hcju/isabelleps huggingface dataset:
lookup_monom_mult_plus: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and c :: "'f"   and t :: "'d"   and p :: "'c \<Rightarrow>\<^sub>0 'f"   and v :: "'c" assumes "term_powerprod pair_of_term term_of_pair" shows "lookup (term_powerprod.monom_mult pair_of_term term_of_pair c t p) (term_powerprod.splus pair_of_term term_of_pair t v) = c * lookup p v"
integrable: shows "set_integrable lborel (einterval a b) (\<lambda>x. g' x *\<^sub>R f (g x))"
assms_4: shows "(sol solves_ode (\<lambda>a. ODE_sem I ODE)) {0..t} {x. mk_v I ODE (sol 0, b) x \<in> fml_sem I \<phi>}"
consistent_extend': fixes A :: "'b set"   and S :: "(('c, 'b) fm list \<times> 'b) set"   and f :: "nat \<Rightarrow> ('c, 'b) fm list \<times> 'b"   and n :: "nat" assumes "infinite UNIV"   and "finite A"   and "consistent A S"   and "finite (\<Union> (block_nominals ` S))" shows "consistent A (extend A S f n)"
smult_r_distr: fixes a :: "'a"   and x :: "nat \<Rightarrow> 'a"   and y :: "nat \<Rightarrow> 'a" assumes "a \<in> carrier R"   and "x \<in> carrier P"   and "y \<in> carrier P" shows "a \<odot>\<^bsub>P\<^esub> (x \<oplus>\<^bsub>P\<^esub> y) = a \<odot>\<^bsub>P\<^esub> x \<oplus>\<^bsub>P\<^esub> a \<odot>\<^bsub>P\<^esub> y"
compact_Un: fixes s :: "'c set"   and t :: "'c set" assumes "compact s"   and "compact t" shows "compact (s \<union> t)"
path_from_toI: fixes G :: "('c, 'd) Graph_scheme"   and xs :: "'c list"   and v :: "'c"   and w :: "'c" assumes "Graph G"   and "Graph.walk G xs \<and> distinct xs"   and "xs \<noteq> []"   and "hd xs = v"   and "last xs = w" shows "v \<leadsto>xs\<leadsto>\<^bsub>G\<^esub> w"
brk_rel_def: fixes R :: "('a \<times> 'b) set" shows "brk_rel R \<equiv> {((False, x), False, y) |x y. (x, y) \<in> R} \<union> {((True, x), False, y) |x y. True}"
composable_char\<^sub>P\<^sub>B\<^sub>H: fixes \<nu> :: "'a"   and \<mu> :: "'a" shows "(\<nu> \<star> \<mu> \<noteq> null) = (arr \<mu> \<and> arr \<nu> \<and> src \<nu> = trg \<mu>)"
intensional_rews_3: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and x :: "'e \<Rightarrow> 'c"   and y :: "'e \<Rightarrow> 'd"   and w :: "'e" shows "(f<x, y>) w \<equiv> f (x w) (y w)"
zhmsetmset_inject: fixes x :: "zhmultiset"   and y :: "zhmultiset" shows "(zhmsetmset x = zhmsetmset y) = (x = y)"
irreducible_connect_field: fixes f :: "'b poly" shows "irreducible\<^sub>d f = irreducible f"
perm_bool: fixes pi :: "('a \<times> 'a) list"   and b :: "bool" shows "pi \<bullet> b = b"
DynProcStaticSpec: fixes P :: "'d set"   and S :: "'d set"   and init :: "'d \<Rightarrow> 'd"   and P' :: "'e \<Rightarrow> 'd set"   and Q' :: "'e \<Rightarrow> 'd set"   and return :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> 'd set"   and A' :: "'e \<Rightarrow> 'd set"   and A :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'g set"   and c :: "'d \<Rightarrow> 'd \<Rightarrow> ('d, 'f, 'g) com"   and Q :: "'d set"   and p :: "'d \<Rightarrow> 'f" assumes "P \<subseteq> {s \<in> S. \<exists>Z. init s \<in> P' Z \<and> (\<forall>\<tau>. \<tau> \<in> Q' Z \<longrightarrow> return s \<tau> \<in> R s \<tau>) \<and> (\<forall>\<tau>. \<tau> \<in> A' Z \<longrightarrow> return s \<tau> \<in> A)}"   and "\<forall>s t. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(R s t) c s t Q,A"   and "\<forall>s\<in>S. \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(P' Z) Call (p s) (Q' Z),(A' Z)" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P dynCall init p return c Q,A"
allAllocationsVarCharacterization: fixes a :: "('c \<times> 'd set) set"   and N :: "'c set"   and \<Omega> :: "'d set" shows "(a \<in> allAllocations'' N \<Omega>) = (a \<in> allocationsUniverse \<and> Domain a \<subseteq> N \<and> \<Union> (Range a) = \<Omega>)"
unit_imp_no_prime_divisors: fixes x :: "'a"   and p :: "'a" assumes "is_unit x"   and "prime_elem p" shows "\<not> p dvd x"
coherent_Hcomp: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src\<^sub>C :: "'a \<Rightarrow> 'a"   and trg\<^sub>C :: "'a \<Rightarrow> 'a"   and V :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and H :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<a> :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and \<i> :: "'d \<Rightarrow> 'd"   and src :: "'d \<Rightarrow> 'd"   and trg :: "'d \<Rightarrow> 'd"   and E :: "'a \<Rightarrow> 'd"   and t :: "'a local.term"   and u :: "'a local.term" assumes "Coherence.evaluation_map C src\<^sub>C trg\<^sub>C V H \<a> \<i> src trg E"   and "bicategorical_language.Arr C src\<^sub>C trg\<^sub>C t"   and "bicategorical_language.Arr C src\<^sub>C trg\<^sub>C u"   and "bicategorical_language.Src src\<^sub>C t = bicategorical_language.Trg trg\<^sub>C u"   and "V (Coherence.evaluation_map.eval V H \<a> \<i> src trg E (bicategorical_language.red C src\<^sub>C trg\<^sub>C (bicategorical_language.Cod C src\<^sub>C trg\<^sub>C t))) (Coherence.evaluation_map.eval V H \<a> \<i> src trg E t) = V (Coherence.evaluation_map.eval V H \<a> \<i> src trg E (bicategorical_language.Nmlize C t)) (Coherence.evaluation_map.eval V H \<a> \<i> src trg E (bicategorical_language.red C src\<^sub>C trg\<^sub>C (bicategorical_language.Dom C src\<^sub>C trg\<^sub>C t)))"   and "V (Coherence.evaluation_map.eval V H \<a> \<i> src trg E (bicategorical_language.red C src\<^sub>C trg\<^sub>C (bicategorical_language.Cod C src\<^sub>C trg\<^sub>C u))) (Coherence.evaluation_map.eval V H \<a> \<i> src trg E u) = V (Coherence.evaluation_map.eval V H \<a> \<i> src trg E (bicategorical_language.Nmlize C u)) (Coherence.evaluation_map.eval V H \<a> \<i> src trg E (bicategorical_language.red C src\<^sub>C trg\<^sub>C (bicategorical_language.Dom C src\<^sub>C trg\<^sub>C u)))" shows "V (Coherence.evaluation_map.eval V H \<a> \<i> src trg E (bicategorical_language.red C src\<^sub>C trg\<^sub>C (bicategorical_language.Cod C src\<^sub>C trg\<^sub>C (t \<^bold>\<star> u)))) (Coherence.evaluation_map.eval V H \<a> \<i> src trg E (t \<^bold>\<star> u)) = V (Coherence.evaluation_map.eval V H \<a> \<i> src trg E (bicategorical_language.Nmlize C (t \<^bold>\<star> u))) (Coherence.evaluation_map.eval V H \<a> \<i> src trg E (bicategorical_language.red C src\<^sub>C trg\<^sub>C (bicategorical_language.Dom C src\<^sub>C trg\<^sub>C (t \<^bold>\<star> u))))"
