hcju/isabelleps huggingface dataset:
vp_split_3: fixes n :: "'node"   and as :: "'edge list"   and a :: "'edge"   and as' :: "'edge list"   and n' :: "'node" assumes "n -as @ a # as'\<rightarrow>\<^sub>\<surd>* n'" shows "targetnode a -as'\<rightarrow>\<^sub>\<surd>* n'"
times_bit_def: fixes a :: "bit"   and b :: "bit" shows "a * b = Abs_bit (Rep_bit a \<and> Rep_bit b)"
assms_3: shows "F \<in> local.set a \<rightarrow> local.set b"
asubg_test: fixes H :: "'a set" assumes "H \<subseteq> carrier A"   and "H \<noteq> {}"   and "\<forall>a\<in>H. \<forall>b\<in>H. a \<plusminus> -\<^sub>a b \<in> H" shows "A +> H"
assms_3: shows "\<forall>x\<in>set xs. key x \<in> {mi..ma}"
simps: fixes a :: "('c, 'd) ipurge_rec"   and X :: "('c, 'd) ipurge_rec" shows "(a \<in> ipurge_tr_rev_t_set X) = (a = X \<or> (\<exists>I D u x xs ys. a = \<lparr>Pol = I, Map = D, Dom = u, In = xs, Out = ys @ [x]\<rparr> \<and> \<lparr>Pol = I, Map = D, Dom = u, In = x # xs, Out = ys\<rparr> \<in> ipurge_tr_rev_t_set X \<and> D x \<in> sources I D u (x # xs)) \<or> (\<exists>I D u x xs ys. a = \<lparr>Pol = I, Map = D, Dom = u, In = ipurge_tr I D (D x) xs, Out = ys\<rparr> \<and> \<lparr>Pol = I, Map = D, Dom = u, In = x # xs, Out = ys\<rparr> \<in> ipurge_tr_rev_t_set X \<and> D x \<notin> sources I D u (x # xs)))"
FrR: shows "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"
interp_ins_maybe_grow3: fixes xs :: "('b \<times> 'b option) list"   and ys :: "('b \<times> 'b option) list" assumes "insert_ops (xs @ ys)" shows "\<exists>A\<subseteq>set (map fst ys). set (interp_ins (xs @ ys)) = set (interp_ins xs) \<union> A"
rw3: shows "IArray (map (\<lambda>x. IArray.of_fun (\<lambda>i. A $ mod_type_class.from_nat x $ mod_type_class.from_nat i) CARD('c)) [0..<CARD('b)]) !! mod_type_class.to_nat a !! mod_type_class.to_nat j = A $ a $ j"
valid_abstraction_def: fixes A :: "('a \<times> ('b, 'd) cconstraint \<times> ('b set \<times> 'a) pmf) set \<times> ('a \<Rightarrow> ('b, 'd) cconstraint)"   and X :: "'b set"   and k :: "'b \<Rightarrow> 'd" shows "PTA.valid_abstraction A X k \<equiv> (\<forall>(x, m)\<in>PTA.clkp_set A. m \<le> k x \<and> x \<in> X \<and> m \<in> \<nat>) \<and> PTA.collect_clkvt A \<subseteq> X \<and> finite X"
munionM_assoc: fixes x :: "'m"   and y :: "'m"   and z :: "'m" shows "munionM (munionM x y) z = munionM x (munionM y z)"
openin_INT2: fixes I :: "'c set"   and T :: "'d topology"   and U :: "'c \<Rightarrow> 'd set" assumes "finite I"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)" shows "openin T (\<Inter> (U ` I))"
mcont_ltake1: fixes xs :: "'c llist" shows "mcont Sup (\<le>) lSup (\<sqsubseteq>) (\<lambda>n. ltake n xs)"
inverse_positive_iff_positive: fixes a :: "'a" shows "((0::'a) < inverse a) = ((0::'a) < a)"
A: fixes a1 :: "'a"   and b1 :: "'a"   and a2 :: "'b"   and b2 :: "'b"   and c1 :: "'a"   and c2 :: "'b" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "dist a1 c1 = dist a2 c2" shows "angle a1 b1 c1 = angle a2 b2 c2"
entails_strengthen_post: fixes P :: "'b \<Rightarrow> real"   and t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real"   and Q :: "'b \<Rightarrow> real"   and R :: "'b \<Rightarrow> real" assumes "P \<tturnstile> t Q"   and "healthy t"   and "sound R"   and "Q \<tturnstile> R"   and "sound Q" shows "P \<tturnstile> t R"
