hcju/isabelleps huggingface dataset:
store_preserved_mem: fixes s :: "s"   and \<S> :: "s_context"   and s' :: "s"   and i :: "nat"   and mem' :: "mem"   and orig_mem :: "mem" assumes "store_typing s \<S>"   and "s' = s\<lparr>s.mem := (s.mem s)[i := mem']\<rparr>"   and "mem_size orig_mem \<le> mem_size mem'"   and "s.mem s ! i = orig_mem" shows "store_typing s' \<S>"
absorb_omega_exp: fixes x :: "ordinal"   and a :: "ordinal" assumes "x < \<omega> ** a" shows "x + \<omega> ** a = \<omega> ** a"
Abs_finfun_inverse: fixes y :: "'d \<Rightarrow> 'e" assumes "y \<in> finfun" shows "finfun_apply (Abs_finfun y) = y"
e_type_const: fixes e :: "e"   and \<S> :: "s_context"   and \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list"   and \<C>' :: "t_context" assumes "is_const e"   and "\<S>\<bullet>\<C> \<turnstile> [e] : ts _> ts'" shows "\<exists>t. ts' = ts @ [t] \<and> \<S>\<bullet>\<C>' \<turnstile> [e] : [] _> [t]"
sig_inv_set_closed_mult_scalar: fixes r :: "'t \<Rightarrow>\<^sub>0 'b"   and j :: "nat"   and p :: "'a \<Rightarrow>\<^sub>0 'b" assumes "r \<in> sig_inv_set' j" shows "p \<odot> r \<in> sig_inv_set' j"
hyps_2: fixes s :: "'a regex" assumes "local.safe_regex Futu Lax r2"   and "s \<in> lpd test i r2" shows "local.safe_regex Futu Lax s"
accI: fixes x :: "'g"   and r :: "('g \<times> 'g) set" assumes "\<And>y. (y, x) \<in> r \<Longrightarrow> y \<in> Wellfounded.acc r" shows "x \<in> Wellfounded.acc r"
summable_cmult_iff: fixes c :: "'b"   and f :: "nat \<Rightarrow> 'b" shows "summable (\<lambda>n. c * f n) = (c = (0::'b) \<or> summable f)"
PCF_lr_PE_delta: shows "PCF_lr (unlr PE.delta)"
1: fixes t :: "('fun, ('fun, 'atom) Term.term \<times> nat) Term.term" assumes "t \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t set C" shows "\<exists>\<delta>. wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>) \<and> t \<in> set C \<cdot>\<^sub>s\<^sub>e\<^sub>t \<delta>"
\<tau>move2_NewArrayD: fixes P :: "'a prog"   and h :: "'heap"   and stk :: "'addr val list"   and T :: "ty"   and e :: "'addr expr1"   and pc :: "nat"   and xcp :: "'addr option" assumes "\<tau>move2 P h stk (newA T\<lfloor>e\<rceil>) pc xcp"   and "pc < length (compE2 e)" shows "\<tau>move2 P h stk e pc xcp"
check_precond_r1_def: fixes s\<^sub>e :: "real"   and v\<^sub>e :: "real"   and a\<^sub>e :: "real"   and s\<^sub>o :: "real"   and v\<^sub>o :: "real"   and a\<^sub>o :: "real"   and \<delta> :: "real" shows "check_precond_r1 s\<^sub>e v\<^sub>e a\<^sub>e s\<^sub>o v\<^sub>o a\<^sub>o \<delta> = (s\<^sub>e < s\<^sub>o \<and> 0 \<le> v\<^sub>e \<and> 0 \<le> v\<^sub>o \<and> a\<^sub>e < 0 \<and> a\<^sub>o < 0 \<and> 0 < \<delta> \<and> \<delta> \<le> - v\<^sub>o / a\<^sub>o)"
div_by_1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b" assumes "class.semidom_divide divide plus minus zero times one" shows "divide a one = a"
pr_joint_monic: fixes a :: "'a"   and b :: "'a"   and h :: "'a"   and h' :: "'a" assumes "seq (pr0 a b) h"   and "pr0 a b \<cdot> h = pr0 a b \<cdot> h'"   and "pr1 a b \<cdot> h = pr1 a b \<cdot> h'" shows "h = h'"
inf_lower: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set"   and x :: "'a" assumes "weak_complete_lattice L"   and "A \<subseteq> carrier L"   and "x \<in> A" shows "\<Sqinter>\<^bsub>L\<^esub>A \<sqsubseteq>\<^bsub>L\<^esub> x"
