hcju/isabelleps huggingface dataset:
open_segment_translation: fixes c :: "'c"   and a :: "'c"   and b :: "'c" shows "open_segment (c + a) (c + b) = (+) c ` open_segment a b"
real_of_rat_def: fixes x :: "rat" shows "real_of x = real_of_rat x"
lookup_empty: fixes k :: "'c" shows "lookup Mapping.empty k = None"
vec_nth_cases: fixes y :: "'b \<Rightarrow> 'c" assumes "y \<in> UNIV" obtains x :: "('c, 'b) vec" where "y = ($) x"
assms_5: shows "fs' = fs[i := fs ! i - c \<cdot>\<^sub>v fs ! j]"
analz_insert_partition: fixes S :: "msg set"   and x :: "msg" assumes "analz S \<subseteq> synth (analz (S \<inter> payload \<union> S \<inter> valid \<union> S \<inter> range LtK \<union> Tags))"   and "x \<in> payload \<union> valid \<union> range LtK" shows "analz (insert x S) \<subseteq> synth (analz (insert x S \<inter> payload \<union> insert x S \<inter> valid \<union> insert x S \<inter> range LtK \<union> Tags))"
quasinorm_subset_Norm_eNorm: fixes f :: "'b"   and N1 :: "'b quasinorm"   and N2 :: "'b quasinorm"   and C :: "real" assumes "f \<in> space\<^sub>N N1 \<Longrightarrow> Norm N2 f \<le> C * Norm N1 f"   and "N1 \<subseteq>\<^sub>N N2"   and "0 < C" shows "eNorm N2 f \<le> ennreal C * eNorm N1 f"
simps_10: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term" shows "\<^bold>\<lfloor>\<^bold>\<a>\<^sup>-\<^sup>1\<^bold>[t, u, v\<^bold>]\<^bold>\<rfloor> = \<^bold>\<lfloor>t\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> \<^bold>\<lfloor>u\<^bold>\<rfloor> \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> \<^bold>\<lfloor>v\<^bold>\<rfloor>"
trm_rep_init_def: fixes t :: "'a trm" shows "trm_rep t = (\<lambda>S. if subterm_reduction_applicable_aux S t then subterm_reduction_aux S t else get_min t (set_of_candidate_values S t))"
is_cupcake_all_envE: fixes env :: "v sem_env" assumes "is_cupcake_all_env env" obtains v :: "(char list \<times> v) list" and c :: "(char list, char list, nat \<times> tid_or_exn) namespace" where "env = make_sem_env (Bind v []) c" and "c = static_cenv" and "list_all (is_cupcake_value \<circ> snd) v"
closest_point_refl: fixes S :: "'b set"   and x :: "'b" assumes "closed S"   and "S \<noteq> {}" shows "(closest_point S x = x) = (x \<in> S)"
d: shows "0 < s' \<and> stp \<le> stpa"
reduction_signs_is_get_signs: fixes p :: "real poly"   and qs :: "real poly list"   and signs :: "rat list list"   and subsets :: "nat list list"   and m :: "rat mat" shows "reduction_signs p qs signs subsets m = get_signs (reduce_system p (qs, m, subsets, signs))"
assms: shows "\<forall>x. P x \<longrightarrow> f x = g x"
finite_block_nominals: fixes block :: "('c, 'b) fm list \<times> 'b" shows "finite (block_nominals block)"
snd_set_split: fixes ps :: "'e"   and pth :: "('f \<times> 'e) list" assumes "ps \<in> snd ` set pth" obtains pth1 :: "('f \<times> 'e) list" and pth2 :: "('f \<times> 'e) list" and e :: "'f \<times> 'e" where "pth = pth1 @ [e] @ pth2" and "snd e = ps" and "ps \<notin> snd ` set pth1"
