hcju/isabelleps huggingface dataset:
tranclD: fixes x :: "'a"   and y :: "'a"   and R :: "('a \<times> 'a) set" assumes "(x, y) \<in> R\<^sup>+" shows "\<exists>z. (x, z) \<in> R \<and> (z, y) \<in> R\<^sup>*"
TC_rank_lt: fixes a :: "V"   and b :: "V" assumes "a \<sqsubset> b" shows "rank a < rank b"
has_domain_iff_arr: fixes f :: "'c \<times> 'c" shows "(CopxC.domains f \<noteq> {}) = CopxC.arr f"
one_tensor_is_id: fixes A :: "complex Matrix.mat" shows "A \<otimes>\<^bsup>1\<^esup> = A"
closure_of_minimal: fixes S :: "'b set"   and T :: "'b set"   and X :: "'b topology" assumes "S \<subseteq> T"   and "closedin X T" shows "X closure_of S \<subseteq> T"
two_pred_simp: fixes c :: "one\<^sub>\<bottom>" shows "two_pred\<cdot>c = (if c \<sqsubseteq> once then none else c)"
assms: shows "\<forall>n. enat n \<le> llength xs \<longrightarrow> i \<le> n \<longrightarrow> \<not> P (lnth xs n)"
image_closure_subset: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "continuous_on (closure S) f"   and "closed T"   and "f ` S \<subseteq> T" shows "f ` closure S \<subseteq> T"
is_iso_dghm_is_arr_isomorphism_1: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>" shows "inv_dghm \<FF> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<AA>"
assms: shows "(\<gamma>' \<UU>\<^sub>c \<gamma>) t n"
tendsto_def: fixes "open" :: "'c set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and l :: "'c"   and F :: "'d filter" assumes "class.topological_space open" shows "filterlim f (topological_space.nhds open l) F = (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> (\<forall>\<^sub>F x in F. f x \<in> S))"
sys_xcpts_cases: fixes C :: "char list"   and P :: "char list \<Rightarrow> bool" assumes "C \<in> sys_xcpts"   and "P NullPointer"   and "P OutOfMemory"   and "P ClassCast"   and "P NoClassDefFoundError"   and "P IncompatibleClassChangeError"   and "P NoSuchFieldError"   and "P NoSuchMethodError" shows "P C"
toProcess_states_nonempty: fixes sidx :: "nat"   and p :: "proc"   and ss :: "(integer \<times> edge list) iarray"   and a :: "nat"   and n :: "String.literal"   and l :: "(String.literal, nat) assoc_list"   and idx :: "nat"   and r :: "edgeIndex \<times> procArg list \<times> varDecl list" assumes "toProcess sidx p = (ss, a, n, l, idx, r)" shows "0 < IArray.length ss"
ev_introL: fixes \<phi> :: "('state \<times> 'aprop set) stream list \<Rightarrow> bool"   and \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "ev \<phi> (map stl \<pi>l)" shows "ev \<phi> \<pi>l"
closest_pair_simps: fixes ps :: "(int \<times> int) list" assumes "1 < length ps" shows "closest_pair ps = (let (uu_, p) = closest_pair_rec (mergesort fst ps) in p)"
set_intros_1: fixes x21 :: "'c"   and x22 :: "'c llist" shows "x21 \<in> lset (LCons x21 x22)"
C1: fixes y :: "name"   and R :: "ccs" assumes "y \<sharp> R" shows "(R, \<lparr>\<nu>y\<rparr>R) \<in> Rel"
must_syncE: fixes t :: "'t"   and x :: "'x1"   and m :: "'m1" assumes "t \<turnstile> \<langle>x, m\<rangle> \<wrong>1" obtains taa :: "('l, 't, 'x1, 'm1, 'w, 'o) thread_action" and x' :: "'x1" and m' :: "'m1" and s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" where "r1.r_syntax t x m taa x' m'" and "r1.actions_ok s t taa" and "m = shr s"