rep_eq: fixes x :: "'b \<Rightarrow> 'a upred" shows "\<lbrakk>ushEx x\<rbrakk>\<^sub>e = (\<lambda>A. \<exists>xa. \<lbrakk>x xa\<rbrakk>\<^sub>e A)"
smult_distrib_right: fixes R :: "'a set"   and smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and r :: "'a"   and s :: "'a"   and m :: "'b" assumes "RModule R smult M"   and "r \<in> R"   and "s \<in> R"   and "m \<in> M" shows "smult (r + s) m = smult r m + smult s m"
P_has_glbs: fixes x :: "'a"   and y :: "'a" assumes "x \<in> P"   and "y \<in> P" shows "\<exists>b. glbound_in_of P x y b"
shift_simps_1: fixes xs :: "'a list"   and s :: "'a stream" shows "shd (xs @- s) = (if xs = [] then shd s else hd xs)"
m'_shd: fixes N :: "'a stream measure"   and M :: "'a stream measure"   and y :: "'a" assumes "R N M" shows "distr (m' N M y) (count_space UNIV) shd = measure_pmf (return_pmf y)"
sup_bot_right: fixes x :: "'a" shows "x + 0 = x"
parPres: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and Rel' :: "(ccs \<times> ccs) set"   and R :: "ccs" assumes "P \<leadsto>\<^sup>^<Rel> Q"   and "(P, Q) \<in> Rel"   and "\<And>S T U. (S, T) \<in> Rel \<Longrightarrow> (S \<parallel> U, T \<parallel> U) \<in> Rel'" shows "P \<parallel> R \<leadsto>\<^sup>^<Rel'> Q \<parallel> R"
subst_fm_trans_commute: fixes u :: "tm"   and name :: "name"   and A :: "fm" shows "subst_dbfm (trans_tm [] u) name (trans_fm [] A) = trans_fm [] (A(name::=u))"
foldl_digit_correct: fixes f :: "'d \<Rightarrow> 'e \<times> 'f \<Rightarrow> 'd"   and \<sigma> :: "'d"   and d :: "('e, 'f) Digit" shows "foldl_digit f \<sigma> d = List.foldl f \<sigma> (digitToList d)"
measure_while_option_Some: fixes P :: "'a \<Rightarrow> bool"   and b :: "'a \<Rightarrow> bool"   and c :: "'a \<Rightarrow> 'a"   and f :: "'a \<Rightarrow> nat"   and s :: "'a" assumes "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> P (c s) \<and> f (c s) < f s"   and "P s" shows "\<exists>t. while_option b c s = Some t"
lookup_HSem_heap: fixes y :: "Vars.var"   and h :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> Value" assumes "y \<in> domA h" shows "(\<lbrace>h\<rbrace>\<rho>) y = \<lbrakk> the (map_of h y) \<rbrakk>\<^bsub>\<lbrace>h\<rbrace>\<rho>\<^esub>"
Gmult0: fixes v :: "'v" assumes "v \<in> V" shows "(0::'g) *\<cdot> v = v"
fromCard_def: fixes A :: "'c set"   and r :: "('d \<times> 'd) set"   and k :: "'d" shows "fromCard A r k \<equiv> SOME b. b \<in> A \<and> toCard A r b = k"
from_hma\<^sub>v_def: fixes v :: "('c, 'd) Finite_Cartesian_Product.vec" shows "from_hma\<^sub>v v = Matrix.vec CARD('d) (\<lambda>i. v $h Bij_Nat.from_nat i)"
sup_absorb1: fixes sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and y :: "'e"   and x :: "'e" assumes "class.semilattice_sup sup less_eq less"   and "less_eq y x" shows "sup x y = x"
dom_override_the: fixes x :: "'b"   and G2 :: "'b \<Rightarrow> 'c option"   and G1 :: "'b \<Rightarrow> 'c option" shows "x \<in> dom G2 \<longrightarrow> (G1 ++ G2) x = G2 x"
subset_iff: fixes A :: "'a set"   and B :: "'a set" shows "order.greater_eq B A = (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
adds_def: fixes b :: "'c"   and a :: "'c" shows "(b adds a) = (\<exists>k. a = b + k)"