oracle\<^sub>n_def: fixes \<eta> :: "nat" shows "oracle\<^sub>n \<eta> = family_oracle (\<lambda>_. oracle\<^sub>1 \<eta>)"
assms_7: shows "local.span (set Ls) = local.span X"
FactGroup_iso_set: assumes "h ` carrier G = carrier H" shows "(\<lambda>X. the_elem (h ` X)) \<in> Group.iso (G Mod kernel G H h) H"
strict2: fixes x :: "'a state \<times> 'a state \<Rightarrow> Sequence('b)" shows "x->prepend\<^sub>S\<^sub>e\<^sub>q(invalid) = invalid"
assms_1: shows "f \<longlonglongrightarrow> l"
ins2_ins_2: fixes n :: "nat"   and d :: "nat"   and x :: "'a"   and t :: "'a tree"   and t' :: "'a tree" assumes "ins n d x t = Bal t'" shows "ins2 n d x t = Bal2 t'"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow>\<^sub>C b\<guillemotright>" shows "\<guillemotleft>trg\<^sub>C f : trg\<^sub>C a \<Rightarrow>\<^sub>C trg\<^sub>C b\<guillemotright>"
th0: shows "(1 - fps_X) $ 0 \<noteq> (0::'a)"
filterlim_iff: fixes f :: "'e \<Rightarrow> 'f"   and F2 :: "'f filter"   and F1 :: "'e filter" shows "filterlim f F2 F1 = (\<forall>P. eventually P F2 \<longrightarrow> (\<forall>\<^sub>F x in F1. P (f x)))"
extremum_unique: fixes a :: "'b list" shows "a \<le>s \<epsilon> = (a = \<epsilon>)"
subspace_finrank: fixes M :: "'v set"   and N :: "'v set" assumes "R_finrank M"   and "N \<subseteq> M" shows "R_finrank N"
autoref_Let: fixes Ra :: "('b \<times> 'e) set"   and Rr :: "('d \<times> 'f) set" shows "(Let, Let) \<in> Ra \<rightarrow> (Ra \<rightarrow> Rr) \<rightarrow> Rr"
satL_cong: fixes l :: "('fsym, 'psym) lit"   and \<xi>1 :: "Preliminaries.var \<Rightarrow> 'univ"   and \<xi>2 :: "Preliminaries.var \<Rightarrow> 'univ" assumes "\<And>x. x \<in> varsL l \<Longrightarrow> \<xi>1 x = \<xi>2 x" shows "satL \<xi>1 l = satL \<xi>2 l"
c_tr_futures: fixes step :: "'c \<Rightarrow> 'a \<Rightarrow> 'c"   and out :: "'c \<Rightarrow> 'a \<Rightarrow> 'b"   and s\<^sub>0 :: "'c"   and xs :: "'a list"   and ys :: "'a list" shows "(c_tr step out (foldl step s\<^sub>0 xs) ys, {(x, p). p \<noteq> out (foldl step (foldl step s\<^sub>0 xs) ys) x}) \<in> futures (c_process step out s\<^sub>0) (c_tr step out s\<^sub>0 xs)"
SUP_Sup_eq: fixes S :: "'c set set" shows "(\<Squnion>i\<in>S. (\<lambda>x. x \<in> i)) = (\<lambda>x. x \<in> \<Union> S)"
disc_eq_case_2: fixes option :: "'e option" shows "(option \<noteq> None) = (case option of None \<Rightarrow> False | Some uu_ \<Rightarrow> True)"
I_K: shows "I = C \<circ> K \<circ> C"
wca_on_V_axiom_on: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set" assumes "wca_on A f"   and "f_range_on A f"   and "decisive_on A f" shows "V_axiom_on A f"
co_c2: shows "connected (path_image c2 - {pathstart c2, pathfinish c2})"
