hcju/isabelleps huggingface dataset:
is_rreq_def: fixes \<xi> :: "'c state_scheme" shows "is_rreq \<xi> \<equiv> case msg \<xi> of Rreq hops' dip' dsn' dsk' oip' osn' sip' \<Rightarrow> {\<xi>\<lparr>hops := hops', dip := dip', dsn := dsn', dsk := dsk', oip := oip', osn := osn', sip := sip'\<rparr>} | _ \<Rightarrow> {}"
iwfH: fixes c :: "'form"   and \<Gamma> :: "'form fset"   and s :: "'subst"   and i :: "nat"   and fc :: "('form, 'rule) graph_node \<Rightarrow> nat \<Rightarrow> 'subst \<Rightarrow> 'form fset \<times> 'form \<Rightarrow> bool" assumes "c |\<notin>| ass_forms"   and "c |\<in>| \<Gamma>"   and "c = subst s (freshen i anyP)" shows "iwf fc (HNode i s []) (\<Gamma> \<turnstile> c)"
le_log_iff: fixes b :: "real"   and x :: "real"   and y :: "real" assumes "1 < b"   and "0 < x" shows "(y \<le> log b x) = (b powr y \<le> x)"
bex_one_point2: fixes A :: "'d set"   and a :: "'d"   and P :: "'d \<Rightarrow> bool" shows "(\<exists>x\<in>A. a = x \<and> P x) = (a \<in> A \<and> P a)"
FvarsT_VarD: fixes x :: "'var"   and y :: "'var" assumes "x \<in> FvarsT (Var y)"   and "y \<in> var" shows "x = y"
cases: fixes x :: "nat \<times> atom" obtains k :: "nat" where "x = (k__, Less 0 0)"   | k :: "nat" and j :: "nat" where "x = (ka__, Less 0 (Suc j__))"   | k :: "nat" and i :: "nat" where "x = (kb__, Less (Suc i__) 0)"   | k :: "nat" and i :: "nat" and j :: "nat" where "x = (kc__, Less (Suc ia__) (Suc ja__))"   | k :: "nat" where "x = (kd__, Eq 0 0)"   | k :: "nat" and v :: "nat" where "x = (ke__, Eq 0 (Suc v__))"   | k :: "nat" and v :: "nat" where "x = (kf__, Eq (Suc v) 0)"   | k :: "nat" and i :: "nat" and j :: "nat" where "x = (k, Eq (Suc i) (Suc j))"
inverts_mat_symm: fixes A :: "'b mat"   and n :: "nat"   and B :: "'b 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"
fps_compose_divide: fixes g :: "'a fps"   and f :: "'a fps"   and h :: "'a fps" assumes "g dvd f"   and "fps_nth h 0 = (0::'a)" shows "f oo h = (f / g oo h) * (g oo h)"
triangle_right_implies_left: assumes "Cop.H f \<epsilon> \<cdot> Cop.\<a> f g f \<cdot> Cop.H \<eta> f = Cop.runit' f \<cdot> Cop.lunit f" shows "Cop.H \<epsilon> g \<cdot> Cop.\<a>' g f g \<cdot> Cop.H g \<eta> = Cop.lunit' g \<cdot> Cop.runit g"
fps_dvd_code: fixes x :: "'b fps"   and y :: "'b fps" shows "(x dvd y) = (y = 0 \<or> x \<noteq> 0 \<and> subdegree x \<le> subdegree y)"
assms_2: fixes x :: "'a" shows "norm (f x) \<le> b * norm x"
indicator_inter_arith: fixes A :: "'b set"   and B :: "'b set"   and x :: "'b" shows "indicator (A \<inter> B) x = indicator A x * indicator B x"
pairwiseD: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a set"   and x :: "'a"   and y :: "'a" assumes "pairwise R S"   and "x \<in> S"   and "y \<in> S"   and "x \<noteq> y" shows "R x y"
normal: fixes t :: "'t"   and x :: "'x" assumes "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "\<not> final x"   and "wset s t = None" shows "t \<turnstile> \<langle>x, shr s\<rangle> \<wrong> \<and> (\<forall>LT. t \<turnstile> \<langle>x, shr s\<rangle> LT \<wrong> \<longrightarrow> (\<exists>lt\<in>LT. must_wait s t lt (dom (thr s))))"
absolutely_integrable_min: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "f absolutely_integrable_on S"   and "g absolutely_integrable_on S" shows "(\<lambda>x. \<Sum>i\<in>Basis. min (f x \<bullet> i) (g x \<bullet> i) *\<^sub>R i) absolutely_integrable_on S"
