hcju/isabelleps huggingface dataset:
simps_3: fixes xs :: "('a \<times> bool) list"   and x :: "'a \<times> bool"   and y :: "'a \<times> bool"   and ys :: "('a \<times> bool) list" shows "prappend_signed_list (xs @ [x]) (y # ys) = (if y = flip_signed x then prappend_signed_list xs ys else xs @ x # y # ys)"
lub_eqI: fixes M :: "'a set"   and l :: "'a" assumes "M <<| l" shows "lub M = l"
quantifier_change_order_once: fixes m :: "nat"   and P :: "nat \<Rightarrow> nat \<Rightarrow> bool" assumes "\<forall>j<m. \<exists>no. \<forall>n\<ge>no. P n j" shows "\<exists>no. \<forall>j<m. \<forall>n\<ge>no. P n j"
Let_1: fixes x :: "Vars.var"   and \<rho> :: "Vars.var \<Rightarrow> 'Value" assumes "x \<in> domA as" shows "\<lbrakk> the (map_of as x) \<rbrakk>\<^bsub>\<rho>\<^esub> = \<lbrakk> the (map_of as x) \<rbrakk>\<^bsub>\<rho> f|` fv (the (map_of as x))\<^esub>"
T: shows "VectorSpaceHom (\<currency>\<currency>) indV (\<sharp>\<star>) (\<psi> T)"
gFreshGAbsRev_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd, 'e) Recursion.model_scheme" shows "gFreshGAbsRev MOD \<equiv> \<forall>ys y xs x s X' X. isInBar (xs, s) \<and> wls s X' \<and> gWls MOD s X \<longrightarrow> gFreshAbs MOD ys y (Abs xs x X') (gAbs MOD xs x X' X) \<longrightarrow> ys = xs \<and> y = x \<or> gFresh MOD ys y X' X"
ile_add1: fixes n :: "enat"   and m :: "enat" shows "n \<le> n + m"
dinv: shows "\<forall>s. I s \<longrightarrow> (\<forall>X\<in>Sols (\<lambda>t. f) (\<lambda>s. {t. 0 \<le> t}) UNIV 0 s. \<forall>t\<ge>0. (\<forall>\<tau>. 0 \<le> \<tau> \<and> \<tau> \<le> t \<longrightarrow> G (X \<tau>)) \<longrightarrow> I (X t))"
to_mregex_progress: fixes m :: "modality"   and g :: "safety"   and r :: "Formula.formula Regex.regex"   and mr :: "mregex"   and \<phi>s :: "Formula.formula list"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and P :: "String.literal \<Rightarrow> nat option"   and j :: "nat" assumes "safe_regex m g r"   and "to_mregex r = (mr, \<phi>s)" shows "progress_regex \<sigma> P r j = (if \<phi>s = [] then j else MIN \<phi>\<in>set \<phi>s. Monitor.progress \<sigma> P \<phi> j)"
kernel_is_ideal: shows "ideal (a_kernel R (S Quot P) ((+>\<^bsub>S\<^esub>) P \<circ> h)) R"
set_iterator_filter_foldli_conv: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "set_iterator_filter P (foldli xs) = foldli (filter P xs)"
Domainp_eq_on: fixes X :: "'a set" shows "Domainp (eq_on X) = (\<lambda>x. x \<in> X)"
tab_simps_5: shows "cod \<rho> = (g \<star> f\<^sup>*) \<star> f"
minus_mult_commute: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and a :: "'d"   and b :: "'d" assumes "class.ring plus zero minus uminus times" shows "times (uminus a) b = times a (uminus b)"
orthogonal_iff_vangle: fixes u :: "'b"   and v :: "'b" shows "orthogonal u v = (vangle u v = pi / 2)"
rep_eq: fixes x :: "int poly" shows "poly.coeff (of_int_poly x) = (\<lambda>xa. of_int (poly.coeff x xa))"
subdag_trans: fixes t :: "dag"   and s :: "dag"   and r :: "dag" assumes "subdag t s"   and "subdag s r" shows "subdag t r"