inv'_1: shows "gs.lin_indpt_list (map of_int_hom.vec_hom fs')"
ring_simprules_24: fixes R :: "('e, 'f) ring_scheme"   and x :: "'e" assumes "ring R"   and "x \<in> carrier R" shows "\<zero>\<^bsub>R\<^esub> \<otimes>\<^bsub>R\<^esub> x = \<zero>\<^bsub>R\<^esub>"
x_def: shows "x = (\<chi>i. if i = undefined then 1 else 0)"
round_up_le1: fixes x :: "real"   and prec :: "int" assumes "x \<le> 1"   and "0 \<le> prec" shows "round_up prec x \<le> 1"
simps_4: fixes np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and a :: "'d node_action"   and i :: "nat"   and p :: "net_tree"   and s :: "'c net_state"   and v :: "nat"   and va :: "'c"   and vb :: "nat set" shows "net_ip_action np a i p s (NodeS v va vb) = True"
chain_total: fixes A :: "'a set set"   and C :: "'a set set"   and x :: "'a set"   and y :: "'a set" assumes "subset.chain A C"   and "x \<in> C"   and "y \<in> C" shows "(\<subset>)\<^sup>=\<^sup>= x y \<or> (\<subset>)\<^sup>=\<^sup>= y x"
1: shows "\<guillemotleft>\<Lambda> a b c (local.comp (eval b c) (HF'.prod h b)) : a \<rightarrow> exp b c\<guillemotright>"
rem_UnPath_cycle: fixes v :: "'v"   and ps :: "('v \<times> 'w \<times> 'v) list"   and v' :: "'v" assumes "finite E"   and "finite V"   and "is_trail v ps v'"   and "v = v'" shows "num_of_odd_nodes (rem_unPath ps G) = num_of_odd_nodes G"
rsplit_l: fixes f :: "int \<Rightarrow> MIR.num \<Rightarrow> fm"   and a :: "MIR.num" assumes "\<forall>n s. numbound0 s \<longrightarrow> isrlfm (f n s)" shows "isrlfm (rsplit f a)"
statically_complete: fixes B :: "'a"   and N :: "'a set" assumes "B \<in> Bot"   and "saturated N"   and "N \<Turnstile> {B}" shows "\<exists>B'\<in>Bot. B' \<in> N"
simps: fixes a1 :: "('a, 'b) Proc"   and a2 :: "('a, 'b) Proc"   and Enc :: "'a \<Rightarrow> 'b" shows "((a1, a2) \<in> encoding.indRelL Enc) = (\<exists>S. a1 = TargetTerm (Enc S) \<and> a2 = SourceTerm S)"
ty_dom_inclusion: fixes X :: "tyvrs"   and T :: "ty"   and \<Gamma> :: "binding list" assumes "TVarB X T \<in> set \<Gamma>" shows "X \<in> ty_dom \<Gamma>"
simps_2: fixes wtFsym :: "'e \<Rightarrow> bool"   and wtPsym :: "'f \<Rightarrow> bool"   and arOf :: "'e \<Rightarrow> 'g list"   and resOf :: "'e \<Rightarrow> 'g"   and intT :: "'g \<Rightarrow> 'h \<Rightarrow> bool"   and intF :: "'e \<Rightarrow> 'h list \<Rightarrow> 'h"   and intP :: "'f \<Rightarrow> 'h list \<Rightarrow> bool"   and \<xi> :: "Preliminaries.var \<Rightarrow> 'h"   and a :: "('e, 'f) atm" assumes "Struct wtFsym wtPsym arOf resOf intT intF intP" shows "Struct.satL intF intP \<xi> (Neg a) = (\<not> Struct.satA intF intP \<xi> a)"
scaleR_right_mono: fixes a :: "real"   and b :: "real"   and x :: "'a" assumes "a \<le> b"   and "(0::'a) \<le> x" shows "a *\<^sub>R x \<le> b *\<^sub>R x"
measurable_buffon_set': fixes l :: "real"   and d :: "real" shows "buffon_set' l d \<in> sets borel"
mult_strict_left_mono: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less a b"   and "less zero c" shows "less (times c a) (times c b)"
