hcju/isabelleps huggingface dataset:
insert_id_unique: fixes e1 :: "'a \<times> 'b \<times> bool"   and e2 :: "'a \<times> 'b \<times> bool"   and i1 :: "'a"   and n1 :: "'a option"   and i :: "nat"   and i2 :: "'a"   and n2 :: "'a option"   and j :: "nat" assumes "fst e1 = fst e2"   and "Broadcast (i1, Insert e1 n1) \<in> set (history i)"   and "Broadcast (i2, Insert e2 n2) \<in> set (history j)" shows "Insert e1 n1 = Insert e2 n2"
le_imp_inv: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.add_linorder plus minus zero less_eq less"   and "less_eq a b" shows "b = plus a (minus b a)"
test_assoc1: fixes P :: "('a \<times> 'a set) set"   and R :: "('a \<times> 'a set) set"   and S :: "('a \<times> 'a set) set" assumes "P \<subseteq> 1\<^sub>\<sigma>" shows "R \<cdot> P \<cdot> S = R \<cdot> (P \<cdot> S)"
cont_dw: fixes w :: "complex" assumes "w \<in> U" shows "F w contour_integrable_on linepath a b"
numeral_less_iff: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and m :: "num"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less (numeral.numeral one plus m) (numeral.numeral one plus n) = (m < n)"
Const_mult: fixes a :: "'a"   and b :: "'a" shows "Const (a * b) = Const a * Const b"
singleton: fixes f :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and z :: "'k"   and x :: "'k" assumes "comm_monoid_mset f z" shows "comm_monoid_mset.F f z {#x#} = x"
simps_1: fixes div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and H :: "'a mat"   and P :: "'a mat"   and K :: "'a mat" shows "reduce_column_aux div_op [] H (P, K) = (P, K)"
in_roofed_essentialD: fixes x :: "'k"   and \<Gamma> :: "('k, 'l) web_scheme"   and S :: "'k set" assumes "x \<in> RF\<^bsub>\<Gamma>\<^esub> S"   and "essential \<Gamma> (B \<Gamma>) S x" shows "x \<in> S"
Dom_dom: fixes f :: "('a, 'a set) Maps.arr" assumes "Maps.arr f" shows "Maps.Dom (Maps.dom f) = Maps.Dom f"
eventuallyI: fixes P :: "'i \<Rightarrow> bool"   and F :: "'i filter" assumes "\<And>x. P x" shows "eventually P F"
quot_of_fract_quot_to_fract: fixes x :: "'b \<times> 'b" shows "quot_of_fract (quot_to_fract x) = normalize_quot x"
D: shows "D \<in> carrier_mat 2 (dim_col A)"
AllN_6: shows "\<forall>x. insertion (nth_default 0 ((drop z L)[var := x])) a = Av"
CALL: fixes \<Phi> :: "((Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool) \<Rightarrow> (Var \<Rightarrow> Val) \<times> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set" assumes "({Sec (FIX \<Phi>)} \<union> G) \<rhd> body : Sec (\<Phi> (FIX \<Phi>))"   and "Monotone \<Phi>" shows "G \<rhd> Call : Sec (FIX \<Phi>)"
rep_eq: fixes x :: "'b poly" shows "count (poly_roots_mset x) = (\<lambda>xa. if x = 0 then 0 else order xa x)"
divisors_int_pos_3: shows "divisors_int_pos 0 = []"
prop_nth_sdrop: fixes j :: "nat"   and P :: "'a \<Rightarrow> bool"   and \<omega> :: "'a stream" assumes "\<forall>i\<ge>j. P (\<omega> !! i)" shows "\<forall>i. P (sdrop j \<omega> !! i)"
partial_function_definitions_spmf: shows "partial_function_definitions (ord_spmf (=)) lub_spmf"
LN_2: shows "[e] = vs @ [Label n es' lfilledk] @ es''"