bit_minus_1_iff: fixes n :: "nat" shows "bit (- (1::'a)) n = ((2::'a) ^ n \<noteq> (0::'a))"
simps_6: fixes x2 :: "int"   and x1 :: "int" shows "Intv' x2 \<noteq> Const' x1"
map_graph_fn_graphI: fixes G :: "('b, 'd) labeled_graph"   and f :: "'d \<Rightarrow> 'c" shows "graph (map_graph_fn G f)"
simps_26: shows "UnlimitedNatural \<noteq> OclVoid"
e: fixes i :: "nat" assumes "i < length I" shows "e i \<in>\<^sub>r I ! i - a ! i"
h: shows "- (c / b) = (sqrt (aa\<^sup>2 - 4 * a * ba) - aa) / (2 * a)"
inext_nth_imirror_iprev_nth_conv2: fixes I :: "nat set"   and n :: "nat" assumes "finite I"   and "I \<noteq> {}" shows "mirror_elem (imirror I \<leftarrow> n) I = I \<rightarrow> n"
q_def: fixes x :: "nat" shows "q x = c_pair x 1"
is_process_REP_Skip2: shows "is_process ({[]} \<times> {X. tick \<notin> X} \<union> {(s, X). s = [tick]}, {})"
update_simps_1: fixes env :: "('f, 'g, 'h) env" shows "update [] None env = env"
altc_sample_option_def: fixes altc_sample :: "'k \<Rightarrow> ('l \<Rightarrow> 'j) \<Rightarrow> 'j"   and p :: "'k"   and f :: "'l \<Rightarrow> ('i, 'j) optionT" shows "altc_sample_option altc_sample p f = OptionT (altc_sample p (\<lambda>x. run_option (f x)))"
induct: fixes P :: "('b, 'a) stateful_strand_step list \<Rightarrow> ('a \<Rightarrow> ('b, 'a) Term.term) \<Rightarrow> (('b, 'a) Term.term \<times> ('b, 'a) Term.term) list \<Rightarrow> bool"   and a0 :: "('b, 'a) stateful_strand_step list"   and a1 :: "'a \<Rightarrow> ('b, 'a) Term.term"   and a2 :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) list" assumes "\<And>I D. P [] I D"   and "\<And>t s A I D. P A I (List.insert (t \<cdot> I, s \<cdot> I) D) \<Longrightarrow> P (insert\<langle>t,s\<rangle> # A) I D"   and "\<And>t s A I D. P A I (removeAll (t \<cdot> I, s \<cdot> I) D) \<Longrightarrow> P (delete\<langle>t,s\<rangle> # A) I D"   and "\<And>v A I D. P A I D \<Longrightarrow> P (send\<langle>v\<rangle> # A) I D"   and "\<And>v A I D. P A I D \<Longrightarrow> P (receive\<langle>v\<rangle> # A) I D"   and "\<And>v va vb A I D. P A I D \<Longrightarrow> P (\<langle>v: va \<doteq> vb\<rangle> # A) I D"   and "\<And>v va vb A I D. P A I D \<Longrightarrow> P (\<langle>v: va \<in> vb\<rangle> # A) I D"   and "\<And>v va vb A I D. P A I D \<Longrightarrow> P (\<forall>v\<langle>\<or>\<noteq>: va \<or>\<notin>: vb\<rangle> # A) I D" shows "P a0 a1 a2"
simps_3: fixes C :: "ty"   and e :: "('f, 'g, 'h) exp"   and A :: "'f set option" shows "\<D> (Cast C e) A = \<D> e A"
leq_refl: fixes x :: "'k" shows "x \<le> x"
verit_comp_simplify1_3: fixes b' :: "'c"   and a' :: "'c" shows "(\<not> b' \<le> a') = (a' < b')"
simpeq: fixes vs :: "'b list"   and bs :: "'b list"   and t :: "tm" shows "Ifm vs bs (simpeq t) = Ifm vs bs (Eq t)"
distinct_1: fixes x21 :: "'a"   and x22 :: "'a list" shows "[] \<noteq> x21 # x22"
classes_above_blank: fixes C :: "char list"   and P :: "'b prog"   and P' :: "'b prog" assumes "classes_above P C \<inter> classes_changed P P' = {}" shows "blank P C = blank P' C"
