hcju/isabelleps huggingface dataset:
mono: fixes a :: "real"   and b :: "real" assumes "a < b" shows "f b < f a"
changing_atoms_works: fixes P :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'b"   and phi :: "'a formula"   and M :: "'a \<Rightarrow> bool" assumes "\<And>x. P x \<Longrightarrow> (f \<circ> g) x = x"   and "\<forall>x\<in>atoms phi. P x" shows "M \<Turnstile> phi = M \<circ> f \<Turnstile> map_formula g phi"
tick_count_sub: fixes f :: "nat \<Rightarrow> nat"   and sub :: "'b run"   and r :: "'b run"   and c :: "clock"   and n :: "nat" assumes "dilating f sub r" shows "tick_count sub c n = tick_count r c (f n)"
simps_3: fixes uu :: "'a" shows "(Some uu \<sqsubseteq> None) = False"
distinct_map_getOneIp_obtain: fixes v :: "'a wordinterval"   and xs :: "'a wordinterval list" assumes "v \<in> set xs"   and "distinct (map getOneIp xs)" shows "\<exists>s_repr. map_of (zip (map getOneIp xs) xs) s_repr = Some v"
maxsimp_vertex_conv_special_coset: fixes S :: "'b set"   and w :: "'b"   and X :: "'b set" assumes "CoxeterComplex S"   and "w \<in> \<langle>S\<rangle>"   and "X \<in> dual_order.poset_simplex_map (PreCoxeterSystem.special_cosets S) {w}" shows "\<exists>s\<in>S. X = w +o \<langle>S - {s}\<rangle>"
va2: fixes \<nu> :: "variable \<Rightarrow> real" assumes "Uvariation \<nu> (F 0) {RVar x, DVar x}" shows "term_sem J \<eta> \<nu> = term_sem I \<theta> \<nu>"
lookup_keys: fixes t :: "('a, 'b) RBT.rbt" shows "dom (RBT.lookup t) = set (RBT.keys t)"
insert_subsetI: fixes x :: "'e"   and A :: "'e set"   and X :: "'e set" assumes "x \<in> A"   and "X \<subseteq> A" shows "insert x X \<subseteq> A"
assms_2: shows "Q\<^sub>1 \<sqsubseteq> Q\<^sub>2"
wf_TVar: fixes \<Gamma> :: "binding list"   and i :: "nat"   and T :: "POPLmarkRecord.type" assumes "\<Gamma>\<langle>i\<rangle> = \<lfloor>TVarB T\<rfloor>" shows "\<Gamma> \<turnstile>\<^sub>w\<^sub>f TVar i"
s_mono: fixes u :: "real"   and t :: "real" assumes "u \<le> t"   and "0 \<le> u" shows "other.s u \<le> other.s t"
stream_rel_def: fixes R :: "('c \<times> 'd) set" shows "\<langle>R\<rangle>stream_rel \<equiv> {(x, y). stream_all2 (\<lambda>x y. (x, y) \<in> R) x y}"
Un_empty: fixes A :: "'c set"   and B :: "'c set" shows "(A \<union> B = {}) = (A = {} \<and> B = {})"
tau_alpha_zero: fixes x :: "'a" shows "\<tau> (\<nu> x) = (0::'a)"
is_path_simps_2: fixes v :: "'v"   and v' :: "'v" shows "is_path v [(v, v')] v' = ((v, v') \<in> E)"
simps_5: fixes Fstd_empty :: "'a"   and Fstd_get :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) Std.instr fundef option"   and Fstd_add :: "'a \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e) Std.instr fundef \<Rightarrow> 'a"   and Fstd_to_list :: "'a \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e) Std.instr fundef) list"   and Finca_empty :: "'f"   and Finca_get :: "'f \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef option"   and Finca_add :: "'f \<Rightarrow> 'b \<Rightarrow> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef \<Rightarrow> 'f"   and Finca_to_list :: "'f \<Rightarrow> ('b \<times> ('c, 'd, 'b, 'e, 'g) Inca.instr fundef) list"   and heap_empty :: "'h"   and heap_get :: "'h \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c option"   and heap_add :: "'h \<Rightarrow> 'd \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'h"   and heap_to_list :: "'h \<Rightarrow> (('d \<times> 'c) \<times> 'c) list"   and is_true :: "'c \<Rightarrow> bool"   and is_false :: "'c \<Rightarrow> bool"   and \<OO>\<pp> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<AA>\<rr>\<ii>\<tt>\<yy> :: "'e \<Rightarrow> nat"   and \<II>\<nn>\<ll>\<OO>\<pp> :: "'g \<Rightarrow> 'c list \<Rightarrow> 'c"   and \<II>\<nn>\<ll> :: "'e \<Rightarrow> 'c list \<Rightarrow> 'g option"   and \<II>\<ss>\<II>\<nn>\<ll> :: "'g \<Rightarrow> 'c list \<Rightarrow> bool"   and \<DD>\<ee>\<II>\<nn>\<ll> :: "'g \<Rightarrow> 'e"   and op :: "'e" assumes "std_inca_simulation Fstd_empty Fstd_get Fstd_add Fstd_to_list Finca_empty Finca_get Finca_add Finca_to_list heap_empty heap_get heap_add heap_to_list is_true is_false \<OO>\<pp> \<AA>\<rr>\<ii>\<tt>\<yy> \<II>\<nn>\<ll>\<OO>\<pp> \<II>\<nn>\<ll> \<II>\<ss>\<II>\<nn>\<ll> \<DD>\<ee>\<II>\<nn>\<ll>" shows "std_inca_simulation.norm_instr \<DD>\<ee>\<II>\<nn>\<ll> (Inca.instr.IOp op) = Std.instr.IOp op"
intros: fixes J :: "'i set"   and X :: "('i \<Rightarrow> 'a) set" assumes "finite J"   and "J \<subseteq> I"   and "X \<in> sets (Pi\<^sub>M J (\<lambda>_. borel))" shows "emb I J X \<in> generator"
nat_of_times_sat: fixes x :: "'a sat"   and y :: "'a sat" shows "nat_of (x * y) = min (nat_of x * nat_of y) LENGTH('a)"
RP_sound_new_statement: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" assumes "chain (\<leadsto>RP) Sts"   and "{#} \<in> N_of_state (Liminf_state Sts) \<union> P_of_state (Liminf_state Sts) \<union> Q_of_state (Liminf_state Sts)" shows "N_of_state (lhd Sts) \<union> P_of_state (lhd Sts) \<union> Q_of_state (lhd Sts) \<TTurnstile>\<G>e {{#}}"
fpairs_eqI: fixes r :: "V" assumes "\<And>x. x \<in>\<^sub>\<circ> r \<Longrightarrow> \<exists>a b. x = [a, b]\<^sub>\<circ>" shows "fpairs r = r"
\<rho>s: shows "\<rho>s = tl (renamings_apart (DA # CAs))"
aux_ineq_with_card_2: fixes A :: "nat set"   and g :: "nat \<Rightarrow> nat" assumes "finite A" shows "- int (card A) \<le> (\<Sum>k\<in>A. (- 1) ^ g k)"
inf_bot_def: fixes x :: "'a bot"   and y :: "'a bot" shows "inf x y = (case x of Value v \<Rightarrow> case y of Value v' \<Rightarrow> Value (inf v v') | Bot \<Rightarrow> Bot | Bot \<Rightarrow> Bot)"
path_component_mono: fixes S :: "'a set"   and T :: "'a set"   and x :: "'a" assumes "S \<subseteq> T" shows "path_component_set S x \<subseteq> path_component_set T x"
int_pos_mult_monor: fixes n :: "int"   and m :: "int" assumes "0 < n"   and "0 \<le> n * m" shows "0 \<le> m"
assms_3: shows "(state0, state) \<in> transitionRelation F0 decisionVars"
mu_fusion_2: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and l :: "'b \<Rightarrow> 'b"   and u :: "'b \<Rightarrow> 'b"   and h :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "class.order less_eq less"   and "ord.galois less_eq l u"   and "ord.isotone less_eq h"   and "order.has_least_prefixpoint less_eq g"   and "order.has_least_fixpoint less_eq h"   and "ord.lifted_less_eq less_eq (l \<circ> g) (h \<circ> l)" shows "less_eq (l (order.the_least_prefixpoint less_eq g)) (order.the_least_fixpoint less_eq h)"
