hcju/isabelleps huggingface dataset:
induct: fixes x :: "('d, 'e) trm"   and P :: "('d, 'e) trm \<Rightarrow> bool" assumes "dsafe x"   and "\<And>i. P (Var i)"   and "\<And>r. P (Const r)"   and "\<And>args i. \<lbrakk>\<And>i. dsafe (args i); \<And>i. P (args i)\<rbrakk> \<Longrightarrow> P ($f i args)"   and "\<And>\<theta>\<^sub>1 \<theta>\<^sub>2. \<lbrakk>dsafe \<theta>\<^sub>1; P \<theta>\<^sub>1; dsafe \<theta>\<^sub>2; P \<theta>\<^sub>2\<rbrakk> \<Longrightarrow> P (Plus \<theta>\<^sub>1 \<theta>\<^sub>2)"   and "\<And>\<theta>\<^sub>1 \<theta>\<^sub>2. \<lbrakk>dsafe \<theta>\<^sub>1; P \<theta>\<^sub>1; dsafe \<theta>\<^sub>2; P \<theta>\<^sub>2\<rbrakk> \<Longrightarrow> P (Times \<theta>\<^sub>1 \<theta>\<^sub>2)"   and "\<And>\<theta>. dfree \<theta> \<Longrightarrow> P (Differential \<theta>)"   and "\<And>i. P ($' i)" shows "P x"
that: fixes A\<^sub>F :: "name list"   and \<Psi>\<^sub>F :: "'a" assumes "F = \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>"   and "distinct A\<^sub>F"   and "A\<^sub>F \<sharp>* C" shows "thesis"
uminus_closed'': shows "\<forall>a\<in>U. -\<^sub>o\<^sub>w a \<in> U"
finite_fset: fixes S :: "'a fset" shows "finite (fset S)"
ideal_inv1_closed: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and x :: "'c" assumes "Ring R"   and "ideal R I"   and "x \<in> I" shows "-\<^sub>a\<^bsub>R\<^esub> x \<in> I"
simps_15: fixes mem' :: "(word8 \<Rightarrow> word36 \<Rightarrow> word8 option) \<Rightarrow> word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "mem_update mem' \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem' mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr>"
meta_mp: fixes P :: "prop"   and thesis :: "prop" assumes "PROP P \<Longrightarrow> PROP thesis"   and "PROP P" shows "PROP thesis"
rep_eq: fixes x :: "'b set" shows "Rep_ttree (singles x) = {xs. \<forall>x\<in>x. length (filter (\<lambda>x'. x' = x) xs) \<le> 1}"
intros_3: fixes l :: "nat"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com" assumes "l \<turnstile> c\<^sub>1"   and "l \<turnstile> c\<^sub>2" shows "l \<turnstile> c\<^sub>1;; c\<^sub>2"
Scan_items_le: fixes T :: "(symbol \<times> character list) set"   and l :: "nat"   and k :: "nat"   and P :: "item set" assumes "\<And>t. t \<in> T \<Longrightarrow> length (chars_of_token t) \<le> l" shows "Scan T k (items_le k P) \<subseteq> items_le (k + l) (Scan T k P)"
prod_nonneg: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'a" assumes "\<forall>a\<in>A. (0::'a) \<le> f a" shows "(0::'a) \<le> prod f A"
R1: fixes u :: "'a set" assumes "u \<in> nhds x" shows "y \<in> u"
mapWithIndex_Nil: fixes f :: "nat \<Rightarrow> 'b \<Rightarrow> 'a" shows "mapWithIndex f [] = []"
invariance_of_domain_gen: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "open S"   and "continuous_on S f"   and "inj_on f S"   and "DIM('b) \<le> DIM('a)" shows "open (f ` S)"
nat_cut_greater_finite_iff: fixes I :: "nat set"   and t :: "nat" shows "finite (I \<down>> t) = finite I"
b: shows "({0..<m} - (X \<union> Y)) \<inter> (X \<union> Y) = {}"
quot_of_fract_1: shows "quot_of_fract 1 = (1::'b, 1::'b)"
