hcju/isabelleps huggingface dataset:
abs_eq: fixes xa :: "nat"   and x :: "int" shows "take_bit xa (Word.Word x) = Word.Word (take_bit (min LENGTH('b) xa) x)"
intro: fixes R :: "('h, 'i) ring_scheme"   and M :: "('h, 'j, 'k) module_scheme"   and N :: "'j set" assumes "module R M"   and "N \<subseteq> carrier M"   and "\<And>v w. \<lbrakk>v \<in> N; w \<in> N\<rbrakk> \<Longrightarrow> v \<oplus>\<^bsub>M\<^esub> w \<in> N"   and "\<zero>\<^bsub>M\<^esub> \<in> N"   and "\<And>c v. \<lbrakk>c \<in> carrier R; v \<in> N\<rbrakk> \<Longrightarrow> c \<odot>\<^bsub>M\<^esub> v \<in> N" shows "LinearCombinations.submodule R N M"
space: fixes i :: "'a" assumes "i \<in> I" shows "space (M i) = space (N i)"
ord_isom_segment_segment: fixes D :: "('c, 'd) Order_scheme"   and E :: "('e, 'f) Order_scheme"   and f :: "'c \<Rightarrow> 'e"   and a :: "'c" assumes "Order D"   and "Order E"   and "ord_isom D E f"   and "a \<in> carrier D" shows "ord_isom (Iod D (segment D a)) (Iod E (segment E (f a))) (restrict f (carrier (Iod D (segment D a))))"
ax_Init_Object: fixes G :: "prog"   and A :: "'b triple set"   and P :: "'b assn" assumes "wf_prog G" shows "G,A\<turnstile>{Normal (supd (init_class_obj G Object) .; P\<leftarrow>\<diamondsuit> \<and>. Not \<circ> initd Object)} .Init Object. {P \<and>. initd Object}"
assms: fixes r :: "('a \<times> 'a) set" assumes "r \<in> R" shows "downset_on (Field r) p"
HInit_def: fixes rmhist :: "state \<Rightarrow> PrIds \<Rightarrow> histState"   and p :: "PrIds" shows "HInit rmhist p = (rmhist!p = #histA)"
diff_diff_commute_ennreal: fixes a :: "ennreal"   and b :: "ennreal"   and c :: "ennreal" shows "a - b - c = a - c - b"
dg_1_is_arrD_1: fixes \<aa> :: "V"   and \<ff> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>dg_1 \<aa> \<ff>\<^esub> b" shows "a = \<aa>"
is_le_relE: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and rel :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "ord.is_le_rel less_eq less rel" obtains "rel = (=)"   | "rel = less_eq"   | "rel = less"
split: fixes P :: "'c \<Rightarrow> bool"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'd \<Rightarrow> 'c"   and point :: "'d point" shows "P (case point of Infinity \<Rightarrow> f1 | Point x xa \<Rightarrow> f2 x xa) = ((point = Infinity \<longrightarrow> P f1) \<and> (\<forall>x21 x22. point = Point x21 x22 \<longrightarrow> P (f2 x21 x22)))"
vp_pow_ideal: fixes v :: "'a \<Rightarrow> ant"   and n :: "nat" assumes "valuation K v" shows "ideal (Vr K v) (vp K v \<^bsup>\<diamondsuit>Vr K v n\<^esup>)"
type_subst_identity: fixes X :: "tyvrs"   and T :: "ty"   and U :: "ty" assumes "X \<sharp> T" shows "T[X \<mapsto> U]\<^sub>\<tau> = T"
sublinear_wp_SetPC: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> real"   and p :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>s a. a \<in> supp (P s) \<Longrightarrow> sublinear (wp (p a))"   and "\<And>s. sum (P s) (supp (P s)) \<le> 1"   and "\<And>s a. 0 \<le> P s a"   and "\<And>s. finite (supp (P s))" shows "sublinear (wp (SetPC p P))"
cases: fixes x :: "nat option" obtains "x = None"   | n :: "nat" where "x = Some n"
Goal: fixes P :: "pi"   and Q :: "pi"   and R :: "pi" assumes "P \<leadsto><weakBisim> Q"   and "Q \<leadsto><weakBisim> R"   and "P \<approx> Q" shows "P \<leadsto><weakBisim> R"