idinfo_init: fixes s :: "(agent \<times> msg) set"   and n :: "nat"   and X :: "msg" assumes "s\<^sub>0 \<Turnstile> s"   and "n \<notin> bad_id_password \<union> bad_id_pubkey \<union> bad_id_shakey" shows "\<langle>n, X\<rangle> \<notin> spied s"
ipset_from_cidr_wordlength: fixes ip :: "'b word" shows "ipset_from_cidr ip LENGTH('b) = {ip}"
used_Nil: shows "used [] = (\<Union>B. parts (initState B))"
tabulate_aux2: fixes M :: "(lit, lit list) mapping"   and v :: "lit"   and cnfi :: "(lit \<times> lit) list" assumes "distinct (\<alpha>sl M v)" shows "distinct (\<alpha>sl (fold ins_edges cnfi M) v)"
intros_1: fixes F :: "'b formula"   and G :: "'b formula" shows "(F::'b formula) \<^bold>\<rightarrow> (G::'b formula) \<^bold>\<rightarrow> F \<in> AX0"
map_def: shows "E\<^sub>S.map \<equiv> \<lambda>f. if arr f then E\<^sub>S.E.map f else D.null"
delete_not_domA: fixes x :: "'d"   and \<Gamma> :: "('d \<times> 'e) list" assumes "x \<notin> domA \<Gamma>" shows "delete x \<Gamma> = \<Gamma>"
iso_iff: shows "(h \<in> Group.iso G H) = (carrier H \<subseteq> h ` carrier G \<and> (\<forall>x\<in>carrier G. h x = \<one>\<^bsub>H\<^esub> \<longrightarrow> x = \<one>))"
infs_suffix: fixes A :: "'b set"   and w :: "'b stream" shows "infs A w = (\<forall>u v. w = u @- v \<longrightarrow> sset v \<inter> A \<noteq> {})"
lrev_lrev_body_eq: shows "lrev = fix\<cdot>lrev_body"
RepAbs_matrix: fixes x :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "\<exists>m. \<forall>j i. m \<le> j \<longrightarrow> x j i = (0::'b)"   and "\<exists>n. \<forall>j i. n \<le> i \<longrightarrow> x j i = (0::'b)" shows "Rep_matrix (Abs_matrix x) = x"
Completeness: fixes p :: "(symbol \<times> character list) list" assumes "p \<in> ll" shows "\<exists>\<alpha>. pvalid_with p (finished_item \<alpha>) 0 [] \<and> finished_item \<alpha> \<in> \<II>"
c_add: shows "c_add \<le> (2 * n + 2 * i + 7) * i"
assms_3: shows "set_borel_measurable borel (einterval A C) f"
primroot_is_primroot: fixes x :: "'b list" assumes "x \<noteq> \<epsilon>" shows "x \<in>\<^sub>p \<rho> x *"
vert_line_integral_zero: shows "one_chain_line_integral F {i} (two_chain_vertical_boundary two_chain) = 0"
split_comp_eq: fixes f :: "'d \<Rightarrow> 'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'd" shows "(\<lambda>u. f (g (fst u)) (snd u)) = (\<lambda>(x, y). f (g x) y)"
composition_of_tangle_diagrams: fixes x :: "wall"   and y :: "wall" assumes "is_tangle_diagram x"   and "is_tangle_diagram y"   and "domain_wall y = codomain_wall x" shows "is_tangle_diagram (x \<circ> y)"
v0_conforms: assumes "P $ n \<in> VV p" shows "\<sigma> (P $ n) = P'.w0"
c: shows "Esigma (h * M) = Esigma h * Esigma M"
lossless_xor_funct: fixes A :: "bool \<times> bool"   and B :: "bool \<times> bool" shows "lossless_spmf (xor_funct A B)"
emptyE: fixes a :: "'k"   and P :: "bool" assumes "a \<in> {}" shows "P"
all_processes_snapshotted_in_final_state: fixes channel :: "nat \<Rightarrow> ('d \<times> 'd) option"   and trans :: "'d \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'f \<Rightarrow> bool"   and init :: "('d, 'e, 'f) configuration"   and final :: "('d, 'e, 'f) configuration"   and t :: "('d, 'e, 'f) event list"   and p :: "'d" assumes "computation channel trans send recv init final"   and "distributed_system.trace channel trans send recv init t final" shows "ps final p \<noteq> None"