soundD_mem: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and I :: "('sf, 'sc, 'sz) interp" assumes "sound (SG, C)"   and "is_interp I"   and "\<And>\<phi>. List.member SG \<phi> \<Longrightarrow> seq_sem I \<phi> = UNIV" shows "seq_sem I C = UNIV"
DomAppendUnion: fixes a :: "('a \<times> 'b) list"   and ab :: "('a \<times> 'b) list" shows "Dom (a @ ab) = Dom a \<union> Dom ab"
key_gen_def: shows "key_gen = spmf_of_set (nlists UNIV len)"
iso_assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "S.ide a"   and "S.ide b"   and "S.ide c" shows "S.iso (M\<^sub>\<omega>.assoc a b c)"
Cons_4: shows "gallery (C # (E # Es) @ [D])"
restrictedRange: fixes P :: "('d \<times> 'c) set"   and X :: "'d set" shows "Range (P || X) = P `` X"
prime_hd_non_zero: fixes p :: "nat"   and q :: "nat" assumes "prime p"   and "prime q" shows "hd (nat_to_bv (p * q)) \<noteq> \<zero>"
eq: fixes A :: "'a set" assumes "A \<in> sets M" shows "T -` T' -` A \<inter> T -` space M' \<inter> space M = A"
absorb2: fixes b :: "'d"   and a :: "'d" assumes "b \<le> a" shows "inf a b = b"
split: fixes P :: "'r \<Rightarrow> bool"   and f1 :: "'s \<Rightarrow> 'r"   and f2 :: "'t \<Rightarrow> 'u \<Rightarrow> 'r"   and generat :: "('s, 't, 'u) generat" shows "P (case generat of Pure x \<Rightarrow> f1 x | IO x xa \<Rightarrow> f2 x xa) = ((\<forall>x1. generat = Pure x1 \<longrightarrow> P (f1 x1)) \<and> (\<forall>x21 x22. generat = IO x21 x22 \<longrightarrow> P (f2 x21 x22)))"
hyps: fixes ys :: "'a memref list"   and \<S> :: "nat \<Rightarrow> bool option" assumes "(all_shared xs \<union> all_unshared xs \<union> all_acquired xs) \<inter> (all_shared ys \<union> all_unshared ys \<union> all_acquired ys) = {}" shows "share xs (share ys \<S>) = share ys (share xs \<S>)"
no_microstep_guard: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and l :: "'d"   and g :: "'b \<Rightarrow> 'b set"   and p :: "('b, 'c, 'a, 'd) seqp"   and q :: "('b, 'c, 'a, 'd) seqp" shows "\<not> ({l}\<langle>g\<rangle> p) \<leadsto>\<^bsub>\<Gamma>\<^esub> q"
Map_comp: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr"   and g :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f"   and "arr g"   and "Dom g = Cod f" shows "Map (g \<cdot> f) = compose (Dom f) (Map g) (Map f)"
simplex: fixes S :: "'a set" assumes "finite S"   and "(0::'a) \<notin> S" shows "convex hull insert (0::'a) S = {y. \<exists>u. (\<forall>x\<in>S. 0 \<le> u x) \<and> sum u S \<le> 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = y}"
sel_2: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "stl (siterate f x) = siterate f (f x)"
nat_pow_mult1l: fixes x :: "'a"   and n :: "nat" assumes "x \<in> carrier G" shows "x \<otimes> x [^] n = x [^] Suc n"
image_atLeastZeroLessThan_integer: fixes u :: "integer" assumes "0 \<le> u" shows "{0..<u} = of_nat ` {..<nat_of_integer u}"