S: shows "subset_dfs A (nfa_startnode A) = (bd, ls)"
not_stable_on_cases: fixes ds :: "'d set"   and X :: "'x set" assumes "\<not> stable_on ds X" obtains "\<not> individually_rational_on ds X"   | "\<not> stable_no_blocking_on ds X"
bhv2cnf_mono: fixes n :: "nat"   and n' :: "nat"   and c :: "'id"   and t :: "nat \<Rightarrow> cnf" assumes "n \<le> n'" shows "\<^bsub>c\<^esub>\<up>\<^bsub>t\<^esub>n \<le> \<^bsub>c\<^esub>\<up>\<^bsub>t\<^esub>n'"
order_tendsto_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and "open" :: "'b set \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'b"   and x :: "'b"   and F :: "'d filter" assumes "class.order_topology less_eq less open" shows "filterlim f (topological_space.nhds open x) F = ((\<forall>l. less l x \<longrightarrow> (\<forall>\<^sub>F x in F. less l (f x))) \<and> (\<forall>u. less x u \<longrightarrow> (\<forall>\<^sub>F x in F. less (f x) u)))"
widen_acom_Com: fixes c2 :: "'a Abs_State.st option acom"   and c1 :: "'a Abs_State.st option acom"   and X :: "char list set" assumes "strip c2 = strip c1"   and "c1 \<in> Com X"   and "c2 \<in> Com X" shows "c1 \<nabla>\<^sub>c c2 \<in> Com X"
AR_eq_absolute_extensor: fixes S :: "'a set" shows "AR S = (\<forall>f U T. continuous_on T f \<longrightarrow> f ` T \<subseteq> S \<longrightarrow> closedin (top_of_set U) T \<longrightarrow> (\<exists>g. continuous_on U g \<and> g ` U \<subseteq> S \<and> (\<forall>x\<in>T. g x = f x)))"
vanishesD: fixes X :: "nat \<Rightarrow> rat"   and r :: "rat" assumes "Real.vanishes X"   and "0 < r" shows "\<exists>k. \<forall>n\<ge>k. \<bar>X n\<bar> < r"
is_aligned_add_less_t2n: fixes p :: "'b word"   and n :: "nat"   and d :: "'b word"   and m :: "nat" assumes "is_aligned p n"   and "d < 2 ^ n"   and "n \<le> m"   and "p < 2 ^ m" shows "p + d < 2 ^ m"
list_emb_primrec: shows "list_emb = (\<lambda>uu uua uuaa. rec_list (\<lambda>P. List.null) (\<lambda>y ys ysa P xs. case xs of [] \<Rightarrow> True | x # xs \<Rightarrow> if P x y then ysa P xs else ysa P (x # xs)) uuaa uu uua)"
markedElementsTo_def: fixes e :: "'b"   and t :: "('b \<times> bool) list" shows "markedElementsTo e t = markedElements (prefixToElement e t)"
comp_right_subdist_inf: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<sqinter> z) \<le> x \<sqinter> y \<sqinter> (x \<sqinter> z)"
boolean_axioms_hold: fixes A :: "fmla"   and e :: "(name, nat) finfun" assumes "A \<in> boolean_axioms" shows "eval_fmla e A"
filter_def: fixes P :: "'d \<Rightarrow> bool"   and A :: "'d set" shows "Set.filter P A = {a \<in> A. P a}"
extensions_def: shows "extensions \<equiv> {L. field L \<and> indexed_const \<in> ring_hom R L \<and> Ball (carrier L) carrier_coeff \<and> (\<forall>\<P>\<in>carrier L. \<forall>P\<in>carrier (poly_ring R). \<forall>i. \<not> index_free \<P> (P, i) \<longrightarrow> \<X>\<^bsub>(P, i)\<^esub> \<in> carrier L \<and> ring.eval L (\<sigma> P) \<X>\<^bsub>(P, i)\<^esub> = \<zero>\<^bsub>L\<^esub>)}"
sub_sup_demorgan: fixes x :: "'a"   and y :: "'a" shows "- (- x \<squnion> - y) = - - x \<sqinter> - - y"
