hcju/isabelleps huggingface dataset:
app_iso_inv: fixes hom :: "('c, 'd, 'e, 'f) digraph_isomorphism" assumes "H.digraph_isomorphism hom" shows "app_iso (H.inv_iso hom) (app_iso hom (app_iso hom G)) = app_iso hom G"
interval_lebesgue_integral_diff_2: fixes M :: "real measure"   and a :: "ereal"   and b :: "ereal"   and f :: "real \<Rightarrow> 'a"   and g :: "real \<Rightarrow> 'a" assumes "interval_lebesgue_integrable M a b f"   and "interval_lebesgue_integrable M a b g" shows "LINT x=a..b|M. (f x - g x) = interval_lebesgue_integral M a b f - interval_lebesgue_integral M a b g"
assms_2: shows "(\<box>(\<lambda>s. Q s \<longrightarrow> R s)) \<sigma>"
setops\<^sub>s\<^sub>s\<^sub>t_mono: fixes M :: "('a, 'b) stateful_strand_step list"   and N :: "('a, 'b) stateful_strand_step list" assumes "set M \<subseteq> set N" shows "setops\<^sub>s\<^sub>s\<^sub>t M \<subseteq> setops\<^sub>s\<^sub>s\<^sub>t N"
span_sum: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'f set"   and f :: "'f \<Rightarrow> 'e"   and S :: "'e set" assumes "module scale"   and "\<And>x. x \<in> A \<Longrightarrow> f x \<in> module.span scale S" shows "sum f A \<in> module.span scale S"
intro: fixes G :: "('c, 'd) Graph_scheme" assumes "E\<^bsub>G\<^esub> \<subseteq> V\<^bsub>G\<^esub> \<times> V\<^bsub>G\<^esub>" shows "Digraph G"
check_method_access_def: fixes G :: "prog"   and accC :: "qtname"   and statT :: "ref_ty"   and mode :: "inv_mode"   and sig :: "sig"   and a' :: "val"   and s :: "state" shows "check_method_access G accC statT mode sig a' s = (let invC = invocation_class mode (snd s) a' statT; dynM = the (dynlookup G statT invC sig) in abupd (error_if (\<not> G \<turnstile>Methd sig dynM in invC dyn_accessible_from accC) AccessViolation) s)"
foldl_assoc: fixes b :: "('c \<Rightarrow> 'c) \<Rightarrow> ('c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c \<Rightarrow> 'c"   and y :: "'c \<Rightarrow> 'c"   and zs :: "('c \<Rightarrow> 'c) list" assumes "\<And>f g h. b f (b g h) = b (b f g) h" shows "foldl b (b x y) zs = b x (foldl b y zs)"
cINF_union: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and B :: "'c set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_below less_eq (f ` A)"   and "B \<noteq> {}"   and "preorder.bdd_below less_eq (f ` B)" shows "Inf (f ` (A \<union> B)) = inf (Inf (f ` A)) (Inf (f ` B))"
fc1: shows "f ` domain c1 \<subseteq> domain c2"
contD: fixes luba :: "'b set \<Rightarrow> 'b"   and orda :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and lubb :: "'c set \<Rightarrow> 'c"   and ordb :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c"   and Y :: "'b set" assumes "cont luba orda lubb ordb f"   and "Complete_Partial_Order.chain orda Y"   and "Y \<noteq> {}" shows "f (luba Y) = lubb (f ` Y)"
multi_arith_prog_def: fixes dims :: "nat"   and start :: "int"   and steps :: "nat \<Rightarrow> nat"   and c :: "nat \<Rightarrow> nat" shows "multi_arith_prog dims start steps c = start + int (\<Sum>i<dims. c i * steps i)"
abs_eq: fixes x :: "'a" shows "cblinfun_scaleC_left x = CBlinfun (\<lambda>y. y *\<^sub>C x)"