i_Equiv_Exec_Stream_expand_shrink_output_set_eq: fixes k1 :: "nat"   and k2 :: "nat"   and equiv_states :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and localState1 :: "'f \<Rightarrow> 'd"   and c1 :: "'f"   and localState2 :: "'g \<Rightarrow> 'e"   and c2 :: "'g"   and A :: "'h set"   and input_fun1 :: "'h \<Rightarrow> 'i message_af"   and input_fun2 :: "'h \<Rightarrow> 'j message_af"   and output_fun1 :: "'f \<Rightarrow> 'k message_af"   and output_fun2 :: "'g \<Rightarrow> 'k message_af"   and trans_fun1 :: "'i message_af \<Rightarrow> 'f \<Rightarrow> 'f"   and trans_fun2 :: "'j message_af \<Rightarrow> 'g \<Rightarrow> 'g"   and input :: "nat \<Rightarrow> 'h" assumes "0 < k1"   and "0 < k2"   and "equiv_states (localState1 c1) (localState2 c2)"   and "\<And>input0 m. \<lbrakk>set input0 \<subseteq> A; m \<in> A\<rbrakk> \<Longrightarrow> Equiv_Exec m equiv_states localState1 localState2 input_fun1 input_fun2 output_fun1 output_fun2 trans_fun1 trans_fun2 k1 k2 (f_Exec_Comp trans_fun1 (map input_fun1 input0 \<odot> k1) c1) (f_Exec_Comp trans_fun2 (map input_fun2 input0 \<odot> k2) c2)"   and "range input \<subseteq> A" shows "(output_fun1 \<circ> i_Exec_Comp_Stream trans_fun1 ((input_fun1 \<circ> input) \<odot> k1) c1) \<div> k1 = (output_fun2 \<circ> i_Exec_Comp_Stream trans_fun2 ((input_fun2 \<circ> input) \<odot> k2) c2) \<div> k2"
idinfo_mpair_snd: fixes Y :: "msg"   and n :: "nat"   and X :: "msg"   and s :: "(agent \<times> msg) set" assumes "(Spy, \<lbrace>Y, \<langle>n, X\<rangle>\<rbrace>) \<in> s"   and "s\<^sub>0 \<Turnstile> s" shows "\<langle>n, X\<rangle> \<in> spied s"
ls: shows "\<Union> (A ` {..<k}) \<lless> A k"
s_2: fixes n :: "nat"   and i :: "nat" shows "f (s n i) \<in> A (Suc n)"
length_def: fixes as :: "'d iarray" shows "IArray.length as = length (IArray.list_of as)"
lmap_llist_of: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "lmap f (llist_of xs) = llist_of (map f xs)"
in_quotientP_imp_in_rel: fixes P :: "'c \<Rightarrow> bool"   and \<phi> :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and X :: "'c set"   and x :: "'c"   and y :: "'c" assumes "equivP P \<phi>"   and "(P /// \<phi>) X"   and "x \<in> X"   and "y \<in> X" shows "\<phi> x y"
hb_consistent_prefix_suffix_exists: fixes ys :: "'a list"   and xs :: "'a list"   and x :: "'a" assumes "hb_consistent ys"   and "hb_consistent (xs @ [x])"   and "{x} \<union> set xs = set ys"   and "distinct (x # xs)"   and "distinct ys" shows "\<exists>prefix suffix. ys = prefix @ x # suffix \<and> concurrent_set x suffix"
LtK_Exp: fixes X :: "ltkey"   and X' :: "msg"   and Y' :: "msg" shows "LtK X \<noteq> Exp X' Y'"
gt_ex: fixes less_eq :: "'m \<Rightarrow> 'm \<Rightarrow> bool"   and less :: "'m \<Rightarrow> 'm \<Rightarrow> bool"   and x :: "'m" assumes "class.no_top less_eq less" shows "\<exists>y. less x y"
rev_min_pm1: fixes a :: "nat"   and b :: "nat" shows "a - b + min b a = a"
B: fixes as :: "('b poly \<times> nat) set"   and p :: "'b poly"   and n :: "nat" assumes "monic_factorization as p" shows "monic_factorization.B p n = (\<Prod>(a, i)\<in>as - UNIV \<times> {0..<n}. a)"