singleCombinators_insort: fixes x :: "('b, 'c) Combinators"   and xs :: "('b, 'c) Combinators list"   and l :: "'b set list" assumes "singleCombinators [x]"   and "singleCombinators xs" shows "singleCombinators (FWNormalisationCore.insort x xs l)"
last_conv_nth: fixes xs :: "'c list" assumes "xs \<noteq> []" shows "last xs = xs ! (|xs| - 1)"
not_case: fixes k :: "nat" assumes "\<not> coprime k n" shows "separable k = (gauss_sum k = 0)"
wf_eq_append': fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and S' :: "('d, 'c) strand_step list"   and t' :: "('d, 'c) Term.term"   and a :: "poscheckvariant"   and t :: "('d, 'c) Term.term" assumes "wf\<^sub>s\<^sub>t V (S @ S')"   and "fv t' \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> V" shows "wf\<^sub>s\<^sub>t V (S @ \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S')"
substitution_code_correct: fixes g :: "'g" assumes "redundant g" shows "substitution g = substitution_code g (chooseNext' g)"
map_to_set_def: fixes m :: "'g \<Rightarrow> 'h option" shows "map_to_set m = {(k, v). m k = Some v}"
rel_compp_Grp: fixes R1 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'e \<Rightarrow> bool" shows "rel_prod R1 R2 = (BNF_Def.Grp {x. Basic_BNFs.fsts x \<subseteq> {(x, y). R1 x y} \<and> Basic_BNFs.snds x \<subseteq> {(x, y). R2 x y}} (map_prod fst fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. Basic_BNFs.fsts x \<subseteq> {(x, y). R1 x y} \<and> Basic_BNFs.snds x \<subseteq> {(x, y). R2 x y}} (map_prod snd snd)"
plus_eint_simps_3: fixes q :: "eint" shows "q + \<infinity> = \<infinity>"
pw_ebind_1: fixes m :: "('c + 'e) nres"   and f :: "'e \<Rightarrow> ('c + 'd) nres" shows "nofail (ebind m f) = (nofail m \<and> (\<forall>x. inres m (Inr x) \<longrightarrow> nofail (f x)))"
partition_on_part_characteristic: fixes A :: "'c set"   and P :: "'c set set"   and X :: "'c set"   and x :: "'c" assumes "partition_on A P"   and "X \<in> P"   and "x \<in> X" shows "X = {y. \<exists>X\<in>P. x \<in> X \<and> y \<in> X}"
converges_to_def: fixes f :: "nat \<Rightarrow> rat \<times> rat"   and x :: "real" shows "converges_to f x \<equiv> (\<forall>n. in_interval (f n) x \<and> sub_interval (f (Suc n)) (f n)) \<and> (\<forall>eps>0. \<exists>n l r. f n = (l, r) \<and> real_of_rat r - real_of_rat l \<le> eps)"
simps_5: fixes f1 :: "'f"   and f2 :: "'g \<Rightarrow> 'g list \<Rightarrow> 'f"   and x21 :: "'g"   and x22 :: "'g list" shows "(case x21 # x22 of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f2 x21 x22"
rewrite_negated_src_ports_not_has_disc_negated: fixes m :: "'b common_primitive match_expr" assumes "normalized_nnf_match m" shows "\<not> has_disc_negated is_Src_Ports False (rewrite_negated_src_ports m)"
WrRequest_def: fixes ch :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and p :: "PrIds"   and l :: "Locs"   and v :: "Vals" shows "WrRequest ch p l v \<equiv> Calling ch p \<and> arg<ch!p> = #(write l v)"
ordinal_order_trans: fixes x :: "ordinal"   and y :: "ordinal"   and z :: "ordinal" assumes "x \<le> y"   and "y \<le> z" shows "x \<le> z"
game_multi_def: fixes \<A> :: "unit \<Rightarrow> (unit, 'x + 'x \<times> 'hash, 'hash + unit) gpv" shows "game_multi \<A> = map_spmf (\<lambda>key. (key, (), {})) key_gen \<bind> (\<lambda>(c_o, c_a, s). exec_gpv (\<dagger>(oracle_hash c_o) \<oplus>\<^sub>O eval_oracle c_o c_a) (\<A> c_a) (False, s) \<bind> (\<lambda>(uu_, b, uu_). return_spmf b))"