has_integral_alt: fixes f :: "'c \<Rightarrow> 'd"   and y :: "'d"   and i :: "'c set" shows "(f has_integral y) i = (if \<exists>a b. i = cbox a b then (f has_integral y) i else \<forall>e>0. \<exists>B>0. \<forall>a b. ball (0::'c) B \<subseteq> cbox a b \<longrightarrow> (\<exists>z. ((\<lambda>x. if x \<in> i then f x else (0::'d)) has_integral z) (cbox a b) \<and> norm (z - y) < e))"
T_eq_T': fixes K :: "'a \<Rightarrow> 'a pmf"   and s :: "'a" shows "MC_syntax.T K s = MC_syntax.T' K (K s)"
assms_2: fixes kv :: "'c"   and x :: "'e" shows "flat_ge (f kv x) (f' kv x)"
pure_def: fixes f :: "('a, 'b, 'c) prog"   and h :: "'a" shows "pure f h = (h \<turnstile> ok f \<longrightarrow> h \<turnstile> f \<rightarrow>\<^sub>h h)"
pid_step_def: shows "pid_step \<equiv> stutter_extend_edges UNIV pid_gstep"
coset_mult_one: fixes G :: "('a, 'b) monoid_scheme"   and M :: "'a set" assumes "Group.group G"   and "M \<subseteq> carrier G" shows "M #>\<^bsub>G\<^esub> \<one>\<^bsub>G\<^esub> = M"
veq_structure_eq_right: fixes t :: "value"   and u :: "value" assumes "veq_structure t u"   and "vno_abs u" shows "t = u"
nat_pow_closed: fixes x :: "nat \<Rightarrow> int"   and n :: "nat" assumes "x \<in> carrier Zp" shows "x [^] n \<in> carrier Zp"
cases: fixes x :: "nat \<times> ('ekey \<times> 'dkey \<times> 'cipher list) option \<times> unit" obtains \<eta> :: "nat" and uu_ :: "unit" where "x = (\<eta>'__, None, uu_)"   | \<eta> :: "nat" and ekey :: "'ekey" and rest :: "'dkey \<times> 'cipher list" and uv_ :: "unit" where "x = (\<eta>, Some (ekey, rest), uv_)"
f_def: fixes a :: "'a" shows "f a = (if a \<in> A then g' (THE X. a \<in> X \<and> X \<in> P) else undefined)"
HAInitStates_InitState_trancl: fixes S :: "'d"   and HA :: "'g \<Rightarrow> ('d, 'e, 'f) hierauto"   and ST :: "'g"   and A :: "('d, 'e, 'f) seqauto" assumes "S \<in> HAInitStates (HA ST)"   and "A \<in> the (CompFun (HA ST) S)" shows "(S, InitState A) \<in> (Restr (ChiRel (HA ST)) (HAInitStates (HA ST)))\<^sup>+"
bij_inv_eq_iff: fixes p :: "'c \<Rightarrow> 'd"   and x :: "'c"   and y :: "'d" assumes "bij p" shows "(x = inv p y) = (p x = y)"
cosh_real_at_bot: shows "filterlim cosh at_top at_bot"
foldl_LTLAnd_prop_entailment_abs: fixes S :: "'c ltl set"   and i :: "'c ltl_prop_equiv_quotient"   and xs :: "'c ltl_prop_equiv_quotient list" shows "S \<up>\<Turnstile>\<^sub>P foldl and_abs i xs = (S \<up>\<Turnstile>\<^sub>P i \<and> (\<forall>y\<in>set xs. S \<up>\<Turnstile>\<^sub>P y))"
lemma2a': fixes p :: "bool list"   and q :: "bool list"   and L :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b option \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" assumes "(p, q) \<in> (eps L)\<^sup>*" shows "(True # p, True # q) \<in> (eps (RegExp2NAe.conc L R))\<^sup>*"
char_poly_def: fixes A :: "'b mat" shows "char_poly A = det (char_poly_matrix A)"
ds1: fixes k :: "nat" assumes "k < n" shows "matrix_sum d (\<lambda>k. M k * A * adjoint (M k)) k \<in> carrier_mat d d"
Let_1: fixes f :: "'b \<Rightarrow> 'a" shows "Let (1::'b) f = f (1::'b)"