tr_vars_disj: fixes A' :: "('fun, 'var) strand_step list"   and A :: "('fun, 'var) stateful_strand_step list"   and D :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list" assumes "A' \<in> set (tr A D)"   and "\<forall>(t, t')\<in>set D. (fv t \<union> fv t') \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}"   and "fv\<^sub>s\<^sub>s\<^sub>t A \<inter> bvars\<^sub>s\<^sub>s\<^sub>t A = {}" shows "fv\<^sub>s\<^sub>t A' \<inter> bvars\<^sub>s\<^sub>t A' = {}"
power_Suc2: fixes a :: "'a"   and n :: "nat" shows "a ^ Suc n = a ^ n * a"
map_const_spmf_of_set: fixes A :: "'b set"   and c :: "'c" assumes "finite A"   and "A \<noteq> {}" shows "map_spmf (\<lambda>_. c) (spmf_of_set A) = return_spmf c"
arcs_cases: fixes a :: "nat"   and b :: "nat"   and i :: "nat"   and j :: "nat"   and xs :: "nat list" assumes "(a, b) \<in> set (arcs i j xs)"   and "xs \<noteq> []" shows "(\<exists>ys. xs = b # ys \<and> a = i) \<or> (\<exists>ys. xs = ys @ [a] \<and> b = j) \<or> (\<exists>c d ys. (a, b) \<in> set (arcs c d ys) \<and> xs = c # ys @ [d])"
hom2: shows "homeomorphism (cball a r \<inter> T) (cball a r \<inter> T) f2 g2"
simps_2: fixes f :: "'c \<Rightarrow> 'b err"   and x :: "'c" shows "strict f (OK x) = f x"
continuous_blinfun_matrix: fixes F :: "'d filter"   and f :: "'d \<Rightarrow> 'e \<Rightarrow>\<^sub>L 'f"   and j :: "'e"   and i :: "'f" assumes "continuous F f" shows "continuous F (\<lambda>x. blinfun_apply (f x) j \<bullet> i)"
strict_iff_order: fixes a :: "'a list"   and b :: "'a list" shows "strict_border a b = (border a b \<and> a \<noteq> b)"
star_slide_var: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and one :: "'c"   and star :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "class.left_kleene_algebra plus times less_eq less one star" shows "times (star x) x = times x (star x)"
disc_def: fixes H :: "ocircline" shows "disc H = {z. in_ocircline H z}"
l8: fixes x :: "'a" shows "x \<le> - - x \<sqinter> x"
simps_1: fixes x :: "'a list" shows "list_diff x [] = x"
inject_4: fixes x4 :: "hfm"   and y4 :: "hfm" shows "(HNeg x4 = HNeg y4) = (x4 = y4)"
not_eq_extremum: fixes a :: "'a list" shows "(\<not> a = []) = strict_prefix [] a"
Bseq_bdd_below: fixes X :: "nat \<Rightarrow> real" assumes "Bseq X" shows "bdd_below (range X)"
self_bij_0_k: fixes a :: "nat"   and k :: "nat"   and i :: "nat" assumes "coprime a k"   and "[a * i = 1] (mod k)"   and "0 < k" shows "bij_betw (\<lambda>r. r * a mod k) {0..k - 1} {0..k - 1}"
assms_5: shows "ptr \<notin> set |h \<turnstile> get_component (cast\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r document_ptr)|\<^sub>r"
onorm_le_matrix_component: fixes A :: "((real, 'c) vec, 'b) vec"   and B :: "real" assumes "\<And>i j. \<bar>A $ i $ j\<bar> \<le> B" shows "\<parallel>A\<parallel>\<^sub>o\<^sub>p \<le> real CARD('b) * real CARD('c) * B"
prems: shows "flowsto (X - trap) T CX (\<Union> (insert x F))"
finite_cspan_crepresentation_bounded: fixes B :: "'d set" assumes "finite B"   and "cindependent B" shows "\<exists>D>0. \<forall>\<psi> b. cmod (crepresentation B \<psi> b) \<le> norm \<psi> * D"