special_nsg_G: fixes G :: "('e, 'f) Group_scheme" assumes "Group G" shows "G \<triangleright> carrier G"
instantiation_: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and \<psi> :: "\<o>" assumes "[\<^bold>\<exists>\<alpha>. \<phi> \<alpha> in v]"   and "\<And>\<alpha>. [\<phi> \<alpha> in v] \<Longrightarrow> [\<psi> in v]" shows "[\<psi> in v]"
SourcesS8_L2: shows "Sources level2 sS8 = {sS7}"
simps_3: fixes l :: "'a list"   and P :: "pol"   and x :: "nat"   and Q :: "pol" shows "Ipol l (PX P x Q) = Ipol l P \<otimes> head l [^] x \<oplus> Ipol (drop 1 l) Q"
q_Eq_def: fixes x :: "hf"   and y :: "hf" shows "q_Eq x y \<equiv> \<langle>htuple 2, x, y\<rangle>"
shift_bounds_Suc_ivl: fixes g :: "nat \<Rightarrow> 'b"   and m :: "nat"   and n :: "nat" shows "sum g {Suc m..<Suc n} = (\<Sum>i = m..<n. g (Suc i))"
LR: assumes "UCAST('a \<rightarrow> 'b) f < b" shows "unat f < unat b"
simps_1: fixes x :: "'e"   and p :: "'f \<Rightarrow> 'e"   and D :: "'f \<Rightarrow> 'g"   and E :: "'h \<Rightarrow> 'g"   and q :: "'h \<Rightarrow> 'e" assumes "x \<in> range p" shows "con_comp_map D E p q x = Some (D (inv p x))"
sample_put: fixes p :: "'p pmf"   and s :: "'s"   and m :: "'p \<Rightarrow> 'm" shows "sample p (\<lambda>x. put s (m x)) = put s (sample p m)"
CHORun_def: fixes A :: "('a, 'b, 'c, 'd) CHOAlgorithm_scheme"   and rho :: "nat \<Rightarrow> 'a \<Rightarrow> 'b"   and HOs :: "nat \<Rightarrow> 'a \<Rightarrow> 'a set"   and coords :: "nat \<Rightarrow> 'a \<Rightarrow> 'a" shows "CHORun A rho HOs coords \<equiv> CSHORun A rho HOs HOs coords"
T2: fixes w :: "'a"   and w' :: "'a"   and \<theta> :: "'a"   and u :: "'a"   and \<theta>' :: "'a"   and \<beta> :: "'a" assumes "ide w"   and "ide w'"   and "\<guillemotleft>\<theta> : s\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<theta>' : s\<^sub>0 \<star> w' \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<beta> : s\<^sub>1 \<star> w \<Rightarrow> s\<^sub>1 \<star> w'\<guillemotright>"   and "(s \<star> \<theta>) \<cdot> \<a>[s, s\<^sub>0, w] \<cdot> (\<sigma> \<star> w) = ((s \<star> \<theta>') \<cdot> \<a>[s, s\<^sub>0, w'] \<cdot> (\<sigma> \<star> w')) \<cdot> \<beta>" shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w \<Rightarrow> w'\<guillemotright> \<and> \<beta> = s\<^sub>1 \<star> \<gamma> \<and> \<theta> = \<theta>' \<cdot> (s\<^sub>0 \<star> \<gamma>)"
intros: fixes "is" :: "int list"   and i :: "int" shows "lformula0 (presb.Eq is i 0)"
bmod_auxl: fixes g :: "'d \<Rightarrow> 'c"   and z :: "'d"   and y :: "'e"   and F :: "('e \<Rightarrow> 'c) set" shows "{x \<cdot> g z |x. \<exists>f. x = f y \<and> f \<in> F} = {f y \<cdot> g z |f. f \<in> F}"
P'_def: shows "P' = assign_post P x e |` {v. stable \<S> v}"
hyps: shows "diag_block_mat (As @ Bs) = four_block_mat (diag_block_mat As) (0\<^sub>m (dim_row (diag_block_mat As)) (dim_col (diag_block_mat Bs))) (0\<^sub>m (dim_row (diag_block_mat Bs)) (dim_col (diag_block_mat As))) (diag_block_mat Bs)"