ctr_1: fixes n :: "enat"   and xs :: "'b llist" assumes "n = 0 \<or> lnull xs" shows "ltake n xs = LNil"
cont_absc_1: fixes cfg :: "('s \<times> ('c \<Rightarrow> real)) cfg"   and s' :: "'s \<times> ('c \<Rightarrow> real)" assumes "cfg \<in> MDP.valid_cfg"   and "s' \<in> set_pmf (action cfg)" shows "cont (absc cfg) (abss s') = absc (cont cfg s')"
timpls_transformable_to_iff_in_timpl_closure: fixes TI' :: "('b set \<times> 'b set) list"   and TI :: "('b set \<times> 'b set) list"   and s :: "(('c, 'd, 'b) prot_fun, 'e) Term.term"   and t :: "(('c, 'd, 'b) prot_fun, 'e) Term.term" assumes "set TI' = {(a, b). (a, b) \<in> (set TI)\<^sup>+ \<and> a \<noteq> b}" shows "timpls_transformable_to TI' s t = (t \<in> timpl_closure s (set TI))"
size_union: fixes M :: "'c multiset"   and N :: "'c multiset" shows "size (M + N) = size M + size N"
wf: fixes \<omega> :: "'a" assumes "\<omega> \<in> \<Omega>" shows "\<exists>n. P n \<omega>"
EWHILEIT_def: fixes I :: "'d \<Rightarrow> bool"   and c :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> ('c + 'd) nres"   and s :: "'d" shows "EWHILEIT I c f s \<equiv> WHILE\<^sub>T\<^bsup>\<lambda>x. case x of Inl x \<Rightarrow> True | Inr s \<Rightarrow> I s\<^esup> (\<lambda>x. case x of Inl x \<Rightarrow> False | Inr s \<Rightarrow> c s) (\<lambda>s. ASSERT (\<not> isl s) \<bind> (\<lambda>_. Let (projr s) f)) (Inr s)"
vball_vifunion: fixes I :: "V"   and f :: "V \<Rightarrow> V"   and P :: "V \<Rightarrow> bool" shows "(\<forall>z\<in>elts (\<Union>\<^sub>\<circ> (VLambda I f `\<^sub>\<circ> I)). P z) = (\<forall>x\<in>elts I. \<forall>z\<in>elts (f x). P z)"
poly_mat_def: fixes p :: "'b poly"   and A :: "('b, 'c) sq_matrix" shows "poly_mat p A = (\<Sum>i\<le>degree p. coeff p i *\<^sub>S A ^ i)"
state: shows "Partial_Cost_Model.steps init (take (Suc x) qs) (take (Suc x) (take x Strat @ (fst (Strat ! x), take l (snd (Strat ! x)) @ drop (Suc l) (snd (Strat ! x))) # drop (Suc x) Strat)) = Partial_Cost_Model.steps init (take (Suc x) qs) (take (Suc x) Strat)"
subst_fv_dom_img_subset_set: fixes M :: "('b, 'a) Term.term set"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "fv\<^sub>s\<^sub>e\<^sub>t M \<subseteq> subst_domain \<theta>" shows "fv\<^sub>s\<^sub>e\<^sub>t (M \<cdot>\<^sub>s\<^sub>e\<^sub>t \<theta>) \<subseteq> range_vars \<theta>"
of_weak_ranking_Nil: shows "of_weak_ranking [] = (\<lambda>_ _. False)"
nb: shows "trace_parallel_plan_strips I (ops # \<pi>) = I # trace_parallel_plan_strips (execute_parallel_operator I ops) \<pi>"
ipurge_tr_aux_single_event: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and x :: "'b" shows "ipurge_tr_aux I D U [x] = (if \<exists>v\<in>U. (v, D x) \<in> I then [] else [x])"
simps_1: fixes uu :: "poscheckvariant"   and t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term" shows "tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<langle>uu: t \<doteq> t'\<rangle> = ((\<exists>\<delta>. Unifier \<delta> t t') \<longrightarrow> \<Gamma> t = \<Gamma> t')"
simps_3: fixes I :: "('a, 'b, 'c) interp"   and t1 :: "('a, 'c) trm"   and t2 :: "('a, 'c) trm"   and v :: "(real, 'c) vec" shows "sterm_sem I (Plus t1 t2) v = sterm_sem I t1 v + sterm_sem I t2 v"