less_eq_list_induct: fixes x1 :: "'d list"   and x2 :: "'d list"   and Pa :: "'d list \<Rightarrow> 'd list \<Rightarrow> bool" assumes "x1 \<preceq> x2"   and "\<And>ys. Pa [] ys"   and "\<And>xs ys y. \<lbrakk>xs \<preceq> ys; Pa xs ys\<rbrakk> \<Longrightarrow> Pa xs (y # ys)"   and "\<And>x y xs ys. \<lbrakk>x = y; xs \<preceq> ys; Pa xs ys\<rbrakk> \<Longrightarrow> Pa (x # xs) (y # ys)" shows "Pa x1 x2"
abs_eq: fixes x :: "int" shows "unsigned (Word.Word x) = of_nat (nat (take_bit LENGTH('c) x))"
"1_6": fixes t :: "real" assumes "0 < t"   and "t < return_time p1 x0" shows "flow0 x0 t \<in> CX"
\<alpha>_limit_crossings: fixes a :: "'a"   and b :: "'a"   and x :: "'a"   and p :: "'a" assumes "transversal_segment a b"   and "{..0} \<subseteq> existence_ivl0 x"   and "\<alpha>_limit_point x p"   and "p \<in> {a<--<b}" obtains s :: "nat \<Rightarrow> real" where "s \<longlonglongrightarrow>\<^bsub>\<^esub> -\<infinity>" and "(flow0 x \<circ> s) \<longlonglongrightarrow> p" and "\<forall>\<^sub>F n in sequentially. flow0 x (s n) \<in> {a<--<b} \<and> s n \<in> existence_ivl0 x"
inj_on_cong: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "\<And>a. a \<in> A \<Longrightarrow> f a = g a" shows "inj_on f A = inj_on g A"
orbits_2: shows "pmf (sds R10) c = pmf (sds R10) b"
comm_group_def: fixes G :: "('b, 'c) monoid_scheme" shows "comm_group G \<equiv> Group.comm_monoid G \<and> Group.group G"
subdeg_sum: shows "fls_subdegree f + fls_subdegree g = 0"
restr_delete_twist: fixes A :: "'a set"   and a :: "'a"   and ps :: "('a \<times> 'b) list" shows "restrictA A (delete a ps) = delete a (restrictA A ps)"
a0i_agreement_def: shows "a0i_agreement \<equiv> {s. \<forall>A B M. a0n_state.signals s (Commit A B M) \<le> a0n_state.signals s (Running A B M)}"
linear_injective_imp_surjective: fixes f :: "('a, 'b) vec \<Rightarrow> ('a, 'c) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "inj f"   and "vec.dim UNIV = vec.dim UNIV" shows "surj f"
vec_lambda_inverse: fixes y :: "'b \<Rightarrow> 'c" assumes "y \<in> UNIV" shows "($) (vec_lambda y) = y"
simps_1: fixes xs :: "'c llist"   and ys :: "'d llist" shows "lnull (lzip xs ys) = (lnull xs \<or> lnull ys)"
gauss_cnj_mult_self: fixes z :: "gauss_int" shows "gauss_cnj z * z = of_nat (gauss_int_norm z)"
compat_wf: fixes r :: "('a \<times> 'a) set"   and r' :: "('b \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b" assumes "compat r r' f"   and "wf r'" shows "wf r"
hyps_1: fixes xs' :: "'a list" assumes "k = length xs - length xs'"   and "vs @ xs \<in> TS M2 M1 \<Omega> V m i"   and "mcp (vs @ xs) V vs"   and "prefix xs' xs" shows "vs @ xs' \<in> TS M2 M1 \<Omega> V m i"
dist_pos_lt: fixes x :: "'b"   and y :: "'b" assumes "x \<noteq> y" shows "0 < dist x y"
set_intros_17: fixes yo :: "'b"   and x132 :: "'b ltlc"   and x131 :: "'b ltlc" assumes "yo \<in> atoms_ltlc x132" shows "yo \<in> atoms_ltlc (x131 W\<^sub>c x132)"
n2_type1: fixes t1 :: "'b bro"   and h :: "nat"   and t2 :: "'b bro"   and a :: "'b" assumes "t1 \<in> Um h"   and "t2 \<in> B h" shows "n2 t1 a t2 \<in> T (Suc h)"
Big_step_if_D: fixes s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int"   and c :: "com" assumes "(s, t) \<in> D c" shows "(s, t) \<in> Big_step c"