step_store: fixes F :: "'fenv_inca"   and f :: "'fun"   and fd :: "('dyn, 'var, 'fun, 'op, 'opinl) Inca.instr fundef"   and pc :: "nat"   and x :: "'var"   and H :: "'henv"   and y :: "'dyn"   and d :: "'dyn"   and H' :: "'henv"   and \<Sigma> :: "'dyn list"   and st :: "('fun, 'dyn) frame list" assumes "Finca_get F f = Some fd"   and "pc < length (body fd)"   and "body fd ! pc = Inca.instr.IStore x"   and "heap_add H (x, y) d = H'" shows "Sinca.step (State F H (Frame f pc (y # d # \<Sigma>) # st)) (State F H' (Frame f (Suc pc) \<Sigma> # st))"
P_LCons: shows "P = LCons w0 (ltl P)"
nc: shows "\<forall>i<length A. length (A ! i) = nc"
rep_eq: fixes x :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and xa :: "('c, 'a) uexpr"   and xb :: "('d, 'a) uexpr" shows "\<lbrakk>bop x xa xb\<rbrakk>\<^sub>e = (\<lambda>b. x (\<lbrakk>xa\<rbrakk>\<^sub>e b) (\<lbrakk>xb\<rbrakk>\<^sub>e b))"
mult_eq: fixes x :: "nat"   and y :: "nat" shows "x \<otimes>\<^bsub>R\<^esub> y = x * y mod n"
induct: fixes P :: "'a act list \<Rightarrow> 'a tree23 \<Rightarrow> bool"   and a0 :: "'a act list"   and a1 :: "'a tree23" assumes "\<And>t. P [] t"   and "\<And>k x as t. P as (add0 k x t) \<Longrightarrow> P (Add k x # as) t"   and "\<And>k as t. P as (del0 k t) \<Longrightarrow> P (Del k # as) t" shows "P a0 a1"
select_convs_2: fixes states :: "'b \<Rightarrow> 'c"   and msgs :: "nat \<Rightarrow> 'a message list"   and process_snapshot :: "'b \<Rightarrow> 'c option"   and channel_snapshot :: "nat \<Rightarrow> 'a list \<times> recording_state"   and more :: "'d" shows "msgs \<lparr>states = states, msgs = msgs, process_snapshot = process_snapshot, channel_snapshot = channel_snapshot, \<dots> = more\<rparr> = msgs"
wf_term_iff_term_ok': fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and t :: "Core.term" shows "wf_term \<Sigma> t = term_ok' \<Sigma> t"
invariant: fixes E :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "Graph_Invariant_Strong E P"   and "E a b" shows "P b"
intros_11: fixes \<C> :: "t_context"   and ts :: "t list"   and ts' :: "t list" shows "\<C> \<turnstile> [Unreachable] : ts _> ts'"
FAILURES_def: fixes P :: "('b event list \<times> 'b event set) set \<times> 'b event list set" shows "FAILURES P = fst P"
set_ConsD: fixes y :: "'b"   and x :: "'b"   and xs :: "'b list" assumes "y \<in> ran\<^sub>l (x # xs)" shows "y = x \<or> y \<in> ran\<^sub>l xs"
assms_11: shows "b\<^sup>\<infinity> = b\<^sup>\<star>"
mult_cancellation_half: fixes a :: "V"   and x :: "V"   and r :: "V"   and y :: "V"   and s :: "V" assumes "a * x + r \<le> a * y + s"   and "r \<sqsubset> a"   and "s \<sqsubset> a" shows "x \<le> y"
asym: fixes b :: "'a list"   and a :: "'a list" assumes "strict_prefix b a"   and "strict_prefix a b" shows "False"
divmod_int_unique_1: fixes k :: "int"   and l :: "int"   and q :: "int"   and r :: "int" assumes "eucl_rel_int k l (q, r)" shows "k div l = q"