union_is_single: fixes M :: "'k multiset"   and N :: "'k multiset"   and a :: "'k" shows "(M + N = {#a#}) = (M = {#a#} \<and> N = {#} \<or> M = {#} \<and> N = {#a#})"
matchC_LC_sym: fixes theta :: "(('test, 'atom, 'choice) cmd \<times> ('test, 'atom, 'choice) cmd) set"   and c :: "('test, 'atom, 'choice) cmd"   and d :: "('test, 'atom, 'choice) cmd" assumes "sym theta"   and "matchC_LC theta c d" shows "matchC_LC theta d c"
opp_opp: fixes a :: "'a"   and b :: "'a"   and p :: "'a point" assumes "on_curve a b p" shows "opp (opp p) = p"
index_one_alt_bl_not_exist: fixes blv :: "'a set"   and p :: "'a set"   and bl :: "'a set" assumes "\<Lambda> = 1"   and "blv \<in># \<B>"   and "p \<subseteq> blv"   and "card p = 2"   and "bl \<in># remove1_mset blv \<B>" shows "\<not> p \<subseteq> bl"
lq_the: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>p v" shows "u\<inverse>\<^sup>>v = (THE z. u \<cdot> z = v)"
f6: shows "uint w + (4294967296 + - 1 * uint w) mod 4294967296 \<le> 4294967296"
t_in_type: shows "(x, right r) \<in> {r. fst r \<le> snd r}"
GIrrRep_repset_def: shows "GIrrRep_repset \<equiv> 0 \<union> set (remisodups FG_constituents)"
\<iota>_mapsto: shows "\<iota> \<in> UNIV \<rightarrow> S.Univ"
imirror_iff: fixes n :: "nat"   and I :: "nat set" shows "(n \<in> imirror I) = (\<exists>x\<in>I. n = iMin I + Max I - x)"
cone_decomp_indets_2: fixes T :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and ps :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and X :: "'x set"   and h :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and U :: "'x set" assumes "cone_decomp T ps"   and "T \<subseteq> P[X]"   and "(h, U) \<in> set ps"   and "h \<noteq> 0" shows "U \<subseteq> X"
igAbsIPresIGWlsSTR_imp_igAbsIPresIGWls: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('g, 'h, 'd, 'e, 'i, 'j, 'k, 'l, 'm) model_scheme" assumes "FixSyn TYPE('a) Delta"   and "FixSyn.igAbsIPresIGWlsSTR Delta MOD" shows "FixSyn.igAbsIPresIGWls Delta MOD"
assms_2: shows "f n < k \<and> k < f (Suc n)"
veval'_wellformed: fixes C_info :: "(name, dt_def) fmap"   and \<Gamma> :: "(name, value) fmap"   and t :: "sterm"   and v :: "value" assumes "special_constants.veval' C_info \<Gamma> t v"   and "pre_strong_term_class.wellformed t"   and "wellformed_venv \<Gamma>" shows "vwellformed v"
Bfun_metric_def: fixes f :: "'c \<Rightarrow> 'd"   and F :: "'c filter" shows "Bfun f F = (\<exists>y K. 0 < K \<and> (\<forall>\<^sub>F x in F. dist (f x) y \<le> K))"
g_bounded: fixes x :: "'a" shows "norm (g x) \<le> B * norm x"
length_map: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "'b list" shows "|map f xs| = |xs|"
liftI_Suc: fixes a :: "nat"   and A :: "nat set" assumes "a \<in> A" shows "Suc a \<in> lift A"
cdif: shows "d\<inverse> O f \<subseteq> ov\<inverse> \<union> s\<inverse> \<union> d\<inverse>"
semiring_1_cancel_ow_def: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" shows "semiring_1_cancel_ow U plus minus zero one times \<equiv> (semiring_ow U plus times \<and> cancel_comm_monoid_add_ow U plus minus zero) \<and> zero_neq_one_ow U one zero \<and> monoid_mult_ow U one times"
