hcju/isabelleps huggingface dataset:
simps: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and ws :: "nat \<Rightarrow> nat"   and J :: "nat \<Rightarrow> ('c, 'b) justifying_execution" shows "(P \<turnstile> (E, ws) justified_by J) = (is_commit_sequence E J \<and> justification_well_formed P J \<and> committed_subset_actions J \<and> happens_before_committed P E J \<and> sync_order_committed P E J \<and> value_written_committed P E J \<and> write_seen_committed ws J \<and> uncommitted_reads_see_hb P J \<and> committed_reads_see_committed_writes ws J \<and> external_actions_committed P J \<and> wf_action_translations E J)"
comp_inj_on: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set"   and g :: "'e \<Rightarrow> 'f" assumes "inj_on f A"   and "inj_on g (f ` A)" shows "inj_on (g \<circ> f) A"
id_matrix_def: fixes nr :: "nat"   and nc :: "nat" shows "id_matrix nr nc = mat nr nc (\<lambda>(r, c). if r = c then 1 else 0)"
quotientI: fixes x :: "'c"   and A :: "'c set"   and r :: "('c \<times> 'c) set" assumes "x \<in> A" shows "r `` {x} \<in> A // r"
hyps: assumes "xs \<noteq> []"   and "ys \<noteq> [] \<longrightarrow> last xs = hd ys" shows "changes (xs @ ys) = changes (xs @ tl ys)"
WbisT_coind: fixes indis :: "('a \<times> 'a) set"   and theta :: "(('b, 'c) com \<times> ('b, 'c) com) set"   and tval :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'a \<Rightarrow> 'a" assumes "PL_Indis indis"   and "sym theta"   and "theta \<subseteq> PL_Indis.WretrT tval aval indis (theta \<union> PL_Indis.WbisT tval aval indis)" shows "theta \<subseteq> PL_Indis.WbisT tval aval indis"
iarray_cong: fixes n :: "nat"   and m :: "nat"   and f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> 'a" assumes "n = m"   and "\<And>i. i < m \<Longrightarrow> f i = g i" shows "IArray.of_fun f n = IArray.of_fun g m"
finite_field_factorization_main_uint64: fixes p :: "int"   and f :: "int poly"   and c :: "int"   and fs :: "int poly list" assumes "poly_mod_prime p"   and "p \<le> 4294967295"   and "finite_field_factorization_main p (finite_field_ops64 (uint64_of_int p)) f = (c, fs)"   and "poly_mod.square_free_m p f" shows "poly_mod.unique_factorization_m p f (c, mset fs) \<and> c \<in> {0..<p} \<and> (\<forall>fi\<in>set fs. set (coeffs fi) \<subseteq> {0..<p})"
sum_cong_aux: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "sum f A = sum g A"
intro: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and default_node_properties :: "'d" assumes "SecurityInvariant_preliminaries sinvar"   and "SecurityInvariant_IFS_axioms sinvar default_node_properties" shows "SecurityInvariant_IFS sinvar default_node_properties"
dependent_live_vars_same_pathsI: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Def :: "'b \<Rightarrow> 'd set"   and Use :: "'b \<Rightarrow> 'd set"   and state_val :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and Exit :: "'b"   and dyn_control_dependence :: "'b \<Rightarrow> 'b \<Rightarrow> 'a list \<Rightarrow> bool"   and V :: "'d"   and n' :: "'b"   and as :: "'a list" assumes "DynPDG sourcenode targetnode kind valid_edge Entry Def Use state_val Exit dyn_control_dependence"   and "V \<in> Use n'"   and "\<forall>as' a as''. as = as' @ a # as'' \<longrightarrow> \<not> DynPDG.DynPDG_edge sourcenode targetnode valid_edge Def Use dyn_control_dependence (sourcenode a) (Some V) (a # as'') n'"   and "as \<noteq> [] \<longrightarrow> n' = last (CFG.targetnodes targetnode as)" shows "(V, as, as) \<in> DynPDG.dependent_live_vars sourcenode targetnode valid_edge Def Use dyn_control_dependence n'"
