hcju/isabelleps huggingface dataset:
map_G_rel_cong: fixes L1 :: "'p \<Rightarrow> 'q \<Rightarrow> bool"   and L2 :: "'r \<Rightarrow> 's \<Rightarrow> bool"   and Co1 :: "'t \<Rightarrow> 'u \<Rightarrow> bool"   and Co2 :: "'v \<Rightarrow> 'w \<Rightarrow> bool"   and Contra1 :: "'x \<Rightarrow> 'y \<Rightarrow> bool"   and Contra2 :: "'z \<Rightarrow> 'aa \<Rightarrow> bool"   and xf :: "('p, 'r, 't, 'v, 'x, 'z, 'ab) G"   and yf :: "('q, 's, 'u, 'w, 'y, 'aa, 'ab) G"   and L1' :: "'ac \<Rightarrow> 'ad \<Rightarrow> bool"   and x :: "'p \<Rightarrow> 'ac"   and y :: "'q \<Rightarrow> 'ad"   and L2' :: "'ae \<Rightarrow> 'af \<Rightarrow> bool"   and xa :: "'r \<Rightarrow> 'ae"   and ya :: "'s \<Rightarrow> 'af"   and Co1' :: "'ag \<Rightarrow> 'ah \<Rightarrow> bool"   and xb :: "'t \<Rightarrow> 'ag"   and yb :: "'u \<Rightarrow> 'ah"   and Co2' :: "'ai \<Rightarrow> 'aj \<Rightarrow> bool"   and xc :: "'v \<Rightarrow> 'ai"   and yc :: "'w \<Rightarrow> 'aj"   and Contra1' :: "'ak \<Rightarrow> 'al \<Rightarrow> bool"   and xd :: "'ak \<Rightarrow> 'x"   and yd :: "'al \<Rightarrow> 'y"   and Contra2' :: "'am \<Rightarrow> 'an \<Rightarrow> bool"   and xe :: "'am \<Rightarrow> 'z"   and ye :: "'an \<Rightarrow> 'aa" assumes "rel_G L1 L2 Co1 Co2 Contra1 Contra2 xf yf"   and "\<And>x y. L1 x y \<Longrightarrow> L1' (x x) (y y)"   and "\<And>x y. L2 x y \<Longrightarrow> L2' (xa x) (ya y)"   and "\<And>x y. Co1 x y \<Longrightarrow> Co1' (xb x) (yb y)"   and "\<And>x y. Co2 x y \<Longrightarrow> Co2' (xc x) (yc y)"   and "\<And>x y. Contra1' x y \<Longrightarrow> Contra1 (xd x) (yd y)"   and "\<And>x y. Contra2' x y \<Longrightarrow> Contra2 (xe x) (ye y)" shows "rel_G L1' L2' Co1' Co2' Contra1' Contra2' (map_G x xa xb xc xd xe xf) (map_G y ya yb yc yd ye yf)"
Sup_eq_Inf: fixes A :: "'a set" shows "\<Sqinter> A = \<Squnion> {b. \<forall>a\<in>A. b \<le> a}"
proj_tuple_join_mask_empty: fixes as :: "'b option list"   and n :: "nat" assumes "length as = n" shows "proj_tuple (join_mask n {}) as = replicate n None"
second_vertices_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set" assumes "DisjointPaths G v0 v1 paths" shows "DisjointPaths.second_vertices paths \<equiv> DisjointPaths.second_vertex ` paths"
space_subgroup_props_1: fixes K :: "'a set"   and n :: "nat"   and E :: "'a set" assumes "subfield K R"   and "R.dimension n K E" shows "E \<subseteq> carrier R"
rep_eq: fixes x :: "('f, 'e) generator" shows "generator (tl_trans x) = tl_raw (generator x)"
Lyndon_fac_emp: shows "LynFac \<epsilon> = \<epsilon>"
r_e2frame_def: shows "r_e2frame \<equiv> Cn 1 r_hd [r_e2stack]"
wf_prec_L: shows "minimal_element (\<sqsubset>L) UNIV"
Per_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p" shows "Per A B C \<equiv> \<exists>C'. B Midpoint C C' \<and> Cong A C A C'"
IH: assumes "rga_ops xs" shows "distinct (interp_rga xs)"
SN_on_imp_acc: fixes r :: "('b \<times> 'b) set"   and x :: "'b" assumes "SN_on {(y, z). (z, y) \<in> r} {x}" shows "x \<in> Wellfounded.acc r"
D2_UNIV: shows "UNIV = set [D1, D2]"
prob_space: fixes i :: "'i" shows "prob i (space (M i)) = 1"