secrets_are_homogeneous: fixes Sec :: "('fun, 'var) Term.term set"   and P :: "('fun, 'var) Term.term \<Rightarrow> bool"   and s :: "('fun, 'var) Term.term"   and \<A> :: "('b strand_label \<times> ('fun, 'var) strand_step) list" assumes "\<forall>s\<in>Sec. P s \<longrightarrow> (\<forall>s'\<in>subterms s. {} \<turnstile>\<^sub>c s' \<or> s' \<in> Sec)"   and "s \<in> Sec"   and "P s" shows "homogeneous\<^sub>l\<^sub>s\<^sub>t s \<A> Sec"
simps: fixes a1 :: "('a, 'b, 'c) configuration"   and a2 :: "('a, 'b, 'c) event list"   and a3 :: "('a, 'b, 'c) configuration" shows "trace a1 a2 a3 = ((\<exists>c. a1 = c \<and> a2 = [] \<and> a3 = c) \<or> (\<exists>c ev c' t c''. a1 = c \<and> a2 = ev # t \<and> a3 = c'' \<and> c \<turnstile> ev \<mapsto> c' \<and> trace c' t c''))"
distinct_rotate: fixes n :: "nat"   and xs :: "'a list" shows "distinct (rotate n xs) = distinct xs"
ang_sym: fixes a :: "'p \<Rightarrow> 'p \<Rightarrow> 'p \<Rightarrow> bool"   and A :: "'p"   and B :: "'p"   and C :: "'p" assumes "QCongA a"   and "a A B C" shows "a C B A"
M: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "\<bar>f x\<bar> \<le> M"
zero_le_square: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.linordered_ring plus zero minus uminus less_eq less abs times" shows "less_eq zero (times a a)"
l3_inv10_def: shows "l3_inv10 \<equiv> {s. secret s \<subseteq> payload}"
comm_monoid_set_combine: shows "comm_monoid_set (combine (\<^bold>*)) Mapping.empty"
Allowed_eq: fixes X :: "'c program set"   and init :: "'c set"   and acts :: "('c \<times> 'c) set set" assumes "safety_prop X" shows "Allowed (mk_program (init, acts, \<Union> (Acts ` X))) = X"
lexmax_elt: fixes r :: "('a \<times> 'a) set"   and x :: "'a"   and \<sigma> :: "'a list" assumes "trans r"   and "x \<in> set \<sigma>"   and "x \<notin># r|\<sigma>|" shows "\<exists>y. (x, y) \<in> r \<and> y \<in># r|\<sigma>|"
birkhoff_sum_1_2: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a" shows "birkhoff_sum f 1 x = f x"
path_connectedin_subset_topspace: fixes X :: "'a topology"   and S :: "'a set" assumes "path_connectedin X S" shows "S \<subseteq> topspace X"
Grp_def: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "BNF_Def.Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)"
CC': shows "Ssubst (\<Gamma>C, \<Delta>C) \<sigma> = (\<Gamma>C', \<Delta>C')"
uminus_galois_var2: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "(f \<circ> \<partial> = g) = (g \<circ> \<partial> = f)"
IH: fixes is' :: "nat list" assumes "is' \<lhd> dims (tensors_from_net (witness_l (tl rs)) $ 0)" shows "lookup (tensors_from_net (witness_l (tl rs)) $ 0) is' = (if nths is' (Collect even) = nths is' {n. odd n} \<and> (\<forall>i\<in>set is'. i < last (butlast (tl rs))) then 1 else 0)"
intros_2: fixes \<phi>s :: "Formula.formula list"   and \<kappa> :: "mregex \<Rightarrow> mregex"   and \<kappa>' :: "Formula.formula Regex.regex \<Rightarrow> Formula.formula Regex.regex"   and mr :: "mregex" assumes "ok_rctxt \<phi>s \<kappa> \<kappa>'" shows "ok_rctxt \<phi>s (\<lambda>t. \<kappa> (MTimes mr t)) (\<lambda>t. \<kappa>' (regex.Times (from_mregex mr \<phi>s) t))"