iso_iff_group_isomorphisms: fixes G :: "('b, 'c) monoid_scheme"   and f :: "'b \<Rightarrow> 'd"   and H :: "('d, 'e) monoid_scheme" assumes "Group.group G" shows "(f \<in> Group.iso G H) = (\<exists>g. group_isomorphisms G H f g)"
order_def: fixes f :: "'b perm" shows "order f = card \<circ> orbit f"
siftDown_Node: fixes t :: "'c Tree"   and v :: "'c"   and l :: "'c Tree"   and r :: "'c Tree" assumes "t = T v l r" shows "\<exists>l' v' r'. siftDown t = T v' l' r' \<and> v \<le> v'"
Diff_eq_empty_iff_mset: fixes A :: "'c multiset"   and B :: "'c multiset" shows "(A - B = {#}) = (A \<subseteq># B)"
domIff: fixes a :: "'c"   and m :: "'c \<Rightarrow> 'd option" shows "(a \<in> dom m) = (m a \<noteq> \<bottom>)"
Zero_ring1: assumes "\<not> zeroring R" shows "1\<^sub>r \<noteq> \<zero>"
le_numeral_extra_2: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "\<not> less_eq one zero"
has_integral_dominated_convergence: fixes f :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and y :: "nat \<Rightarrow> 'b"   and S :: "'a set"   and h :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> 'b"   and x :: "'b" assumes "\<And>k. (f k has_integral y k) S"   and "h integrable_on S"   and "\<And>k. \<forall>x\<in>S. norm (f k x) \<le> h x"   and "\<forall>x\<in>S. (\<lambda>k. f k x) \<longlonglongrightarrow> g x"   and "y \<longlonglongrightarrow> x" shows "(g has_integral x) S"
sqn_update_another: fixes ip :: "nat"   and dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dsn :: "nat"   and dsk :: "k"   and flag :: "f"   and hops :: "nat"   and nhip :: "nat" assumes "ip \<noteq> dip" shows "sqn (update rt dip (dsn, dsk, flag, hops, nhip)) ip = sqn rt ip"
snd_not_one_fst: fixes a :: "('f, 'e) OrdSum" assumes "snd (Rep_OrdSum a) \<noteq> (1::'e)" shows "fst (Rep_OrdSum a) = (1::'f)"
mkVarChannel_pState_inv: fixes prog :: "program"   and p :: "pState"   and g :: "'a gState_scheme"   and v :: "varDecl"   and upd :: "((String.literal, variable) assoc_list \<Rightarrow> (String.literal, variable) assoc_list) \<Rightarrow> 'a gState_scheme \<times> pState \<Rightarrow> 'a gState_scheme \<times> pState" assumes "pState_inv prog p"   and "cl_inv (g, p)"   and "\<And>k v' cs. toVariable g p v = (k, v', cs) \<Longrightarrow> cl_inv (upd (lm.update k v') (g, p))"   and "\<And>k v' cs. toVariable g p v = (k, v', cs) \<Longrightarrow> pState_inv prog (snd (upd (lm.update k v') (g, p)))" shows "pState_inv prog (snd (mkVarChannel v upd g p))"
product_category_axioms: shows "product_category (\<cdot>) (\<cdot>)"
cong_mod_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and 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 modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and b :: "'a"   and c :: "'a"   and a :: "'a" assumes "class.unique_euclidean_semiring divide plus minus zero times one modulo euclidean_size division_segment" shows "unique_euclidean_semiring.cong modulo b (modulo c a) a = unique_euclidean_semiring.cong modulo b c a"
d_def: shows "d = (\<lambda>i n. \<integral>\<^sup>+x\<in>D n. f ((T ^^ i) x)\<partial>M)"