Gauss_Jordan_in_ij_det_P_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and i :: "'d"   and j :: "'c" shows "Gauss_Jordan_in_ij_det_P A i j = (let n = LEAST n. A $ n $ j \<noteq> (0::'b) \<and> i \<le> n in (if i = n then (1::'b) / A $ i $ j else - (1::'b) / A $ n $ j, Gauss_Jordan_in_ij A i j))"
simps_4: fixes f1 :: "nat \<Rightarrow> 'b"   and f2 :: "'b"   and nat :: "nat" shows "(case enat nat of enat x \<Rightarrow> f1 x | \<infinity> \<Rightarrow> f2) = f1 nat"
point_replication_number_def: fixes B :: "'a set multiset"   and x :: "'a" shows "B rep x \<equiv> int (size (filter_mset ((\<in>) x) B))"
SUP_plus: fixes A :: "'a set set"   and f :: "'a set \<Rightarrow> real"   and g :: "'a set \<Rightarrow> real" assumes "A \<noteq> {}"   and "bdd_above (f ` A)"   and "bdd_above (g ` A)" shows "(SUP x\<in>A. f x + g x) \<le> \<Squnion> (f ` A) + \<Squnion> (g ` A)"
Keys_to_list_sorted_wrt_aux: fixes ts :: "'t list"   and ps :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "sorted_wrt (\<succ>\<^sub>t) ts" shows "sorted_wrt (\<succ>\<^sub>t) (fold (\<lambda>p. merge_wrt (\<succ>\<^sub>t) (keys_to_list p)) ps ts)"
measurable_If_restrict_space_iff: fixes M :: "'c measure"   and P :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and N :: "'d measure" assumes "{x \<in> space M. P x} \<in> sets M" shows "((\<lambda>x. if P x then f x else g x) \<in> M \<rightarrow>\<^sub>M N) = (f \<in> restrict_space M {x. P x} \<rightarrow>\<^sub>M N \<and> g \<in> restrict_space M {x. \<not> P x} \<rightarrow>\<^sub>M N)"
get_element_by_id_def: shows "get_element_by_id \<equiv> l_get_element_by\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_element_by_id first_in_tree_order get_attribute"
lin_indpt_list_def: fixes n :: "nat"   and fs :: "'a vec list" shows "cof_vec_space.lin_indpt_list n fs = (set fs \<subseteq> carrier_vec n \<and> distinct fs \<and> \<not> module.lin_dep class_ring (module_vec TYPE('a) n) (set fs))"
maxsimp_map_into_image: fixes x :: "'a set" assumes "folding.chamber x" shows "SimplicialComplex.maxsimp (f \<turnstile> X) (f ` x)"
vec_def: fixes x :: "'b" shows "vec x = (\<chi>i. x)"
bij_betw_swap_iff: fixes x :: "'d"   and A :: "'d set"   and y :: "'d"   and f :: "'d \<Rightarrow> 'e"   and B :: "'e set" assumes "x \<in> A"   and "y \<in> A" shows "bij_betw (Fun.swap x y f) A B = bij_betw f A B"
map_acom_If: fixes f :: "'c \<Rightarrow> 'b"   and c :: "'c acom"   and b :: "bexp"   and c1' :: "'b acom"   and c2' :: "'b acom"   and S' :: "'b" shows "(ACom.map_acom f c = IF b THEN c1' ELSE c2' {S'}) = (\<exists>S c1 c2. c = IF b THEN c1 ELSE c2 {S} \<and> ACom.map_acom f c1 = c1' \<and> ACom.map_acom f c2 = c2' \<and> S' = f S)"
minusinf_inf: fixes p :: "fm"   and bbs :: "bool list"   and bs :: "int list" assumes "iszlfm p"   and "d_\<beta> p 1" shows "\<exists>z. \<forall>x<z. Ifm bbs (x # bs) (minusinf p) = Ifm bbs (x # bs) p"
lock_ok_lasD: fixes ls :: "('c, 'd) locks"   and t :: "'d"   and las :: "'c lock_actions"   and l :: "'c" assumes "lock_ok_las ls t las" shows "lock_actions_ok (ls $ l) t (las $ l)"