intro: fixes Q :: "heap \<Rightarrow> bool"   and lookup_st :: "'c \<Rightarrow> (heap, 'd option) state"   and lookup :: "'c \<Rightarrow> 'd option Heap"   and update_st :: "'c \<Rightarrow> 'd \<Rightarrow> (heap, unit) state"   and update :: "'c \<Rightarrow> 'd \<Rightarrow> unit Heap" assumes "rel_fun (=) (heap_mem_defs.rel_state Q (=)) lookup_st lookup"   and "rel_fun (=) (rel_fun (=) (heap_mem_defs.rel_state Q (=))) update_st update" shows "dp_heap_axioms Q lookup lookup_st update update_st"
t: shows "finite t \<and> f C1_differentiable_on s - t"
pred_True: shows "pred_prod (\<lambda>_. True) (\<lambda>_. True) = (\<lambda>_. True)"
exec_move_BinOpI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e2 :: "'addr expr1"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and pc :: "nat"   and xcp :: "'addr option"   and ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and e1 :: "'addr expr1"   and bop :: "bop"   and v :: "'addr val" assumes "exec_move ci P t e2 h (stk, loc, pc, xcp) ta h' (stk', loc', pc', xcp')" shows "exec_move ci P t (e1 \<guillemotleft>bop\<guillemotright> e2) h (stk @ [v], loc, length (compE2 e1) + pc, xcp) ta h' (stk' @ [v], loc', length (compE2 e1) + pc', xcp')"
strong_sum_intro: fixes x :: "i"   and z :: "i"   and y :: "i" shows "(P x z \<and> P y z) \<and> (\<forall>w. P w z \<longrightarrow> O w x \<or> O w y) \<longrightarrow> x \<oplus> y = z"
Cons_1: assumes "distincts xss"   and "xs \<in> set xss" shows "lists_succ (xs # remove1 xs xss) = lists_succ xss"
top_plus: fixes x :: "'a" shows "1 ; x\<^sup>+ = 1 ; x"
simps_1: fixes c :: "'b poly"   and n :: "nat"   and p :: "'b poly" shows "degreen (CN c n p) = (\<lambda>m. if n = m then 1 + degreen p n else 0)"
interval_lebesgue_integral_le_eq: fixes a :: "ereal"   and b :: "ereal"   and M :: "real measure"   and f :: "real \<Rightarrow> 'a" assumes "a \<le> b" shows "interval_lebesgue_integral M a b f = set_lebesgue_integral M (einterval a b) f"
gap_Suc_2: fixes b :: "nat"   and i :: "nat"   and t :: "nat"   and k :: "nat" assumes "b = block i t"   and "k = gap i t"   and "change_on_1 b k" shows "gap i (Suc t) = e_length b"
steps_z_alt: fixes X :: "'d set"   and A :: "('e \<times> ('f, 'g) cconstraint \<times> 'h \<times> 'f list \<times> 'e) set \<times> ('e \<Rightarrow> ('f, 'g) cconstraint)"   and l :: "'e"   and Z :: "('f \<Rightarrow> 'g) set"   and l' :: "'e"   and Z' :: "('f \<Rightarrow> 'g) set"   and l'' :: "'e"   and Z'' :: "('f \<Rightarrow> 'g) set" assumes "AlphaClosure X"   and "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l', Z'\<rangle>"   and "A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto> \<langle>l'', Z''\<rangle>" shows "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l'', Z''\<rangle>"
distinct_21: fixes x3 :: "'c" shows "NotifyAll x3 \<noteq> Notified"
lunit_char_1: fixes a :: "'c" assumes "C.ide a" shows "\<guillemotleft>\<l>\<^sub>C[a] : \<I>\<^sub>C \<otimes>\<^sub>C a \<rightarrow>\<^sub>C a\<guillemotright>"
right_total_mat_rel: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "right_total R" shows "right_total (mat_rel R)"