simps: fixes f :: "bool \<Rightarrow> String.literal \<Rightarrow> integer option \<Rightarrow> 'b \<Rightarrow> 'c gState_scheme \<Rightarrow> pState \<Rightarrow> 'a"   and gl :: "bool"   and v :: "String.literal"   and idx :: "expr option"   and argm :: "'b"   and g :: "'c gState_scheme"   and p :: "pState" shows "liftVar f (VarRef gl v idx) argm g p = f gl v (map_option (exprArith g p) idx) argm g p"
simps_2: fixes R :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and w :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> 'd \<Rightarrow> 'f"   and x :: "'e"   and h :: "'e"   and l :: "'e list" shows "wlp R w g f x (h # l) = (if R x h then g (f (w x) (wlp R w g f h l)) (wlp R w g f x l) else wlp R w g f x l)"
map_sel_1: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b stream" shows "shd (smap f a) = f (shd a)"
fmdrop_set_empty: fixes A :: "'c set" shows "fmdrop_set A fmempty = fmempty"
transversal_segment_reverse: fixes x :: "'a"   and y :: "'a" assumes "transversal_segment x y" shows "transversal_segment y x"
Empty_WordInterval_set_eq: shows "wordinterval_to_set Empty_WordInterval = {}"
assms_2: shows "\<pi> j = ipd (\<pi> k)"
mult_monos_4: fixes c :: "'a"   and a :: "'a"   and d :: "'a"   and b :: "'a" assumes "c \<le> a"   and "d \<le> b"   and "c \<le> (0::'a)"   and "b \<le> (0::'a)" shows "a * b \<le> c * d"
induct: fixes P :: "'b Tree \<Rightarrow> bool"   and a0 :: "'b Tree" assumes "P E"   and "\<And>v. P (T v E E)"   and "\<And>v va vb vc. P (T va vb vc) \<Longrightarrow> P (T v E (T va vb vc))"   and "\<And>v va vb vc. P (T va vb vc) \<Longrightarrow> P (T v (T va vb vc) E)"   and "\<And>v va vb vc vd ve vf. \<lbrakk>P (T vd ve vf); P (T va vb vc)\<rbrakk> \<Longrightarrow> P (T v (T va vb vc) (T vd ve vf))"   and "\<And>v vd ve vf va vb vc. \<lbrakk>P (T va vb vc); P (T vd ve vf)\<rbrakk> \<Longrightarrow> P (T v (T vd ve vf) (T va vb vc))" shows "P a0"
l9_18: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and Y :: "'a"   and P :: "'a"   and A :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet X Y P"   and "Tarski_neutral_dimensionless.Col Bet A B P" shows "Tarski_neutral_dimensionless.TS Bet X Y A B = (Bet A P B \<and> \<not> Tarski_neutral_dimensionless.Col Bet X Y A \<and> \<not> Tarski_neutral_dimensionless.Col Bet X Y B)"
T2: fixes w :: "'a"   and w' :: "'a"   and \<theta> :: "'a"   and u :: "'a"   and \<theta>' :: "'a"   and \<beta> :: "'a" assumes "ide w"   and "ide w'"   and "\<guillemotleft>\<theta> : tab\<^sub>0 f \<star> w \<rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<theta>' : tab\<^sub>0 f \<star> w' \<rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<beta> : tab\<^sub>1 f \<star> w \<rightarrow> tab\<^sub>1 f \<star> w'\<guillemotright>"   and "(f \<star> \<theta>) \<cdot> \<a>[f, tab\<^sub>0 f, w] \<cdot> (f.tab \<star> w) = ((f \<star> \<theta>') \<cdot> \<a>[f, tab\<^sub>0 f, w'] \<cdot> (f.tab \<star> w')) \<cdot> \<beta>" shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w \<rightarrow> w'\<guillemotright> \<and> \<beta> = tab\<^sub>1 f \<star> \<gamma> \<and> \<theta> = \<theta>' \<cdot> (tab\<^sub>0 f \<star> \<gamma>)"
igSwapAbs_termMOD: shows "igSwapAbs termMOD = swapAbs"
ChiRel_OneAncestor2: fixes S :: "'a"   and A :: "('a, 'b, 'c) hierauto" assumes "S \<in> HAStates A"   and "S \<notin> States (HARoot A)" shows "\<exists>!T. (T, S) \<in> ChiRel A"