inverse_nonnegative_iff_nonnegative: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and a :: "'c" assumes "class.linordered_field divide inverse times one plus zero minus uminus less_eq less sgn abs" shows "less_eq zero (inverse a) = less_eq zero a"
is_processT2: fixes P :: "'a process" shows "\<forall>s X. (s, X) \<in> F P \<longrightarrow> front_tickFree s"
tan_minus: fixes x :: "'a" shows "tan (- x) = - tan x"
prefix_def: fixes xs :: "'a list"   and zs :: "'a list" shows "(xs \<le> zs) = ((xs, zs) \<in> genPrefix Id)"
mult_commute: fixes a :: "'b \<Rightarrow> bool"   and b :: "'b \<Rightarrow> bool" shows "(a \<and>* b) = (b \<and>* a)"
not_dvd_imp_multiplicity_0: 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 p :: "'a"   and x :: "'a" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize"   and "\<not> dvd.dvd times p x" shows "factorial_semiring.multiplicity times one p x = 0"
mult_closed: fixes U :: "'c set"   and V :: "'c set"   and x :: "'c"   and a :: "real" assumes "Subspace.subspace U V"   and "x \<in> U" shows "a \<cdot> x \<in> U"
set_child_nodes_writes: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and set_child_nodes :: "(_) object_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and ptr :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "l_set_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr set_child_nodes set_child_nodes_locs" shows "writes (set_child_nodes_locs ptr) (set_child_nodes ptr children) h h'"
conv_2: shows "g \<turnstile> b-bs\<rightarrow>n"
singular_chain_diff: fixes p :: "nat"   and X :: "'b topology"   and a :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int"   and b :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" assumes "singular_chain p X a"   and "singular_chain p X b" shows "singular_chain p X (a - b)"
override_A_def: fixes m2 :: "'a \<Rightarrow> 'b decision option"   and m1 :: "'a \<Rightarrow> 'b decision option" shows "m2 \<Oplus>\<^sub>A m1 = (\<lambda>x. case m1 x of \<bottom> \<Rightarrow> m2 x | \<lfloor>allow a\<rfloor> \<Rightarrow> \<lfloor>allow a\<rfloor> | \<lfloor>deny a\<rfloor> \<Rightarrow> case m2 x of \<bottom> \<Rightarrow> \<lfloor>deny a\<rfloor> | \<lfloor>allow b\<rfloor> \<Rightarrow> \<lfloor>allow b\<rfloor> | \<lfloor>deny \<alpha>\<rfloor> \<Rightarrow> \<lfloor>deny a\<rfloor>)"
distinct_6: fixes x2 :: "nat" shows "p \<noteq> c x2"
simps_2: fixes Delta :: "('b, 'c, 'd, 'e, 'f) signature"   and MOD :: "('b, 'c, 'd, 'e, 'f, 'a, 'g, 'h) model"   and delta :: "'f"   and qinp :: "'b \<Rightarrow> ('b, 'c, 'd, 'a, 'f) qTerm option"   and qbinp :: "'c \<Rightarrow> ('b, 'c, 'd, 'a, 'f) qAbs option" assumes "FixSyn TYPE('a) Delta" shows "FixSyn.qInit MOD (qOp delta qinp qbinp) = igOp MOD delta (lift (FixSyn.qInit MOD) qinp) (lift (FixSyn.qInitAbs MOD) qbinp)"
discI: fixes list :: "'g list"   and x21 :: "'g"   and x22 :: "'g list" assumes "list = x21 # x22" shows "list \<noteq> []"
G: shows "simAbs (runJP k_mkAlgAuto (t \<leadsto> s) a) = sim_equiv_class a (t \<leadsto> s)"