product_run_embed_fst: fixes \<delta>\<^sub>1 :: "'a \<Rightarrow> 'c \<Rightarrow> 'a"   and \<delta>\<^sub>2 :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and q\<^sub>1 :: "'a"   and q\<^sub>2 :: "'b"   and w :: "nat \<Rightarrow> 'c"   and i :: "nat" shows "run\<^sub>t \<delta>\<^sub>1 \<times> \<delta>\<^sub>2 (q\<^sub>1, q\<^sub>2) w i \<in> embed_transition_fst (run\<^sub>t \<delta>\<^sub>1 q\<^sub>1 w i)"
finite_Union: fixes A :: "'h set set" assumes "finite A"   and "\<And>M. M \<in> A \<Longrightarrow> finite M" shows "finite (\<Union> A)"
assms_6: shows "f \<bullet> \<alpha> = \<zero>"
out_subseqs1_set_def: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c"   and head :: "'d"   and tail :: "'d iarray"   and prevs :: "(nat \<times> 'c) list" shows "my_subseqs.out_subseqs1_set f head tail prevs \<equiv> (f head \<circ> snd) ` set prevs \<union> \<Union> {my_subseqs.out_subseqs2_set f tail v [0..<i] |v i. (i, v) \<in> set prevs}"
SUP_eq: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and bot :: "'d"   and top :: "'d"   and A :: "'e set"   and B :: "'f set"   and f :: "'e \<Rightarrow> 'd"   and g :: "'f \<Rightarrow> 'd" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> \<exists>j\<in>B. less_eq (f i) (g j)"   and "\<And>j. j \<in> B \<Longrightarrow> \<exists>i\<in>A. less_eq (g j) (f i)" shows "Sup (f ` A) = Sup (g ` B)"
irrefl: fixes a :: "'a" shows "\<not> a \<succ> a"
inv_renaming_cancel_r_list: fixes rs :: "'s list" assumes "is_renaming_list rs" shows "rs \<odot>s map inv_renaming rs = replicate (length rs) id_subst"
assms: shows "{y1, y2} \<subseteq> Var ys"
init_no_grey_refs: fixes s :: "'g process_name \<Rightarrow> ('i, 'j, 'k, ('f, 'g, 'h, 'e) Model.local_state) com list \<times> 'j option \<times> ('f, 'g, 'h, 'e) Model.local_state" assumes "gc_W \<lparr>GST = s, HST = []\<rparr>\<down> = {}"   and "\<forall>m. W (\<lparr>GST = s, HST = []\<rparr>\<down> (mutator m)) = {}"   and "sys_W \<lparr>GST = s, HST = []\<rparr>\<down> = {}"   and "gc_ghost_honorary_grey \<lparr>GST = s, HST = []\<rparr>\<down> = {}"   and "\<forall>m. ghost_honorary_grey (\<lparr>GST = s, HST = []\<rparr>\<down> (mutator m)) = {}"   and "sys_ghost_honorary_grey \<lparr>GST = s, HST = []\<rparr>\<down> = {}" shows "no_grey_refs \<lparr>GST = s, HST = []\<rparr>\<down>"
eint_def: fixes n :: "int" shows "eint n = Abs_eint (Some n)"
vpair_neq_snd: fixes a :: "V"   and b :: "V" shows "\<langle>a, b\<rangle> \<noteq> b"
induct: fixes P :: "symbol list \<Rightarrow> (nat \<times> symbol \<times> symbol list) list \<Rightarrow> symbol list \<Rightarrow> bool"   and a0 :: "symbol list"   and a1 :: "(nat \<times> symbol \<times> symbol list) list"   and a2 :: "symbol list" assumes "\<And>a b. P a [] b"   and "\<And>a d D b. (\<And>x. P x D b) \<Longrightarrow> P a (d # D) b" shows "P a0 a1 a2"
src_hcomp: fixes \<mu> :: "'c"   and \<nu> :: "'c" assumes "C.hseq \<mu> \<nu>" shows "src\<^sub>C (\<mu> \<star>\<^sub>C \<nu>) = src\<^sub>C \<nu>"
constr_rep_3: fixes x2 :: "ccs"   and x1 :: "ccs" shows "Rep_ccs (x2 \<oplus> x1) = Sum_Rep (Rep_ccs x2) (Rep_ccs x1)"
True_True_eps_starI: fixes p :: "bool list"   and r :: "bool list"   and A :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" assumes "(p, r) \<in> (eps A)\<^sup>*" shows "(True # p, True # r) \<in> (eps (RegExp2NAe.star A))\<^sup>*"