applied_rule_rev_def: fixes C :: "'b \<Rightarrow> 'c \<Rightarrow> 'd option"   and a :: "'c"   and x :: "'b list" shows "applied_rule_rev C a x = applied_rule C a (rev x)"
prems_1: shows "\<forall>s. P s \<longrightarrow> pre ({I} WHILE b DO C) Q s"
mono_Omega_fun: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "mono f" shows "mono (Omega_fun f g)"
translation_bijective: fixes a :: "'a" assumes "a \<in> G" shows "bij_betw (a)\<^sub>L G G"
poly_mult_eq_zero_disj: fixes p :: "'a list"   and q :: "'a list"   and x :: "'a" shows "(poly (p *** q) x = (0::'a)) = (poly p x = (0::'a) \<or> poly q x = (0::'a))"
rel_interior_affine_hull: fixes S :: "'b set" shows "rel_interior (affine hull S) = affine hull S"
hd_filter_in_list: fixes P :: "'e \<Rightarrow> bool"   and xs :: "'e list" assumes "filter P xs \<noteq> []" shows "hd (filter P xs) \<in> set xs"
satisfies_properties_sys2: shows "satisfies_properties p qs2 subsets2 signs2 matrix2"
gr0_conv_Suc: fixes n :: "nat" shows "(0 < n) = (\<exists>m. n = Suc m)"
assms_2: assumes "nofail a"   and "nofail b" shows "b \<le>\<^sub>n m"
monad_reader_def: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and ask :: "('c \<Rightarrow> 'b) \<Rightarrow> 'b" shows "monad_reader return bind ask \<equiv> monad return bind \<and> monad_reader_axioms bind ask"
some_trg_some_src: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "some_trg (some_src \<mu>) = some_src \<mu>"
zero_less_double_add_iff_zero_less_single_add: 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 a :: "'a" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less zero (plus a a) = less zero a"
ce_rel_to_set_inj: shows "inj ce_rel_to_set"
inR1: fixes \<Gamma> :: "'a formula multiset"   and G :: "'a formula"   and H :: "'a formula"   and \<Delta> :: "'a formula multiset" assumes "\<Gamma> \<Rightarrow> G, H, \<Delta>" shows "\<Gamma> \<Rightarrow> H, G, \<Delta>"
Throw: fixes \<Gamma> :: "'e \<Rightarrow> ('f, 'e, 'g) com option"   and \<Theta> :: "('f set \<times> 'e \<times> 'f set \<times> 'f set) set"   and F :: "'g set"   and A :: "'f set"   and Q :: "'f set" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>A Throw Q,A"
isUIPApplyExplainUIP: fixes state :: "State"   and F0 :: "Literal list list" assumes "applyExplainUIP_dom state"   and "InvariantUniq (getM state)"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantCEntailed (getConflictFlag state) F0 (getC state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCnCharacterization (getCn state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))"   and "InvariantEquivalentZL (getF state) (getM state) F0"   and "getConflictFlag state"   and "0 < currentLevel (getM state)" shows "let state' = applyExplainUIP state in isUIP (opposite (getCl state')) (getC state') (getM state')"
mset_set_insert_eq: fixes E :: "'c set"   and x :: "'c" assumes "finite E" shows "mset_set (E \<union> {x}) \<subseteq># mset_set E + {#x#}"
desargues_config_2D_rkABCP: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>" shows "rk {A, B, C, P} = 3"
wf_prec_FL: shows "minimal_element (\<sqsubset>) UNIV"
splitFace_holds_minGraphProps': fixes g' :: "graph"   and v :: "nat"   and a :: "nat"   and f' :: "face"   and vs :: "nat list" assumes "pre_splitFace g' v a f' vs"   and "minGraphProps' g'" shows "minGraphProps' (snd (snd (splitFace g' v a f' vs)))"