Float_num_4: shows "real_of_float (Float 1 (- 1)) = 1 / 2"
cp_name_name1: fixes pi1 :: "(name \<times> name) list"   and pi2 :: "(name \<times> name) list"   and x :: "'b" shows "pi1 \<bullet> pi2 \<bullet> x = (pi1 \<bullet> pi2) \<bullet> pi1 \<bullet> x"
surjective: fixes F :: "('a \<times> 'b) set"   and f :: "'a \<Rightarrow> 'b"   and A :: "'a" assumes "su_rel_fun F f" obtains B :: "'b" where "(A, B) \<in> F"
evaluate_determ_2: fixes ck :: "bool"   and env :: "v sem_env"   and s :: "'b state"   and es :: "exp0 list"   and r2a :: "'b state \<times> (v list, v) result"   and r2b :: "'b state \<times> (v list, v) result" assumes "evaluate_list ck env s es r2a"   and "evaluate_list ck env s es r2b" shows "r2a = r2b"
simps_4: fixes v :: "('b, 'c) Combinators"   and va :: "('b, 'c) Combinators"   and y :: "('b, 'c) Combinators list" shows "removeShadowRules2 (v \<oplus> va # y) = v \<oplus> va # removeShadowRules2 y"
these_not_empty_eq: fixes B :: "'a option set" shows "(Option.these B \<noteq> {}) = (B \<noteq> {} \<and> B \<noteq> {None})"
hyps_1: shows "F, \<Gamma> \<Rightarrow>\<^sub>n"
assms_1: shows "theta \<subseteq> ZOretr (theta \<union> ZObis)"
eq_Sym: fixes a :: "cmsg"   and b :: "cmsg" assumes "eq a b" shows "eq b a"
project_act_def: fixes h :: "'c \<times> 'd \<Rightarrow> 'e"   and act :: "('e \<times> 'e) set" shows "project_act h act = {(x, x'). \<exists>y y'. (h (x, y), h (x', y')) \<in> act}"
abs_eq: shows "inf_sls = Abs_semilattice_set inf"
a1: shows "r = memory_write_asi asi addr byte_mask data s \<and> r = Some s' \<and> get_S (cpu_reg_val PSR s) = 0"
reachable_induced: fixes G :: "('c, 'd) pre_digraph"   and S :: "'c set"   and u :: "'c"   and v :: "'c" assumes "max_reachable_set G S"   and "u \<in> S"   and "v \<in> S"   and "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v" shows "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> v"
min_dist_succ: fixes v :: "nat"   and v' :: "nat"   and v'' :: "nat" assumes "connected v v'"   and "(v', v'') \<in> E" shows "min_dist v v'' \<le> Suc (min_dist v v')"
regular_closed_top: fixes 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" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "top = uminus (uminus top)"
assms_3: shows "D = {w \<in> space M. Y w \<in> space N \<and> P (expl_cond_expect M Y X w)}"
outstanding_non_write_non_vol_reads_drop_disj: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'a"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'a memref list"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'a"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'a memref list"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "j < length ts\<^sub>s\<^sub>b"   and "i \<noteq> j"   and "ts\<^sub>s\<^sub>b ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "ts\<^sub>s\<^sub>b ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "outstanding_refs is_Write\<^sub>s\<^sub>b (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>i) \<inter> outstanding_refs is_non_volatile_Read\<^sub>s\<^sub>b (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j) = {}"