"3_5": fixes x :: "'b"   and y :: "'b" assumes "x \<in> set (x__ # xs)"   and "y \<in> set (y__ # ys)" shows "r2 (f x) (f y) = r1 x y"
arbitrary_union_of_alt: fixes Q :: "'d set \<Rightarrow> bool"   and S :: "'d set" shows "(arbitrary union_of Q) S = (\<forall>x\<in>S. \<exists>U. Q U \<and> x \<in> U \<and> U \<subseteq> S)"
simps: fixes D :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and S :: "(nat \<times> ('sf, 'sc, 'sz) step) list" shows "proof_result (D, S) = deriv_result (start_proof D) S"
override_on_def: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "override_on f g A = (\<lambda>a. if a \<in> A then g a else f a)"
clo: fixes U :: "'b set" assumes "closedin (top_of_set T) U" shows "closedin (top_of_set S) {x \<in> S. f x \<subseteq> U}"
decode_plan'_def: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and i :: "nat" shows "decode_plan' \<Pi> \<A> i \<equiv> let ops = \<Pi>\<^sub>\<O>; vs = map (\<lambda>op. Operator i (index ops op)) (remdups ops) in map (\<lambda>v. case v of Operator x k \<Rightarrow> ops ! k) (filter \<A> vs)"
Gcd_0_iff: fixes A :: "'a set" shows "(Gcd A = (0::'a)) = (A \<subseteq> {0::'a})"
eNorm_cmult: fixes N :: "'c quasinorm"   and c :: "real"   and x :: "'c" shows "eNorm N (c *\<^sub>R x) = ennreal \<bar>c\<bar> * eNorm N x"
cfun_map_beta: fixes a :: "'c \<rightarrow> 'd"   and b :: "'e \<rightarrow> 'b"   and f :: "'d \<rightarrow> 'e"   and x :: "'c" shows "cfun_map\<cdot>a\<cdot>b\<cdot>f\<cdot>x = b\<cdot>(f\<cdot>(a\<cdot>x))"
inverts_mat_symm: fixes A :: "'a mat"   and n :: "nat"   and B :: "'a mat" assumes "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "inverts_mat A B" shows "inverts_mat B A"
hom_mult: fixes x :: "rat"   and y :: "rat" shows "of_rat (x * y) = of_rat x * of_rat y"
reduction_def: fixes q :: "nat"   and \<A> :: "'d \<Rightarrow> (unit, 'b + 'a, 'c + unit) gpv"   and c_a :: "'d" shows "guessing_many_one.reduction q \<A> c_a = lift_spmf (spmf_of_set {..<q}) \<bind> (\<lambda>j_star. inline_stop (plus_intercept_stop guessing_many_one.process_call guessing_many_one.process_guess) (\<A> c_a) (Inr j_star) \<bind> (\<lambda>(uu_, s). Generative_Probabilistic_Value.Done (projl s)))"
strict_coboundedI2: fixes c :: "'d"   and b :: "'d"   and a :: "'d" assumes "c < b" shows "c < max a b"
division_ring_divide_zero: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.division_ring divide inverse one times plus zero minus uminus" shows "divide a zero = zero"
poly_const: fixes \<alpha> :: "'d \<Rightarrow> 'c"   and a :: "'c" shows "eval_poly \<alpha> (poly_const a) = a"
l_4: shows "(\<lambda>x. ereal (l x)) \<longlonglongrightarrow> a"
permute_eqvt: fixes p :: "perm"   and q :: "perm"   and x :: "'a" shows "p \<bullet> q \<bullet> x = (p \<bullet> q) \<bullet> p \<bullet> x"
subst2_fresh_switch: fixes \<phi> :: "'fmla"   and t :: "'trm"   and s :: "'trm"   and x :: "'var"   and y :: "'var" assumes "\<phi> \<in> fmla"   and "t \<in> trm"   and "s \<in> trm"   and "x \<in> var"   and "y \<in> var"   and "x \<noteq> y"   and "x \<notin> FvarsT s"   and "y \<notin> FvarsT t" shows "subst (subst \<phi> s y) t x = subst (subst \<phi> t x) s y"