simps: fixes a1 :: "('fenv_std, 'c, 'fun) prog"   and a2 :: "('fenv_std, 'c, ('fun, 'd) frame) state" shows "Sstd.load a1 a2 = (\<exists>F main fd H. a1 = Prog F H main \<and> a2 = State F H [Frame main 0 []] \<and> Fstd_get F main = Some fd \<and> arity fd = 0)"
inf_left_commute: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
runit_char_1: fixes f :: "'a" assumes "S.ide f" shows "\<guillemotleft>\<r>[f] : R f \<Rightarrow>\<^sub>S f\<guillemotright>"
genby_uminus_add_closed: fixes g :: "'b"   and S :: "'b set"   and h :: "'b" assumes "g \<in> \<langle>S\<rangle>"   and "h \<in> \<langle>S\<rangle>" shows "- g + h \<in> \<langle>S\<rangle>"
\<Phi>F: fixes n :: "nat" shows "\<Phi> (F n) (Eps (\<Phi> (F n)))"
simps_7: fixes x3 :: "'c"   and x1 :: "'a" shows "In3 x3 \<noteq> In1 x1"
dvd_right_imp_subdegree_le: fixes g :: "'a fps"   and f :: "'a fps" assumes "\<exists>k. g = k * f"   and "g \<noteq> 0" shows "subdegree f \<le> subdegree g"
p_comp_pp: fixes x :: "'a"   and y :: "'a" shows "- (x * - - y) = - (x * y)"
CFG_SSA_defs_3: fixes g :: "'g"   and n :: "'node" shows "phiUses g n \<equiv> \<Union>n'\<in>set (successors g n). \<Union>v'\<in>phiDefs g n'. snd ` Set.filter (\<lambda>(n'', v). n'' = n) (set (zip (predecessors g n') (the (phis g (n', v')))))"
lformula_nFOR: fixes \<phi>s :: "('a, 'k) aformula list" shows "lformula (nFOR \<phi>s) = Ball (set \<phi>s) lformula"
neg_lincomb: fixes rs :: "'f list"   and ms :: "'v list" assumes "set rs \<subseteq> UNIV"   and "set ms \<subseteq> V" shows "- (rs \<bullet>\<cdot> ms) = map uminus rs \<bullet>\<cdot> ms"
rqs_cost_exp_Suc': fixes n :: "nat" shows "rqs_cost_exp (Suc n) = real n + 2 / real (Suc n) * sum rqs_cost_exp {..n}"
REFPHI: fixes it' :: "'Sa set"   and it :: "'S set"   and \<sigma>' :: "'b"   and \<sigma> :: "'a" assumes "it' = \<alpha> ` it"   and "it \<subseteq> S"   and "it' \<subseteq> S'"   and "\<Phi>' it' \<sigma>'"   and "\<Phi>'' it \<sigma> it' \<sigma>'"   and "(\<sigma>, \<sigma>') \<in> R" shows "\<Phi> it \<sigma>"
vec_of_list_Cons: fixes a :: "'a"   and as :: "'a list" shows "vec_of_list (a # as) = vCons a (vec_of_list as)"
Theorem_9_stable_Xd1_Xd2: shows "Theorem_9.stable_on UNIV {X3.Xd1, X3.Xd2}"
comp_assoc: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and h :: "'c"   and g :: "'c"   and f :: "'c" assumes "category C" shows "C (C h g) f = C h (C g f)"
f_tendsto: fixes x :: "real" shows "f \<midarrow>x\<rightarrow> f x"
initial_def: fixes cfg :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'c \<times> 'd \<times> nat set \<times> ('e \<Rightarrow> 'f option)) list \<times> 'g \<times> (nat \<Rightarrow> bool option)" shows "initial cfg \<equiv> let (ts, m, \<S>) = cfg in unowned_shared \<S> ts \<and> (\<forall>i<length ts. let (p, is, \<theta>, x, \<D>, \<O>, \<R>) = ts ! i in \<R> = Map.empty)"
cont_comp: fixes luba :: "'f set \<Rightarrow> 'f"   and orda :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and lubb :: "'g set \<Rightarrow> 'g"   and ordb :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and g :: "'f \<Rightarrow> 'g"   and lubc :: "'h set \<Rightarrow> 'h"   and ordc :: "'h \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'g \<Rightarrow> 'h" assumes "mcont luba orda lubb ordb g"   and "cont lubb ordb lubc ordc f" shows "cont luba orda lubc ordc (f \<circ> g)"