nsum_mem: fixes A :: "('e, 'f) aGroup_scheme"   and n :: "nat"   and f :: "nat \<Rightarrow> 'e" assumes "aGroup A"   and "\<forall>j\<le>n. f j \<in> carrier A" shows "\<Sigma>\<^sub>e A f n \<in> carrier A"
gT: fixes x :: "'a" assumes "x \<in> T"   and "b \<bullet> x = 0" shows "g x \<in> T"
equivclp_sym: fixes r :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d"   and y :: "'d" assumes "equivclp r x y" shows "equivclp r y x"
simps_5: fixes v :: "'Var"   and va :: "'AExp" shows "has_annotated_stop (v \<leftarrow> va) = False"
cases: fixes a :: "('sequent \<times> 'rule) tree" assumes "local.wf a" obtains t :: "('sequent \<times> 'rule) tree" where "a = t" and "snd (tree.root t) \<in> R" and "effStep (tree.root t) ((fst \<circ> tree.root) |`| cont t)" and "\<And>t'. t' |\<in>| cont t \<Longrightarrow> local.wf t'"
rep_eq: fixes x :: "real_int" shows "left x = fst (Rep_real_int x)"
sup_union_left2: fixes x :: "'k"   and N :: "'k multiset"   and M :: "'k multiset" assumes "x \<in># N" shows "add_mset x M \<union># N = add_mset x (M \<union># (N - {#x#}))"
vars_of_instances: fixes t :: "'b trm"   and \<sigma> :: "('b \<times> 'b trm) list" shows "vars_of (t \<lhd> \<sigma>) = \<Union> {V. \<exists>x. x \<in> vars_of t \<and> V = vars_of (Var x \<lhd> \<sigma>)}"
pot_comp_pos: fixes u :: "'a"   and v :: "'a"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> order" assumes "snd (rep_nat_term u) < snd (rep_nat_term v)" shows "pot_comp cmp u v = Lt"
lfp_unfold: fixes f :: "'d \<Rightarrow> 'd" assumes "mono f" shows "lfp f = f (lfp f)"
iMOD_infinite: fixes m :: "nat"   and r :: "nat" assumes "0 < m" shows "infinite [ r, mod m ]"
span_add1: fixes U :: "'c set"   and v :: "'c"   and w :: "'c" assumes "U \<subseteq> carrier V"   and "v \<in> local.module.span U"   and "w \<in> local.module.span U" shows "v \<oplus>\<^bsub>V\<^esub> w \<in> local.module.span U"
lens_comp_left_id: fixes X :: "'b \<Longrightarrow> 'c" shows "\<Sigma> ;\<^sub>L X = X"
enc_respects_binary_pred_iff_source_target_rel_respects_binary_pred_encR: fixes Pred :: "('procS, 'procT) Proc \<Rightarrow> 'a \<Rightarrow> bool" shows "enc_respects_binary_pred Pred = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> rel_respects_binary_pred Rel Pred)"
log_sep: fixes s :: "(agent \<times> msg) set"   and s' :: "(agent \<times> msg) set"   and X :: "msg"   and Y :: "msg"   and Z :: "msg" assumes "s\<^sub>0 \<Turnstile> s"   and "s' = insert (Spy, X) (insert (Spy, Y) s) \<and> (Spy, \<lbrace>X, Y\<rbrace>) \<in> s" shows "key_sets Z (crypts {Z. Log Z = X}) \<subseteq> key_sets Z (crypts (Log -` spied s)) \<and> key_sets Z (crypts {Z. Log Z = Y}) \<subseteq> key_sets Z (crypts (Log -` spied s))"
submodule_is_module: fixes R :: "('b, 'c) ring_scheme"   and M :: "('b, 'd, 'e) module_scheme"   and N :: "'d set" assumes "module R M"   and "LinearCombinations.submodule R N M" shows "module R (M\<lparr>carrier := N\<rparr>)"