Inf_eqI: 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 :: "'d set"   and x :: "'d" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "\<And>i. i \<in> A \<Longrightarrow> less_eq x i"   and "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> less_eq y i) \<Longrightarrow> less_eq y x" shows "Inf A = x"
eval\<^sub>G_prop_entailment: fixes S :: "'a ltl set"   and \<phi> :: "'a ltl" shows "S \<Turnstile>\<^sub>P eval\<^sub>G S \<phi> = S \<Turnstile>\<^sub>P \<phi>"
basic_prop_3: fixes v :: "i" shows "[Contingent (??.TAO_9_PLM.PLM.q\<^sub>0\<^sup>-) in v]"
a_set_mode_def: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "set_mode shadow_root_ptr = put_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr mode_update"
edge_rev_in_arcs: fixes a :: "'b \<times> 'b" shows "(edge_rev (to_map' (snd G_list) xss) a \<in> arcs (with_proj (list_digraph G_list))) = (a \<in> arcs (with_proj (list_digraph G_list)))"
MGU_sym: fixes \<sigma> :: "('b \<times> 'b trm) list"   and s :: "'b trm"   and t :: "'b trm" assumes "MGU \<sigma> s t" shows "MGU \<sigma> t s"
eq: shows "[g ^ totient (p ^ k) = 1 + t * p ^ k] (mod p ^ Suc k)"
rephrase2: shows "fst (concl_of \<iota>) = concl_of (to_F \<iota>)"
that_1: fixes b :: "'a" assumes "x = (K$ b)" shows "thesis"
space_stat: fixes C :: "'s set" shows "space (stat C) = UNIV"
all_accessible: fixes r :: "('k \<times> 'k) set" assumes "wf r" shows "\<forall>M. M \<in> Wellfounded.acc (mult1 r)"
bounded_clinear_add: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "bounded_clinear f"   and "bounded_clinear g" shows "bounded_clinear (\<lambda>x. f x + g x)"
c_def: shows "c = - 2 / 3 * (real n + 1 / 2)"
assms_2: shows "\<forall>\<^sub>F x in at_top. 0 < l x"
replay'_sound_pre: fixes thy :: "((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)) \<times> Core.term set"   and Hs :: "Core.term list"   and ns :: "variable set"   and P :: "proofterm"   and vs :: "(variable \<times> typ) list"   and res :: "Core.term" assumes "wf_theory thy"   and "\<And>x. x \<in> set Hs \<Longrightarrow> term_ok thy x"   and "\<And>x. x \<in> set Hs \<Longrightarrow> typ_of x = Some propT"   and "finite ns"   and "fst ` FV (set Hs) \<subseteq> ns"   and "fst ` fv_Proof P \<subseteq> ns"   and "fst ` set vs \<subseteq> ns"   and "replay' thy vs ns Hs P = Some res" shows "thy,set Hs \<turnstile> res"
adjoint_pair_def: fixes f :: "'d"   and g :: "'d" shows "D.adjoint_pair f g \<equiv> \<exists>\<eta> \<epsilon>. adjunction_in_bicategory (\<cdot>\<^sub>D) (\<star>\<^sub>D) \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D f g \<eta> \<epsilon>"
component_connected: fixes g :: "'b ugraph"   and r :: "'b" shows "Undirected_Graph.connected (component_of g r)"
assms: shows "\<forall>s. s \<in> wa_invar WAA \<and> s \<notin> wa_cond WAA \<longrightarrow> P s"
that: assumes "vsv f"   and "\<D>\<^sub>\<circ> f = I"   and "vsv a"   and "\<D>\<^sub>\<circ> a = I"   and "vsv b"   and "\<D>\<^sub>\<circ> b = I"   and "\<And>i. i \<in>\<^sub>\<circ> I \<Longrightarrow> f\<lparr>i\<rparr> : a\<lparr>i\<rparr> \<mapsto>\<^bsub>\<AA> i\<^esub> b\<lparr>i\<rparr>" shows "thesis"
inverse_squares_sums': shows "(\<lambda>n. 1 / (real n)\<^sup>2) sums (pi\<^sup>2 / 6)"