PBQ: shows "(P, S, Q) = diagonal_to_Smith_PQ A bezout"
w1_def: shows "w1 = take (n - length (u @ v)) w"
whiletrue: fixes BMap :: "'a \<Rightarrow> bool"   and E :: "'b \<Rightarrow> ('c \<Rightarrow> 'a) \<Rightarrow> 'a"   and b :: "'b"   and m :: "'c \<Rightarrow> 'a"   and c :: "('b, 'c) MWLfCom" assumes "BMap (E b m) = True" shows "((while b do c od, m), [c;while b do c od], m) \<in> MWLf_semantics.MWLfSteps_det E BMap"
r_coset_def: fixes G :: "('e, 'f) monoid_scheme"   and H :: "'e set"   and a :: "'e" shows "H #>\<^bsub>G\<^esub> a = (\<Union>h\<in>H. {h \<otimes>\<^bsub>G\<^esub> a})"
isometryD_2: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "isometry f" shows "dist (f x) (f y) = dist x y"
normal_vector_6: fixes W :: "'a vec set" assumes "finite W"   and "Suc (card W) = n"   and "lin_indpt W"   and "W \<subseteq> carrier_vec n" shows "normal_vector W \<notin> W"
ide_char: fixes a :: "'a" shows "Right.ide a = (Right.arr a \<and> ide a)"
read_actionsp_read_actions_eq: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist" shows "read_actionsp E = (\<lambda>x. x \<in> read_actions E)"
resCap_cf_refine: fixes p :: "(nat \<times> nat) list" shows "resCap_cf f.cf p = f.resCap p"
comp_arr_inv: fixes f :: "'a"   and g :: "'a" assumes "C.inverse_arrows f g" shows "C f g = C.dom g"
ide_MkIde: fixes A :: "'a" assumes "A \<in> Collect obj" shows "Maps.ide (Maps.MkIde A)"
f: shows "vec_of_list u \<bullet> vec_of_list v = (\<Sum>i = 0..<length v. u ! i * v ! i)"
sq_mtx_norm_le_sum_col: fixes A :: "'a sq_mtx" shows "\<parallel>A\<parallel> \<le> (\<Sum>i\<in>UNIV. \<parallel>\<c>\<o>\<l> i A\<parallel>)"
Dom_terminal: fixes t :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "terminal t" shows "Dom t = {DOWN t}"
learnLess_def: fixes C :: "Literal list" shows "learnLess C \<equiv> {(F1, F2). C el F1 \<and> \<not> C el F2}"
wf_rcv_fv_single: fixes V :: "'d set"   and t :: "('e, 'd) Term.term"   and S' :: "('e, 'd) strand_step list" assumes "wf\<^sub>s\<^sub>t V (receive\<langle>t\<rangle>\<^sub>s\<^sub>t # S')" shows "fv t \<subseteq> V"
assms_2: shows "is_leveln_digit n pr \<and> is_measured_digit pr"
plus_Box_strict1: fixes y :: "'a Box" shows "\<bottom> + y = \<bottom>"
isomorphic_relative_contractible_space_imp_homology_groups: fixes X :: "'a topology"   and Y :: "'b topology"   and S :: "'a set"   and T :: "'b set"   and p :: "int" assumes "contractible_space X"   and "contractible_space Y"   and "S \<subseteq> topspace X"   and "T \<subseteq> topspace Y"   and "(S = {}) = (T = {})"   and "\<And>p. relative_homology_group p X S \<cong> relative_homology_group p Y T" shows "homology_group p (subtopology X S) \<cong> homology_group p (subtopology Y T)"
db_defs_13: shows "vars \<equiv> (inp, mid, out, q1, q2)"
insert_absorb: fixes a :: "'b"   and A :: "'b set" assumes "a \<in> A" shows "a \<triangleright> A = A"