Min_eq_iff: fixes A :: "'a set"   and m :: "'a" assumes "finite A"   and "A \<noteq> {}" shows "(linorder.Min (\<sqsubseteq>) A = m) = (m \<in> A \<and> (\<forall>a\<in>A. m \<sqsubseteq> a))"
sup_idem: fixes x :: "'c \<times> 'd" shows "sup_prod x x = x"
fv: fixes \<phi> :: "'fmla" assumes "set (map snd txs) \<subseteq> var"   and "set (map fst txs) \<subseteq> trm"   and "set [subst \<phi> s y] \<subseteq> fmla"   and "\<phi> \<in> set [subst \<phi> s y]" shows "set (getFrN (map snd txs) (map fst txs) [subst \<phi> s y] (length txs)) \<inter> Fvars \<phi> = {}"
B_Un_1: shows "basis_in (X \<union> Y) B_Un"
parts_sign_start: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and n :: "nat"   and A :: "nat" assumes "s\<^sub>0 \<Turnstile> s"   and "s \<turnstile> s'"   and "Sign n A \<in> parts (used s')"   and "Sign n A \<notin> parts (used s)" shows "A = Auth_PriK n"
comp_in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>S b\<guillemotright>"   and "\<guillemotleft>g : b \<Rightarrow>\<^sub>S c\<guillemotright>" shows "\<guillemotleft>g \<cdot>\<^sub>S f : a \<Rightarrow>\<^sub>S c\<guillemotright>"
grid_level_d: fixes d :: "nat"   and b :: "(nat \<times> int) list"   and p :: "(nat \<times> int) list" assumes "d < length b"   and "p \<in> grid b {d}"   and "p \<noteq> b" shows "lv b d < lv p d"
arrE: fixes f :: "('obj, 'obj) concrete_category.arr" assumes "arr f" obtains "f \<noteq> Null" and "Dom f \<in> Obj" and "Cod f \<in> Obj" and "Map f \<in> (if Dom f = Cod f then {Dom f} else {})"
filter_valid_lists': fixes xs :: "Vars.var list"   and S :: "Vars.var set"   and P :: "Vars.var \<Rightarrow> bool"   and G :: "CoCalls" assumes "xs \<in> valid_lists {x' \<in> S. P x'} G" shows "xs \<in> filter P ` valid_lists S G"
distinct_1: fixes x1 :: "variable"   and x2 :: "object" shows "term.VAR x1 \<noteq> term.CONST x2"
mult_float_intervalI: fixes x :: "real"   and A :: "float interval"   and y :: "real"   and B :: "float interval"   and prec :: "nat" assumes "x \<in>\<^sub>r A"   and "y \<in>\<^sub>r B" shows "x * y \<in>\<^sub>r mult_float_interval prec A B"
proj_measurable_singleton: fixes A :: "'d set"   and M :: "'e \<Rightarrow> 'd measure"   and i :: "'e"   and I :: "'e set" assumes "A \<in> sets (M i)" shows "(\<lambda>x. (x)\<^sub>F i) -` A \<inter> space (Pi\<^sub>F {I} M) \<in> sets (Pi\<^sub>F {I} M)"
good1: shows "is_interp (local.adjoint I \<sigma> \<nu>)"
select_convs_6: fixes cLengthF :: "rat"   and cMassF :: "rat"   and cTimeF :: "rat"   and cCurrentF :: "rat"   and cTemperatureF :: "rat"   and cAmountF :: "rat"   and cIntensityF :: "rat"   and more :: "'c" shows "cAmountF \<lparr>cLengthF = cLengthF, cMassF = cMassF, cTimeF = cTimeF, cCurrentF = cCurrentF, cTemperatureF = cTemperatureF, cAmountF = cAmountF, cIntensityF = cIntensityF, \<dots> = more\<rparr> = cAmountF"
assms_3: shows "\<guillemotleft>g : G.map\<^sub>0 a \<rightarrow>\<^sub>D G.map\<^sub>0 a'\<guillemotright>"
value_flow: fixes \<Delta> :: "('a, 'b) network_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and x :: "'a" assumes "countable_network \<Delta>"   and "flow \<Delta> f"   and "\<And>y. edge \<Delta> (source \<Delta>) y = (y = x)" shows "value_flow \<Delta> f = f (source \<Delta>, x)"
type_definition_ennreal': shows "type_definition enn2ereal e2ennreal {x. 0 \<le> x}"