prio_pop_min_def: fixes q :: "'b \<Rightarrow> 'c option" shows "prio_pop_min q \<equiv> ASSERT (dom q \<noteq> {}) \<bind> (\<lambda>_. SPEC (\<lambda>(e, w, q'). q' = q(e := None) \<and> q e = Some w \<and> (\<forall>e' w'. q e' = Some w' \<longrightarrow> w \<le> w')))"
new_thread_memory: fixes t :: "'t"   and s :: "'x2 \<times> 'm2"   and ta :: "('l, 't, 'x2, 'm2, 'w, 'o) thread_action"   and s' :: "'x2 \<times> 'm2"   and t' :: "'t"   and x :: "'x2"   and m :: "'m2" assumes "t \<turnstile> s -2-ta\<rightarrow> s'"   and "NewThread t' x m \<in> set \<lbrace>ta\<rbrace>\<^bsub>t\<^esub>" shows "m = snd s'"
mvt_very_simple: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and f' :: "real \<Rightarrow> real \<Rightarrow> real" assumes "a \<le> b"   and "\<And>x. \<lbrakk>a \<le> x; x \<le> b\<rbrakk> \<Longrightarrow> (f has_derivative f' x) (at x within {a..b})" shows "\<exists>x\<in>{a..b}. f b - f a = f' x (b - a)"
smult_diff_left: fixes a :: "'b"   and b :: "'b"   and p :: "'b poly" shows "smult (a - b) p = smult a p - smult b p"
card_Un_disjoint: fixes A :: "'a set"   and B :: "'a set" assumes "finite A"   and "finite B"   and "A \<inter> B = {}" shows "card (A \<union> B) = card A + card B"
boolean_algebra_axioms: shows "boolean_algebra (\<^bold>\<sqinter>) (\<^bold>\<squnion>) compl \<zero> \<one>"
tpa_states: fixes tp :: "(action \<times> nat) list"   and as :: "nat"   and ap :: "abc_inst list" assumes "tp = concat (take as (tms_of ap))"   and "as \<le> length ap" shows "start_of (layout_of ap) as = Suc (length tp div 2)"
closure_of_hull: fixes S :: "'c set"   and X :: "'c topology" assumes "S \<subseteq> topspace X" shows "X closure_of S = closedin X hull S"
finalGraph_faces: fixes g :: "graph" assumes "final g" shows "finals g = faces g"
map_FGco_def: fixes l1 :: "'n \<Rightarrow> 'aa"   and co1 :: "'o \<Rightarrow> 'ab"   and co2 :: "'q \<Rightarrow> 'ad"   and co3 :: "'r \<Rightarrow> 'ae"   and co4 :: "'s \<Rightarrow> 'af"   and co5 :: "'p \<Rightarrow> 'ac"   and co6 :: "'w \<Rightarrow> 'ai"   and contra1 :: "'ag \<Rightarrow> 't"   and contra2 :: "'ah \<Rightarrow> 'u"   and contra3 :: "'aj \<Rightarrow> 'x"   and contra4 :: "'ak \<Rightarrow> 'y" shows "map_FGco l1 co1 co2 co3 co4 co5 co6 contra1 contra2 contra3 contra4 = map_F l1 co1 co5 (map_G co1 co2 co3 co4 contra1 contra2) co3 co6 contra1 contra3 contra4"
wf_nodes_le: shows "wf {(y, x). nodes_le x y \<and> x \<noteq> y}"
rep_eq: shows "coeff 1 = (\<lambda>n. of_bool (n = 0))"
lborel_def: shows "lborel = distr (Pi\<^sub>M Basis (\<lambda>b. interval_measure (\<lambda>x. x))) borel (\<lambda>f. \<Sum>b\<in>Basis. f b *\<^sub>R b)"
dom_subid_aux2: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and domain_op :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.domain_near_semiring plus times domain_op less_eq less" shows "less_eq (times (domain_op x) y) y"
bl1_finalThrow: fixes h :: "'heap"   and t :: "'thread_id"   and a :: "'addr"   and xs :: "'addr val list" assumes "hconf h"   and "preallocated h" shows "bisim1_list1 t h (Throw a, xs) [] \<lfloor>a\<rfloor> []"