signed_of_int: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and uminus :: "'k \<Rightarrow> 'k"   and n :: "int" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.signed one plus zero minus (word_of_int n) = ring_1.of_int one plus zero minus (signed_take_bit (LENGTH('l) - Suc 0) n)"
pseudo_proj_True_imp_False: fixes n :: "nat"   and w :: "bool stream"   and x :: "bool stream" assumes "pseudo_proj_True n w = pseudo_proj_True n x" shows "pseudo_proj_False n w = pseudo_proj_False n x"
G_eq: fixes cf :: "('test, 'atom, 'choice) cmd \<times> 'state" shows "G cf = (if proper (fst cf) then {cont_eff cf i |i. i < brn (fst cf) \<and> 0 < wt (fst cf) (snd cf) i} else {cf})"
uminus_eqvt: fixes p :: "perm"   and q :: "perm" shows "p \<bullet> - q = - (p \<bullet> q)"
assms: shows "matrix_idempotent_semiring.coreflexive g"
ORRIGHT2_elim: fixes a :: "coname"   and M :: "trm"   and B :: "ty"   and C :: "ty" assumes "<a>:M \<in> ORRIGHT2 (B OR C) (\<parallel><C>\<parallel>)" obtains a' :: "coname" and M' :: "trm" where "M = OrR2 <a'>.M' a" and "fic (OrR2 <a'>.M' a) a" and "<a'>:M' \<in> \<parallel><C>\<parallel>"
arr_hcomp: fixes \<mu> :: "('a E.term, 'a) arr"   and \<nu> :: "('a E.term, 'a) arr" assumes "arr \<mu>"   and "arr \<nu>"   and "src \<mu> = trg \<nu>" shows "arr (\<mu> \<star> \<nu>)"
delete_def: fixes x :: "'c"   and t :: "('c \<times> 'd) tree23" shows "Tree23_Map.delete x t = treeD (Tree23_Map.del x t)"
composite_span_axioms_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b span_data"   and T :: "'b span_data" shows "composite_span_axioms C S T \<equiv> partial_magma.cod C (Leg0 S) = partial_magma.cod C (Leg1 T)"
maxsimpchain_obtain_pmaxsimpchain: fixes x :: "'a set"   and y :: "'a set"   and xs :: "'a set list" assumes "x \<noteq> y"   and "gallery (x # xs @ [y])" shows "\<exists>ys. order.greater_eq (set xs) (set ys) \<and> order.greater_eq (length xs) (length ys) \<and> pgallery (x # ys @ [y])"
face_face_op_def: fixes g :: "graph" shows "face_face_op g \<equiv> |faces g| \<noteq> 2 \<longrightarrow> (\<forall>f\<in>\<F> g. \<forall>f'\<in>\<F> g. f \<noteq> f' \<longrightarrow> \<E> f \<noteq> (\<E> f')\<inverse>)"
rel_S_witness'1: fixes L1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and L2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and Co1 :: "'e \<Rightarrow> 'o \<Rightarrow> bool"   and Co1' :: "'o \<Rightarrow> 'g \<Rightarrow> bool"   and Co2 :: "'h \<Rightarrow> 'p \<Rightarrow> bool"   and Co2' :: "'p \<Rightarrow> 'i \<Rightarrow> bool"   and Contra1 :: "'j \<Rightarrow> 'q \<Rightarrow> bool"   and Contra1' :: "'q \<Rightarrow> 'k \<Rightarrow> bool"   and Contra2 :: "'l \<Rightarrow> 'r \<Rightarrow> bool"   and Contra2' :: "'r \<Rightarrow> 'm \<Rightarrow> bool"   and x :: "('a, 'c, 'e, 'h, 'j, 'l, 'n) S"   and y :: "('b, 'd, 'g, 'i, 'k, 'm, 'n) S"   and tytok :: "('a \<times> ('a \<times> 'b) \<times> 'b \<times> 'c \<times> ('c \<times> 'd) \<times> 'd \<times> 'n) itself" assumes "rel_S L1 L2 (Co1 OO Co1') (Co2 OO Co2') (Contra1 OO Contra1') (Contra2 OO Contra2') x y"   and "rel_S_neg_distr_cond' Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' tytok" shows "rel_S (\<lambda>x (x', y). x' = x \<and> L1 x y) (\<lambda>x (x', y). x' = x \<and> L2 x y) Co1 Co2 Contra1 Contra2 x (rel_S_witness' L1 L2 Co1 Co1' Co2 Co2' Contra1 Contra1' Contra2 Contra2' (x, y))"