Z_to_M: fixes M :: "('f, 'a, 'g) Module_scheme"   and f :: "'h \<Rightarrow> 'f"   and e :: "'h"   and g :: "'h \<Rightarrow> 'f" assumes "R module M"   and "f \<in> mHom R (Zm R e) M"   and "g \<in> mHom R (Zm R e) M" shows "f = g"
assms_4: shows "\<exists>b\<in>pin (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>). length (bc (\<sigma>\<^bsub>tid\<^esub>t \<langle>tid \<leftarrow> t\<rangle>\<^bsub>n\<^esub>)) < length b"
assms_2: fixes p :: "'a"   and q :: "'a" assumes "\<not> prime p" shows "P p q"
greedy_conforming_path_properties_4: fixes G :: "('c, 'd) ParityGame_scheme"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and \<sigma>' :: "'c \<Rightarrow> 'c" assumes "ParityGame G"   and "v0 \<in> V\<^bsub>G\<^esub>"   and "ParityGame.strategy G p \<sigma>"   and "ParityGame.strategy G p** \<sigma>'" shows "Digraph.maximal_path G (ParityGame.greedy_conforming_path G p \<sigma> \<sigma>' v0)"
partial_span: fixes i :: "nat" assumes "i \<le> length (map of_int_hom.vec_hom fs')" shows "gs.span (gs2.gso ` {0..<i}) = gs.span (set (take i (map of_int_hom.vec_hom fs')))"
isCont_real_root: fixes x :: "real"   and n :: "nat" shows "isCont (root n) x"
INF_empty: fixes f :: "'c \<Rightarrow> 'b" shows "\<Squnion> (f ` {}) = \<bottom>"
thetaSeqWD_def: shows "thetaSeqWD \<equiv> {(c1 ;; c2, d1 ;; d2) |c1 c2 d1 d2. c1 \<approx>w d1 \<and> discr c2 \<and> discr d2}"
sel_8: fixes x1 :: "bool"   and x2 :: "bool"   and x3 :: "bool"   and x4 :: "bool"   and x5 :: "bool"   and x6 :: "bool"   and x7 :: "bool"   and x8 :: "bool" shows "digit7 (Char x1 x2 x3 x4 x5 x6 x7 x8) = x8"
IH: shows "\<forall>ys. length ys < length xs' \<longrightarrow> (\<forall>x. (\<forall>xa. xa \<in> set ys \<longrightarrow> (\<forall>xb. xb \<in> set x \<longrightarrow> f xa \<noteq> f xb)) \<longrightarrow> local.group (ys @ x) = local.group ys @ local.group x)"
dehomogenize_ideal: fixes x :: "'b"   and F :: "(('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set" shows "dehomogenize x ` ideal F = ideal (dehomogenize x ` F) \<inter> P[- {x}]"
remove_preserves_known_ptrs: fixes h :: "(_) heap"   and child :: "(_) node_ptr"   and h' :: "(_) heap" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> remove child \<rightarrow>\<^sub>h h'" shows "known_ptrs h'"
gallery_def: fixes xs :: "'w set set list" shows "gallery xs \<equiv> Ball (set xs) chamber \<and> adjacentchain xs"
ex_gt_imp_less_multiset: fixes N :: "'a multiset"   and M :: "'a multiset" assumes "\<exists>y. y \<in># N \<and> (\<forall>x. x \<in># M \<longrightarrow> x < y)" shows "M < N"
pair_measure_eqI: fixes M1 :: "'d measure"   and M2 :: "'e measure"   and M :: "('d \<times> 'e) measure" assumes "sigma_finite_measure M1"   and "sigma_finite_measure M2"   and "sets (M1 \<Otimes>\<^sub>M M2) = sets M"   and "\<And>A B. \<lbrakk>A \<in> sets M1; B \<in> sets M2\<rbrakk> \<Longrightarrow> emeasure M1 A * emeasure M2 B = emeasure M (A \<times> B)" shows "M1 \<Otimes>\<^sub>M M2 = M"
find_exponent_1: fixes p :: "int"   and bnd :: "int" assumes "1 < p" shows "bnd < p ^ find_exponent p bnd"
compress_normalize_input_interfaces_nnf: fixes m :: "'b common_primitive match_expr"   and m' :: "'b common_primitive match_expr" assumes "normalized_nnf_match m"   and "compress_normalize_input_interfaces m = Some m'" shows "normalized_nnf_match m'"