ACI_norm_flatten: fixes r :: "'b rexp" shows "\<guillemotleft>r\<guillemotright> = flatten PLUS (ACI_norm ` toplevel_summands r)"
resolution_sound: fixes F :: "char list \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "char list \<Rightarrow> 'a list \<Rightarrow> bool"   and C\<^sub>1 :: "fterm literal set"   and C\<^sub>2 :: "fterm literal set"   and L\<^sub>1 :: "fterm literal set"   and L\<^sub>2 :: "fterm literal set"   and \<sigma> :: "char list \<Rightarrow> fterm" assumes "eval\<^sub>c F G C\<^sub>1"   and "eval\<^sub>c F G C\<^sub>2"   and "applicable C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>" shows "eval\<^sub>c F G (resolution C\<^sub>1 C\<^sub>2 L\<^sub>1 L\<^sub>2 \<sigma>)"
valid_store_sops: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "\<forall>sop\<in>store_sops is. valid_sop sop"
respection_of_binary_predicates_and_closures_2: fixes Rel :: "('a \<times> 'a) set"   and Pred :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "rel_respects_binary_pred Rel Pred" shows "rel_respects_binary_pred (symcl Rel) Pred"
sorted_wrt_rev: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and xs :: "'d list" shows "sorted_wrt P (rev xs) = sorted_wrt (\<lambda>x y. P y x) xs"
cond_override_def: fixes cond :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and old :: "'c \<Rightarrow> 'd option"   and new :: "'c \<Rightarrow> 'd option" shows "cond_override cond old new = (\<lambda>k. case new k of None \<Rightarrow> old k | Some new_val \<Rightarrow> case old k of None \<Rightarrow> Some new_val | Some old_val \<Rightarrow> if cond new_val old_val then Some new_val else Some old_val)"
factgroup_finite: assumes "finite (carrier G)" shows "finite (rcosets H)"
Int_insert_left_if0: fixes a :: "'a"   and C :: "'a set"   and B :: "'a set" assumes "a \<notin> C" shows "insert a B \<inter> C = B \<inter> C"
simps_2: fixes V :: "'c set"   and t :: "('d, 'c) Term.term"   and S :: "('d, 'c) stateful_strand_step list" shows "wf'\<^sub>s\<^sub>s\<^sub>t V (receive\<langle>t\<rangle> # S) = (fv t \<subseteq> V \<and> wf'\<^sub>s\<^sub>s\<^sub>t V S)"
exec_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 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> \<Rightarrow> Stuck" shows "\<Gamma>\<turnstile> \<langle>call init p return c,Normal s\<rangle> \<Rightarrow> Stuck"
uint_plus_simple_iff: fixes x :: "'b word"   and y :: "'b word" shows "(x \<le> x + y) = (uint (x + y) = uint x + uint y)"
distinct_1: fixes x2 :: "'a" shows "infty.Infty \<noteq> Num x2"
ide_is_iso: fixes a :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "ide a" shows "local.iso a"
count_mset_set_leq: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "count (mset_set A) x \<le> 1"
positively_invariant_y_upper: shows "g.positively_invariant {p. snd p - 751 / 100 \<le> 0}"
\<Gamma>'_sel_1: shows "edge \<Gamma>' = edge \<Gamma>"
unit_vec_def: fixes n :: "nat"   and i :: "nat" shows "unit_vec n i = vec n (\<lambda>j. if j = i then 1::'a else (0::'a))"