addrs_def: fixes P :: "'b ref set" shows "addrs P = {a. Ref a \<in> P}"
subset_eq: fixes A :: "'a set"   and B :: "'a set" shows "(A \<subseteq> B) = (\<forall>x\<in>A. s2p B x)"
insert: fixes g :: "'c \<Rightarrow> 'a"   and x :: "'c"   and A :: "'c multiset" shows "\<Sum>\<^sub># (image_mset g (add_mset x A)) = g x + \<Sum>\<^sub># (image_mset g A)"
Units_inv_comm: fixes x :: "'a mat"   and y :: "'a mat" assumes "x \<otimes>\<^bsub>ring_mat TYPE('a) n undefined\<^esub> y = \<one>\<^bsub>ring_mat TYPE('a) n undefined\<^esub>"   and "x \<in> Units (ring_mat TYPE('a) n undefined)"   and "y \<in> Units (ring_mat TYPE('a) n undefined)" shows "y \<otimes>\<^bsub>ring_mat TYPE('a) n undefined\<^esub> x = \<one>\<^bsub>ring_mat TYPE('a) n undefined\<^esub>"
canonical_retraction_comp_simplex_im_subset: fixes A :: "'a set set"   and B :: "'a set set"   and C :: "'a set"   and D :: "'a set" assumes "A \<in> \<A>"   and "B \<in> \<A>"   and "chamber C"   and "chamber D"   and "C \<in> A"   and "D \<in> B" shows "order.greater_eq A ((canonical_retraction A C \<circ> canonical_retraction B D) \<turnstile> X)"
lnull_imp_lfinite: fixes xs :: "'c llist" assumes "lnull xs" shows "lfinite xs"
method_exit_cases: fixes n :: "'b LDCFG_node" assumes "local.CFGExit.method_exit n" obtains "n = NewExit"   | a :: "'b LDCFG_node \<times> ('c, 'd, 'e, 'f) edge_kind \<times> 'b LDCFG_node" and Q :: "('c \<Rightarrow> 'd option) \<times> 'e \<Rightarrow> bool" and f :: "('c \<Rightarrow> 'd option) \<Rightarrow> ('c \<Rightarrow> 'd option) \<Rightarrow> 'c \<Rightarrow> 'd option" and p :: "'f" where "n = src a" and "lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit a" and "knd a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"
simps_2: shows "pass\<^sub>2 \<langle>\<rangle> = \<langle>\<rangle>"
not_waiting_iff: fixes w :: "'a wait_set_status option" shows "(\<not> waiting w) = (w = None \<or> (\<exists>w'. w = \<lfloor>PostWS w'\<rfloor>))"
same_bdt_var: fixes lt1 :: "dag"   and p1 :: "ref"   and rt1 :: "dag"   and var :: "ref \<Rightarrow> nat"   and bdt1 :: "bdt"   and lt2 :: "dag"   and p2 :: "ref"   and rt2 :: "dag" assumes "bdt (Node lt1 p1 rt1) var = Some bdt1"   and "bdt (Node lt2 p2 rt2) var = Some bdt1" shows "var p1 = var p2"
zet_image_mem: fixes A :: "'a set"   and g :: "'a \<Rightarrow> 'b" assumes "A \<in> zet" shows "g ` A \<in> zet"
sorted_on_sort_on: fixes r :: "('a \<times> 'a) set"   and xs :: "'a list" assumes "Linear_order r"   and "set xs \<subseteq> Field r" shows "sorted_on r (sort_on r xs)"
i_contract_def: fixes f :: "'d \<Rightarrow> 'e"   and R :: "('d, 'f) Ring_scheme"   and R' :: "('e, 'g) Ring_scheme"   and J :: "'e set" shows "i_contract f R R' J = invim f (carrier R) J"
perm_rev: fixes xs :: "'b list" shows "rev xs <~~> xs"
intro: fixes G :: "('c, 'd) pre_digraph" assumes "fin_digraph G"   and "sym_digraph G" shows "pseudo_graph G"
