H.face_cycle_set b = S"
pause_env_def: fixes pause :: "'k \<Rightarrow> ('l \<Rightarrow> 'j) \<Rightarrow> 'j"   and out :: "'k"   and c :: "'l \<Rightarrow> ('i, 'j) envT" shows "pause_env pause out c = EnvT (\<lambda>r. pause out (\<lambda>i. run_env (c i) r))"
lift_irreducible_terms: fixes T :: "'a trm set"   and C :: "'a literal set"   and D :: "'a literal set"   and \<sigma> :: "('a \<times> 'a trm) list"   and E :: "'a trm set"   and \<theta> :: "('a \<times> 'a trm) list"   and \<eta> :: "('a \<times> 'a trm) list"   and C' :: "'a literal set" assumes "T = get_trms C (dom_trms (subst_cl D \<sigma>) (subst_set E \<sigma>)) Ground"   and "\<sigma> \<doteq> \<theta> \<lozenge> \<eta>" shows "\<exists>T'. subst_set T' \<eta> \<subseteq> T \<and> T' = get_trms C' (dom_trms (subst_cl D \<theta>) (subst_set E \<theta>)) FirstOrder"
that: fixes Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> !P \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q"   and "\<Psi> \<rhd> Q \<sim> P' \<parallel> !P" shows "thesis"
sets_UNIV: fixes A :: "'a set" shows "A \<in> sets (count_space UNIV)"
n_nS12: fixes s :: "'s" assumes "s \<in> S"   and "s \<notin> S1"   and "s \<notin> S2" shows "n s = 0"
lift_character_lower_character: fixes \<chi> :: "'a \<Rightarrow> complex" assumes "\<chi> \<in> characters (G\<lparr>carrier := adjoin G H a\<rparr>)" shows "lift_character (\<lambda>x. if x \<in> H then \<chi> x else 0, \<chi> a) = \<chi>"
differentiable_bound: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'b \<Rightarrow> 'c"   and B :: "real"   and x :: "'b"   and y :: "'b" assumes "convex S"   and "\<And>x. x \<in> S \<Longrightarrow> (f has_derivative f' x) (at x within S)"   and "\<And>x. x \<in> S \<Longrightarrow> onorm (f' x) \<le> B"   and "x \<in> S"   and "y \<in> S" shows "norm (f x - f y) \<le> B * norm (x - y)"
distributed_marginal_eq_joint2: fixes T :: "'d measure"   and S :: "'e measure"   and Y :: "'a \<Rightarrow> 'd"   and Py :: "'d \<Rightarrow> ennreal"   and X :: "'a \<Rightarrow> 'e"   and Pxy :: "'e \<times> 'd \<Rightarrow> ennreal" assumes "sigma_finite_measure T"   and "sigma_finite_measure S"   and "distributed M T Y Py"   and "distributed M (S \<Otimes>\<^sub>M T) (\<lambda>x. (X x, Y x)) Pxy" shows "AE y in T. Py y = \<integral>\<^sup>+ x. Pxy (x, y) \<partial>S"
h1: shows "sortedTree h t1 \<longrightarrow> setOf (remove h e t1) = setOf t1 - eqs h e"
proots_half_proots_upper: fixes a :: "complex"   and b :: "complex"   and p :: "complex poly" assumes "a \<noteq> b"   and "p \<noteq> 0" shows "proots_half p a b = proots_upper (p \<circ>\<^sub>p [:a, b - a:])"
FORK_def: fixes i :: "nat" shows "FORK i = (\<mu> x. picks i i \<rightarrow> putsdown i i \<rightarrow> x [+] picks ((i + 1) mod N) i \<rightarrow> putsdown ((i + 1) mod N) i \<rightarrow> x)"
hs_dfs_cond_def: fixes S :: "('a, unit) HashMap.hashmap \<times> 'a list" shows "hs_dfs_cond S \<equiv> let (Q, W) = S in W \<noteq> []"
bilinear_lmul: fixes h :: "'a \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "real"   and x :: "'a"   and y :: "'b" assumes "bilinear h" shows "h (c *\<^sub>R x) y = c *\<^sub>R h x y"
polyhedron_hyperplane: fixes a :: "'b"   and b :: "real" shows "polyhedron {x. a \<bullet> x = b}"