simps_1: fixes red :: "('d \<times> nat) rgraph"   and black :: "('d, 'e, 'f) lts"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and init_conf :: "('e, 'f) conf"   and confs :: "'d \<times> nat \<Rightarrow> ('e, 'f) conf"   and marked :: "'d \<times> nat \<Rightarrow> bool"   and strengthenings :: "'d \<times> nat \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool"   and more :: "'g" shows "red \<lparr>red = red, black = black, subs = subs, init_conf = init_conf, confs = confs, marked = marked, strengthenings = strengthenings, \<dots> = more\<rparr> = red"
simps_4: fixes tst :: "'a"   and c :: "('a, atom) com" shows "Example_PL.SC_discr (while tst do c) = Example_PL.SC_discr c"
i_append_eq_i_append_conv2_aux: fixes xs :: "'b list"   and f :: "nat \<Rightarrow> 'b"   and ys :: "'b list"   and g :: "nat \<Rightarrow> 'b" assumes "xs \<frown> f = ys \<frown> g"   and "length xs \<le> length ys" shows "\<exists>zs. xs @ zs = ys \<and> f = zs \<frown> g"
mono_gamma_o: fixes S1 :: "(char list \<Rightarrow> 'av) option"   and S2 :: "(char list \<Rightarrow> 'av) option" assumes "S1 \<le> S2" shows "\<gamma>\<^sub>o S1 \<subseteq> \<gamma>\<^sub>o S2"
gHom_mem: fixes F :: "('e, 'f) Group_scheme"   and G :: "('g, 'h) Group_scheme"   and f :: "'e \<Rightarrow> 'g"   and x :: "'e" assumes "Group F"   and "Group G"   and "f \<in> gHom F G"   and "x \<in> carrier F" shows "f x \<in> carrier G"
degree_0_iff: fixes P :: "'b poly" shows "(degree P = 0) = (\<exists>a. P = [:a:])"
Gromov_hyperbolic_subset_def: fixes delta :: "real"   and A :: "'b set" shows "Gromov_hyperbolic_subset delta A = (\<forall>x\<in>A. \<forall>y\<in>A. \<forall>z\<in>A. \<forall>t\<in>A. dist x y + dist z t \<le> max (dist x z + dist y t) (dist x t + dist y z) + 2 * delta)"
Collect_minI: fixes k :: "'a"   and I :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "k \<in> I"   and "P k" shows "\<exists>x\<in>I. P x \<and> (\<forall>y\<in>I. y < x \<longrightarrow> \<not> P y)"
rtrancl_is_path: fixes u :: "'a"   and v :: "'a"   and E :: "('a \<times> 'a) set" assumes "(u, v) \<in> E\<^sup>*" obtains l :: "'a list" where "path E u l v"
transp_onI: fixes A :: "'a set"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>x \<in> A; y \<in> A; z \<in> A; P x y; P y z\<rbrakk> \<Longrightarrow> P x z" shows "transp_on P A"
assms_1: shows "n -as\<rightarrow>\<^sub>\<surd>* n'"
fSum_def: shows "fSum \<equiv> fsum (\<lambda>x. x)"
Cod_UP: fixes \<mu> :: "'a" assumes "B.arr \<mu>" shows "Cod (UP \<mu>) = \<^bold>\<langle>B.cod \<mu>\<^bold>\<rangle>"
uint_0_iff: fixes x :: "'a word" shows "(uint x = 0) = (x = 0)"
class_add_conf: fixes P :: "jvm_prog"   and h :: "heap"   and v :: "val"   and T :: "ty"   and C :: "char list"   and cdec :: "jvm_method class" assumes "P,h \<turnstile> v :\<le> T"   and "\<not> is_class P C" shows "class_add P (C, cdec),h \<turnstile> v :\<le> T"
distinct_nth_eq_iff: fixes xs :: "'a list"   and x :: "nat"   and y :: "nat" assumes "distinct xs"   and "x < length xs"   and "y < length xs" shows "(xs ! x = xs ! y) = (x = y)"
eq_ideal_eq_val: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and y :: "'a" assumes "valuation K v"   and "x \<in> carrier (Vr K v)"   and "y \<in> carrier (Vr K v)"   and "Vr K v \<diamondsuit>\<^sub>p x = Vr K v \<diamondsuit>\<^sub>p y" shows "v x = v y"