tranclp_def: shows "tranclp \<equiv> \<lambda>r. lfp (\<lambda>p x1 x2. (\<exists>a b. x1 = a \<and> x2 = b \<and> r a b) \<or> (\<exists>a b c. x1 = a \<and> x2 = c \<and> p a b \<and> r b c))"
best_worst_indiff_all_indiff_1: fixes b :: "'a pmf"   and w :: "'a pmf" assumes "b \<in> local.best"   and "w \<in> local.worst"   and "b \<approx>[\<R>] w" shows "\<forall>e\<in>local.\<P>. e \<approx>[\<R>] w"
length_compl_rev_induct: fixes P :: "'f list \<Rightarrow> bool"   and l :: "'f list" assumes "P []"   and "\<And>l e. (\<And>ll. length ll \<le> length l \<Longrightarrow> P ll) \<Longrightarrow> P (l @ [e])" shows "P l"
finite_UNIV_inj_surj: fixes f :: "'c \<Rightarrow> 'c" assumes "finite UNIV"   and "inj f" shows "surj f"
qSwapAll_simps_1: fixes zs :: "'c"   and x :: "'d"   and y :: "'d"   and zs' :: "'c"   and z :: "'d" shows "qVar zs' z #[[x \<and> y]]_zs = qVar zs' (z @zs'[x \<and> y]_zs)"
iso_classes_iff: fixes G :: "('c, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme" assumes "Group.group G"   and "Group.group H" shows "(\<exists>\<phi>. \<phi> \<in> Group.iso G H) = (group.iso_class G = group.iso_class H)"
trm_prm_simp_1: fixes \<pi> :: "'b prm" shows "\<pi> \<cdot> Unit = Unit"
lm092: fixes f :: "('c \<times> 'd) set"   and X :: "'c set" assumes "runiq f"   and "runiq (f\<inverse>)" shows "Range (f outside X) \<subseteq> Range f - f `` X"
dist_triangle3: fixes x :: "'a"   and y :: "'a"   and a :: "'a" shows "dist x y \<le> dist a x + dist a y"
cfRead_def: fixes cfile :: "CFile"   and byteIndex :: "nat" shows "cfRead cfile byteIndex \<equiv> if byteIndex < fileSize cfile then let i = byteIndex div blockSize; j = byteIndex mod blockSize in Some (readCArray (readCArray (data cfile) i) j) else None"
rhs: fixes x :: "name"   and s :: "Lambda_Z.term"   and t :: "Lambda_Z.term" shows "subst x s\<^sup>\<bullet> t\<^sup>\<bullet> \<rightarrow>\<^sub>\<beta>\<^sup>* (subst x s t)\<^sup>\<bullet>"
read_only_share_unacquired: fixes \<O> :: "nat set"   and S :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list"   and a :: "nat" assumes "\<O> \<inter> read_only S = {}"   and "weak_sharing_consistent \<O> sb"   and "a \<in> read_only (share sb S)" shows "a \<notin> acquired True sb \<O>"
LocalLexing_axioms: shows "LocalLexing \<NN> \<TT> \<RR> \<SS> Lex Sel"
entail_unions: fixes I :: "'a set"   and N :: "'g set"   and Ni :: "'a \<Rightarrow> 'g set" shows "(\<forall>i\<in>I. N \<Turnstile>G Ni i) = (N \<Turnstile>G \<Union> (Ni ` I))"
lhd_inf_llist: fixes f :: "nat \<Rightarrow> 'c" shows "lhd (inf_llist f) = f 0"
n: shows "(tsl s, tsl') \<in> \<langle>Rk\<rangle>list_rel"
assms_4: shows "D \<cdot> \<sigma> \<subseteq># C"
VcompNml_HcompNml: fixes t :: "'a local.term"   and u :: "'a local.term"   and v :: "'a local.term"   and w :: "'a local.term" assumes "Nml t"   and "Nml u"   and "Nml v"   and "Nml w"   and "VSeq t v"   and "VSeq u w"   and "Src v = Trg w" shows "(t \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u) \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> (v \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> w) = t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> v \<^bold>\<lfloor>\<^bold>\<star>\<^bold>\<rfloor> u \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> w"