no_proots_line_code: fixes p :: "complex poly"   and st :: "complex"   and tt :: "complex" shows "no_proots_line p st tt = (if poly p st \<noteq> 0 \<and> poly p tt \<noteq> 0 then let pc = p \<circ>\<^sub>p [:st, tt - st:]; pR = map_poly Re pc; pI = map_poly Im pc; g = gcd pR pI in if changes_itv_smods 0 1 g (pderiv g) = 0 then True else False else False)"
open_dist: fixes S :: "'b set" shows "open S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
enforce_option_eq_None_iff: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c option" shows "(enforce_option P x = None) = (\<forall>a. x = Some a \<longrightarrow> \<not> P a)"
to_nat_less_card: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and Rep :: "'a \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'a"   and j :: "'a" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "mod_type.to_nat Rep j < CARD('a)"
group_l_invI: assumes "\<And>x. x \<in> carrier (units_of R) \<Longrightarrow> \<exists>y\<in>carrier (units_of R). y \<otimes>\<^bsub>units_of R\<^esub> x = \<one>\<^bsub>units_of R\<^esub>" shows "Group.group (units_of R)"
simps_7: fixes t :: "'a local.term" shows "Cod \<^bold>\<r>\<^bold>[t\<^bold>] = Cod t"
cases: fixes \<Gamma> :: "'b \<Rightarrow> ('c, 'd, 'b, 'e) seqp"   and a1 :: "('c, 'd, 'b, 'e) seqp"   and a2 :: "('c, 'd, 'b, 'e) seqp" assumes "a1 \<leadsto>\<^bsub>\<Gamma>\<^esub> a2" obtains p1 :: "('c, 'd, 'b, 'e) seqp" and p2 :: "('c, 'd, 'b, 'e) seqp" where "a1 = p1__ \<oplus> p2__" and "a2 = p1__"   | p1 :: "('c, 'd, 'b, 'e) seqp" and p2 :: "('c, 'd, 'b, 'e) seqp" where "a1 = p1 \<oplus> p2" and "a2 = p2"   | pn :: "'b" where "a1 = call(pn)" and "a2 = \<Gamma> pn"
nat_power_functions: fixes n :: "nat" shows "nat_power n \<in> carrier (function_ring (carrier R) R)"
uminus_Bit_eq: fixes k :: "int"   and b :: "bool" shows "- k BIT b = (- k - of_bool b) BIT b"
simps_1: fixes f :: "'d \<Rightarrow> 'c"   and P :: "'d" shows "ACom.map_acom f (SKIP {P}) = SKIP {f P}"
disc_rfr_proc_positive: fixes r :: "real"   and n :: "nat"   and w :: "'b" assumes "- 1 < r" shows "0 < disc_rfr_proc r n w"
bounded: fixes w :: "'c" assumes "w \<in> S" shows "f w < bd"
integrable: shows "set_integrable lborel D (\<lambda>(x, y). P x * (1 - y) ^ n / (1 - (1 - x * y) * w))"
norm_unsigned_Cons0: fixes bs :: "bit list" shows "norm_unsigned (\<zero> # bs) = norm_unsigned bs"
k: fixes a :: "'c"   and b :: "'c" assumes "a \<in> A"   and "b \<in> A"   and "f a = f b" shows "g a = g b"
space_tree_sigma: fixes M :: "'d measure" shows "space (tree_sigma M) = trees (space M)"
var_subterm_trms\<^sub>s\<^sub>s\<^sub>t_is_vars\<^sub>s\<^sub>s\<^sub>t: fixes x :: "'d"   and A :: "('c, 'd) stateful_strand_step list" assumes "Var x \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t trms\<^sub>s\<^sub>s\<^sub>t A" shows "x \<in> vars\<^sub>s\<^sub>s\<^sub>t A"
map_to_set_def: fixes m :: "'e \<Rightarrow> 'f option" shows "map_to_set m = {(k, v). m k = Some v}"
finite_bind: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b set" assumes "finite S"   and "\<forall>x\<in>S. finite (f x)" shows "finite (Set.bind S f)"