cobounded2: fixes b :: "'b multiset"   and a :: "'b multiset" shows "b \<subseteq># a \<union># b"
frac_in_Ints_iff: fixes x :: "'a" shows "(frac x \<in> \<int>) = (x \<in> \<int>)"
inf_of_two_exists: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "\<exists>s. is_glb L s {x, y}"
floor_one: shows "\<lfloor>1::'a\<rfloor> = 1"
DIM_eq_def: fixes n :: "nat" shows "DIM_precond TYPE('a) n = (DIM('a) = n)"
intra_SDG_path_valid_SDG_node_1: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> ('c, 'd, 'e, 'f) edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and get_proc :: "'b \<Rightarrow> 'f"   and get_return_edges :: "'a \<Rightarrow> 'a set"   and procs :: "('f \<times> 'c list \<times> 'c list) list"   and Main :: "'f"   and Exit :: "'b"   and Def :: "'b \<Rightarrow> 'c set"   and Use :: "'b \<Rightarrow> 'c set"   and ParamDefs :: "'b \<Rightarrow> 'c list"   and ParamUses :: "'b \<Rightarrow> 'c set list"   and n :: "'b SDG_node"   and ns :: "'b SDG_node list"   and n' :: "'b SDG_node" assumes "SDG sourcenode targetnode kind valid_edge Entry get_proc get_return_edges procs Main Exit Def Use ParamDefs ParamUses"   and "SDG.intra_SDG_path sourcenode targetnode kind valid_edge get_proc get_return_edges procs Exit Def Use ParamDefs ParamUses n ns n'" shows "SDG.valid_SDG_node sourcenode targetnode kind valid_edge procs n"
matrix_to_iarray_minus: fixes A :: "(('c, 'd) vec, 'e) vec"   and B :: "(('c, 'd) vec, 'e) vec" shows "matrix_to_iarray (A - B) = matrix_to_iarray A - matrix_to_iarray B"
zero_vec_Suc: fixes n :: "nat" shows "0\<^sub>v (Suc n) = vCons (0::'a) (0\<^sub>v n)"
chamber_image_closer: fixes D :: "'a set"   and B :: "'a set"   and Ds :: "'a set list" assumes "D \<in> folding.\<C> - f \<turnstile> folding.\<C>"   and "B \<in> f \<turnstile> folding.\<C>"   and "B \<noteq> f ` D"   and "folding.gallery (B # Ds @ [D])" shows "\<exists>Cs. folding.gallery (B # Cs @ [f ` D]) \<and> order.greater (length Ds) (length Cs)"
pr_algo_lts_length_bound: fixes p :: "pr_operation list"   and f' :: "nat \<times> nat \<Rightarrow> 'a"   and l' :: "nat \<Rightarrow> nat" assumes "((f, l), p, f', l') \<in> trcl pr_algo_lts" shows "length p \<le> pr_algo_len_bound"
saturated_ordered_sel_res_empty_sel: fixes atom_ordering :: "('a \<times> 'a) set"   and Sel :: "'a set"   and S :: "'a Literal set set" assumes "propositional_atoms atom_ordering"   and "saturated_binary_rule (propositional_atoms.ordered_sel_resolvent atom_ordering Sel) S" shows "saturated_binary_rule (propositional_atoms.ordered_resolvent atom_ordering) (empty_selected_part Sel S)"
prems_2: shows "card {q. prime q \<and> q < p} = n"
assms_2: shows "invariantsHoldInState stateA F0 decisionVars"
apartment_trivial_morphism: fixes X :: "'a set set" assumes "X \<in> \<A>" shows "ChamberComplexMorphism X X id"
zorder_exist: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex" assumes "isolated_singularity_at f z"   and "not_essential f z"   and "\<exists>\<^sub>F w in at z. f w \<noteq> 0" shows "zor_poly f z z \<noteq> 0 \<and> (\<exists>r>0. zor_poly f z holomorphic_on cball z r \<and> (\<forall>w\<in>cball z r - {z}. f w = zor_poly f z w * (w - z) powr of_int (zorder f z) \<and> zor_poly f z w \<noteq> 0))"