simps: fixes a1 :: "'a set"   and a2 :: "'a set"   and R :: "('a \<times> 'a) set" shows "((a1, a2) \<in> max_ext R) = (\<exists>X Y. a1 = X \<and> a2 = Y \<and> finite X \<and> finite Y \<and> Y \<noteq> {} \<and> (\<forall>x. x \<in> X \<longrightarrow> (\<exists>xa\<in>Y. (x, xa) \<in> R)))"
closure_translation: fixes a :: "'b"   and s :: "'b set" shows "closure ((+) a ` s) = (+) a ` closure s"
22: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> y \<squnion> x \<sqinter> z \<squnion> x \<sqinter> (y \<squnion> z) = x \<sqinter> (y \<squnion> z)"
e_inverse: fixes x :: "'e" shows "PRJ('e)\<cdot>(EMB('e)\<cdot>x) = x"
constant_rep_point_not_0: fixes x :: "'a" assumes "x \<in> \<V>" shows "\<B> rep x \<noteq> 0"
OAlist_tc_sorted_domain_def: fixes xs :: "('c, 'd) oalist_tc" shows "OAlist_tc_sorted_domain xs = map fst (list_of_oalist_tc xs)"
basisdef: fixes G :: "'b list set" shows "\<BB> G = {x.  x \<in>B G }"
rep_eq: fixes x :: "'a Bonk_Schramm_extension"   and xa :: "'a Bonk_Schramm_extension" shows "dist x xa = extended_distance (rep_Bonk_Schramm_extension x) (rep_Bonk_Schramm_extension xa)"
assms_3: shows "\<forall>\<^sub>F x in at_top. f x - 1 = 0"
in_set_conv_decomp_last: fixes x :: "'g"   and xs :: "'g list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
stable_pair_on_Cd_match: fixes ds :: "'d set"   and XD_XH :: "'x set \<times> 'x set"   and d :: "'d" assumes "stable_pair_on ds XD_XH"   and "d \<in> ds" shows "Cd d (match XD_XH) = Cd d (fst XD_XH)"
insert: fixes A :: "'e set"   and x :: "'e" assumes "finite A"   and "x \<notin> A" shows "card (insert x A) = Suc (card A)"
carrier_eq: shows "carrier \<inter> A = {x. restrict_relation A le x x}"
matrix_matrix_mult_def: fixes m :: "(('a, 'd) vec, 'c) vec"   and m' :: "(('a, 'b) vec, 'd) vec" shows "m ** m' \<equiv> \<chi>i j. \<Sum>k\<in>UNIV. m $ i $ k * m' $ k $ j"
lexp_lex: fixes order :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and ys :: "'b list" shows "lexp order xs ys = ((xs, ys) \<in> lex {(x, y). order x y})"
Bseq_eq_bounded: fixes f :: "nat \<Rightarrow> real"   and a :: "real"   and b :: "real" assumes "range f \<subseteq> {a..b}" shows "Bseq f"
MPair_def: shows "MPair \<equiv> (rep_msg ---> rep_msg ---> abs_msg) MPAIR"
assms: shows "A' \<in> set (tr\<^sub>p\<^sub>c A D)"
of_real_stieltjes_gamma: fixes k :: "nat" shows "of_real (stieltjes_gamma k) = stieltjes_gamma k"
simps_15: fixes f1 :: "'b"   and f2 :: "'b"   and f3 :: "'b"   and f4 :: "'b" shows "(case Protected of Private \<Rightarrow> f1 | Package \<Rightarrow> f2 | Protected \<Rightarrow> f3 | Public \<Rightarrow> f4) = f3"
smult_distrib_left_diff: fixes r :: "'r"   and m :: "'m"   and n :: "'m" assumes "r \<in> R"   and "m \<in> M"   and "n \<in> M" shows "r \<cdot> (m - n) = r \<cdot> m - r \<cdot> n"
scale_right_imp_eq: fixes x :: "'a"   and a :: "real"   and b :: "real" assumes "x \<in> S"   and "x \<noteq> (0::'a)"   and "a *\<^sub>R x = b *\<^sub>R x" shows "a = b"
YX: fixes U :: "'b set" assumes "openin Y U" shows "openin X {x \<in> topspace X. f x \<in> U}"
F_g_one: fixes g :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "aggregation.sum_0 (\<lambda>x. g x + zero) A = aggregation.sum_0 g A"
Conj_eq_iff: fixes xset1 :: "('b, 'c, 'd) formula set['b]"   and xset2 :: "('b, 'c, 'd) formula set['b]" assumes "finite (supp xset1)"   and "finite (supp xset2)" shows "(Conj xset1 = Conj xset2) = (xset1 = xset2)"