Schur1: fixes G :: "'a set"   and smult :: "('b, 'a) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and V :: "'c set"   and smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd"   and W :: "'d set"   and T :: "'c \<Rightarrow> 'd" assumes "IrrFinGroupRepresentation G smult V"   and "IrrFinGroupRepresentation G smult' W"   and "FGModuleHom G smult V smult' T"   and "T ` V \<subseteq> W"   and "T ` V \<noteq> 0" shows "FGModuleIso G smult V smult' T W"
1: shows "S.arr (\<nu> \<star> \<mu>)"
zero_ik_interchange: shows "is_zero_row_upt_k (mod_type_class.from_nat i) k (interchange_rows A (mod_type_class.from_nat i) (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> mod_type_class.from_nat i \<le> n))"
otherE: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and I :: "'e set"   and \<sigma> :: "'e \<Rightarrow> 'd"   and \<sigma>' :: "'e \<Rightarrow> 'd"   and R :: "('e \<Rightarrow> 'd) \<Rightarrow> ('e \<Rightarrow> 'd) \<Rightarrow> bool" assumes "other P I \<sigma> \<sigma>'"   and "\<lbrakk>\<forall>i\<in>I. \<sigma>' i = \<sigma> i; \<forall>j. j \<notin> I \<longrightarrow> P (\<sigma> j) (\<sigma>' j)\<rbrakk> \<Longrightarrow> R \<sigma> \<sigma>'" shows "R \<sigma> \<sigma>'"
e: fixes C :: "'a set set" assumes "C \<noteq> {}"   and "subset.chain \<B> C" shows "B \<subseteq> \<Union> C"
l: shows "(\<lambda>n. C * dist (u n) (v n) - dist (f (u n)) (f (v n))) \<longlonglongrightarrow> C * dist x y - dist (f x) (f y)"
flow_cleanup: fixes i :: "nat" shows "\<exists>h\<le>f_i i. flow \<Delta> h \<and> value_flow \<Delta> h = value_flow \<Delta> (f_i i) \<and> (\<forall>x. d_IN h x \<le> value_flow \<Delta> h)"
abs_mult_less: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sgn :: "'a \<Rightarrow> 'a"   and abs :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'a"   and c :: "'a"   and b :: "'a"   and d :: "'a" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "less (abs a) c"   and "less (abs b) d" shows "less (times (abs a) (abs b)) (times c d)"
getTvars_def: fixes \<sigma>l :: "'tp list" shows "getTvars \<sigma>l \<equiv> map Var (getVars \<sigma>l)"
ennreal_mult_less_top: fixes a :: "ennreal"   and b :: "ennreal" shows "(a * b < top) = (a = 0 \<or> b = 0 \<or> a < top \<and> b < top)"
path_terminal_path_from: fixes c :: "'form"   and "is" :: "nat list" assumes "c \<in> set conclusions"   and "is \<in> it_paths (it' c)" shows "path (c, 0 # is) (c, []) (terminal_path_from c is)"
has_field_derivative_inverse_strong: fixes f :: "'a \<Rightarrow> 'a"   and f' :: "'a"   and x :: "'a"   and S :: "'a set"   and g :: "'a \<Rightarrow> 'a" assumes "(f has_field_derivative f') (at x)"   and "f' \<noteq> (0::'a)"   and "open S"   and "x \<in> S"   and "continuous_on S f"   and "\<And>z. z \<in> S \<Longrightarrow> g (f z) = z" shows "(g has_field_derivative inverse f') (at (f x))"
intros_2: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and \<Gamma> :: "'a \<Rightarrow> 'd set option"   and \<S> :: "'a set \<times> 'a set"   and P :: "'d set" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False" shows "sifum_types.has_type ev\<^sub>B aexp_vars bexp_vars \<C>_vars bexp_neg assign_post dma_type \<Gamma> \<S> P Skip \<Gamma> \<S> P"
flpPseudoConsensus_axioms: shows "flpPseudoConsensus trans sends start"
distinct_8: fixes x51 :: "term"   and x52 :: "term"   and x1 :: "name" shows "x51 $ x52 \<noteq> Const x1"