InitConf_HAStates2: fixes S :: "'d"   and A :: "('d, 'e, 'f) hierauto" assumes "S \<in> InitConf A" shows "S \<in> HAStates A"
d_delta_arctan_upper_41_pos: fixes x :: "real" assumes "x \<noteq> 0" shows "0 < diff_delta_arctan_upper_41 x"
a_get_child_nodes_tups_def: shows "CD.a_get_child_nodes_tups = [(is_element_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, CD.get_child_nodes\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>e\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r), (is_character_data_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, CD.get_child_nodes\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r), (is_document_ptr\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r, CD.get_child_nodes\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r \<circ> the \<circ> cast\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>d\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r)]"
not_USUP: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'a upred" shows "(\<not> (\<Squnion> i \<in> A \<bullet> P i)) = (\<Sqinter> i \<in> A \<bullet> \<not> P i)"
simps_13: fixes Vs :: "char list list"   and e :: "char list exp"   and c :: "char list exp" shows "compE\<^sub>1 Vs (while (e) c) = while (compE\<^sub>1 Vs e) compE\<^sub>1 Vs c"
abelian_monoidE_2: fixes R :: "('c, 'd) ring_scheme" assumes "abelian_monoid R" shows "\<zero>\<^bsub>R\<^esub> \<in> carrier R"
iIN_minus: fixes x :: "nat"   and n :: "nat"   and d :: "nat"   and k :: "nat" assumes "x \<in> [n\<dots>,d]"   and "k \<le> x - n" shows "x - k \<in> [n\<dots>,d]"
nxt_subset_atLeast: fixes m :: "nat"   and n :: "nat"   and N :: "nat set" assumes "m \<le> n" shows "nxt N n \<subseteq> {m..}"
simps_6: fixes uw :: "((char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool) \<times> char list set"   and E' :: "(char list \<Rightarrow> int) \<Rightarrow> char list \<Rightarrow> int"   and E :: "(char list \<Rightarrow> int) \<Rightarrow> nat"   and x :: "char list set \<times> (char list \<Rightarrow> char list set)"   and b :: "bexp"   and C :: "acom" shows "time ({(uw, E', E, x)} WHILE b DO C) = E"
cdf_cts_step_1: fixes x :: "real"   and y :: "real" assumes "x < y" shows "cdf M x \<le> expectation (cts_step x y)"
emb_def: fixes f :: "'f \<Rightarrow> 'a"   and D :: "'f set"   and x :: "'f" shows "emb f D x = (if x \<in> D then f x else \<zero>\<^bsub>K\<^esub>)"
rc: shows "root_cond (cf_pos_poly (poly_mult_rat r1 p2), l, r) (real_of_rat r1 * real_of_1 (p2, l2, r2))"
preserves_modes_mem_mds\<^sub>A_simp: fixes mds\<^sub>A :: "Mode \<Rightarrow> 'Var set"   and mds\<^sub>C :: "Mode \<Rightarrow> 'Var set" assumes "\<forall>m. id ` mds\<^sub>A m = range id \<inter> mds\<^sub>C m" shows "mds\<^sub>A = gen_refine.mds\<^sub>A_of mds\<^sub>C"
clock_def: fixes state :: "'b state" shows "clock state = (case state of make_state x1 x2 x3 x4 x5 \<Rightarrow> x1)"
bn_eqvt: fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a"   and bn :: "'a \<Rightarrow> atom set"   and p :: "perm"   and \<alpha> :: "'a" assumes "class.bn permute bn" shows "p \<bullet> bn \<alpha> = bn (permute p \<alpha>)"