gauge_integral_Fubini_universe_x_2: fixes f :: "'b \<times> 'c \<Rightarrow> 'd" assumes "integrable lborel f"   and "(\<lambda>x. integral UNIV (\<lambda>y. f (x, y))) \<in> borel_measurable lborel" shows "(\<lambda>x. integral UNIV (\<lambda>y. f (x, y))) integrable_on UNIV"
abs_summable_on_nat_iff': fixes f :: "nat \<Rightarrow> 'd" shows "(f abs_summable_on UNIV) = summable (\<lambda>n. norm (f n))"
neighbors_adj: fixes u :: "nat" shows "no u \<subseteq> adjacent_nodes u"
d'_1: shows "\<forall>t\<in>S_Min_Le. t \<le> d'"
option_admissible: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "option.admissible (\<lambda>f. \<forall>x y. f x = Some y \<longrightarrow> P x y)"
meh_simp: fixes s' :: "char list \<Rightarrow> int"   and t' :: "char list \<Rightarrow> int" shows "(SKIP, s') \<Rightarrow> t' = (s' = t')"
vifintersectionD: fixes a :: "V"   and I :: "V"   and f :: "V \<Rightarrow> V"   and i :: "V" assumes "a \<in>\<^sub>\<circ> \<Inter>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)"   and "i \<in>\<^sub>\<circ> I" shows "a \<in>\<^sub>\<circ> f i"
init_item_end: fixes r :: "symbol \<times> symbol list"   and k :: "nat" shows "item_end (init_item r k) = k"
prems_2: shows "(oid, ref) \<in> set (xs @ [x])"
inject_3: fixes x41 :: "char list list"   and x42 :: "nat"   and y41 :: "char list list"   and y42 :: "nat" shows "(Throwing x41 x42 = Throwing y41 y42) = (x41 = y41 \<and> x42 = y42)"
map_gpv_conv_bind: fixes f :: "'h \<Rightarrow> 'e"   and x :: "('h, 'f, 'g) gpv" shows "map_gpv f (\<lambda>x. x) x = x \<bind> (\<lambda>x. Generative_Probabilistic_Value.Done (f x))"
swap_in_def: fixes S :: "('b \<times> 'b) set"   and x :: "'b \<times> 'b" shows "swap_in S x = (if x \<in> S then prod.swap x else x)"
bl2wc_Bk_0: fixes m :: "nat" shows "bl2wc (Bk \<up> m) = 0"
integral_nonneg: fixes f :: "'a \<Rightarrow> real"   and S :: "'a set" assumes "f integrable_on S"   and "\<And>x. x \<in> S \<Longrightarrow> 0 \<le> f x" shows "0 \<le> integral S f"
separating_TER_plus_current: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a"   and y :: "'a"   and p :: "'a list" assumes "current \<Gamma> f"   and "wave \<Gamma> f"   and "current (quotient_web \<Gamma> f) g"   and "wave (quotient_web \<Gamma> f) g"   and "x \<in> RF\<^bsub>\<Gamma>\<^esub> (TER\<^bsub>\<Gamma>\<^esub> f)"   and "y \<in> B \<Gamma>"   and "path \<Gamma> x p y" shows "(\<exists>z\<in>set p. z \<in> TER\<^bsub>\<Gamma>\<^esub> (plus_current f g)) \<or> x \<in> TER\<^bsub>\<Gamma>\<^esub> (plus_current f g)"
compl_mono: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c"   and y :: "'c" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top"   and "less_eq x y" shows "less_eq (uminus y) (uminus x)"
l11_24: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.InAngle Bet P A B C" shows "Tarski_neutral_dimensionless.InAngle Bet P C B A"
D': shows "D' \<in> sets (borel \<Otimes>\<^sub>M borel \<Otimes>\<^sub>M borel)"