sorted_distinct_set_unique: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and ys :: "'c list" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs"   and "distinct xs"   and "linorder.sorted less_eq ys"   and "distinct ys"   and "set xs = set ys" shows "xs = ys"
star_back_loop_mu: fixes y :: "'a"   and z :: "'a" shows "\<mu> (\<lambda>x. x * y \<squnion> z) = z * y\<^sup>\<star>"
add_Pair: fixes a :: "'d"   and b :: "'e"   and c :: "'d"   and d :: "'e" shows "(a, b) + (c, d) = (a + c, b + d)"
Vfrom_trans: fixes A :: "V"   and x :: "V"   and X :: "V"   and i :: "V" assumes "Transset A"   and "x \<in>\<^sub>\<circ> X"   and "X \<in>\<^sub>\<circ> Vfrom A i" shows "x \<in>\<^sub>\<circ> Vfrom A i"
Nil: fixes r :: "('g \<times> 'h) set" shows "([], []) \<in> listrel r"
theor: fixes Aa :: "'a vec set"   and Ba :: "'a vec set" assumes "Aa \<subseteq> carrier_vec n"   and "Ba \<subseteq> carrier_vec n"   and "C = Aa + Ba"   and "Aa \<noteq> {}" shows "\<exists>bndB. Ba \<subseteq> Bounded_vec bndB"
Image_absorb_rtrancl: fixes A :: "('b \<times> 'b) set"   and B :: "'b set"   and C :: "'b set" assumes "trans A"   and "refl_on B A"   and "C \<subseteq> B" shows "A\<^sup>* `` C = A `` C"
oproj_embed: fixes r :: "('c \<times> 'c) set"   and s :: "('d \<times> 'd) set"   and f :: "'c \<Rightarrow> 'd" assumes "Well_order r"   and "Well_order s"   and "oproj r s f" shows "\<exists>g. embed s r g"
set_eqI': fixes A :: "'a set"   and B :: "'a set" assumes "\<And>x. x \<in> A \<Longrightarrow> x \<in> B"   and "\<And>x. x \<in> B \<Longrightarrow> x \<in> A" shows "A = B"
Case_In0: fixes c :: "('c, 'd) Old_Datatype.node set \<Rightarrow> 'b"   and d :: "('c, 'd) Old_Datatype.node set \<Rightarrow> 'b"   and M :: "('c, 'd) Old_Datatype.node set" shows "Case c d (In0 M) = c M"
hyps: fixes i :: "nat" shows "i < length v * row_length N \<and> mat nr (length N) N \<longrightarrow> row (vec_mat_Tensor v N) i = local.times (v ! (i div row_length N)) (row N (i mod row_length N))"
hyps_3: shows "consistent (Node w t\<^sub>1 t\<^sub>2)"
assms_5: shows "moebius_pt M1 z2 = w2"
assms_1: fixes op :: "char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat" assumes "op \<in> set (ast_problem.ast\<delta> prob)" shows "is_standard_operator op"
inj_on_image_mem_iff: fixes f :: "'c \<Rightarrow> 'd"   and B :: "'c set"   and a :: "'c"   and A :: "'c set" assumes "inj_on f B"   and "a \<in> B"   and "A \<subseteq> B" shows "(f a \<in> f ` A) = (a \<in> A)"
orthogonal_transformation_octo_congruence: fixes q :: "octo" assumes "norm q = 1" shows "orthogonal_transformation (\<lambda>x. HIm (Octonions.cnj q * Hv x * q))"
Diff: fixes a :: "'c set"   and b :: "'c set" assumes "a \<in> sigma_sets UNIV (range greaterThan)"   and "b \<in> sigma_sets UNIV (range greaterThan)" shows "a - b \<in> sigma_sets UNIV (range greaterThan)"
lists_n_lepoll_self: fixes A :: "'b set"   and n :: "nat" assumes "infinite A" shows "{l \<in> lists A. length l = n} \<lesssim> A"
return_cong: fixes A :: "'a measure"   and B :: "'a measure"   and x :: "'a" assumes "sets A = sets B" shows "return A x = return B x"
simps_3: fixes x1 :: "'g"   and x21 :: "'h"   and x22 :: "'i" shows "Pure x1 \<noteq> IO x21 x22"