assms_1: shows "w\<^sub>1 \<le> w\<^sub>2"
order_def: fixes S :: "('d, 'e) monoid_scheme" shows "order S = card (carrier S)"
f: shows "has_bochner_integral lborel (\<lambda>x. indicat_real {0..} x *\<^sub>R f x) x"
wls_Abs_vsubst_all: fixes s :: "'sort"   and X :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and X' :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set"   and xs :: "'varSort"   and x :: "'var"   and x' :: "'var" assumes "wls s X"   and "wls s X'" shows "(Abs xs x X = Abs xs x' X') = (\<forall>y. X #[y // x]_xs = X' #[y // x']_xs)"
induct: fixes P :: "'c match_expr list \<Rightarrow> bool"   and a0 :: "'c match_expr list" assumes "P []"   and "\<And>e. P [e]"   and "\<And>e v va. P (v # va) \<Longrightarrow> P (e # v # va)" shows "P a0"
simp_boolean_subset: fixes x :: "'b boolean_subset" shows "\<exists>y. Rep_boolean_subset x = - y"
sel: fixes x :: "'c \<Rightarrow> 'd" shows "run_state (StateT x) = x"
abs_eq: fixes xa :: "int"   and x :: "int" shows "(Word.Word xa \<le> Word.Word x) = (take_bit LENGTH('k) xa \<le> take_bit LENGTH('k) x)"
alwaysE: fixes P :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "(\<box>P) \<sigma>" obtains "P (\<sigma> |\<^sub>s i)"
quicksort_pre_def: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and h :: "'d \<Rightarrow> 'c"   and xs0 :: "'d list"   and lo :: "nat"   and hi :: "nat"   and xs :: "'d list" shows "quicksort_pre R h xs0 lo hi xs \<equiv> lo \<le> hi \<and> hi < length xs \<and> mset xs = mset xs0"
atLeastAtMost_singleton': fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.order less_eq less"   and "a = b" shows "ord.atLeastAtMost less_eq a b = {a}"
simps_1: fixes x :: "rat" shows "radicals (Const x) = {}"
lens_indep_get: fixes x :: "'a \<Longrightarrow> 'b"   and y :: "'c \<Longrightarrow> 'b"   and \<sigma> :: "'b"   and v :: "'c" assumes "x \<bowtie> y" shows "get\<^bsub>x\<^esub> (put\<^bsub>y\<^esub> \<sigma> v) = get\<^bsub>x\<^esub> \<sigma>"
rem_implicit_pres_ops_enabled_2: fixes prob :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list"   and name :: "char list"   and s :: "nat \<Rightarrow> nat option" assumes "\<And>op. op \<in> set (ast_problem.ast\<delta> prob) \<Longrightarrow> consistent_pres_op op"   and "ast_problem.enabled (rem_implicit_pres_ops prob) name s" shows "ast_problem.enabled prob name s"
descartes_sign_real: fixes p :: "real poly" assumes "p \<noteq> 0"   and "all_roots_real p" shows "int (proots_count p {x. 0 < x}) = changes (coeffs p)"
balance_left_rbt_sorted: fixes l :: "('a, 'c) rbt"   and r :: "('a, 'c) rbt"   and k :: "'a"   and v :: "'c" assumes "rbt_sorted l"   and "rbt_sorted r"   and "l |\<guillemotleft> k"   and "k \<guillemotleft>| r" shows "rbt_sorted (balance_left l k v r)"
rel_sel: fixes R1 :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and R2 :: "'i \<Rightarrow> 'j \<Rightarrow> bool"   and R3 :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and a :: "('g, 'i, 'k) generat"   and b :: "('h, 'j, 'l) generat" shows "rel_generat R1 R2 R3 a b = (is_Pure a = is_Pure b \<and> (is_Pure a \<longrightarrow> is_Pure b \<longrightarrow> R1 (generat.result a) (generat.result b)) \<and> (\<not> is_Pure a \<longrightarrow> \<not> is_Pure b \<longrightarrow> R2 (generat.output a) (generat.output b) \<and> R3 (continuation a) (continuation b)))"