QP: shows "((\<forall>nm\<in>incoming n'. nm < name n') \<and> name n' \<le> name (fst (n, ns)) \<and> (incoming n' \<inter> incoming (fst (n, ns)) \<noteq> {} \<longrightarrow> new n' \<subseteq> old (fst (n, ns)) \<union> new (fst (n, ns)))) \<and> (\<forall>nd\<in>ns. \<forall>nm\<in>incoming n'. nm \<in> incoming nd \<longrightarrow> new n' \<subseteq> old nd)"
inverse_arrows_sym: fixes f :: "'d"   and g :: "'d" shows "D.inverse_arrows f g = D.inverse_arrows g f"
simps_3: fixes M :: "'b trm"   and N :: "'b trm" shows "vars_of (M \<cdot> N) = vars_of M \<union> vars_of N"
increasing5: fixes as :: "'a list"   and bs :: "'a list" assumes "increasing (as @ bs)" shows "increasing bs"
subspace_col_space: fixes A :: "(('a, 'c) vec, 'b) vec" shows "vec.subspace (col_space A)"
inv_fg_closure1: fixes l :: "(bool \<times> 'd) list" assumes "canceled l" shows "canceled (inv_fg l)"
VarNonOccForm_dbfm: fixes A :: "dbfm"   and i :: "name"   and e :: "(name, hf) finfun" assumes "wf_dbfm A"   and "atom i \<sharp> A" shows "VarNonOccForm (q_Var i) \<lbrakk>quot_dbfm A\<rbrakk>e"
dist_fs_cvec_def: fixes z1 :: "complex \<times> complex"   and z2 :: "complex \<times> complex" shows "dist_fs_cvec z1 z2 = (let (z1x, z1y) = z1; (z2x, z2y) = z2; num = (z1x * z2y - z2x * z1y) * (cnj z1x * cnj z2y - cnj z2x * cnj z1y); den = (z1x * cnj z1x + z1y * cnj z1y) * (z2x * cnj z2x + z2y * cnj z2y) in 2 * sqrt (Re num / Re den))"
compact_eq_Bolzano_Weierstrass: fixes S :: "'b set" shows "compact S = (\<forall>T. infinite T \<and> T \<subseteq> S \<longrightarrow> (\<exists>x\<in>S. x islimpt T))"
smallo_imp_abs_less_real: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "f \<in> o[F](g)"   and "\<forall>\<^sub>F x in F. 0 < g x" shows "\<forall>\<^sub>F x in F. \<bar>f x\<bar> < g x"
continuous_within_subset: fixes x :: "'c"   and s :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and t :: "'c set" assumes "continuous (at x within s) f"   and "t \<subseteq> s" shows "continuous (at x within t) f"
hyps: shows "xs \<noteq> [] \<longrightarrow> x \<in> dom (Cp (list2FWpolicy xs)) \<longrightarrow> Cp (list2FWpolicy (b @ xs)) x = Cp (list2FWpolicy xs) x"
FOREACHcd_rule: fixes S\<^sub>0 :: "'c set"   and I :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>\<^sub>0 :: "'d"   and c :: "'d \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd nres"   and \<Phi> :: "'d \<Rightarrow> bool" assumes "finite S\<^sub>0"   and "I {} S\<^sub>0 \<sigma>\<^sub>0"   and "\<And>S1 S2 x \<sigma>. \<lbrakk>S\<^sub>0 = insert x (S1 \<union> S2); I S1 (insert x S2) \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> SPEC (I (insert x S1) S2)"   and "\<And>S1 S2 \<sigma>. \<lbrakk>S\<^sub>0 = S1 \<union> S2; I S1 S2 \<sigma>; \<not> c \<sigma>\<rbrakk> \<Longrightarrow> \<Phi> \<sigma>"   and "\<And>\<sigma>. \<lbrakk>I S\<^sub>0 {} \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> \<Phi> \<sigma>" shows "FOREACHcd S\<^sub>0 c f \<sigma>\<^sub>0 \<le> SPEC \<Phi>"
beta_upright_def: fixes va :: "'b zmultiset"   and vb :: "'b zmultiset" shows "beta_upright va vb = (\<forall>t. 0 < zcount va t \<longrightarrow> (\<exists>s<t. zcount va s < 0 \<or> zcount vb s < 0))"