iwlsFSwSTR_def: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'c, 'd, 'e, 'f) model_scheme" shows "iwlsFSwSTR MOD \<equiv> igWlsAllDisj MOD \<and> igWlsAbsIsInBar MOD \<and> igConsIPresIGWlsSTR MOD \<and> igSwapAllIPresIGWlsAllSTR MOD \<and> igFreshClsSTR MOD \<and> igSwapClsSTR MOD \<and> igAbsCongSSTR MOD"
inf_absorb2: fixes y :: "'k multiset"   and x :: "'k multiset" assumes "y \<subseteq># x" shows "x \<inter># y = y"
in_set_nthsD: fixes x :: "'b"   and xs :: "'b list"   and I :: "nat set" assumes "x \<in> set (nths xs I)" shows "x \<in> set xs"
span_add_eq: fixes x :: "'b"   and S :: "'b set"   and y :: "'b" assumes "x \<in> span S" shows "(x + y \<in> span S) = (y \<in> span S)"
order_refl_type: fixes \<tau> :: "'a OCL_Types.type" shows "\<tau> \<le> \<tau>"
product_nn_integral_singleton: fixes f :: "real \<Rightarrow> ennreal"   and i :: "'b" assumes "f \<in> borel_measurable (interval_measure (\<lambda>x. x))" shows "\<integral>\<^sup>+ x. f (x i) \<partial>Pi\<^sub>M {i} (\<lambda>_. interval_measure (\<lambda>x. x)) = integral\<^sup>N (interval_measure (\<lambda>x. x)) f"
step: fixes r :: "'b sequent list \<times> 'b sequent"   and R :: "('b sequent list \<times> 'b sequent) set" assumes "r \<in> R"   and "fst r \<noteq> []"   and "\<forall>p\<in>set (fst r). p \<in> derivable' R" shows "snd r \<in> derivable' R"
aff_dim_translation_eq: fixes a :: "'b"   and S :: "'b set" shows "aff_dim ((+) a ` S) = aff_dim S"
nu_star: fixes x :: "'a" assumes "\<And>x. x\<^sup>\<star> \<cdot> x\<^sup>\<star> \<le> x\<^sup>\<star>"   and "\<And>x y z. x \<cdot> (y \<cdot> z) = x \<cdot> y \<cdot> z" shows "\<nu> x\<^sup>\<star> \<cdot> (1\<^sub>\<sigma> + \<tau> x) = x\<^sup>\<star>"
case1: assumes "x \<le> y" shows "t_ins_aux \<lparr>t_type.folding = True, item = x, subtrees = Branch y xt yr # ts\<rparr> \<in> t_ins_set \<lparr>t_type.folding = True, item = x, subtrees = Branch y xt yr # ts\<rparr>"
abs_minus_cancel: fixes abs :: "'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and uminus :: "'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "abs (uminus a) = abs a"
epsclo_UN: fixes B :: "'b \<Rightarrow> hf set"   and A :: "'b set" shows "epsclo (\<Union> (B ` A)) = (\<Union>x\<in>A. epsclo (B x))"
homotopic_join_subpaths: fixes g :: "real \<Rightarrow> 'a"   and s :: "'a set"   and u :: "real"   and v :: "real"   and w :: "real" assumes "path g"   and "path_image g \<subseteq> s"   and "u \<in> {0..1}"   and "v \<in> {0..1}"   and "w \<in> {0..1}" shows "homotopic_paths s (subpath u v g +++ subpath v w g) (subpath u w g)"
rell_G_def: fixes L1 :: "'b \<Rightarrow> 'j \<Rightarrow> bool"   and L2 :: "'d \<Rightarrow> 'k \<Rightarrow> bool" shows "rell_G L1 L2 = rel_G L1 L2 (=) (=) (=) (=)"
table_of_mapconst_NoneI: fixes t :: "('b \<times> 'a) list"   and k :: "'b"   and c :: "'c" assumes "table_of t k = None" shows "table_of (map (\<lambda>(k, x). (k, c, x)) t) k = None"
nexts_code_SPEC: fixes prog :: "program"   and g :: "gState"   and g' :: "gState" assumes "gState_inv prog g"   and "program_inv prog"   and "g' \<in> ls.\<alpha> (nexts_code prog g)" shows "(g, g') \<in> gState_progress_rel prog"
prices_of: fixes S :: "'c set"   and A :: "'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> real" shows "prices (discrete_market_of S A) = A"