lsb_odd: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and lsb :: "'b \<Rightarrow> bool" assumes "class.lsb plus minus zero times one divide modulo bit lsb" shows "lsb = (\<lambda>a. \<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
bij_betw_respects_range_permutation: fixes A :: "'a set"   and B :: "'b set" shows "(\<lambda>f. bij_betw f A B) respects range_permutation A B"
D_dest: fixes x :: "'a"   and m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and i :: "nat"   and j :: "nat"   and k :: "nat" assumes "x = D m i j k" shows "x \<in> {len m i j xs |xs. set xs \<subseteq> {0..Suc k} \<and> i \<notin> set xs \<and> j \<notin> set xs \<and> distinct xs}"
update_tableau_id: fixes s :: "('i, 'a) state"   and x :: "nat"   and c :: "'a" assumes "\<triangle> (\<T> s)"   and "\<nabla> s"   and "x \<notin> lvars (\<T> s)" shows "\<T> (update x c s) = \<T> s"
initPsi_def: shows "initPsi \<equiv> $pc1 = #a \<and> $pc2 = #a \<and> $x = #0 \<and> $y = #0 \<and> $sem = #1"
Var: fixes V :: "char list"   and s :: "char list \<Rightarrow> val option" shows "\<lbrakk>Var V\<rbrakk>s = s V"
powsum_split_var3: fixes x :: "'a"   and m :: "nat"   and n :: "nat" shows "x\<^bsub>0\<^esub>\<^bsup>m + Suc n\<^esup> = x\<^bsub>0\<^esub>\<^bsup>m\<^esup> + x\<^bsub>0 + Suc m\<^esub>\<^bsup>n\<^esup>"
det3_zero_translate_plus': fixes a :: "real \<times> real"   and b :: "real \<times> real"   and c :: "real \<times> real" shows "coll a (a + b) (a + c) = coll 0 b c"
ccompare_finite_2_def: shows "ccompare \<equiv> Some compare"
restrict_extensional: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set" shows "restrict f A \<in> extensional A"
insert_remove: fixes A :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and x :: "'d" assumes "finite A" shows "prod g (insert x A) = g x * prod g (A - {x})"
l3_dy_valid_refines_dy_fake_chan_Un: fixes M :: "msg" assumes "M \<in> valid" shows "{R23s \<inter> UNIV \<times> (l3_inv5 \<inter> l3_inv6 \<inter> l3_inv7 \<inter> l3_inv8 \<inter> l3_inv9)} \<Union> (range l2_dy_fake_chan), l3_dy M {> R23s}"
llength_inf_llist: fixes f :: "nat \<Rightarrow> 'c" shows "llength (inf_llist f) = \<infinity>"
vsv_vimage_vsingleton: fixes a :: "V"   and A :: "V"   and f :: "V \<Rightarrow> V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (VLambda A f)" shows "VLambda A f `\<^sub>\<circ> set {a} = set {VLambda A f\<lparr>a\<rparr>}"
may_acquire_allE: fixes ls :: "('a, 'b) locks"   and t :: "'b"   and ln :: "'a \<Rightarrow>f nat" assumes "may_acquire_all ls t ln" obtains "\<forall>l. 0 < ln $ l \<longrightarrow> may_lock (ls $ l) t"
hom_0: fixes x :: "'a poly" assumes "pCons (0::'a) x = 0" shows "x = 0"
cases: fixes x :: "('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> ('a literal list \<times> nat) list \<times> nat" obtains N :: "('a literal list \<times> nat) list" and P :: "('a literal list \<times> nat) list" and Q :: "('a literal list \<times> nat) list" and n :: "nat" where "x = (N, P, Q, n)"
timesI: fixes a :: "V"   and A :: "V"   and b :: "V"   and B :: "V" assumes "a \<in>\<^sub>\<circ> A"   and "b \<in>\<^sub>\<circ> B" shows "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> A \<times>\<^sub>\<circ> B"