weak_ranking_aux_Union: fixes carrier :: "'b set"   and le :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "finite_total_preorder_on carrier le"   and "A \<subseteq> carrier" shows "\<Union> (set (finite_total_preorder_on.weak_ranking_aux carrier le A)) = A"
vD_or_iD: fixes ip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and P :: "(nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option) \<Rightarrow> nat \<Rightarrow> bool" assumes "ip \<in> kD rt"   and "ip \<in> vD rt \<Longrightarrow> P rt ip"   and "ip \<in> iD rt \<Longrightarrow> P rt ip" shows "P rt ip"
take_cols_subsets_og_cols_R: fixes subsets :: "(nat list \<times> nat list) list"   and signs :: "rat list list"   and p :: "real poly"   and qs :: "real poly list" assumes "length subsets = length signs"   and "invertible_mat (M_mat_R signs subsets)"   and "p \<noteq> 0"   and "well_def_signs (length qs) signs"   and "distinct signs"   and "set (characterize_consistent_signs_at_roots p qs) \<subseteq> set signs"   and "satisfy_equation_R p qs subsets signs" shows "set (take_indices (cols (M_mat_R signs subsets)) (find_nonzeros_from_input_vec (solve_for_lhs_R p qs subsets (M_mat_R signs subsets)))) \<subseteq> set (cols (M_mat_R signs subsets))"
exhaust_sel: fixes prod :: "'h \<times> 'i" obtains "prod = (fst prod, snd prod)"
Inl_inject: fixes x :: "'a"   and y :: "'a" assumes "Inl x = Inl y" shows "x = y"
pred_cof_def: fixes L :: "'c set"   and b :: "'b option"   and b' :: "'b option"   and l :: "'a" shows "pred_cof L b b' l = (\<forall>s p. s \<notin> L \<and> p \<notin> L \<and> s \<noteq> p \<longrightarrow> P s p b b' l)"
binop_type_det: fixes op :: "super_binop + boolean_binop + numeric_binop + string_binop + collection_binop"   and k :: "call_kind"   and \<tau> :: "'a OCL_Types.type"   and \<sigma> :: "'a OCL_Types.type"   and \<rho>\<^sub>1 :: "'a OCL_Types.type"   and \<rho>\<^sub>2 :: "'a OCL_Types.type" assumes "binop_type op k \<tau> \<sigma> \<rho>\<^sub>1"   and "binop_type op k \<tau> \<sigma> \<rho>\<^sub>2" shows "\<rho>\<^sub>1 = \<rho>\<^sub>2"
hcomp_reassoc_2: fixes \<tau> :: "'b"   and \<mu> :: "'b"   and \<nu> :: "'b" assumes "D.arr \<tau>"   and "D.arr \<mu>"   and "D.arr \<nu>"   and "src\<^sub>D \<tau> = trg\<^sub>D \<mu>"   and "src\<^sub>D \<mu> = trg\<^sub>D \<nu>" shows "\<tau> \<star>\<^sub>D \<mu> \<star>\<^sub>D \<nu> = \<a>\<^sub>D[D.cod \<tau>, D.cod \<mu>, D.cod \<nu>] \<cdot>\<^sub>D ((\<tau> \<star>\<^sub>D \<mu>) \<star>\<^sub>D \<nu>) \<cdot>\<^sub>D \<a>\<^sub>D\<^sup>-\<^sup>1[D.dom \<tau>, D.dom \<mu>, D.dom \<nu>]"
irreflI: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" assumes "r \<subseteq> A \<times> A"   and "\<And>x. x \<in> A \<Longrightarrow> (x, x) \<notin> r" shows "RPRs.irrefl A r"
widen_Array: fixes P :: "'c prog"   and T :: "ty"   and U :: "ty" shows "P \<turnstile> T \<le> U\<lfloor>\<rceil> = (T = NT \<or> (\<exists>V. T = V\<lfloor>\<rceil> \<and> P \<turnstile> V \<le> U))"
assms_1: shows "emeasure M {x \<in> space M. P x} = 1"