poly_shift_subring_closed: fixes S :: "'a set"   and g :: "nat \<Rightarrow> 'a" assumes "subring S R"   and "g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))" shows "poly_shift g \<in> carrier (UP (R\<lparr>carrier := S\<rparr>))"
le: shows "(x = 0) = (int_of_integer x = 0)"
ccpo: fixes ord :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and lb :: "'f set \<Rightarrow> 'f" assumes "partial_function_definitions ord lb" shows "class.ccpo lb ord (mk_less ord)"
WSE_1: shows "Ball (set ms\<^sub>1) valid_node"
is_lb_insert: fixes x :: "'a"   and A :: "'a set"   and y :: "'a" shows "insert x A >| y = (y \<sqsubseteq> x \<and> A >| y)"
simps_1: fixes uu :: "'a \<Rightarrow> 'b tm list \<Rightarrow> bool"   and uv :: "('b, 'a) fm \<Rightarrow> bool" shows "boolean uu uv \<^bold>\<bottom> = False"
neq_le_trans: fixes a :: "'a list"   and b :: "'a list" assumes "a \<noteq> b"   and "prefix a b" shows "strict_prefix a b"
sngs: fixes z :: "complex" assumes "Re z \<noteq> 0 \<or> 0 < Im z" shows "(exp z = 1) = (z \<in> sngs)"
assms: fixes F :: "atom fm"   and xs :: "real list" shows "eval F xs = eval (step F) xs"
rec: fixes m :: "'node" assumes "v \<notin> defs g n"   and "predecessors g n = [m]"   and "m \<in> set (\<alpha>n g)"   and "v' = lookupDef g m v" shows "thesis"
exhaust: fixes y :: "\<langle>'d\<rangle>\<^sub>\<bottom>" obtains "y = None"   | x2 :: "'d" where "y = \<lfloor>x2\<rfloor>"
oseqp_sos_subreachable: fixes OA :: "((nat \<Rightarrow> 'e) \<times> ('e, 'f, 'g, 'h) seqp, 'f seq_action) automaton"   and \<Gamma> :: "'g \<Rightarrow> ('e, 'f, 'g, 'h) seqp"   and i :: "nat"   and ANY :: "'e \<Rightarrow> 'e \<Rightarrow> bool" assumes "automaton.trans OA = oseqp_sos \<Gamma> i" shows "subreachable OA (other ANY {i}) {i}"
a1: shows "low_equal s1 s2 \<and> get_delayed_pool s1 = [] \<and> get_delayed_pool s2 = []"
simps_7: fixes a :: "floatarith"   and b :: "floatarith"   and vs :: "real list" shows "interpret_floatarith (Min a b) vs = min (interpret_floatarith a vs) (interpret_floatarith b vs)"
lenlex_transI: fixes r :: "('g \<times> 'g) set" assumes "trans r" shows "trans (lenlex r)"
rel_cases: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "('e, 'a, 'c, 'f) converter"   and b :: "('e, 'b, 'd, 'f) converter" assumes "rel_converter' R1 R2 a b" obtains xa :: "'e \<Rightarrow> ('a \<times> ('e, 'a, 'c, 'f) converter, 'c, 'f) gpv" and ya :: "'e \<Rightarrow> ('b \<times> ('e, 'b, 'd, 'f) converter, 'd, 'f) gpv" where "a = Converter xa" and "b = Converter ya" and "rel_fun (=) (rel_gpv (rel_prod R1 (rel_converter' R1 R2)) R2) xa ya"
Un_eq_UN: fixes A :: "'c set"   and B :: "'c set" shows "A \<union> B = (\<Union>b. if b then A else B)"
assms: shows "\<exists>P'. P \<Longrightarrow>\<^sub>l \<alpha> \<prec> P' \<and> (P', Q') \<in> Rel"
expand_init_def: shows "expand_init \<equiv> 0::'b"
tmbound_split0: fixes t :: "tm"   and vs :: "'b list" assumes "tmbound 0 t" shows "\<lparr>fst (split0 t)\<rparr>\<^sub>p\<^bsup>vs\<^esup> = (0::'b)"
im_sheaf_def: fixes V :: "'c set" shows "local.im_sheaf V \<equiv> \<O>\<^sub>X (f \<^sup>\<inverse> X V)"