stake_siterate: fixes n :: "nat"   and f :: "'b \<Rightarrow> 'b"   and x :: "'b" shows "stake n (siterate f x) = map (\<lambda>n. (f ^^ n) x) [0..<n]"
sorted_fst_foldr_Cons: fixes f :: "'c \<Rightarrow> 'a"   and as :: "'c list"   and P :: "'c \<Rightarrow> 'c list \<Rightarrow> bool" assumes "sorted (map f as)" shows "sorted (map f (foldr (\<lambda>x xs. if P x xs then x # xs else xs) as []))"
inV: fixes t :: "'a"   and u :: "real" assumes "t \<in> T"   and "0 \<le> u"   and "u \<le> 1" shows "(u * a t, t) \<in> V"
hyp_port_i_for_def: fixes t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and "is" :: "nat list"   and f :: "'form" shows "hyp_port_i_for t is f = fst (snd (hyp_port_for' t is f))"
F1bd'_Card_order: fixes r1 :: "('d \<times> 'd) set"   and r2 :: "('e \<times> 'e) set" shows "Card_order (r1 +c r2)"
WTrt_binop_simps_16: fixes P :: "'b prog"   and T1 :: "ty"   and T2 :: "ty"   and T :: "ty" shows "(P \<turnstile> T1\<guillemotleft>ShiftRightZeros\<guillemotright>T2 : T) = (T1 = Integer \<and> T2 = Integer \<and> T = Integer)"
approx_minus_iff: fixes x :: "'a star"   and y :: "'a star" shows "(x \<approx> y) = (x - y \<approx> 0)"
T_merge_nneg: fixes t1 :: "'b tree"   and t2 :: "'b tree" shows "0 \<le> T_merge t1 t2"
is_Inf_binary: fixes x :: "'c"   and y :: "'c"   and inf :: "'c" shows "is_Inf {x, y} inf = is_inf x y inf"
subst_Bool_def2: fixes P :: "'b assn"   and b :: "bool"   and Y :: "vals"   and s :: "state"   and Z :: "'b" shows "(P\<leftarrow>=b) Y s Z = (\<exists>v. P (In1 v) s Z \<and> (normal s \<longrightarrow> the_Bool v = b))"
from_decomp_def: fixes x :: "'a poly"   and y :: "'a poly"   and n :: "nat" shows "from_decomp x y n = fps_of_poly x / fps_of_poly y ^ n"
ivl_split: shows "split_float_interval ivl = (ivl1, ivl2)"
assms: shows "\<forall>\<^sub>F x in nhds x. g x = h x"
size_pratt_le: fixes c :: "pratt list"   and d :: "real" assumes "\<forall>x\<in>set c. Pratt_Certificate.size_pratt x \<le> d" shows "size_cert c \<le> real (length c) * (1 + d)"
Recv_given_channel: fixes ev :: "('p, 's, 'm) event"   and cid :: "nat"   and p :: "'p"   and q :: "'p"   and c :: "('p, 's, 'm) configuration" assumes "isRecv ev"   and "getId ev = cid"   and "channel cid = Some (p, q)"   and "can_occur ev c" shows "\<exists>s s' m. ev = Recv cid q p s s' m"
cp0': fixes self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1"   and a2 :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2"   and \<tau> :: "'\<AA> state \<times> '\<AA> state" shows "f self a1 a2 \<tau> = f (\<lambda>_. self \<tau>) (\<lambda>_. a1 \<tau>) (\<lambda>_. a2 \<tau>) \<tau>"
countable: shows "countable \<^bold>E\<^bsub>quotient_web \<Gamma> f\<^esub>"
semiring_Gcd: 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 modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b" assumes "class.normalization_euclidean_semiring divide plus minus zero times one modulo euclidean_size unit_factor normalize" shows "class.semiring_Gcd one zero times (normalization_euclidean_semiring.gcd zero modulo normalize) (normalization_euclidean_semiring.lcm divide zero times modulo normalize) (normalization_euclidean_semiring.Gcd zero times euclidean_size normalize) (normalization_euclidean_semiring.Lcm zero times euclidean_size normalize) divide plus minus unit_factor normalize"