gauge_def: fixes \<gamma> :: "'c \<Rightarrow> 'c set" shows "gauge \<gamma> = (\<forall>x. x \<in> \<gamma> x \<and> open (\<gamma> x))"
leg: fixes w :: "complex" assumes "w \<in> cball \<xi> d" shows "cmod x \<le> cmod (g w)"
simps: fixes I :: "'at set"   and S :: "'at Literal set set" shows "I \<Turnstile> S = (\<forall>C. C \<in> S \<longrightarrow> I \<Turnstile> C)"
slice_kind_Pred_empty_obs_nearer_SOME: fixes a :: "'edge"   and S :: "'node SDG_node set"   and Q :: "('var \<Rightarrow> 'val option) \<Rightarrow> bool"   and mex :: "'node"   and x :: "nat" assumes "sourcenode a \<notin> \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"   and "kind a = (Q)\<^sub>\<surd>"   and "obs_intra (sourcenode a) \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub> = {}"   and "method_exit mex"   and "get_proc (sourcenode a) = get_proc mex"   and "distance (targetnode a) mex x"   and "distance (sourcenode a) mex (x + 1)"   and "targetnode a = (SOME n'. \<exists>a'. sourcenode a = sourcenode a' \<and> distance (targetnode a') mex x \<and> valid_edge a' \<and> intra_kind (kind a') \<and> targetnode a' = n')" shows "slice_kind S a = (\<lambda>s. True)\<^sub>\<surd>"
bl2nat_append: fixes xs :: "cell list"   and ys :: "cell list" shows "bl2nat (xs @ ys) 0 = bl2nat xs 0 + bl2nat ys (length xs)"
snap_p_2: shows "ps (S t (j + 1)) p \<noteq> None"
discI_5: fixes stateful_strand_step :: "('c, 'd) stateful_strand_step"   and x51 :: "('c, 'd) Term.term"   and x52 :: "('c, 'd) Term.term" assumes "stateful_strand_step = delete\<langle>x51,x52\<rangle>" shows "is_Delete stateful_strand_step"
indis_sym: fixes cval :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and indis :: "('b \<times> 'b) set"   and s :: "'b"   and s' :: "'b" assumes "PL_Indis cval indis"   and "(s, s') \<in> indis" shows "(s', s) \<in> indis"
simps_2: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> bool \<times> bool"   and uu :: "'b"   and uv :: "'b list" shows "lex_ext_unbounded f (uu # uv) [] = (True, True)"
In1_UN1: fixes f :: "'h \<Rightarrow> ('f, 'g) Old_Datatype.node set" shows "In1 (\<Union> (range f)) = (\<Union>x. In1 (f x))"
netglobal_def: fixes P :: "(nat \<Rightarrow> 'g) \<Rightarrow> bool" shows "netglobal P \<equiv> \<lambda>s. P (fst (initmissing (netgmap sr s)))"
complex_i_mult_minus: fixes x :: "complex" shows "\<i> * (\<i> * x) = - x"
space_join: fixes M :: "'c measure measure" shows "space (join M) = space (select_sets M)"
inv_mult_group: fixes x :: "nat"   and y :: "nat" assumes "x \<in> carrier G"   and "y \<in> carrier G" shows "inv\<^bsub>G\<^esub> (x \<otimes>\<^bsub>G\<^esub> y) = inv\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x"
B: shows "affine hull B = affine hull V \<and> \<not> affine_dependent B \<and> int (card B) = aff_dim V + 1"