linear_comb0_1: fixes M :: "('e, 'f, 'g) Module_scheme"   and R :: "('f, 'h) Ring_scheme"   and H :: "'e set"   and s :: "nat \<Rightarrow> 'f"   and n :: "nat"   and m :: "nat \<Rightarrow> 'e" assumes "R module M"   and "H \<subseteq> carrier M"   and "s \<in> {j. j \<le> n} \<rightarrow> {\<zero>\<^bsub>R\<^esub>}"   and "m \<in> {j. j \<le> n} \<rightarrow> H" shows "l_comb R M n s m = \<zero>\<^bsub>M\<^esub>"
gcd_mult_left: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "gcd (times c a) (times c b) = normalize (times c (gcd a b))"
observe_execution_possible_step: fixes e :: "((nat \<times> nat) \<times> transition) fset"   and s :: "nat"   and r :: "nat \<Rightarrow>f value option"   and h :: "String.literal \<times> value list"   and s' :: "nat"   and t :: "transition"   and p :: "value option list"   and r' :: "nat \<Rightarrow>f value option"   and es :: "(String.literal \<times> value list) list"   and obs :: "value option list list" assumes "possible_steps e s r (fst h) (snd h) = {|(s', t)|}"   and "evaluate_outputs t (snd h) r = p"   and "evaluate_updates t (snd h) r = r'"   and "observe_execution e s' r' es = obs" shows "observe_execution e s r (h # es) = p # obs"
rep_tensor: fixes f :: "'c local.term set"   and g :: "'c local.term set" assumes "arr f"   and "arr g" shows "rep (f \<otimes> g) = \<^bold>\<parallel>rep f \<^bold>\<otimes> rep g\<^bold>\<parallel>"
fls_times_nth_eq0: fixes n :: "int"   and f :: "'a fls"   and g :: "'a fls" assumes "n < fls_subdegree f + fls_subdegree g" shows "fls_nth (f * g) n = (0::'a)"
star_unfoldl: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and star :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "class.left_near_kleene_algebra plus times one less_eq less star" shows "less_eq (plus one (times x (star x))) (star x)"
Join_left_commute: fixes A :: "'c program"   and B :: "'c program"   and C :: "'c program" shows "A \<squnion> (B \<squnion> C) = B \<squnion> (A \<squnion> C)"
subspace_sums: fixes S :: "('a, 'b) vec set"   and T :: "('a, 'b) vec set" assumes "vec.subspace S"   and "vec.subspace T" shows "vec.subspace {x + y |x y. x \<in> S \<and> y \<in> T}"
a_redu_OrL_elim: fixes x :: "name"   and M :: "trm"   and y :: "name"   and N :: "trm"   and z :: "name"   and R :: "trm" assumes "OrL x.M y.N z \<longrightarrow>\<^sub>a R" shows "(\<exists>M'. R = OrL x.M' y.N z \<and> M \<longrightarrow>\<^sub>a M') \<or> (\<exists>N'. R = OrL x.M y.N' z \<and> N \<longrightarrow>\<^sub>a N')"
bacc_upper_bound: fixes r :: "('a \<times> 'a) set" shows "bacc r (card UNIV) = \<Union> (range (bacc r))"
Sup_fun_def: fixes A :: "('a \<Rightarrow> 'b) set" shows "\<Squnion>A = (\<lambda>x. SUP f\<in>A. f x)"
prod_list_replicate: fixes n :: "nat"   and a :: "'a" shows "prod_list (replicate n a) = a ^ n"
append_Cons: fixes x :: "'d"   and xs :: "'d list"   and ys :: "'d list" shows "(x # xs) @ ys = x # xs @ ys"
mn_loop_correct': fixes B :: "abc_inst list"   and arity :: "nat"   and fap :: "abc_inst list"   and ft :: "nat"   and fft :: "nat"   and xs :: "nat list"   and x :: "nat"   and far :: "nat"   and f :: "recf"   and anything :: "nat list" assumes "B = [Dec (Suc arity) (length fap + 5), Dec (Suc arity) (length fap + 3), Goto (Suc (length fap)), Inc arity, Goto 0]"   and "ft = max (Suc arity) fft"   and "length xs = arity"   and "\<forall>i\<le>x. \<forall>xc. {\<lambda>nl. nl = xs @ i # 0 \<up> (fft - far) @ xc} fap {\<lambda>nl. nl = xs @ i # rec_exec f (xs @ [i]) # 0 \<up> (fft - Suc far) @ xc}"   and "\<forall>i\<le>x. 0 < rec_exec f (xs @ [i])"   and "rec_ci f = (fap, far, fft)"   and "far = Suc arity" shows "\<exists>stp>x. abc_steps_l (0, xs @ 0 # 0 \<up> (ft - Suc arity) @ anything) (fap @ B) stp = (0, xs @ Suc x # 0 \<up> (ft - Suc arity) @ anything)"