entail_all_bot: fixes Bot :: "'a set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and N :: "'a set" assumes "consequence_relation Bot entails"   and "\<exists>B\<in>Bot. entails N {B}" shows "\<forall>B'\<in>Bot. entails N {B'}"
union: fixes A :: "'a set"   and B :: "'a set" shows "F (A \<union> B) = F A \<^bold>* F B"
set_intros_1: fixes x21 :: "'d"   and x22 :: "'d list" shows "x21 \<in> set (x21 # x22)"
integrable: fixes x :: "'b"   and a' :: "real"   and b :: "real" assumes "x \<in> A"   and "a \<le> a'"   and "a' \<le> b" shows "f x integrable_on {a'..b}"
id_2: shows "t = Tableau (preprocess' (map (Pair ()) nsl) (start_fresh_variable (map (Pair ()) nsl)))"
continuous_on_coordinatewise_then_product: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'b \<Rightarrow> 'd" assumes "\<And>i. continuous_on S (\<lambda>x. f x i)" shows "continuous_on S f"
red_single_def: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c"   and f :: "'a \<Rightarrow>\<^sub>0 'c"   and t :: "'a" shows "punit.red_single p q f t = (f \<noteq> 0 \<and> lookup p (local.punit.splus t (punit.lt f)) \<noteq> (0::'c) \<and> q = p - local.punit.monom_mult (lookup p (local.punit.splus t (punit.lt f)) / punit.lc f) t f)"
1: shows "x\<^sup>T ; r ; r\<^sup>T \<le> x\<^sup>T"
bound: fixes j :: "nat" assumes "k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> \<pi> m" shows "j \<le> k"
multiplicity_le_imp_dvd: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "\<And>p. normalization_semidom.prime zero times one normalize p \<Longrightarrow> factorial_semiring.multiplicity times one p x \<le> factorial_semiring.multiplicity times one p y" shows "dvd.dvd times x y"
simps_4: fixes tst :: "'a"   and c :: "('a, 'atom) com" shows "SC_discr (while tst do c) = SC_discr c"
nat_term_compare: fixes x :: "'b nat_term_order" shows "nat_term_compare x \<in> {cmp. comparator cmp \<and> nat_term_comp cmp}"
app_inv: fixes ok :: "nat"   and f :: "'a list"   and r :: "'a list" assumes "st_inv (App ok f r)" shows "\<exists>x. steps (rem_steps (App ok f r)) (App ok f r) = Done x"
All_None_def: fixes Ts :: "('a ann_com option \<times> 'a set) list" shows "All_None Ts \<equiv> \<forall>(c, q)\<in>set Ts. c = None"
poly_at_bot_at_top: fixes p :: "real poly" assumes "1 \<le> degree p"   and "lead_coeff p < 0" shows "filterlim (poly p) at_bot at_top"
distr_I_eq_M: shows "distr (restrict_space lborel {0<..<1}) borel M.I = M"
sets_T: fixes s :: "'s" shows "T.events s = sets (stream_space (count_space UNIV))"
e: fixes i :: "nat" shows "edges (map_graph (g i) X) \<subseteq> edges Y"
rank_eq_state_rank: fixes x :: "nat"   and n :: "nat" assumes "x \<le> n" shows "\<UU>.rank x n = \<UU>.state_rank (\<UU>.token_run x n) n"
sign_r_pos_nzero_right: fixes c :: "real"   and d :: "real"   and p :: "real poly" assumes "\<forall>x. c < x \<and> x \<le> d \<longrightarrow> poly p x \<noteq> 0"   and "c < d" shows "if sign_r_pos p c then 0 < poly p d else poly p d < 0"
oth_class_taut_1_a: fixes v :: "i"   and \<phi> :: "\<o>" shows "[\<^bold>\<not>(\<phi> \<^bold>& \<^bold>\<not>\<phi>) in v]"