conc_fun_def: fixes R :: "('a \<times> 'b) set"   and m :: "'b nres" shows "\<Down> R m \<equiv> case m of FAILi \<Rightarrow> FAIL | RES X \<Rightarrow> RES (R\<inverse> `` X)"
cut_greater_Max_empty: fixes I :: "'b set"   and t :: "'b" assumes "finite I"   and "Max I \<le> t" shows "I \<down>> t = {}"
swap_cs_Trans_Snapshot: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isTrans ev"   and "isSnapshot ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'" shows "cs e i = cs e' i"
I: shows "P \<Longrightarrow>\<^sub>t P'"
Deduct_with_False_axioms: shows "Deduct_with_False var trm fmla Var FvarsT substT Fvars subst eql cnj imp all exi fls num prv"
wf_eq_fv: fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and t :: "('d, 'c) Term.term"   and t' :: "('d, 'c) Term.term"   and S' :: "('d, 'c) strand_step list" assumes "wf\<^sub>s\<^sub>t V (S @ \<langle>assign: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S')" shows "fv t' \<subseteq> wfvarsoccs\<^sub>s\<^sub>t S \<union> V"
i_append_i_take_i_drop_id: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "f \<Down> n \<frown> f \<Up> n = f"
monom_zero: shows "monom 0 (0::'b) = 0"
convergentE: fixes F :: "'a set set" assumes "F convergent" obtains x :: "'a" where "F \<longlongrightarrow> x" and "x \<in> carrier"
MakeCatComp2: fixes C :: "('g, 'h, 'i) Category_scheme"   and f :: "'h"   and g :: "'h" assumes "f \<approx>>\<^bsub>MakeCat C\<^esub> g" shows "f ;;\<^bsub>MakeCat C\<^esub> g = f ;;\<^bsub>C\<^esub> g"
subst_compose_same: fixes \<phi> :: "'fmla"   and t1 :: "'trm"   and t2 :: "'trm"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm"   and "x \<in> var" shows "subst (subst \<phi> t1 x) t2 x = subst \<phi> (substT t1 t2 x) x"
measurable_on_compose_continuous: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'c" assumes "f measurable_on UNIV"   and "continuous_on UNIV g" shows "g \<circ> f measurable_on UNIV"
eq_span_insert_eq: fixes x :: "'c"   and y :: "'c"   and S :: "'c set" assumes "x - y \<in> ideal S" shows "ideal (insert x S) = ideal (insert y S)"
independent_if_scalars_zero: fixes A :: "'b poly set" assumes "finite A"   and "\<And>f x. \<lbrakk>(\<Sum>x\<in>A. smult (f x) x) = 0; x \<in> A\<rbrakk> \<Longrightarrow> f x = (0::'b)" shows "poly_vs.independent A"
A9: shows "Esigma 69615 = Esigma (3\<^sup>2) * Esigma 7 * Esigma 13 * Esigma 5 * Esigma 17"
mult2_alt_s_single: fixes a :: "'a"   and b :: "'a"   and s :: "('a \<times> 'a) set"   and ns :: "('a \<times> 'a) set" assumes "(a, b) \<in> s" shows "({#a#}, {#b#}) \<in> mult2_alt_s ns s"
rep_eq: fixes x :: "('a, 'b) lifted_boolean_algebra"   and xa :: "('a, 'b) lifted_boolean_algebra" shows "(x \<le> xa) = (\<forall>f. Rep_lifted_boolean_algebra x f \<le> Rep_lifted_boolean_algebra xa f)"
incl_subgroup: fixes J :: "'a set"   and I :: "'a set" assumes "subgroup J (G\<lparr>carrier := generate G {a}\<rparr>)"   and "subgroup I (G\<lparr>carrier := generate G {a}, carrier := J\<rparr>)" shows "subgroup I (G\<lparr>carrier := generate G {a}\<rparr>)"
angle_refl_2: fixes a :: "'b"   and b :: "'b" shows "angle a b b = pi / 2"
AE_T_all_fair: fixes s :: "'s" shows "AE \<omega> in T s. \<forall>(t, t')\<in>SIGMA x:UNIV. set_pmf (K x). fair t t' \<omega>"