r: shows "x - 2 * (x div 2 div y) * y = x mod y + x div y mod 2 * y"
accept_def: shows "accept\<^sub>R' = Alm_all token_succeeds"
length_replace1: fixes x :: "'a"   and y :: "'a"   and xs :: "'a list" shows "|replace x [y] xs| = |xs|"
of_nat_eq_1_iff: fixes x :: "nat" shows "(of_nat x = (1::'b)) = (x = 1)"
bigtheta_const_ln_pow: fixes a :: "real"   and d :: "nat" assumes "0 < a" shows "(\<lambda>x. ln (a * x) ^ d) \<in> \<Theta>(\<lambda>x. ln x ^ d)"
Op: fixes inp :: "'index \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qTerm option"   and binp :: "'bindex \<Rightarrow> ('index, 'bindex, 'varSort, 'var, 'opSym) qAbs option"   and delta :: "'opSym" assumes "liftAll phi inp"   and "liftAll phiAbs binp" shows "phi (qOp delta inp binp)"
collapse: fixes simple_rule :: "'a simple_rule" shows "SimpleRule (match_sel simple_rule) (action_sel simple_rule) = simple_rule"
simps_2: fixes g :: "'g \<Rightarrow> ('e, 'g) step"   and h :: "'h \<Rightarrow> ('f, 'h) step"   and sg :: "'g"   and sh :: "'h"   and a :: "'e" shows "zip_raw g h (sg, sh, Some a) = (case h sh of Done \<Rightarrow> Done | Skip sh' \<Rightarrow> Skip (sg, sh', Some a) | Yield b sh' \<Rightarrow> Yield (a, b) (sg, sh', None))"
simps: fixes a :: "'a sequent list \<times> 'a sequent"   and R :: "('a sequent list \<times> 'a sequent) set" shows "(a \<in> R*) = (\<exists>r seq. a = extendRule seq r \<and> r \<in> R)"
remove1_insort_insert_same: fixes x :: "'a"   and xs :: "'a list" assumes "x \<notin> set xs" shows "remove1 x (insort_insert x xs) = xs"
prems_2: shows "(l # ts) ! i = (p, is, \<theta>, Read\<^sub>s\<^sub>b volatile a t v # sb, \<D>, \<O>, \<R>)"
pow_mat_dim_2: fixes A :: "'b mat"   and k :: "nat" shows "dim_col (A ^\<^sub>m k) = (if k = 0 then dim_row A else dim_col A)"
double_negation: fixes x :: "'a" shows "- - - x = - x"
satPB_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and \<Phi> :: "('fsym, 'psym) lit list set" shows "U.satPB \<xi> \<Phi> \<equiv> \<forall>c\<in>\<Phi>. U.satC \<xi> c"
wlp_arrayinit_eq: fixes \<pi> :: "char list \<Rightarrow> com option"   and x :: "char list"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s :: "char list \<Rightarrow> int \<Rightarrow> int" shows "wlp \<pi> (CLEAR x[]) Q s = Q (s(x := \<lambda>_. 0))"
circ_simulate_right_plus: fixes z :: "'a"   and x :: "'a"   and y :: "'a"   and w :: "'a" shows "z * x \<le> y\<^sup>+ * z \<squnion> w \<longrightarrow> z * x\<^sup>\<star> \<le> y\<^sup>\<star> * (z \<squnion> w * x\<^sup>\<star>)"
ha: shows "\<Gamma> \<turnstile> q \<leftrightarrow> r : T\<^sub>1'\<rightarrow>T\<^sub>2"
null_sym_diff_transitive: fixes A :: "'b set"   and B :: "'b set"   and M :: "'b measure"   and C :: "'b set" assumes "A \<Delta> B \<in> null_sets M"   and "B \<Delta> C \<in> null_sets M"   and "A \<in> sets M"   and "C \<in> sets M" shows "A \<Delta> C \<in> null_sets M"