pdevs_apply_sum_pdevs: fixes f :: "'b \<Rightarrow> 'a pdevs"   and X :: "'b set"   and i :: "nat" shows "pdevs_apply (sum_pdevs f X) i = (\<Sum>x\<in>X. pdevs_apply (f x) i)"
deliver_is_valid: fixes history :: "nat \<Rightarrow> ('c \<times> 'd) event list"   and interp :: "'d \<Rightarrow> 'e \<Rightarrow> 'e option"   and initial_state :: "'e"   and valid_msg :: "'e \<Rightarrow> 'c \<times> 'd \<Rightarrow> bool"   and m :: "'c \<times> 'd"   and i :: "nat" assumes "network_with_constrained_ops history interp initial_state valid_msg"   and "Deliver m \<in> set (history i)" shows "\<exists>j pre state. node_histories.prefix_of_node_history history (pre @ [Broadcast m]) j \<and> network_with_ops.apply_operations interp initial_state pre = Some state \<and> valid_msg state m"
None_in_map_option_set: fixes f :: "'b \<Rightarrow> 'a option set"   and x :: "'b option" shows "(None \<in> map_option_set f x) = (None \<in> set_option x \<bind> f \<or> x = None)"
rel_list_all2_Cons: fixes S :: "('e \<times> 'f) set"   and x :: "'e"   and xs :: "'e list"   and y :: "'f"   and ys :: "'f list" shows "list_all2 (fun_of S) (x # xs) (y # ys) = ((x, y) \<in> S \<and> list_all2 (fun_of S) xs ys)"
atMost_def: fixes u :: "'b word" shows "signed.atMost u \<equiv> {x. x \<le>s u}"
p_def: fixes a :: "real"   and v :: "real"   and s0 :: "real"   and t :: "real" shows "movement.p a v s0 t = s0 + v * t + 1 / 2 * a * t\<^sup>2"
seqI: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VoV.arr f"   and "VoV.arr g"   and "VoV.dom g = VoV.cod f" shows "VoV.seq g f"
face_distance_eq_chamber_distance_map: fixes C :: "'a set"   and D :: "'a set"   and z :: "'a set" assumes "domain.chamber C"   and "domain.chamber D"   and "C \<noteq> D"   and "order.greater_eq C z"   and "codomain.face_distance (f ` z) (f ` D) = domain.face_distance z D"   and "domain.face_distance z D = domain.chamber_distance C D" shows "codomain.face_distance (f ` z) (f ` D) = codomain.chamber_distance (f ` C) (f ` D)"
mHom_mHom: fixes N :: "('g, 'b, 'h) Module_scheme"   and g :: "'a \<Rightarrow> 'g" assumes "R module N"   and "g \<in> mHom R (mdl M (carrier M)) N" shows "g \<in> mHom R M N"
atLeastAtMost_singleton': fixes a :: "'a Word.word"   and b :: "'a Word.word" assumes "a = b" shows "signed.atLeastAtMost a b = {a}"
empty_interior_finite: fixes S :: "'b set" assumes "finite S" shows "interior S = {}"
is_empty_empty: fixes t :: "'c set_rbt" assumes "ID ccompare \<noteq> None" shows "RBT_Set2.is_empty t = (t = RBT_Set2.empty)"
valid_prepend_B_iff: fixes xs :: "color list" shows "valid (B # xs) = valid xs"
var_self: fixes v :: "'a" shows "(\<lambda>w. if w = v then Var v else Var w) = Var"
rtrancl_path_last: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and xs :: "'a list"   and y :: "'a" assumes "rtrancl_path R x xs y"   and "xs \<noteq> []" shows "last xs = y"
abs_eq: fixes x :: "('a, 'b) RBT_Impl.rbt" assumes "eq_onp is_rbt x x" shows "lookup (RBT x) = rbt_lookup x"