rev_below_trans: fixes below :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and y :: "'a"   and z :: "'a"   and x :: "'a" assumes "class.po below"   and "below y z"   and "below x y" shows "below x z"
subst_freshen_assumptions: fixes freshenLC :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'b \<Rightarrow> 'b"   and lconsts :: "'b \<Rightarrow> 'a set"   and closed :: "'b \<Rightarrow> bool"   and subst :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and subst_lconsts :: "'c \<Rightarrow> 'a set"   and subst_renameLCs :: "('a \<Rightarrow> 'a) \<Rightarrow> 'c \<Rightarrow> 'c"   and anyP :: "'b"   and antecedent :: "'d \<Rightarrow> ('b, 'a) antecedent list"   and consequent :: "'d \<Rightarrow> 'b list"   and rules :: "'d stream"   and assumptions :: "'b list"   and conclusions :: "'b list"   and pf :: "'b"   and s :: "'c"   and a :: "nat" assumes "Abstract_Task freshenLC renameLCs lconsts closed subst subst_lconsts subst_renameLCs anyP antecedent consequent rules assumptions conclusions"   and "pf \<in> set assumptions" shows "subst s (Abstract_Formulas.freshen freshenLC renameLCs a pf) = pf"
exponential_distributed_iff: fixes l :: "real"   and X :: "'a \<Rightarrow> real" assumes "0 < l" shows "distributed M lborel X (\<lambda>x. ennreal (exponential_density l x)) = (random_variable borel X \<and> (\<forall>a\<ge>0. prob {x \<in> space M. X x \<le> a} = 1 - exp (- a * l)))"
sorted_Cons: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "all_in_list (x # xs) l"   and "singleCombinators (x # xs)" shows "(FWNormalisationCore.sorted xs l \<and> (\<forall>y\<in>set xs. smaller x y l)) = FWNormalisationCore.sorted (x # xs) l"
LAssRedsVal: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and v :: "val"   and h' :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and l' :: "char list \<Rightarrow> val option"   and 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 V :: "char list"   and T :: "ty"   and v' :: "val" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>Val v,(h', l')\<rangle>"   and "E V = \<lfloor>T\<rfloor>"   and "P \<turnstile> T casts v to v' " shows "P,E \<turnstile> \<langle>V:=e,s\<rangle> \<rightarrow>* \<langle>Val v',(h', l'(V \<mapsto> v'))\<rangle>"
speq: shows "space M = space (fct_gen_subalgebra M N X)"
is_denormal_minus_float: fixes f :: "('a, 'b) IEEE.float" shows "is_denormal (- f) = is_denormal f"
east_west_tangents_1: shows "polar east = proj2_line_abs (vector [- 1, 0, 1])"
naturality: fixes A :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and B :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and G :: "'b \<Rightarrow> 'c"   and \<tau> :: "'b \<Rightarrow> 'c"   and f :: "'b" assumes "natural_transformation A B F G \<tau>"   and "partial_magma.arr A f" shows "B (\<tau> (partial_magma.cod A f)) (F f) = B (G f) (\<tau> (partial_magma.dom A f))"
orbit_eqI_2: fixes z :: "'a"   and f :: "'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a" assumes "z = f y"   and "y \<in> orbit f x" shows "z \<in> orbit f x"
ts'_def_1: shows "\<S>\<bullet>\<C> \<turnstile> vs' : [] _> ts'"