IH: fixes x21 :: "('a btree \<times> 'a) list"   and x22 :: "'a btree" assumes "t = Node x21 x22"   and "nonempty_lasttreebal t"   and "t \<noteq> Leaf" shows "inorder_pair (split_max k t) = inorder t"
standard_open_def: fixes x :: "('a \<times> 'a) set" shows "local.local.standard_open x \<equiv> local.local.spectrum\<setminus>local.local.closed_subsets (local.local.gen_ideal x)"
same_level_path_aux_callstack_Append: fixes cs :: "'edge list"   and as :: "'edge list"   and cs' :: "'edge list" assumes "same_level_path_aux cs as" shows "same_level_path_aux (cs @ cs') as"
wf_cdeclE: fixes G :: "prog"   and C :: "qtname"   and c :: "class" assumes "wf_cdecl G (C, c)" obtains "\<not> is_iface G C" and "\<forall>I\<in>set (superIfs c). is_acc_iface G (pid C) I \<and> (\<forall>s. \<forall>im\<in>DeclConcepts.imethds G I s. ? cm:methd G C s: G\<turnstile>resTy cm\<preceq>resTy im \<and> \<not> is_static cm \<and> accmodi im \<le> accmodi cm)" and "\<forall>f\<in>set (cfields c). wf_fdecl G (pid C) f" and "unique (cfields c)" and "\<forall>m\<in>set (methods c). wf_mdecl G C m" and "unique (methods c)" and "jumpNestingOkS {} (init c)" and "\<exists>A. \<lparr>prg = G, cls = C, lcl = Map.empty\<rparr>\<turnstile> {} \<guillemotright>\<langle>init c\<rangle>\<guillemotright> A" and "\<lparr>prg = G, cls = C, lcl = Map.empty\<rparr>\<turnstile>init c\<Colon>\<surd>" and "ws_cdecl G C (super c)" and "C \<noteq> Object \<longrightarrow> is_acc_class G (pid C) (super c) \<and> (table_of (map (\<lambda>(s, m). (s, C, m)) (methods c)) entails \<lambda>new. \<forall>old sig. (G \<turnstile> qmdecl sig new overrides\<^sub>S qmdecl sig old \<longrightarrow> G\<turnstile>resTy new\<preceq>resTy old \<and> accmodi old \<le> accmodi new \<and> \<not> is_static old) \<and> (G,sig\<turnstile> new hides old \<longrightarrow> accmodi old \<le> accmodi new \<and> is_static old))"
graph_nodes_it_def: fixes \<alpha> :: "'e \<Rightarrow> ('f, 'g) graph"   and invar :: "'e \<Rightarrow> bool"   and nodes_list_it :: "'e \<Rightarrow> ('f list \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'f list \<Rightarrow> 'f list) \<Rightarrow> 'f list \<Rightarrow> 'f list" shows "graph_nodes_it \<alpha> invar nodes_list_it \<equiv> graph \<alpha> invar \<and> graph_nodes_it_axioms \<alpha> invar nodes_list_it"
distinct_set_mset_eq_iff: fixes M :: "'a multiset"   and N :: "'a multiset" assumes "distinct_mset M"   and "distinct_mset N" shows "(set_mset M = set_mset N) = (M = N)"
InitCfgSet: shows "set initCfgList = {x. \<exists>ind<n + 1. x = \<lparr>states = start, msgs = initMsg ind\<rparr>}"
q_mono: fixes a :: "real"   and v :: "real"   and x :: "real"   and y :: "real"   and s0 :: "real" assumes "braking_movement a v"   and "x \<le> y" shows "movement.q v s0 x \<le> movement.q v s0 y"
fin_dim: shows "vectorspace.fin_dim class_ring (vs (subspace_sum W1 W2))"