le_divide_eq_1_pos: fixes a :: "'a"   and b :: "'a" assumes "(0::'a) < a" shows "((1::'a) \<le> b / a) = (a \<le> b)"
is_interval_cbox: fixes a :: "'c"   and b :: "'c" shows "is_interval (cbox a b)"
pmf_eq_iff: fixes M :: "'b pmf"   and N :: "'b pmf" shows "(M = N) = (\<forall>i. pmf M i = pmf N i)"
empty: shows "(if g integrable_on {} then Some (integral {} g) else None) = Some (0::'b)"
represents_scale: fixes B :: "('t \<Rightarrow>\<^sub>0 'c) set"   and r :: "('t \<Rightarrow>\<^sub>0 'c) \<Rightarrow>\<^sub>0 'a \<Rightarrow>\<^sub>0 'c"   and x :: "'t \<Rightarrow>\<^sub>0 'c"   and c :: "'a \<Rightarrow>\<^sub>0 'c" assumes "pmdl.represents B r x" shows "pmdl.represents B (monomial c 0 * r) (c \<odot> x)"
is_shadow_root_kind_def: fixes ptr :: "('m, 'n, 'o, (\<lparr>RShadowRoot.nothing :: unit, mode :: shadow_root_mode, child_nodes :: (_) node_ptr list, \<dots> :: 'p option\<rparr> + 'q) option) RDocument_scheme" shows "is_shadow_root_kind ptr = (cast ptr \<noteq> None)"
C\<pi>\<rho>_def: shows "C\<pi>\<rho> = classical_operator (\<pi> \<circ>\<^sub>m \<rho>)"
dim_col_transfer_rule: fixes A :: "'b mat"   and A' :: "(('b, 'c) Finite_Cartesian_Product.vec, 'd) Finite_Cartesian_Product.vec" assumes "Mod_Type_Connect.HMA_M A A'" shows "dim_col A = CARD('c)"
removeNones_inI: fixes a :: "'a"   and ls :: "'a option list" assumes "Some a \<in> set ls" shows "a \<in> set (removeNones ls)"
assms_1: assumes "\<bar>a\<bar> < \<bar>b\<bar>" shows "thesis"
1: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "{a \<otimes> b} = {b \<otimes> a}"
l_one: fixes x :: "'a \<Rightarrow> complex" assumes "x \<in> carrier C" shows "\<one>\<^bsub>C\<^esub> \<otimes>\<^bsub>C\<^esub> x = x"
prems_1: shows "LeftDerivationLadder (\<alpha> @ \<delta>) D L \<gamma>"
heaps_eq_DagI1: fixes p :: "ref"   and l :: "ref \<Rightarrow> ref"   and r :: "ref \<Rightarrow> ref"   and t :: "dag"   and l' :: "ref \<Rightarrow> ref"   and r' :: "ref \<Rightarrow> ref" assumes "Dag p l r t"   and "\<forall>x\<in>set_of t. l x = l' x \<and> r x = r' x" shows "Dag p l' r' t"
hyps_3: fixes X :: "nat set"   and Y :: "nat set" assumes "finite X"   and "finite Y"   and "X \<inter> Y = {}"   and "S \<inter> (X \<union> Y) = {}"   and "S \<union> X \<union> Y = {0..<m}" shows "card {xs. (\<forall>a\<in>X. xs ! a) \<and> (\<forall>i\<in>Y. \<not> xs ! i) \<and> length xs = m} = 2 ^ (m - card X - card Y)"
decode_plan_length: fixes \<pi> :: "'b strips_operator list list"   and \<Pi> :: "'b strips_problem"   and \<nu> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" assumes "\<pi> = \<Phi>\<inverse> \<Pi> \<nu> t" shows "length \<pi> = t"
prWls_def: fixes g :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> ('varSort \<Rightarrow> 'b \<Rightarrow> 'c) \<Rightarrow> 'c"   and SEM :: "('d, 'e, 'f, 'sort, 'g, 'c, 'h) semDom_scheme" shows "prWls g SEM \<equiv> \<forall>s X val. wls s X \<and> sWlsVal SEM val \<longrightarrow> sWls SEM s (g X val)"