is_projection_on_unique: fixes M :: "'b set"   and \<pi>\<^sub>1 :: "'b \<Rightarrow> 'b"   and \<pi>\<^sub>2 :: "'b \<Rightarrow> 'b" assumes "convex M"   and "is_projection_on \<pi>\<^sub>1 M"   and "is_projection_on \<pi>\<^sub>2 M" shows "\<pi>\<^sub>1 = \<pi>\<^sub>2"
UP_hom_unique: fixes Phi :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'c"   and s :: "'c"   and Psi :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'c"   and p :: "nat \<Rightarrow> 'a" assumes "ring_hom_cring P S Phi"   and "Phi (monom P \<one> (Suc 0)) = s"   and "\<And>r. r \<in> carrier R \<Longrightarrow> Phi (monom P r 0) = h r"   and "ring_hom_cring P S Psi"   and "Psi (monom P \<one> (Suc 0)) = s"   and "\<And>r. r \<in> carrier R \<Longrightarrow> Psi (monom P r 0) = h r"   and "p \<in> carrier P"   and "s \<in> carrier S" shows "Phi p = Psi p"
CONSTRAINT_def: fixes f :: "'g"   and R :: "('h \<times> 'g) set" shows "CONSTRAINT f R \<equiv> True"
simps_1: fixes f :: "nat \<Rightarrow> nat"   and b :: "bool" shows "change_tv f (Det b) = Det b"
BseqI: fixes K :: "real"   and X :: "nat \<Rightarrow> 'b" assumes "0 < K"   and "\<forall>n. norm (X n) \<le> K" shows "Bseq X"
minimal_sequence_to_failure_extending_mcp: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and V :: "'d list set"   and vs :: "('d \<times> 'e) list"   and xs :: "('d \<times> 'e) list" assumes "OFSM M1"   and "OFSM M2"   and "is_det_state_cover M2 V"   and "minimal_sequence_to_failure_extending V M1 M2 vs xs" shows "mcp (map fst (vs @ xs)) V (map fst vs)"
e_1_not_in_A: shows "e_1 \<in> - A"
fmpred_map: fixes P :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c"   and m :: "('b, 'd) fmap" shows "fmpred P (fmmap f m) = fmpred (\<lambda>k v. P k (f v)) m"
truncate_up_le1: fixes x :: "real"   and p :: "nat" assumes "x \<le> 1" shows "truncate_up p x \<le> 1"
that: fixes pth1 :: "(('v \<times> 'outPort) \<times> 'v \<times> 'inPort) list"   and e :: "('v \<times> 'outPort) \<times> 'v \<times> 'inPort"   and pth2 :: "(('v \<times> 'outPort) \<times> 'v \<times> 'inPort) list" assumes "pth = (pth1 @ [e]) @ pth2"   and "path v (fst ps) (pth1 @ [e])"   and "path (fst ps) v' pth2"   and "snd e = ps"   and "ps \<notin> snd ` set pth1" shows "thesis"
isomorphic_unit_right: fixes f :: "'b" assumes "D.ide f" shows "D.isomorphic (f \<star>\<^sub>D src\<^sub>D f) f"
keys_mapping_sum_add: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b mpoly" assumes "finite A" shows "keys (mapping_of (sum f A)) \<subseteq> \<Union> (keys ` mapping_of ` range f)"
run_state_bind: fixes x :: "('c, 'd) state"   and f :: "'d \<Rightarrow> ('c, 'b) state"   and s :: "'c" shows "run_state (x \<bind> f) s = (case run_state x s of (a, s') \<Rightarrow> run_state (f a) s')"
subspace_diff_fun_space: shows "subspace l.diff_fun_space"
set_cases_1: fixes e :: "'a"   and a :: "('a, 'b) hd" assumes "e \<in> syms_hd a" obtains "a = Sym e"
simps_2: shows "Eg2_var\<^sub>C_of_Eg1 buffer = buffer\<^sub>C"
awalk_def: fixes u :: "'a"   and p :: "'b list"   and v :: "'a" shows "H.awalk u p v \<equiv> u \<in> verts H \<and> set p \<subseteq> arcs H \<and> H.cas u p v"