that: fixes w :: "('a, 'b) tm" assumes "s \<unrhd>\<^sub>e\<^sub>m\<^sub>b w"   and "subst \<rho> w \<unrhd>\<^sub>e\<^sub>m\<^sub>b u"   and "is_Sym (head w) \<Longrightarrow> head w = head u \<and> num_args w = num_args u \<and> (\<forall>k<num_args w. args (subst \<rho> w) ! k \<unrhd>\<^sub>e\<^sub>m\<^sub>b args u ! k)" shows "thesis"
conc_def: shows "RegExp2NA.conc = (\<lambda>(ql, dl, fl) (qr, dr, fr). (True # ql, \<lambda>a s. case s of [] \<Rightarrow> {} | left # s \<Rightarrow> if left then True ## dl a s \<union> (if fl s then False ## dr a qr else {}) else False ## dr a s, \<lambda>s. case s of [] \<Rightarrow> False | left # s \<Rightarrow> left \<and> fl s \<and> fr qr \<or> \<not> left \<and> fr s))"
mset_set_set_mset_subseteq: fixes M :: "'b multiset" shows "mset_set (set_mset M) \<subseteq># M"
ex_subseq_reduce_index: fixes n :: "nat"   and k :: "nat" shows "seqseq (n + k) = seqseq n \<circ> fold_reduce n k"
inter_comm_line_line_comm: fixes C :: "Points"   and D :: "Points"   and A :: "Points"   and B :: "Points" assumes "C \<noteq> D"   and "line A B \<noteq> line C D" shows "Pascal_Property.inter (line A B) (line C D) = Pascal_Property.inter (line D C) (line A B)"
gcd_poly_code: fixes p :: "'a poly"   and q :: "'a poly" shows "gcd p q = gcd_poly_code p q"
INT_extend_simps_2: fixes A :: "'b set"   and B :: "'c \<Rightarrow> 'b set"   and C :: "'c set" shows "A \<inter> \<Inter> (B ` C) = (if C = {} then A else \<Inter>x\<in>C. A \<inter> B x)"
bij_betwE: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set" assumes "bij_betw f A B" shows "\<forall>a\<in>A. f a \<in> B"
integral_refl: fixes a :: "'b"   and f :: "'b \<Rightarrow> 'a" shows "integral (cbox a a) f = (0::'a)"
ThrowNull: 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"   and E :: "char list \<Rightarrow> ty option"   and e :: "expr"   and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "P,E \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>null,s\<^sub>1\<rangle>" shows "P,E \<turnstile> \<langle>throw e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer, [NullPointer]),s\<^sub>1\<rangle>"
ex_prefix_of: fixes p :: "'a prefix" shows "\<exists>s. prefix_of p s"
antisymE: fixes cl :: "'b potype"   and a :: "'b"   and b :: "'b" assumes "PO cl"   and "(a, b) \<in> order cl"   and "(b, a) \<in> order cl" shows "a = b"
DN: fixes H :: "'b pl set"   and p :: "'b pl" shows "H |- ((p -> false) -> false) -> p"
ennreal_power: fixes r :: "real"   and n :: "nat" assumes "0 \<le> r" shows "ennreal r ^ n = ennreal (r ^ n)"
subclsPlus_subtype: 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"   and D :: "char list"   and Bs' :: "base list"   and fs' :: "(char list \<times> ty) list"   and ms' :: "(char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list"   and M :: "char list"   and Ts' :: "ty list"   and T' :: "ty"   and m' :: "char list list \<times> expr"   and C :: "char list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "class P D = \<lfloor>(Bs', fs', ms')\<rfloor>"   and "map_of ms' M = \<lfloor>(Ts', T', m')\<rfloor>"   and "(C, D) \<in> (subcls1 P)\<^sup>+"   and "wf_prog wf_md P" shows "\<forall>Bs fs ms Ts T m. class P C = \<lfloor>(Bs, fs, ms)\<rfloor> \<and> map_of ms M = \<lfloor>(Ts, T, m)\<rfloor> \<longrightarrow> Ts' = Ts \<and> P \<turnstile> T \<le> T'"