algebraic_divide_cancel_right: fixes y :: "'a"   and x :: "'a" assumes "algebraic y"   and "y \<noteq> (0::'a)" shows "algebraic (x / y) = algebraic x"
KripkeStructure_ext_def: fixes worlds :: "'g set"   and relations :: "'e \<Rightarrow> ('g \<times> 'g) set"   and valuation :: "'g \<Rightarrow> 'f \<Rightarrow> bool"   and more :: "'h" shows "\<lparr>worlds = worlds, relations = relations, valuation = valuation, \<dots> = more\<rparr> \<equiv> Record.iso_tuple_cons KripkeStructure_ext_Tuple_Iso (Record.iso_tuple_cons Record.tuple_iso_tuple worlds (Record.iso_tuple_cons Record.tuple_iso_tuple relations valuation)) more"
coprime_mult_left_iff: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "algebraic_semidom.coprime times one (times a b) c = (algebraic_semidom.coprime times one a c \<and> algebraic_semidom.coprime times one b c)"
list_update_length: fixes xs :: "'g list"   and x :: "'g"   and ys :: "'g list"   and y :: "'g" shows "(xs @ x # ys)[length xs := y] = xs @ y # ys"
hom_int_pow: fixes x :: "'a"   and n :: "int" assumes "x \<in> carrier G" shows "h (x [^] n) = h x [^]\<^bsub>H\<^esub> n"
le_mult: fixes \<beta> :: "V"   and \<alpha> :: "V" assumes "Ord \<beta>"   and "\<beta> \<noteq> 0" shows "\<alpha> \<subseteq>\<^sub>\<circ> \<alpha> * \<beta>"
assms_7: shows "r\<^sup>T * g\<^sup>\<star> \<le> r\<^sup>T * w\<^sup>\<star>"
INV_init_from_refinement: fixes R :: "('e \<times> 'f) set"   and Ta :: "('e, 'g) TS_scheme"   and Tc :: "('f, 'h) TS_scheme"   and I :: "'f set" assumes "PO_refines R Ta Tc"   and "Range R \<subseteq> I" shows "init Tc \<subseteq> I"
iAll_subset: fixes A :: "nat set"   and B :: "nat set"   and P :: "nat \<Rightarrow> bool" assumes "A \<subseteq> B"   and "\<box> t B. P t" shows "\<box> t A. P t"
intros_5: fixes \<tau> :: "'a OCL_Types.type" shows "to_ordered_collection (Sequence \<tau>) (Sequence \<tau>)"
Nil_in_Prefixes_iff: fixes A :: "'a list set" shows "([] \<in> Prefixes A) = (A \<noteq> {})"
end_smult_comp_comm_left: fixes a :: "'f"   and T :: "'v \<Rightarrow> 'v"   and S :: "'v \<Rightarrow> 'v" shows "a \<cdot>\<cdot> T \<circ> S = a \<cdot>\<cdot> (T \<circ> S)"
seq_doesnt_modify: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and x :: "'Var"   and c' :: "('Var, 'AExp, 'BExp) Stmt" assumes "doesnt_modify c x" shows "doesnt_modify (c ;; c') x"
in_set_spmf: fixes x :: "'h"   and p :: "'h spmf" shows "(x \<in> set_spmf p) = (Some x \<in> set_pmf p)"
asm: shows "steps0 (1, tp) A (Suc n) = (0, tp')"
le_finI: fixes w\<^sub>1 :: "'action list"   and v\<^sub>1 :: "'action list"   and w\<^sub>2 :: "'action list" assumes "w\<^sub>1 @ v\<^sub>1 =\<^sub>F w\<^sub>2" shows "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2"
abs_ln_one_plus_x_minus_x_bound: fixes x :: "real" assumes "\<bar>x\<bar> \<le> 1 / 2" shows "\<bar>ln (1 + x) - x\<bar> \<le> 2 * x\<^sup>2"