SeqForm_imp_wf_dbfm: fixes s :: "hf"   and k :: "hf"   and x :: "hf"   and e :: "(name, hf) finfun" assumes "SeqForm s k x" shows "\<exists>A. wf_dbfm A \<and> x = \<lbrakk>quot_dbfm A\<rbrakk>e"
Un_commute: fixes A :: "'g set"   and B :: "'g set" shows "A \<union> B = B \<union> A"
separated_by_wall_ex_foldpair: fixes H :: "'a set set set"   and C :: "'a set"   and D :: "'a set" assumes "H \<in> (\<Union>(f, g)\<in>foldpairs. {{f \<turnstile> \<C>, g \<turnstile> \<C>}})"   and "separated_by H C D" shows "\<exists>(f, g)\<in>foldpairs. H = {f \<turnstile> \<C>, g \<turnstile> \<C>} \<and> C \<in> f \<turnstile> \<C> \<and> D \<in> g \<turnstile> \<C>"
induct: fixes P :: "pol \<Rightarrow> int \<Rightarrow> bool"   and a0 :: "pol"   and a1 :: "int" assumes "\<And>c' c. P (Pc c') c"   and "\<And>i Pa c. P Pa c \<Longrightarrow> P (Pinj i Pa) c"   and "\<And>Pa i Q c. \<lbrakk>P Pa c; \<And>x xa y. \<lbrakk>x = cfactor Pa c; (xa, y) = x\<rbrakk> \<Longrightarrow> P Q c\<rbrakk> \<Longrightarrow> P (PX Pa i Q) c" shows "P a0 a1"
mono_D13: fixes P :: "'c process"   and Q :: "'c process"   and S :: "'c process" assumes "P \<sqsubseteq> Q" shows "min_elems (D (P `;` S)) \<subseteq> T (Q `;` S)"
g_unchanged: shows "\<forall>a m. a \<notin> G \<longrightarrow> g m a = m a"
is_none_code_2: fixes x :: "'b" shows "Option.is_none \<lfloor>x\<rfloor> = False"
hext_def: fixes typeof_addr :: "'a \<Rightarrow> 'b \<Rightarrow> htype option"   and h :: "'a"   and h' :: "'a" shows "heap_base.hext typeof_addr h h' \<equiv> typeof_addr h \<subseteq>\<^sub>m typeof_addr h'"
subset_insert_iff: fixes A :: "'d set"   and x :: "'d"   and B :: "'d set" shows "(A \<subseteq> insert x B) = (if x \<in> A then A - {x} \<subseteq> B else A \<subseteq> B)"
tauCases: fixes P :: "pi"   and \<alpha> :: "freeRes"   and P' :: "pi"   and Prop :: "freeRes \<Rightarrow> pi \<Rightarrow> bool" assumes "\<tau>.P \<longmapsto> \<alpha> \<prec> P'"   and "Prop (\<tau>) P" shows "Prop \<alpha> P'"
beta_reduces_app_left: fixes A :: "'a trm"   and A' :: "'a trm"   and B :: "'a trm" assumes "A \<rightarrow>\<beta>\<^sup>* A'" shows "App A B \<rightarrow>\<beta>\<^sup>* App A' B"
1: shows "\<guillemotleft>\<Phi> (g, f) \<cdot>\<^sub>D \<eta>' \<cdot>\<^sub>D D.inv (unit (src\<^sub>C f)) : F (src\<^sub>C f) \<Rightarrow>\<^sub>D F (g \<star>\<^sub>C f)\<guillemotright>"
tyenv_eq_sym: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" assumes "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2" shows "mem\<^sub>2 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>1"
Rep_ttree_cases: fixes y :: "'b list set" assumes "y \<in> {xss. [] \<in> xss \<and> downset xss}" obtains x :: "'b ttree" where "y = Rep_ttree x"
I_intP_Text_imp: fixes p :: "'psym"   and al :: "univ list" assumes "polC p = Text"   and "intP p al" shows "I_intP p al"
sem_5b: shows "\<forall>X Y Z. (\<forall>x. (X \<sqinter> Y) x = (X \<sqinter> Z) x) \<longrightarrow> ob X Y = ob X Z"
open_segment_eq_empty': fixes a :: "'a"   and b :: "'a" shows "({} = open_segment a b) = (a = b)"
unique_circline_type_zero: fixes z :: "complex_homo" shows "\<exists>!H. circline_type H = 0 \<and> z \<in> circline_set H"