unitary_intros_5: fixes P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real" assumes "unitary P"   and "unitary Q" shows "unitary (P && Q)"
induct: fixes x1 :: "char list set"   and x2 :: "com"   and x3 :: "char list set"   and P :: "char list set \<Rightarrow> com \<Rightarrow> char list set \<Rightarrow> bool" assumes "D x1 x2 x3"   and "\<And>A. P A SKIP A"   and "\<And>a A x. vars a \<subseteq> A \<Longrightarrow> P A (x ::= a) (insert x A)"   and "\<And>A\<^sub>1 c\<^sub>1 A\<^sub>2 c\<^sub>2 A\<^sub>3. \<lbrakk>D A\<^sub>1 c\<^sub>1 A\<^sub>2; P A\<^sub>1 c\<^sub>1 A\<^sub>2; D A\<^sub>2 c\<^sub>2 A\<^sub>3; P A\<^sub>2 c\<^sub>2 A\<^sub>3\<rbrakk> \<Longrightarrow> P A\<^sub>1 (c\<^sub>1;; c\<^sub>2) A\<^sub>3"   and "\<And>b A c\<^sub>1 A\<^sub>1 c\<^sub>2 A\<^sub>2. \<lbrakk>vars b \<subseteq> A; D A c\<^sub>1 A\<^sub>1; P A c\<^sub>1 A\<^sub>1; D A c\<^sub>2 A\<^sub>2; P A c\<^sub>2 A\<^sub>2\<rbrakk> \<Longrightarrow> P A (IF b THEN c\<^sub>1 ELSE c\<^sub>2) (A\<^sub>1 \<inter> A\<^sub>2)"   and "\<And>b A c A'. \<lbrakk>vars b \<subseteq> A; D A c A'; P A c A'\<rbrakk> \<Longrightarrow> P A (WHILE b DO c) A" shows "P x1 x2 x3"
T_up3_def2: fixes x :: "'a"   and sib :: "'a tree"   and twist :: "bool"   and u :: "'a up2" shows "T_up3 x sib twist u = (case u of Unbal2 t n1 h1 \<Rightarrow> let n2 = size sib; t' = node twist t x sib; h' = h1 + 1; n' = n1 + n2 + 1 in 2 * size sib + 1 + (if bal_i n' h' then 1 else T_bal_tree n' t' + 1) | _ \<Rightarrow> 1)"
distrib: fixes g :: "'d \<Rightarrow> 'c"   and h :: "'d \<Rightarrow> 'c"   and A :: "'d set" shows "sup_monoid.sum (\<lambda>x. g x \<squnion> h x) A = sup_monoid.sum g A \<squnion> sup_monoid.sum h A"
rel_set_eq_eq: fixes A :: "'a set"   and B :: "'a set" assumes "rel_set (=) A B" shows "A = B"
c_ipurge_aux_ipurge_tr: fixes I :: "('e \<times> 'e) set"   and D :: "'f \<Rightarrow> 'e"   and u :: "'e"   and ys :: "'f list"   and U :: "'e set"   and xs :: "'f list" assumes "refl I"   and "\<not> (\<exists>v\<in>sinks I D u ys. \<exists>w\<in>U. (v, w) \<in> I)" shows "c_ipurge_aux I D U (xs @ ipurge_tr I D u ys) = c_ipurge_aux I D U (xs @ ys)"
apos_tna_pos: fixes n :: "ant" assumes "n \<noteq> \<infinity>"   and "0 \<le> n" shows "0 \<le> tna n"
one_dim_apply_is_times_def: fixes \<psi> :: "'a"   and \<phi> :: "'a" shows "\<psi> * \<phi> = ((1::'a) \<bullet>\<^sub>C \<psi> * ((1::'a) \<bullet>\<^sub>C \<phi>)) *\<^sub>C (1::'a)"
bij_imp_bij_inv: fixes f :: "'c \<Rightarrow> 'd" assumes "bij f" shows "bij (inv f)"
assms_4: shows "index (mtf2 n q xs) (xs ! i) = index xs q - n"
row_add_2: fixes i :: "nat"   and A :: "'a mat"   and B :: "'a mat" assumes "i < dim_row A"   and "dim_row B = dim_row A"   and "dim_col B = dim_col A" shows "row (A + B) i = row A i + row B i"
lower_rank_of_eq_upper_rank_of: fixes carrier :: "'b set"   and indep :: "'b set \<Rightarrow> bool"   and \<E> :: "'b set" assumes "matroid carrier indep"   and "\<E> \<subseteq> carrier" shows "indep_system.lower_rank_of indep \<E> = indep_system.upper_rank_of indep \<E>"
crel_vs_success: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c"   and b :: "'d Heap"   and heap :: "heap" assumes "crel_vs R a b"   and "P heap"   and "P heap"   and "state_dp_consistency.cmem heap" shows "success b heap"