r_coset_def: fixes G :: "('a, 'b) monoid_scheme"   and H :: "'a set"   and a :: "'a" shows "H #>\<^bsub>G\<^esub> a = (\<Union>h\<in>H. {h \<otimes>\<^bsub>G\<^esub> a})"
Suc: fixes ds :: "nat list"   and a :: "nat" shows "take n (digit_encode ds a) = digit_encode (take n ds) a"
abs_eq: fixes x :: "nat \<Rightarrow> rat" assumes "eq_onp Abstract_Linear_Poly.inv x x" shows "vars (Abs_linear_poly x) = fun_vars x"
perp_up_in_S: fixes a :: "proj2"   and b :: "proj2"   and l :: "proj2_line" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "proj2_incident b l" shows "perp_up a l \<in> S"
exhaust: fixes y :: "('e, 'f, 'g) Tree" obtains x1 :: "('e, 'f, 'g) Tree set['e]" where "y = tConj x1"   | x2 :: "('e, 'f, 'g) Tree" where "y = tNot x2"   | x3 :: "'f" where "y = tPred x3"   | x41 :: "'g" and x42 :: "('e, 'f, 'g) Tree" where "y = tAct x41 x42"
Trans: shows "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> P'"
monoid_add_hom_axioms: shows "monoid_add_hom (\<lambda>a. [:a:])"
abs_ab: shows "\<bar>sint a * sint b\<bar> < 2 ^ (LENGTH('b) - 1)"
simps_3: fixes x :: "'a"   and y :: "'a" shows "(extB.Val x \<le> extB.Val y) = (x \<le> y)"
Diagonal_to_Smith_row_i_PQ_def: fixes i :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and P :: "(('a, 'b) vec, 'b) vec"   and A :: "(('a, 'c) vec, 'b) vec"   and Q :: "(('a, 'c) vec, 'c) vec" shows "Diagonal_to_Smith_row_i_PQ i bezout (P, A, Q) = diagonal_to_Smith_i_PQ [i + 1..<min (nrows A) (ncols A)] i bezout (P, A, Q)"
row_add_iterate_preserves_greater_than_n: fixes n :: "nat"   and A :: "(('a, 'b) vec, 'c) vec"   and a :: "'c"   and i :: "'c"   and j :: "'b"   and b :: "'b" assumes "n < nrows A"   and "n < mod_type_class.to_nat a" shows "row_add_iterate A n i j $ a $ b = A $ a $ b"
"1_3": shows "(flow0 x0 h, Dflow x0 h o\<^sub>L d0) \<in> P1"
pdevs_val_minus: fixes e :: "nat \<Rightarrow> real"   and f :: "nat \<Rightarrow> real"   and xs :: "'a pdevs" shows "pdevs_val (\<lambda>i. e i - f i) xs = pdevs_val e xs - pdevs_val f xs"
trancl_into_rtrancl: fixes a :: "'a"   and b :: "'a"   and r :: "('a \<times> 'a) set" assumes "(a, b) \<in> r\<^sup>+" shows "(a, b) \<in> r\<^sup>*"
degree_aform_independent_from: fixes d1 :: "nat"   and X :: "'a \<times> 'a pdevs" shows "degree_aform (independent_from d1 X) \<le> d1 + degree_aform X"
farkas: shows "farkas_coefficients_ns (set (concat (map constraint_to_qdelta_constraint csl))) N"
Prev: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and j :: "nat"   and P :: "String.literal \<Rightarrow> nat option"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and n :: "nat"   and R :: "event_data list set"   and \<phi> :: "('msaux, 'muaux) mformula"   and \<phi>' :: "Formula.formula"   and first :: "bool"   and buf :: "event_data option list set list"   and nts :: "nat list"   and I :: "\<I>" assumes "wf_mformula \<sigma> j P V n R \<phi> \<phi>'"   and "first = (j = 0)"   and "list_all2 (\<lambda>i. qtable n (fv \<phi>') (mem_restr R) (\<lambda>v. Formula.sat \<sigma> V (map the v) i \<phi>')) [min (Monitor.progress \<sigma> P \<phi>' j) (j - 1)..