Conj_def: fixes xset :: "('b, 'c, 'd, 'e) formula set['b]" shows "Conj xset = Abs_formula (Conj\<^sub>\<alpha> (map_bset Rep_formula xset))"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'b"   and bool :: "bool" shows "P (case bool of True \<Rightarrow> f1 | False \<Rightarrow> f2) = ((bool = True \<longrightarrow> P f1) \<and> (bool = False \<longrightarrow> P f2))"
M_6: fixes vs :: "nat \<Rightarrow> nat"   and x :: "nat"   and vs' :: "nat \<Rightarrow> nat" assumes "vs 0 = x"   and "vs 1 = 0"   and "((2, vs), 0, vs') \<in> (step M)\<^sup>*" shows "x \<in> A"
set_add_pairs_single_sorted: fixes rel :: "bool \<times> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool \<times> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f \<Rightarrow> bool"   and flag :: "bool"   and gs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and bs :: "(('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list"   and h :: "('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f"   and ps :: "(bool \<times> (('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) \<times> ('a \<Rightarrow>\<^sub>0 'e) \<times> nat \<times> 'f) list" shows "set (punit.add_pairs_single_sorted rel flag gs bs h ps) = set ps \<union> Pair flag ` ({h} \<times> (set gs \<union> set bs))"
hom_power: fixes x :: "int poly"   and n :: "nat" shows "of_int_poly (x ^ n) = of_int_poly x ^ n"
Abs_inverse: fixes m :: "int" assumes "m \<in> {0..<int CARD('a)}" shows "Rep (Abs m) = m"
type: shows "type_definition Rep_bit1 Abs_bit1 {0..<int CARD('a bit1)}"
fun_similar_fmap_bottom: shows "\<bottom> \<triangleleft>\<triangleright>\<^sup>* \<bottom>"
that: fixes d :: "nat" assumes "d = poly_deg p - poly_deg (homogenize x (dehomogenize x p))"   and "punit.monom_mult (1::'b) (monomial d x) (homogenize x (dehomogenize x p)) = p" shows "thesis"
proc_ord_proc_eq_spec: fixes P :: "'a process"   and Q :: "'a process" assumes "P \<sqsubseteq> Q"   and "D P \<subseteq> D Q" shows "P = Q"
step_invariant_weakenE: fixes A :: "('a, 'b) automaton"   and PI :: "'b \<Rightarrow> bool"   and P :: "'a \<times> 'b \<times> 'a \<Rightarrow> bool"   and Q :: "'a \<times> 'b \<times> 'a \<Rightarrow> bool"   and QI :: "'b \<Rightarrow> bool" assumes "A \<TTurnstile>\<^sub>A (PI \<rightarrow>) P"   and "\<And>t. P t \<Longrightarrow> Q t"   and "\<And>a. QI a \<Longrightarrow> PI a" shows "A \<TTurnstile>\<^sub>A (QI \<rightarrow>) Q"
sep_disj_addI3: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and sep_disj :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.sep_algebra plus zero sep_disj"   and "sep_disj (plus x y) z"   and "sep_disj x y" shows "sep_disj x (plus y z)"
simps_1: fixes uu :: "'b \<Rightarrow> 'a" shows "NegPos_map uu [] = []"
qAbs_alphaAbs_qSwap_qAFresh: fixes X :: "('d, 'e, 'varSort, 'var, 'f) qTerm"   and ys :: "'varSort"   and x' :: "'var"   and x :: "'var" assumes "qGood X"   and "qAFresh ys x' X" shows "qAbs ys x X $= qAbs ys x' (X #[[x' \<and> x]]_ys)"
order_iff_strict: fixes a :: "'t \<Rightarrow>\<^sub>0 'c"   and b :: "'t \<Rightarrow>\<^sub>0 'c" shows "(a \<preceq>\<^sub>p b) = (a \<prec>\<^sub>p b \<or> a = b)"