mcont_eSuc: shows "mcont Sup (\<le>) Sup (\<le>) eSuc"
snd_diff: fixes x :: "'e \<times> 'd"   and y :: "'e \<times> 'd" shows "snd (x - y) = snd x - snd y"
interleaving_Nil_implies_eq2: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "interleavings xs ys zs"   and "ys = []" shows "xs = zs"
fps_of_poly_pcompose: fixes q :: "'b poly"   and p :: "'b poly" assumes "coeff q 0 = (0::'b)" shows "fps_of_poly (p \<circ>\<^sub>p q) = fps_of_poly p oo fps_of_poly q"
outstanding_refs_subsets_2: fixes sb :: "'b memref list" shows "outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b sb \<subseteq> outstanding_refs is_Write\<^sub>s\<^sub>b sb"
trm_simps_3: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category_with_terminal_object C"   and "partial_magma.arr C f" shows "partial_magma.cod C (category_with_terminal_object.trm C f) = category_with_terminal_object.some_terminal C"
region_compatible_suntil2: fixes \<phi> :: "'s \<times> ('c \<Rightarrow> real) \<Rightarrow> bool"   and \<psi> :: "'s \<times> ('c \<Rightarrow> real) \<Rightarrow> bool"   and x :: "('s \<times> ('c \<Rightarrow> real)) stream" assumes "(holds \<phi> suntil holds \<psi>) x"   and "pred_stream (\<lambda>s. \<phi> s \<longrightarrow> \<phi> (reps (abss s))) x"   and "pred_stream (\<lambda>s. \<psi> s \<longrightarrow> \<psi> (reps (abss s))) x" shows "(holds (\<lambda>x. \<phi> (reps x)) suntil holds (\<lambda>x. \<psi> (reps x))) (smap abss x)"
not_eq_order_implies_strict: fixes a :: "'k"   and b :: "'k" assumes "a \<noteq> b"   and "a \<le> b" shows "a < b"
f3: shows "Deliver (i, Create i e) \<notin> set pre"
l_new_shadow_root_get_shadow_root_def: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_shadow_root_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<equiv> l_get_shadow_root type_wf get_shadow_root get_shadow_root_locs \<and> l_new_shadow_root_get_shadow_root_axioms get_shadow_root_locs"
vector_smult_component: fixes c :: "'a"   and y :: "('a, 'b) vec"   and i :: "'b" shows "(c *s y) $ i = c * y $ i"
dbl_inc_simps_5: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and k :: "num" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_inc plus one (numeral.numeral one plus k) = numeral.numeral one plus (num.Bit1 k)"
isPath_distD: fixes u :: "nat"   and p :: "(nat \<times> nat) list"   and v :: "nat" assumes "cf.isPath u p v" shows "cf.dist u (length p) v"
assms_2: fixes y :: "'a"   and z :: "'a sort_key" assumes "x = \<lbrakk>NonFinal (y, sort_dir.Right) z\<rbrakk>" shows "p"
fun_pair_ineqs: fixes \<theta> :: "'b \<Rightarrow> ('fun, 'a) Term.term"   and \<delta> :: "'c \<Rightarrow> ('fun, 'b) Term.term"   and d :: "('fun, 'c) Term.term \<times> ('fun, 'c) Term.term"   and \<I> :: "'d \<Rightarrow> ('fun, 'a) Term.term"   and d' :: "('fun, 'd) Term.term \<times> ('fun, 'd) Term.term" assumes "d \<cdot>\<^sub>p \<delta> \<cdot>\<^sub>p \<theta> \<noteq> d' \<cdot>\<^sub>p \<I>" shows "pair d \<cdot> \<delta> \<cdot> \<theta> \<noteq> pair d' \<cdot> \<I>"