<Monitor.progress \<sigma> P \<phi>' j] buf"   and "list_all2 (\<lambda>i t. t = \<tau> \<sigma> i) [min (Monitor.progress \<sigma> P \<phi>' j) (j - 1)..<j] nts" shows "wf_mformula \<sigma> j P V n R (MPrev I \<phi> first buf nts) (formula.Prev I \<phi>')"
inext_less_mono_infin_fin: fixes a :: "nat"   and b :: "nat"   and I :: "nat set" assumes "a < b"   and "a \<in> I"   and "b \<in> I"   and "b \<noteq> Max I \<or> infinite I" shows "inext a I < inext b I"
aform_err_eqI: fixes y :: "real"   and e :: "nat \<Rightarrow> real"   and c :: "real \<times> real pdevs"   and d' :: "real"   and d :: "real" assumes "y \<in> aform_err e (c, d')"   and "d' = d" shows "y \<in> aform_err e (c, d)"
XSVar: shows "\<forall>s. |XS s| <o |UNIV| \<or> finite (XS s)"
CondFalse_reds: fixes b :: "expr"   and c\<^sub>1 :: "com"   and c\<^sub>2 :: "com"   and s :: "char list \<Rightarrow> val option"   and s' :: "char list \<Rightarrow> val option" assumes "\<langle>if (b) c\<^sub>1 else c\<^sub>2,s\<rangle> \<rightarrow>* \<langle>Skip,s'\<rangle>"   and "\<lbrakk>b\<rbrakk>s = Some false" shows "\<langle>c\<^sub>2,s\<rangle> \<rightarrow>* \<langle>Skip,s'\<rangle>"
assms_5: shows "v \<in> nodes g\<^sub>2"
state_memoryless_on: fixes f :: "'a \<Rightarrow> 'a pmf"   and s :: "'a" shows "state (memoryless_on f s) = s"
IH: fixes ys :: "'dyn list" assumes "traverse cast_Dyn xs = Some ys" shows "norm_stack xs = ys"
map_of_update_with_aux': fixes v :: "'a"   and k :: "'b"   and f :: "'a \<Rightarrow> 'a"   and ps :: "('b \<times> 'a) list"   and k' :: "'b" shows "map_of (AList.update_with_aux v k f ps) k' = (map_of ps(k \<mapsto> case map_of ps k of None \<Rightarrow> f v | Some v \<Rightarrow> f v)) k'"
sb_ident: fixes ys :: "'b"   and y :: "'a"   and xs :: "'b"   and x :: "'a" shows "x @xs[y / y]_ys = x"
card_insert_disjoint: fixes A :: "'d set"   and x :: "'d" assumes "finite A"   and "x \<notin> A" shows "card (insert x A) = Suc (card A)"
norm_zero: fixes minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and dist :: "'d \<Rightarrow> 'd \<Rightarrow> real"   and norm :: "'d \<Rightarrow> real"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and uminus :: "'d \<Rightarrow> 'd"   and scaleR :: "real \<Rightarrow> 'd \<Rightarrow> 'd"   and sgn :: "'d \<Rightarrow> 'd"   and uniformity :: "('d \<times> 'd) filter"   and "open" :: "'d set \<Rightarrow> bool" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm zero = 0"
l2_disjoint: fixes d :: "nat"   and b :: "(nat \<times> int) list"   and p :: "(nat \<times> int) list"   and p' :: "(nat \<times> int) list" assumes "d < length b"   and "p \<in> grid b {d}"   and "p' \<in> grid b {d}"   and "p' \<notin> grid p {d}"   and "lv p d \<le> lv p' d" shows "l2_\<phi> (p' ! d) (p ! d) = 0"