snd1: fixes P :: "'a trm"   and P' :: "'a trm" assumes "P \<rightarrow>\<beta> P'" shows "Snd P \<rightarrow>\<beta> Snd P'"
card_of_Times_assoc: fixes A :: "'c set"   and B :: "'d set"   and C :: "'e set" shows "|(A \<times> B) \<times> C| =o |A \<times> B \<times> C|"
emp_partJoin: fixes P :: "'a set set"   and Q :: "'a set set" assumes "{} \<notin> P"   and "{} \<notin> Q" shows "{} \<notin> partJoin P Q"
sqrt_rat_ceiling_def: fixes x :: "rat" shows "sqrt_rat_ceiling x = root_rat_ceiling 2 x"
fps_integral0_one': assumes "inverse (1::'a) = (1::'a)" shows "fps_integral0 1 = fps_X"
Sup_fset_transfer: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "bi_unique A" shows "rel_fun (rel_set (rel_set A)) (rel_set A) (\<lambda>A. if finite (\<Union> A) then \<Union> A else {}) (\<lambda>A. if finite (\<Union> A) then \<Union> A else {})"
Cons_to_snoc: fixes x :: "'b"   and xs :: "'b list" shows "\<exists>ys y. x # xs = ys @ [y]"
isTrue_imp: fixes \<phi> :: "'fmla"   and \<psi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<psi> \<in> fmla"   and "Fvars \<phi> = {}"   and "Fvars \<psi> = {}"   and "isTrue \<phi>"   and "isTrue (imp \<phi> \<psi>)" shows "isTrue \<psi>"
lc_not_0: fixes pair_of_term :: "'e \<Rightarrow> 'f \<times> 'g"   and term_of_pair :: "'f \<times> 'g \<Rightarrow> 'e"   and ord :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and ord_strict :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and ord_term :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and ord_term_strict :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and p :: "'e \<Rightarrow>\<^sub>0 'h" assumes "ordered_term pair_of_term term_of_pair ord ord_strict ord_term ord_term_strict"   and "p \<noteq> 0" shows "ordered_term.lc term_of_pair ord_term p \<noteq> (0::'h)"
assms: shows "low_equal s1 s2 \<and> (fst instr = load_store_type STB \<or> fst instr = load_store_type STH \<or> fst instr = load_store_type ST \<or> fst instr = load_store_type STD) \<and> t1 = snd (fst (store_sub1 instr rd 0 s1)) \<and> t2 = snd (fst (store_sub1 instr rd 0 s2))"
step_10: shows "InvariantWatchCharacterization (getF state') (getWatch1 state') (getWatch2 state') (getM state')"
takeWhile_neq_rev: fixes xs :: "'c list"   and x :: "'c" assumes "distinct xs"   and "x \<in> set xs" shows "takeWhile (\<lambda>y. y \<noteq> x) (rev xs) = rev (tl (dropWhile (\<lambda>y. y \<noteq> x) xs))"
is_prefix_admissible: fixes a :: "(symbol \<times> character list) list"   and b :: "(symbol \<times> character list) list" assumes "is_prefix a b"   and "admissible b" shows "admissible a"
iso_is_retraction: fixes f :: "'c" assumes "C.iso f" shows "C.retraction f"
Diff_single_insert: fixes A :: "'d set"   and x :: "'d"   and B :: "'d set" assumes "A - {x} \<subseteq> B" shows "A \<subseteq> insert x B"
join_comm: fixes x :: "'c"   and y :: "'c" shows "x \<squnion> y = y \<squnion> x"
eval_cases_2: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and v :: "val"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>Val v,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains "e' = Val v" and "s' = s"