l_set_child_nodes_get_tag_name_def: fixes type_wf :: "(_) heap \<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 get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_child_nodes_get_tag_name type_wf set_child_nodes set_child_nodes_locs get_tag_name get_tag_name_locs \<equiv> l_set_child_nodes type_wf set_child_nodes set_child_nodes_locs \<and> l_get_tag_name type_wf get_tag_name get_tag_name_locs \<and> l_set_child_nodes_get_tag_name_axioms set_child_nodes_locs get_tag_name_locs"
assms_3: fixes a :: "int" assumes "b < a" shows "f a = g a"
freshTransition_4: fixes P :: "pi"   and P' :: "pi"   and c :: "name" assumes "P \<Longrightarrow>\<tau> \<prec> P'"   and "c \<sharp> P" shows "c \<sharp> P'"
sorted_list_of_set_sort_remdups: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and xs :: "'k list" assumes "class.linorder less_eq less" shows "linorder.sorted_list_of_set less_eq (set xs) = linorder.sort_key less_eq (\<lambda>x. x) (remdups xs)"
coeffs_map_poly: fixes f :: "'c \<Rightarrow> 'b"   and p :: "'c poly" shows "coeffs (map_poly f p) = strip_while ((=) (0::'b)) (map f (coeffs p))"
freshBinp_def: fixes xs :: "'a"   and x :: "'b"   and binp :: "'c \<Rightarrow> ('d, 'e, 'a, 'b, 'f) qAbs set option" shows "freshBinp xs x binp \<equiv> liftAll (freshAbs xs x) binp"
dim_eq_card: fixes B :: "'b set"   and V :: "'b set" assumes "local.span B = local.span V"   and "local.independent B" shows "local.dim V = card B"
poly_condE: fixes p :: "int poly" assumes "poly_cond p" obtains "irreducible p" and "0 < Polynomial.lead_coeff p" and "root_free p" and "square_free p" and "p \<noteq> 0"
ennreal_spmf_lub_spmf: fixes Y :: "'c spmf set"   and x :: "'c" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y"   and "Y \<noteq> {}" shows "ennreal (spmf (lub_spmf Y) x) = (\<Squnion>p\<in>Y. ennreal (spmf p x))"
inj_var_ran_subst_is_invertible: fixes \<delta> :: "'c \<Rightarrow> ('d, 'c) Term.term"   and t :: "('d, 'c) Term.term" assumes "inj_on \<delta> (fv t)"   and "\<delta> ` fv t \<subseteq> range Var" shows "t = t \<cdot> \<delta> \<circ>\<^sub>s subst_var_inv \<delta> (fv t)"
balance_right_rbt_less: fixes k :: "'a"   and a :: "('a, 'c) rbt"   and b :: "('a, 'c) rbt"   and x :: "'a"   and t :: "'c" assumes "a |\<guillemotleft> k"   and "b |\<guillemotleft> k"   and "x < k" shows "balance_right a x t b |\<guillemotleft> k"
crel_vs_iterate_and_compute: fixes lookup :: "'a \<Rightarrow> ('b, 'c option) state"   and update :: "'a \<Rightarrow> 'c \<Rightarrow> ('b, unit) state"   and P :: "'b \<Rightarrow> bool"   and dp :: "'a \<Rightarrow> 'c"   and R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and g :: "'d \<Rightarrow> 'e"   and f :: "'d \<Rightarrow> ('b, 'f) state"   and x :: "'d"   and xs :: "'d list" assumes "dp_consistency lookup update P"   and "rel_fun (=) (dp_consistency.crel_vs lookup P dp R) g f" shows "dp_consistency.crel_vs lookup P dp R (g x) (iterate_state f xs \<bind> (\<lambda>_. f x))"
rat_number_expand_1: shows "0 = Fract 0 1"
simps_9: fixes f1 :: "'b"   and f2 :: "num \<Rightarrow> 'b"   and f3 :: "num \<Rightarrow> 'b" shows "(case num.One of num.One \<Rightarrow> f1 | num.Bit0 x \<Rightarrow> f2 x | num.Bit1 x \<Rightarrow> f3 x) = f1"
unwinding_in_Ints: fixes z :: "complex" shows "(z - Ln (exp z)) / (complex_of_real (2 * pi) * \<i>) \<in> \<int>"