CACH_INSTR: fixes C :: "Class"   and m :: "Method"   and l :: "int"   and I :: "Instr"   and c :: "Val"   and x :: "Var"   and f :: "Val \<Rightarrow> Val \<Rightarrow> Val"   and g :: "Val \<Rightarrow> Val"   and d :: "Class"   and F :: "Field"   and Mspec :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and Minv :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool"   and Anno :: "(int \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and G :: "((Class \<times> Method \<times> int) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool) \<times> ((Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool)) list"   and n :: "nat" assumes "ins_is C m l I"   and "I \<in> {const c, dup, pop, swap, load x, store x, binop f, unop g, getfield d F, putfield d F, checkcast d}"   and "MST\<down>(C, m) = Some (Mspec, Minv, Anno)"   and "Anno\<down>l = None"   and "derivAssum G C m (l + 1) (Cachera n)" shows "deriv G C m l (Cachera n)"
drinks2_1_invalid: fixes a :: "String.literal"   and b :: "value list"   and r :: "nat \<Rightarrow>f value option" assumes "\<not> (a = STR ''coin'' \<and> length b = 1)"   and "\<not> (a = STR ''vend'' \<and> b = [])" shows "possible_steps drinks2 1 r a b = {||}"
fg': fixes j :: "nat"   and h :: "real"   and x :: "'a" assumes "t j \<le> t1" shows "dist (incr h (t j) x) (incr' h (t j) x) \<le> B * stepsize j ^ p"
star_inductr_var: fixes y :: "'a"   and x :: "'a" assumes "y \<cdot> x \<le> y" shows "y \<cdot> qstar x \<le> y"
prems: shows "\<forall>P\<in>S. uhnf P \<and> valid P"
list_ex_length: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "list_ex P xs = (\<exists>n<length xs. P (xs ! n))"
\<alpha>'_def: shows "\<alpha>' = (\<lambda>m. \<Sum>i<n. of_nat (lookup m i) * \<alpha> i)"
assms_3: shows "g = Cring_Poly.compose R f (monom (UP R) d 1)"
lnull_llist_of_stream: fixes xs :: "'c stream" shows "\<not> lnull (llist_of_stream xs)"
B_def: fixes i :: "nat" shows "B i = map_sq_matrix (\<lambda>p. coeff p i) (adjugate (\<^bold>X - \<^bold>C A))"
composition_series_prefix_closed: fixes i :: "nat" assumes "i \<le> length (take 2 \<GG>)"   and "0 < i" shows "composition_series (G\<lparr>carrier := \<GG> ! 1, carrier := take 2 \<GG> ! (i - 1)\<rparr>) (take i (take 2 \<GG>))"
map_sel: fixes a :: "'a option"   and f :: "'a \<Rightarrow> 'b" assumes "a \<noteq> None" shows "the (map_option f a) = f (the a)"
anonymous: fixes \<pi> :: "'agent \<Rightarrow> 'agent"   and R :: "'agent \<Rightarrow> 'alt \<Rightarrow> 'alt \<Rightarrow> bool" assumes "\<pi> permutes agents"   and "is_pref_profile R" shows "sds (R \<circ> \<pi>) = sds R"
subcls1p_subcls1_eq: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list" shows "subcls1p P = subcls1' P"
nth_subdegree_nonzero: fixes f :: "'a fps" assumes "f \<noteq> 0" shows "fps_nth f (subdegree f) \<noteq> (0::'a)"
g1: shows "0 < \<gamma> \<and> \<gamma> < 1"
eventually_False_sequentially: shows "\<not> (\<forall>\<^sup>\<infinity>n. False)"
linear_surjective_imp_injective: fixes f :: "('b, 'c) vec \<Rightarrow> ('b, 'd) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "surj f"   and "vec.dim UNIV = vec.dim UNIV" shows "inj f"
initiali_aodv: fixes i :: "nat" shows "initiali i (init (opaodv i)) (init (paodv i))"
assms_2: assumes "\<D>\<^sub>\<circ> xs \<in>\<^sub>\<circ> \<omega>" shows "P"
xfR: fixes j :: "nat" shows "xf j \<in> R"