trans: fixes b :: "'c multiset"   and a :: "'c multiset"   and c :: "'c multiset" assumes "b \<subseteq># a"   and "c \<subseteq># b" shows "c \<subseteq># a"
greaterThanLessThan_eq: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" shows "ord.greaterThanLessThan less a b = ord.greaterThan less a \<inter> ord.lessThan less b"
chf_lm_4: fixes A :: "nat set" assumes "chf A \<in> PrimRec1" shows "A \<in> ce_sets"
one_dim_iso_of_complex: fixes c :: "complex" shows "one_dim_iso (of_complex c) = of_complex c"
asym: fixes a :: "'a list"   and b :: "'a list" assumes "strict_prefix a b"   and "strict_prefix b a" shows "False"
app_vdomainI: fixes a :: "V"   and b :: "V"   and r :: "V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> r" shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r"
simps_2: fixes f :: "'c \<Rightarrow> 'c"   and x :: "'c" shows "stl (siterate f x) = siterate f (f x)"
is_glb_glb: fixes M :: "'a set"   and x :: "'a" assumes "M >>| x" shows "M >>| \<Sqinter>M"
\<pi>_mono: fixes x :: "real"   and y :: "real" assumes "x \<le> y" shows "\<pi> x \<le> \<pi> y"
convex_on_linorderI: fixes A :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "\<And>t x y. \<lbrakk>0 < t; t < 1; x \<in> A; y \<in> A; x < y\<rbrakk> \<Longrightarrow> f ((1 - t) *\<^sub>R x + t *\<^sub>R y) \<le> (1 - t) * f x + t * f y" shows "convex_on A f"
RDirProd_list_iso1: fixes R :: "('c, 'd) ring_scheme"   and Rs :: "('c, 'd) ring_scheme list" shows "(\<lambda>(a, as). a # as) \<in> ring_iso (RDirProd R (RDirProd_list Rs)) (RDirProd_list (R # Rs))"
s_top_sub: fixes H :: "'a set"   and K :: "'a set" assumes "G \<guillemotright> H "   and "G \<guillemotright> K " shows "H \<diamondop> K \<subseteq> carrier G"
approximating_bigstep_fun_seq_wf: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and p :: "'c"   and rs\<^sub>1 :: "'b rule list"   and rs\<^sub>2 :: "'b rule list"   and s :: "state" assumes "wf_ruleset \<gamma> p rs\<^sub>1" shows "approximating_bigstep_fun \<gamma> p (rs\<^sub>1 @ rs\<^sub>2) s = approximating_bigstep_fun \<gamma> p rs\<^sub>2 (approximating_bigstep_fun \<gamma> p rs\<^sub>1 s)"
exhaust: fixes y :: "'b interrupt_action" obtains x11 :: "'b" and x12 :: "bool" where "y = IsInterrupted x11 x12"   | x2 :: "'b" where "y = Interrupt x2"   | x3 :: "'b" where "y = ClearInterrupt x3"
simps_4: fixes c :: "'b poly"   and n :: "nat"   and p :: "'b poly"   and c' :: "'b poly"   and n' :: "nat"   and p' :: "'b poly" shows "CN c n p +\<^sub>p CN c' n' p' = (if n < n' then CN (c +\<^sub>p CN c' n' p') n p else if n' < n then CN (CN c n p +\<^sub>p c') n' p' else let cc' = c +\<^sub>p c'; pp' = p +\<^sub>p p' in if pp' = 0\<^sub>p then cc' else CN cc' n pp')"
disjoint_iff: fixes A :: "'g set"   and B :: "'g set" shows "(A \<inter> B = {}) = (\<forall>x. x \<in> A \<longrightarrow> x \<notin> B)"
assign_def: fixes A :: "'b set"   and H :: "(('c, 'b) fm list \<times> 'b) set"   and i :: "'b" shows "assign A H i \<equiv> if \<exists>a. a \<in> A \<and> Nom a at i in' H then proj (hequiv_rel A H) (SOME a. a \<in> A \<and> Nom a at i in' H) else {i}"
merge_pure_and: fixes a :: "bool"   and b :: "bool" shows "\<up> a \<and>\<^sub>A \<up> b = \<up> (a \<and> b)"