ipresCons_def: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'b"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'c"   and MOD :: "('index, 'bindex, 'varSort, 'd, 'opSym, 'var, 'b, 'c, 'e) model_scheme" shows "ipresCons h hA MOD \<equiv> ipresVar h MOD \<and> ipresAbs h hA MOD \<and> ipresOp h hA MOD"
C'_p: shows "instance_of\<^sub>l\<^sub>s C' (C \<cdot>\<^sub>l\<^sub>s \<tau>) \<and> falsifies\<^sub>g G C'"
that: fixes z :: "'a list" assumes "u \<cdot> z = v"   and "z \<noteq> \<epsilon>" shows "thesis"
trivial_imp_cyclic_group: fixes G :: "('e, 'f) monoid_scheme" assumes "trivial_group G" shows "cyclic_group G"
lex_pm_strict_plus_left: fixes s :: "'c \<Rightarrow>\<^sub>0 'd"   and t :: "'c \<Rightarrow>\<^sub>0 'd"   and u :: "'c \<Rightarrow>\<^sub>0 'd" assumes "lex_pm_strict s t"   and "\<And>x y. \<lbrakk>x \<in> keys t; y \<in> keys u\<rbrakk> \<Longrightarrow> x < y" shows "lex_pm_strict (u + s) t"
ltakeWhile_conv_fixp: fixes P :: "'c \<Rightarrow> bool" shows "ltakeWhile P \<equiv> llist.fixp_fun (\<lambda>ltakeWhile xs. case xs of LNil \<Rightarrow> LNil | LCons x xs \<Rightarrow> if P x then LCons x (ltakeWhile xs) else LNil)"
FAccRedsVal: fixes 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)"   and a :: "nat"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and 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 C :: "char list"   and fs :: "char list \<times> char list \<Rightarrow> val option"   and F :: "char list"   and D :: "char list"   and v :: "val" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>addr a,s'\<rangle>"   and "hp s' a = \<lfloor>(C, fs)\<rfloor>"   and "fs (F, D) = \<lfloor>v\<rfloor>" shows "P \<turnstile> \<langle>e\<bullet>F{D},s\<rangle> \<rightarrow>* \<langle>Val v,s'\<rangle>"
OV_inv4_def: shows "OV_inv4 = {s. \<forall>r p q v w. obsv_state.obs s r p = Some v \<and> obsv_state.obs s r q = Some w \<longrightarrow> w = v}"
IH2: fixes s :: "'a net_state"   and a :: "'b node_action"   and s' :: "'a net_state" assumes "(s, a, s') \<in> automaton.trans (pnet np n2)" shows "\<forall>i d di. a \<noteq> i:newpkt(d, di)"
\<eta>o_def: fixes y :: "'d" shows "\<eta>o y = (SOME u. initial_arrow_to_functor (\<cdot>\<^sub>C) (\<cdot>\<^sub>D) G y (Fo y) u)"
f_1_lemma_1: fixes x :: "int" assumes "x \<in> local.S1" shows "local.P x (local.f_1 x)"
mat_transpose_def: fixes f :: "'c \<Rightarrow> 'b \<Rightarrow> 'd" shows "f\<^sup>\<dagger> \<equiv> \<lambda>i j. f j i"
length_aforms_err_to_aforms: fixes i :: "nat"   and XS :: "((real \<times> real pdevs) \<times> real) list" shows "length (aforms_err_to_aforms i XS) = length XS"
bounded_gen_check_axioms: shows "bounded_gen_check (cond2 a b) (Max (set a))"
t: shows "\<bar>t\<bar> \<noteq> \<infinity>"
cos_integer_2pi: fixes n :: "real" assumes "n \<in> \<int>" shows "cos (2 * pi * n) = 1"
assms_1: shows "Lfilled n lholed (vs @ es) LI"
a_l': fixes i :: "nat" assumes "i < length fs" shows "a i i = gso' i"
fmeasurableI2: fixes A :: "'c set"   and M :: "'c measure"   and B :: "'c set" assumes "A \<in> fmeasurable M"   and "B \<subseteq> A"   and "B \<in> sets M" shows "B \<in> fmeasurable M"
E: shows "\<forall>u. u \<in> range D \<and> (\<forall>v\<in>U. (v, u) \<notin> I) \<longrightarrow> (xs, xs') \<in> R u"