abs_fun_finite_supp: fixes x :: "'a"   and a :: "'b" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)"   and "finite (supp x)" shows "finite (supp ([a].x))"
C1: fixes \<Psi>' :: "'b" shows "(\<Psi> \<otimes> \<Psi>', \<alpha>\<cdot>(\<tau>.P), \<alpha>\<cdot>Q) \<in> Rel"
1: shows "rel_restrict E R \<subseteq> E \<inter> UNIV \<times> - R"
density_embed_measure: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'d \<Rightarrow> ennreal"   and M :: "'c measure" assumes "inj f"   and "g \<in> borel_measurable (embed_measure M f)" shows "density (embed_measure M f) g = embed_measure (density M (g \<circ> f)) f"
know_decomposition: fixes PQ :: "specID"   and m :: "KS"   and P :: "specID"   and Q :: "specID" assumes "know PQ m"   and "subcomponents PQ = {P, Q}"   and "correctCompositionIn PQ"   and "correctCompositionLoc PQ" shows "know P m \<or> know Q m"
Write_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and mm :: "state \<Rightarrow> Locs \<Rightarrow> Vals"   and rs :: "state \<Rightarrow> PrIds \<Rightarrow> Vals"   and p :: "PrIds"   and l :: "Locs" shows "Write ch mm rs p l \<equiv> \<exists>v. WriteInner ch mm rs p l v"
finite_\<epsilon>: shows "\<epsilon> \<noteq> \<top>"
simps_1: fixes f\<^sub>1 :: "'b val \<Rightarrow> 'b val"   and f\<^sub>2 :: "'b val \<Rightarrow> 'b val"   and v :: "'b val" shows "F_Plus f\<^sub>1 f\<^sub>2 (val.Right v) = val.Right (f\<^sub>2 v)"
orderE: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" obtains "a = inf_class.inf a b"
p'_stop_zero: shows "ego.p' ego.t_stop = (if a\<^sub>e = 0 then v\<^sub>e else 0)"
P2SQ2: shows "(P2, S', Q2) = diagonal_to_Smith_PQ_JNF D euclid_ext2"
echelon_form_imp_upper_triagular_upt: fixes A :: "(('d, 'e) vec, 'e) vec"   and k :: "nat" assumes "echelon_form A" shows "upper_triangular_upt_k A k"
in_set_butlastD: fixes x :: "'d"   and xs :: "'d list" assumes "x \<in> set (butlast xs)" shows "x \<in> set xs"
interleave_ConsConsE: fixes xs :: "'b list"   and y :: "'b"   and ys :: "'b list"   and z :: "'b"   and zs :: "'b list" assumes "xs \<in> y # ys \<otimes> z # zs" obtains zs :: "'b list" where "xs = y # zsc__" and "zsc__ \<in> ys \<otimes> z # zsb__"   | zs :: "'b list" where "xs = z # zs" and "zs \<in> y # ys \<otimes> zsb__"
resultant_smult_left: fixes c :: "'b"   and f :: "'b poly"   and g :: "'b poly" assumes "c \<noteq> (0::'b)" shows "resultant (Polynomial.smult c f) g = c ^ degree g * resultant f g"
arr_cod_iff_arr: fixes f :: "'a" shows "arr (cod f) = arr f"
le_multiset_empty_left: fixes M :: "'a multiset" assumes "M \<noteq> {#}" shows "{#} < M"
mono: fixes F :: "ordinal \<Rightarrow> ordinal" assumes "normal F" shows "mono F"
hdls: shows "hd (trace n f g xs y) = the (eval (Pr n f g) (y # xs))"
ext_induct: fixes P :: "('b, 'c) arrow_of_spans_data_scheme \<Rightarrow> bool"   and r :: "('b, 'c) arrow_of_spans_data_scheme" assumes "\<And>Chn Dom Cod more. P \<lparr>Chn = Chn, Dom = Dom, Cod = Cod, \<dots> = more\<rparr>" shows "P r"
sub: shows "P \<turnstile> C' \<preceq>\<^sup>* C"
image_sequence_of_inj: fixes enum :: "nat \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw enum {0..<card A} A" shows "sequence_of A enum ` {f \<in> A \<rightarrow>\<^sub>E B. inj_on f A} \<subseteq> {xs. set xs \<subseteq> B \<and> length xs = card A \<and> distinct xs}"