wf_bs_def: fixes bs :: "'a list"   and p :: "poly" shows "wf_bs bs p = (maxindex p \<le> length bs)"
flipped_reflections_def: fixes w :: "'w" shows "flipped_reflections w \<equiv> {t \<in> \<Union>w\<in>W. lconjby w ` S. GroupByPresentationInducedFun.induced_hom S (map (charpair S) ` R) signed_lconjpermutation (the_inv_into (Group.quotient_group (FreeGroup S) \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>) (GroupByPresentationInducedFun.induced_hom S (map (charpair S) ` R) id) (- w)) \<rightarrow> pairtrue t = pairfalse (rconjby w t)}"
Fvars_softSubst: fixes \<phi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var" shows "Fvars (softSubst \<phi> t x) = Fvars \<phi> \<union> FvarsT t - {x}"
simps_1: fixes uu :: "('b \<times> 'b) set"   and uv :: "'a \<Rightarrow> 'b"   and uw :: "'b set" shows "ipurge_tr_aux uu uv uw [] = []"
prime_factorization_mult: 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 unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "y \<noteq> zero" shows "factorial_semiring.prime_factorization zero times one normalize (times x y) = factorial_semiring.prime_factorization zero times one normalize x + factorial_semiring.prime_factorization zero times one normalize y"
keys_dom_lookup: fixes m :: "('b, 'c) mapping" shows "Mapping.keys m = dom (Mapping.lookup m)"
A_power_P: fixes k :: "nat" shows "of_real_hom.mat_hom A ^\<^sub>m k * P = P * J ^\<^sub>m k"
the_Arr_def: fixes obj :: "obj option" shows "the_Arr obj = (SOME (T, k, t). obj = Some \<lparr>tag = Arr T k, values = t\<rparr>)"
F_pos: fixes x :: "'a" shows "0 \<le> F x"
sub_refl: fixes s :: "('c, 'd) tm" shows "sub s s"
dims: shows "nths dims (vars1 \<union> vars2) = dims"
compactin_subspace: fixes X :: "'a topology"   and S :: "'a set" shows "compactin X S = (S \<subseteq> topspace X \<and> compact_space (subtopology X S))"
continuous_mult: fixes F :: "'c filter"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "continuous F f"   and "continuous F g" shows "continuous F (\<lambda>x. f x * g x)"
rep_eq: fixes x :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and xa :: "('b, 'c) RBT.rbt" shows "fold1_keys x xa = rbt_fold1_keys x (rbt.impl_of xa)"
x: shows "x \<in> space (Pi\<^sub>M {0..<n} M)"
lnth_ltl: fixes xs :: "'c llist"   and n :: "nat" assumes "\<not> lnull xs" shows "ltl xs $ n = xs $ Suc n"
ENFsr_offending_flows_exist: shows "set_offending_flows G nP \<noteq> {}"
qfree_plus_inf: fixes \<phi> :: "atom fm" assumes "nqfree \<phi>" shows "qfree (inf\<^sub>+ \<phi>)"
1: shows "\<forall>v\<in>fv s. snd v \<le> max_var_set (fv\<^sub>s\<^sub>e\<^sub>t M)"
poly_induct: fixes p :: "nat \<Rightarrow> 'a"   and Q :: "(nat \<Rightarrow> 'a) \<Rightarrow> bool" assumes "p \<in> carrier P"   and "\<And>p. \<lbrakk>p \<in> carrier P; deg R p = 0\<rbrakk> \<Longrightarrow> Q p"   and "\<And>p. \<lbrakk>\<And>q. \<lbrakk>q \<in> carrier P; deg R q < deg R p\<rbrakk> \<Longrightarrow> Q q; p \<in> carrier P; 0 < deg R p\<rbrakk> \<Longrightarrow> Q p" shows "Q p"
image_eq: fixes g :: "'b \<Rightarrow> 'c"   and A :: "'b set" assumes "inj_on g A" shows "\<Prod>(g ` A) = prod g A"