nsets_eq_empty: fixes A :: "'a set"   and r :: "nat" assumes "finite A"   and "card A < r" shows "[A]\<^bsup>r\<^esup> = {}"
frag_extend_diff: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow>\<^sub>0 int"   and a :: "'d \<Rightarrow>\<^sub>0 int"   and b :: "'d \<Rightarrow>\<^sub>0 int" shows "frag_extend f (a - b) = frag_extend f a - frag_extend f b"
partial_function_definitions_axioms: shows "partial_function_definitions local.tllist_ord local.tSup"
A'_2: shows "(f ^^ m') x = (f ^^ n') x"
med01iar_def: fixes t :: "l1_state" shows "med01iar t \<equiv> \<lparr>signals = signalsResp t\<rparr>"
z2n1_def: shows "z2n1 = z2 * n1"
simps_12: fixes v :: "\<I>"   and va :: "Formula.formula" shows "remove_neg (formula.Prev v va) = formula.Prev v va"
reachable: fixes as :: "'b list"   and r :: "'b rexp" shows "reachable_a as r = {fold delta_a w (init_a r) |w. w \<in> lists (set as)}"
cauchy_imp_bounded: fixes s :: "nat \<Rightarrow> 'b" assumes "Cauchy s" shows "bounded (range s)"
add_comm': fixes times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.pas times D"   and "D x y" shows "times x y = times y x"
map_simp_1: fixes A :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and B :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and F :: "'d \<Rightarrow> 'e"   and G :: "'d \<Rightarrow> 'e"   and H :: "'d \<Rightarrow> 'e"   and \<sigma> :: "'d \<Rightarrow> 'e"   and \<tau> :: "'d \<Rightarrow> 'e"   and f :: "'d" assumes "vertical_composite A B F G H \<sigma> \<tau>"   and "partial_magma.arr A f" shows "vertical_composite.map A B \<sigma> \<tau> f = B (\<tau> (partial_magma.cod A f)) (\<sigma> f)"
distinct_remdups: fixes xs :: "'b list" shows "distinct (remdups xs)"
bind_SE'_def: fixes f :: "'b \<Rightarrow> ('d \<times> 'b) option"   and g :: "'b \<Rightarrow> ('c \<times> 'b) option" shows "f ;- g = _bind_SE _ f g"
that: fixes x :: "'a"   and y :: "'b" shows "A x y \<longrightarrow> A' x y"
lm077: shows "inj_on pseudoAllocation (runiqs \<inter> Pow (UNIV \<times> (UNIV - {{}})))"
incoming'_def: fixes k :: "nat set" shows "incoming' k \<equiv> {(u, v) |u v. u \<notin> k \<and> v \<in> k \<and> (u, v) \<in> E}"
sum_upto_mangoldt_over_id_minus_phi_bounded: shows "(\<lambda>x. sum_upto (\<lambda>d. mangoldt d / real d) x - primes_M x) \<in> O(\<lambda>_. 1)"
slpa_split_1: fixes cs :: "'edge list"   and as :: "'edge list"   and xs :: "'edge list"   and ys :: "'edge list" assumes "same_level_path_aux cs as"   and "as = xs @ ys"   and "upd_cs cs xs = []" shows "same_level_path_aux cs xs"
is_hashmap'_def: fixes m :: "'c \<Rightarrow> 'd option"   and l :: "('c \<times> 'd) list list"   and ht :: "('c, 'd) hashtable" shows "is_hashmap' m l ht = is_hashtable l ht * \<up> (map_of (concat l) = m)"
S_list_not_min_gallery_not_reduced: fixes ss :: "'a permutation list" assumes "ss \<noteq> []"   and "\<not> min_gallery (map (\<lambda>w. w `\<rightarrow> C0) (sums ss))" shows "\<not> reduced_word (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) ss"
process_prop_4_def: fixes P :: "('a list \<times> 'a set) set \<times> 'a list set" shows "process_prop_4 P \<equiv> \<forall>xs x X. (xs, X) \<in> fst P \<longrightarrow> (xs @ [x], {}) \<in> fst P \<or> (xs, insert x X) \<in> fst P"
rep_eq: fixes x :: "'a vector" shows "Rep_vector (- x) = - Rep_vector x"