C: shows "card {i. \<exists>N. 284 = N * 4 \<and> prime i \<and> i dvd N \<and> \<not> i dvd 4} = 1"
check_bounds_id: fixes s :: "('i, 'a) state" assumes "\<not> \<U> s"   and "\<Turnstile>\<^sub>n\<^sub>o\<^sub>l\<^sub>h\<^sub>s s"   and "\<diamond> s"   and "\<triangle> (\<T> s)"   and "\<nabla> s" shows "\<B>\<^sub>i (check s) = \<B>\<^sub>i s"
translation_graph: fixes X :: "'b allegorical_term" shows "graph (translation X)"
lfinite_code_1: shows "lfinite []\<^sub>l = True"
Call1Obj: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and M :: "String.literal"   and es :: "'addr expr1 list" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>e\<bullet>M(es),s\<rangle> -ta\<rightarrow> \<langle>e'\<bullet>M(es),s'\<rangle>"
comp_arr_ide: fixes a :: "'c"   and f :: "'c" assumes "C.ide a"   and "C.seq f a" shows "f \<cdot>\<^sub>C a = f"
matrix_to_iarray_Gram_Schmidt_matrix: fixes A :: "((real, 'a) vec, 'b) vec" shows "matrix_to_iarray (Gram_Schmidt_matrix A) = Gram_Schmidt_matrix_iarrays (matrix_to_iarray A)"
simps_4: fixes a :: "'a"   and \<nu> :: "'a set" shows "af_letter nprop\<^sub>n(a) \<nu> = (if a \<notin> \<nu> then true\<^sub>n else false\<^sub>n)"
4: shows "search_enum (the (eval (Mn 1 r_search_enum) [b])) b \<down>= (if e_length b \<le> the (search b) then 0 else 1)"
ord_p_zero_min: fixes p :: "'t \<Rightarrow>\<^sub>0 'c" shows "0 \<preceq>\<^sub>p p"
ennreal_top_neq_zero: shows "top \<noteq> 0"
r_coset_def: fixes G :: "('d, 'e) monoid_scheme"   and H :: "'d set"   and a :: "'d" shows "H #>\<^bsub>G\<^esub> a = (\<Union>h\<in>H. {h \<otimes>\<^bsub>G\<^esub> a})"
multiplicity_eq_zero_iff: 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 p :: "'b" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "x \<noteq> zero"   and "\<not> dvd.dvd times p one" shows "(factorial_semiring.multiplicity times one p x = 0) = (\<not> dvd.dvd times p x)"
map_simp: fixes f :: "'d" assumes "D.arr f" shows "D.\<ll>' f = D.inv (D.\<ll> (D.cod f)) \<cdot>\<^sub>D D.map f"
ll_on_open_congI: fixes Y :: "('a \<Rightarrow>\<^sub>L 'a) set"   and x0 :: "'a"   and S :: "real set"   and g :: "real \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a \<Rightarrow> 'a \<Rightarrow>\<^sub>L 'a" assumes "UNIV = Y"   and "Y.existence_ivl0 x0 = S"   and "\<And>x t. \<lbrakk>x \<in> Y; t \<in> S\<rbrakk> \<Longrightarrow> Y.vareq x0 t o\<^sub>L x = g t x" shows "ll_on_open S g Y"
intro: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set" assumes "\<And>x y z. \<lbrakk>less_eq x y; less_eq y x; less_eq y z; x \<in> A; y \<in> A; z \<in> A\<rbrakk> \<Longrightarrow> less_eq x z" shows "semiattractive A less_eq"
set_intros_1: fixes x1 :: "'b" shows "x1 \<in> set_trm (Var x1)"
resolution_blocks_subset: fixes S :: "'a set multiset" assumes "S \<in># \<P>" shows "S \<subseteq># \<B>"
set_insort_wrt: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and xs :: "'c list" shows "set (insort_wrt r x xs) = insert x (set xs)"
bind_ask2: fixes m :: "'m"   and f :: "'a fset \<Rightarrow> 'b \<Rightarrow> 'm" shows "bind m (\<lambda>x. ask (f x)) = ask (\<lambda>r. bind m (\<lambda>x. f x r))"