finite_borel_measure_axioms: shows "finite_borel_measure M2"
Cons_1: assumes "a \<in> set xs"   and "b \<in> set xs" shows "(flag, a, b) \<in> set (pairs aps flag xs) \<or> (flag, b, a) \<in> set (pairs aps flag xs)"
coeff_Var1: fixes x :: "nat" shows "coeff (Abstract_Linear_Poly.Var x) x = 1"
simps_8: fixes \<sigma> :: "(String.literal \<times> event_data list) Trace.trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and v :: "event_data list"   and i :: "nat"   and \<phi> :: "formula"   and \<psi> :: "formula" shows "sat \<sigma> V v i (And \<phi> \<psi>) = (sat \<sigma> V v i \<phi> \<and> sat \<sigma> V v i \<psi>)"
commute: fixes a :: "'d multiset"   and b :: "'d multiset" shows "a \<union># b = b \<union># a"
interval_integral_to_infinity_eq: fixes M :: "real measure"   and a :: "real"   and f :: "real \<Rightarrow> 'a" shows "interval_lebesgue_integral M (ereal a) \<infinity> f = set_lebesgue_integral M {a<..} f"
DGaxiom_def: shows "DGaxiom = ([[EvolveODE (OSing vid1 (f1 fid1 vid1)) (p1 vid1 vid1)]]p1 vid2 vid1 \<leftrightarrow> Exists vid2 ([[EvolveODE (OProd (OSing vid1 (f1 fid1 vid1)) (OSing vid2 (Plus (Times (f1 fid2 vid1) (Var vid2)) (f1 fid3 vid1)))) (p1 vid1 vid1)]]p1 vid2 vid1))"
assms_1: shows "(b, a) \<in> \<langle>R\<rangle>nres_rel"
l_known_ptrs_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and known_ptrs :: "(_) heap \<Rightarrow> bool" shows "l_known_ptrs known_ptr known_ptrs \<equiv> ((\<forall>h ptr. known_ptrs h \<longrightarrow> ptr |\<in>| object_ptr_kinds h \<longrightarrow> known_ptr ptr) \<and> (\<forall>h h'. object_ptr_kinds h = object_ptr_kinds h' \<longrightarrow> known_ptrs h = known_ptrs h')) \<and> (\<forall>h' h. object_ptr_kinds h' |\<subseteq>| object_ptr_kinds h \<longrightarrow> known_ptrs h \<longrightarrow> known_ptrs h') \<and> (\<forall>h' h new_ptr. object_ptr_kinds h' = object_ptr_kinds h |\<union>| {|new_ptr|} \<longrightarrow> known_ptr new_ptr \<longrightarrow> known_ptrs h \<longrightarrow> known_ptrs h')"
VUnion_vempty: shows "\<Union>\<^sub>\<circ> []\<^sub>\<circ> = []\<^sub>\<circ>"
prSwapAll_def: fixes g :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'c"   and gA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> ('varSort \<Rightarrow> 'var \<Rightarrow> 'b) \<Rightarrow> 'd"   and SEM :: "('e, 'f, 'g, 'sort, 'h, 'b, 'i) semDom_scheme" shows "prSwapAll g gA SEM \<equiv> prSwap g SEM \<and> prSwapAbs gA SEM"
execp_resumption_def: shows "local.execp_resumption \<equiv> curry (spmf.fixp_fun (\<lambda>execp_resumption (r, s). case r of Resumption.resumption.Done x \<Rightarrow> return_pmf (map_option (\<lambda>a. (a, s)) x) | Resumption.resumption.Pause out c \<Rightarrow> callee s out \<bind> (\<lambda>(input, s'). curry execp_resumption (c input) s')))"
iterate_0: fixes F :: "'a \<rightarrow> 'a"   and x :: "'a" shows "iterate 0\<cdot>F\<cdot>x = x"
cases: fixes x :: "tm \<times> tm \<times> tm \<times> tm" obtains s :: "name" and v :: "tm" and i :: "tm" and t :: "tm" and u :: "tm" and k :: "name" where "atom s \<sharp> (v, i, t, u, k)" and "atom k \<sharp> (v, i, t, u)" and "x = (v, i, t, u)"
equivalence_functor_L: shows "equivalence_functor (\<cdot>) (\<cdot>) L"