OPT: shows "T\<^sub>p [x, y] qs (OPT2 qs [x, y]) = length v div 2"
False_neq_True: fixes P :: "bool" assumes "False = True" shows "P"
pwX: fixes Y :: "'a set" shows "Y \<in> \<F> \<and> Y \<noteq> X \<longrightarrow> X \<inter> Y \<subseteq> K \<and> Y \<inter> X \<subseteq> K"
lookupDef_induct'': fixes g :: "'g"   and n :: "'node"   and v :: "'var"   and val :: "'var \<times> 'node \<times> Def"   and P :: "'node \<Rightarrow> bool" assumes "lookupDef g n v = val"   and "n \<in> set (\<alpha>n g)"   and "\<lbrakk>v \<in> defs g (defNode val); defKind val = SimpleDef\<rbrakk> \<Longrightarrow> P (defNode val)"   and "\<lbrakk>v \<notin> defs g (defNode val); length (predecessors g (defNode val)) \<noteq> 1; defKind val = PhiDef\<rbrakk> \<Longrightarrow> P (defNode val)"   and "\<And>n m. \<lbrakk>v \<notin> defs g n; predecessors g n = [m]; m \<in> set (\<alpha>n g); lookupDef g n v = lookupDef g m v; P m\<rbrakk> \<Longrightarrow> P n" shows "P n"
range_sbintrunc: shows "range (signed_take_bit 63) = {i. - (2 ^ 63) \<le> i \<and> i < 2 ^ 63}"
append_butlast_last_id: fixes xs :: "'a list" assumes "xs \<noteq> []" shows "append (butlast xs) [last xs] = xs"
A_7: shows "v \<notin> \<lbrace>LE x c\<rbrace>"
simps_2: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "R\<^sup>d = (R, invalid_assn R)"
RETURN_ref_SPECD: fixes c :: "'a"   and R :: "('a \<times> 'b) set"   and \<Phi> :: "'b \<Rightarrow> bool" assumes "RETURN c \<le> \<Down> R (SPEC \<Phi>)" obtains a :: "'b" where "(c, a) \<in> R" and "\<Phi> a"
isIntgV_def: fixes v :: "Value" shows "isIntgV v = (case v of intgV i \<Rightarrow> True | _ \<Rightarrow> False)"
atoms_toplevel_summands: fixes s :: "'b rexp" shows "atoms s = \<Union> (atoms ` toplevel_summands s)"
eval_type_soundE: fixes G :: "prog"   and s0 :: "state"   and t :: "Term.term"   and v :: "vals"   and s1 :: "state"   and L :: "lenv"   and accC :: "qtname"   and T :: "tys"   and A :: "assigned" assumes "G\<turnstile>s0 \<midarrow>t\<succ>\<rightarrow> (v, s1)"   and "s0\<Colon>\<preceq>(G, L)"   and "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile>t\<Colon>T"   and "\<lparr>prg = G, cls = accC, lcl = L\<rparr>\<turnstile> dom (locals (snd s0)) \<guillemotright>t\<guillemotright> A"   and "wf_prog G" obtains "s1\<Colon>\<preceq>(G, L)" and "normal s1 \<Longrightarrow> G,L,snd s1\<turnstile>t\<succ>v\<Colon>\<preceq>T" and "error_free s0 = error_free s1"
inducts: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and t :: "'f" assumes "free_thread_id ts t" obtains "ts t = None"
fun_eq_iff: fixes f :: "'h \<Rightarrow> 'i"   and g :: "'h \<Rightarrow> 'i" shows "(f = g) = (\<forall>x. f x = g x)"
s2r_inter_hom_var: fixes A :: "'b set"   and B :: "'b set" shows "s2r (A \<inter> B) = s2r A ; s2r B"
takeWhile_prefix: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "\<exists>ys. takeWhile P xs @ ys = xs"
cring: shows "cring (R\<lparr>carrier := K\<rparr>)"
intros_3: fixes \<tau> :: "'a basic_type" shows "\<tau>[1] \<sqsubset> \<tau>[?]"