pell_mul_trivial_right: fixes z :: "'a \<times> 'a" shows "pell_mul z (1::'a, 0::'a) = z"
null_setsD1: fixes A :: "'b set"   and M :: "'b measure" assumes "A \<in> null_sets M" shows "emeasure M A = 0"
inj_on_implies_all_one: fixes A :: "'c set"   and B :: "'d set"   and F :: "('c \<Rightarrow> 'd) set" assumes "finite A"   and "finite B"   and "F \<in> (A \<rightarrow>\<^sub>E B) // domain_and_range_permutation A B"   and "univ (\<lambda>f. inj_on f A) F" shows "\<forall>n. n \<in># number_partition_of A B F \<longrightarrow> n = 1"
l2_commutative: fixes p :: "nat \<times> int"   and q :: "nat \<times> int" shows "l2_\<phi> p q = l2_\<phi> q p"
fmlookup_inject: fixes x :: "'a \<rightharpoonup>\<^sub>f 'b"   and y :: "'a \<rightharpoonup>\<^sub>f 'b" shows "(fmlookup x = fmlookup y) = (x = y)"
C_in_subset: shows "C \<in> Sup_llist Ns - Liminf_llist Ns"
eqvt: fixes x1 :: "pat"   and x2 :: "ty"   and x3 :: "(name \<times> ty) list"   and pi :: "(name \<times> name) list" assumes "\<turnstile> x1 : x2 \<Rightarrow> x3" shows "\<turnstile> pi \<bullet> x1 : pi \<bullet> x2 \<Rightarrow> pi \<bullet> x3"
the_equality: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "(THE x. P x) = a"
mark_guards_simps_10: fixes f :: "'f" shows "mark_guards f Throw = Throw"
Map_dom: fixes f :: "('a E.term, 'a) arr" assumes "arr f" shows "Map (local.dom f) = EVAL (Dom f)"
nsum_split: fixes n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "\<forall>j\<le>Suc (n + m). f j \<in> carrier M" shows "\<Sigma>\<^sub>e M f Suc (n + m) = \<Sigma>\<^sub>e M f n \<plusminus> \<Sigma>\<^sub>e M cmp f (slide (Suc n)) m"
Fail_HInv3_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "Fail s s' p"   and "HInv3_L s' p q d" shows "HInv3_R s' p q d"
runProc_gState_inv: fixes prog :: "program"   and g :: "'a gState_scheme"   and p :: "pState"   and name :: "String.literal"   and args :: "expr list" assumes "program_inv prog"   and "gState_inv prog g"   and "pState_inv prog p"   and "cl_inv (g, p)" shows "gState_inv prog (fst (runProc name args prog g p))"
closest_point_self: fixes x :: "'a"   and S :: "'a set" assumes "x \<in> S" shows "closest_point S x = x"
seq_proj_inverse: fixes x :: "'b seq" shows "seq_inj (seq_proj x) = x"
trans: fixes b :: "'c list"   and a :: "'c list"   and c :: "'c list" assumes "sublist b a"   and "sublist c b" shows "sublist c a"
xy: shows "infdist w Gxy \<le> 4 * deltaG TYPE('a)"
elims_2: fixes x :: "('a, 'b) Combinators list"   and xa :: "'a set list" assumes "all_in_list x xa" obtains l :: "'a set list" where "x = []" and "xa = l__"   | xb :: "('a, 'b) Combinators" and xs :: "('a, 'b) Combinators list" and l :: "'a set list" where "x = xb # xs" and "xa = l" and "in_list xb l \<and> all_in_list xs l"
cl_max: fixes C :: "'b formula set set"   and K :: "'b formula set"   and S :: "'b formula set" assumes "pcp (C::'b formula set set)"   and "subset_closed (C::'b formula set set)"   and "(K::'b formula set) \<in> (C::'b formula set set)"   and "pcp_lim (C::'b formula set set) (S::'b formula set) \<subseteq> (K::'b formula set)" shows "pcp_lim (C::'b formula set set) (S::'b formula set) = (K::'b formula set)"