push_bit_word_beyond: fixes n :: "nat"   and w :: "'b word" assumes "LENGTH('b) \<le> n" shows "push_bit n w = 0"
vrange_atE: fixes r :: "V"   and b :: "V" assumes "vsv r"   and "b \<in>\<^sub>\<circ> \<R>\<^sub>\<circ> r" obtains a :: "V" where "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" and "r\<lparr>a\<rparr> = b"
open_halfspace_Im_lt: fixes b :: "real" shows "open {z. Im z < b}"
ex_mk_moebius: fixes M :: "moebius" shows "\<exists>a b c d. M = mk_moebius a b c d \<and> mat_det (a, b, c, d) \<noteq> 0"
zero_ts_ord: fixes x :: "'a" shows "o \<sqsubseteq> n x"
Root_CTRL_States_def: shows "Root_CTRL_States = {''CarAudioSystem''}"
assigned_def: fixes V :: "char list"   and e :: "char list exp" shows "assigned V e \<equiv> \<exists>v e'. e = V:=Val v;; e'"
simps_1: fixes b :: "bool"   and m :: "'b \<Rightarrow> 'b trie option" shows "isin (Nd b m) [] = b"
null_char: shows "B.VxV.null = (B.null, B.null)"
inject: fixes x :: "'d atMost \<Rightarrow> 'e atMost \<Rightarrow> 'c"   and ya :: "'d atMost \<Rightarrow> 'e atMost \<Rightarrow> 'c" shows "(Matrix x = Matrix ya) = (x = ya)"
in_indetsI: fixes x :: "'b"   and t :: "'b \<Rightarrow>\<^sub>0 nat"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "x \<in> keys t"   and "t \<in> keys p" shows "x \<in> indets p"
agree_UNIV_eq: fixes \<nu> :: "(real, 'd) vec \<times> (real, 'd) vec"   and \<omega> :: "(real, 'd) vec \<times> (real, 'd) vec" assumes "Vagree \<nu> \<omega> UNIV" shows "\<nu> = \<omega>"
intro: fixes S :: "'b set"   and pls :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<And>a b c. \<lbrakk>a \<in> S; b \<in> S; c \<in> S\<rbrakk> \<Longrightarrow> pls (pls a b) c = pls a (pls b c)"   and "\<And>a b. \<lbrakk>a \<in> S; b \<in> S\<rbrakk> \<Longrightarrow> pls a b \<in> S" shows "semigroup_add_ow S pls"
neg: fixes A :: "fmla"   and i :: "name"   and x :: "trm" shows "(neg A)(i::=x) = neg (A(i::=x))"
resultant_map_poly: fixes p :: "'a poly"   and q :: "'a poly" assumes "degree (map_poly hom p) = degree p"   and "degree (map_poly hom q) = degree q" shows "resultant (map_poly hom p) (map_poly hom q) = hom (resultant p q)"
r_null: fixes R :: "('a, 'b) ring_scheme"   and x :: "'a" assumes "semiring R"   and "x \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = \<zero>\<^bsub>R\<^esub>"
exec_gpv_parametric': fixes S :: "'h \<Rightarrow> 'k \<Rightarrow> bool"   and CALL :: "'i \<Rightarrow> 'l \<Rightarrow> bool"   and R :: "'j \<Rightarrow> 'm \<Rightarrow> bool"   and A :: "'n \<Rightarrow> 'o \<Rightarrow> bool" shows "rel_fun (rel_fun S (rel_fun CALL (rel_spmf (rel_prod R S)))) (rel_fun (rel_gpv'' A CALL R) (rel_fun S (rel_spmf (rel_prod A S)))) exec_gpv exec_gpv"
BFs_2: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[\<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>\<rightarrow> (\<^bold>\<forall>\<alpha>. \<^bold>\<box>\<phi> \<alpha>) in v]"
assms: shows "\<exists>A\<in>unit_disc. set_order A \<phi> \<psi>"
a: shows "a = [:- x, 1::'a:] ^ Suc i"
simps_2: fixes P :: "'c \<Rightarrow> bool" shows "enforce_option P None = None"