Q'''Chain: shows "Q''' \<Longrightarrow>\<^sub>\<tau> Q'"
\<omega>_limit_set_closed: fixes x :: "'a" shows "closed (rev.\<omega>_limit_set x)"
to_bl_def: shows "to_bl \<equiv> map_fun Word.rep id (bin_to_bl LENGTH('b))"
poly_split_eval: fixes m :: "'d monom"   and p :: "('d monom \<times> 'c) list"   and c :: "'c"   and q :: "('d monom \<times> 'c) list"   and \<alpha> :: "'d \<Rightarrow> 'c" assumes "poly_split m p = (c, q)" shows "eval_poly \<alpha> p = eval_monom \<alpha> m * c + eval_poly \<alpha> q"
cases: fixes x :: "(nat \<times> 'a) list" obtains "x = []"   | n :: "nat" and a :: "'a" and ps :: "(nat \<times> 'a) list" where "x = (Suc n, a__) # ps__"   | a :: "'a" and ps :: "(nat \<times> 'a) list" where "x = (0, a) # ps"
IH: assumes "j \<le> n"   and "i'' \<le> i"   and "j'' \<le> j" shows "fwi m n k i j i' j' \<le> fwi m n k i'' j'' i' j'"
pmf_code: fixes p :: "'b pmf"   and x :: "'b" shows "pmf p x = Mapping.lookup_default 0 (mapping_of_pmf p) x"
cut_ge_all_iff: fixes I :: "'b set"   and t :: "'b" shows "(I \<down>\<ge> t = I) = (\<forall>x\<in>I. t \<le> x)"
snoc_7: shows "set xs \<subseteq> set (ops @ [a])"
cases: fixes x :: "('e, 'f, 'g, 'h) SLin_state_scheme \<times> ('i, 'f, 'g, 'j) SLin_state_scheme" obtains s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme" and s2 :: "('i, 'f, 'g, 'j) SLin_state_scheme" where "x = (s1, s2)"
simps_4: fixes t :: "'a local.term"   and u :: "'a local.term" shows "Src (t \<^bold>\<cdot> u) = Src t"
Write0_Sim: fixes P :: "('c, 'd) action"   and S :: "'e \<Rightarrow> 'd"   and Q :: "('c, 'e) action"   and a :: "'c" assumes "P \<preceq>S Q" shows "a \<rightarrow> P \<preceq>S a \<rightarrow> Q"
minimalize_offending_overapprox_subseteq_input: fixes ff :: "('a \<times> 'a) list"   and keeps :: "('a \<times> 'a) list"   and G :: "'a graph"   and nP :: "'a \<Rightarrow> 'b" shows "set (minimalize_offending_overapprox ff keeps G nP) \<subseteq> set ff \<union> set keeps"
c'nh: shows "isnpolyh c' (Suc n')"
DBM_reset'_complete_empty': fixes n :: "nat"   and v :: "'c \<Rightarrow> nat"   and cs :: "'c list"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and d :: "'d" assumes "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c. v c = k)"   and "\<forall>c. 0 < v c \<and> (\<forall>x y. v x \<le> n \<and> v y \<le> n \<and> v x = v y \<longrightarrow> x = y)"   and "\<forall>c\<in>set cs. v c \<le> n"   and "\<forall>u. \<not> u \<turnstile>\<^bsub>v,n\<^esub> M" shows "\<forall>u. \<not> u \<turnstile>\<^bsub>v,n\<^esub> reset' M n cs v d"
bd_F_Cinfinite: fixes r2 :: "(bd_type_F2 \<times> bd_type_F2) set" shows "Cinfinite (dir_image (bd_F1 +c r2) Abs_bd_type_F)"