ccSUP_empty: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "class.countable_complete_lattice Inf Sup inf less_eq less sup bot top" shows "Sup (f ` {}) = bot"
simps: fixes P :: "'d prog"   and ad :: "'addr"   and al :: "addr_loc" shows "vs_type_all P (ad, al) = {v. \<exists>T. P \<turnstile> ad@al : T \<and> P \<turnstile> v :\<le> T}"
QTrans: shows "perm \<bullet> Q \<longmapsto> \<alpha> \<prec> Q'"
isS: shows "statediff \<omega> \<omega>' \<subseteq> - FVF \<phi>"
snd_fst_flip: fixes xy :: "'b \<times> 'a" shows "snd xy = (cc \<circ>\<circ> case_prod) (\<lambda>x y. (y, x)) xy"
c_pair_is_pr: shows "c_pair \<in> PrimRec2"
l3_dy_valid_refines_dy_fake_chan: fixes M :: "msg"   and M' :: "chan" assumes "M \<in> valid"   and "M' \<in> local.abs {M}" shows "{R23s \<inter> UNIV \<times> (l3_inv5 \<inter> l3_inv6 \<inter> l3_inv7 \<inter> l3_inv8 \<inter> l3_inv9)} l2_dy_fake_chan M', l3_dy M {> R23s}"
PO_refines_implies_Range_invariant: fixes R :: "('e \<times> 'f) set"   and Ta :: "('e, 'g) TS_scheme"   and Tc :: "('f, 'h) TS_scheme" assumes "PO_refines R Ta Tc" shows "reach Tc \<subseteq> Range R"
assms_2: shows "(G ===> (=)) \<alpha>e1 \<alpha>e2"
disjoint_fun_syms: shows "(\<Union> (range \<Sigma>n) - \<C>) \<inter> \<C> = {}"
simps: fixes i :: "nat"   and j :: "nat" shows "gs.\<mu> i j = (if j < i then map of_int_hom.vec_hom fs ! i \<bullet> gs.gso j / \<parallel>gs.gso j\<parallel>\<^sup>2 else if i = j then 1 else 0)"
eq2: shows "(\<lambda>X. X \<union> {x \<in> B - B'. f x = X}) ` P = (\<lambda>X. X \<union> {x \<in> B - B'. g x = X}) ` P"
orderE: fixes b :: "'b"   and a :: "'b" assumes "b \<le> a" obtains "a = sup_class.sup a b"
abs_1: fixes abs :: "'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'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.idom_abs_sgn abs sgn times one plus zero minus uminus" shows "abs one = one"
HStartBallot_blocksOf: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc" assumes "HStartBallot s s' p" shows "blocksOf s' q \<subseteq> blocksOf s q \<union> {dblock s' q}"
execn_callStuck: fixes \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and p :: "'e"   and bdy :: "('d, 'e, 'f) com"   and init :: "'d \<Rightarrow> 'd"   and s :: "'d"   and n :: "nat"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'e, 'f) com" assumes "\<Gamma> p = Some bdy"   and "\<Gamma>\<turnstile> \<langle>bdy,Normal (init s)\<rangle> =n\<Rightarrow> Stuck" shows "\<Gamma>\<turnstile> \<langle>call init p return c,Normal s\<rangle> =Suc n\<Rightarrow> Stuck"
simps_3: fixes x :: "'b trm"   and y :: "'b trm" shows "is_a_constant (x \<cdot> y) = False"
language_state: fixes M :: "('d, 'e, 'f) FSM"   and w :: "('d \<times> 'e) list"   and r :: "'f list"   and q :: "'f" assumes "path M (w || r) q"   and "length w = length r" shows "w \<in> LS M q"
exp_conj_def: fixes a :: "'a \<Rightarrow> real"   and b :: "'a \<Rightarrow> real" shows "a && b \<equiv> \<lambda>s. a s .& b s"
Push_Node_inject: fixes i :: "'b + nat"   and m :: "('a, 'b) node"   and j :: "'b + nat"   and n :: "('a, 'b) node" assumes "Push_Node i m = Push_Node j n" obtains "i = j" and "m = n"