singleton_set_is: fixes M :: "'a set"   and y :: "'a" assumes "\<forall>x\<in>M. x = y"   and "M \<noteq> {}" shows "M = {y}"
backward_finite_path_connected: fixes r :: "'a"   and x :: "'a" assumes "backward_finite_path_root r x" shows "connected x"
star_slide: fixes x :: "'a"   and y :: "'a" shows "(x * y)\<^sup>\<star> * x = x * (y * x)\<^sup>\<star>"
hcomp_ide_isomorphic: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "ide f"   and "g \<cong> h"   and "src f = trg g" shows "f \<star> g \<cong> f \<star> h"
le_div_conv: fixes m :: "nat"   and n :: "nat"   and k :: "nat" assumes "0 < m" shows "(n \<le> k div m) = (n * m \<le> k)"
inverse_distributive_1: fixes R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "ring R addition multiplication zero unit"   and "a \<in> R"   and "b \<in> R"   and "c \<in> R" shows "multiplication a (addition b (monoid.inverse R addition zero c)) = addition (multiplication a b) (monoid.inverse R addition zero (multiplication a c))"
axioms_1: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b"   and H :: "'b set" assumes "GroupIso G T H" shows "GroupHom G T"
chain_equality: fixes A :: "'b set" shows "Complete_Partial_Order.chain (=) A = (\<forall>x\<in>A. \<forall>y\<in>A. x = y)"
density_marg_dens_eq: fixes x :: "nat"   and \<rho> :: "nat \<Rightarrow> val" assumes "x \<in> V"   and "\<rho> \<in> space (state_measure V' \<Gamma>)" shows "density (stock_measure (\<Gamma> x)) (marg_dens \<Y> x \<rho>) = distr (dens_ctxt_measure \<Y> \<rho>) (stock_measure (\<Gamma> x)) (\<lambda>\<sigma>. \<sigma> x)"
upto_enum_word: fixes x :: "'b word"   and y :: "'b word" shows "[x .e. y] = map word_of_nat [unat x..<Suc (unat y)]"
compose_surj: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and g :: "'f \<Rightarrow> 'g"   and C :: "'g set" assumes "f \<in> A \<rightarrow> B"   and "surj_to f A B"   and "g \<in> B \<rightarrow> C"   and "surj_to g B C" shows "surj_to (compose A g f) A C"
sets_PiM_eq_proj: fixes I :: "'d set"   and M :: "'d \<Rightarrow> 'e measure" assumes "I \<noteq> {}" shows "sets (Pi\<^sub>M I M) = sets (SUP i\<in>I. vimage_algebra (\<Pi>\<^sub>E i\<in>I. space (M i)) (\<lambda>x. x i) (M i))"
csqrt_of_real_nonneg: fixes x :: "complex" assumes "Im x = 0"   and "0 \<le> Re x" shows "csqrt x = complex_of_real (sqrt (Re x))"
hseq_char: fixes \<mu> :: "('a S.E.term, 'a) S.arr"   and \<nu> :: "('a S.E.term, 'a) S.arr" shows "S.hseq \<mu> \<nu> = (S.arr \<mu> \<and> S.arr \<nu> \<and> S.src \<mu> = S.trg \<nu>)"
\<phi>'_def: shows "\<phi>' \<equiv> instInpP Pinp 0 eN\<tau>"
assms: shows "\<nexists>f. \<forall>i. r (f (Suc i)) (f i)"
addsE: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "plus.adds plus b a" obtains k :: "'b" where "a = plus b k"
fun_upd_eqD: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'e"   and g :: "'d \<Rightarrow> 'e"   and z :: "'e" assumes "f(x := y) = g(x := z)" shows "y = z"
assms: shows "fold (\<lambda>(T, U) subs. Option.bind subs (f T U)) (zip (x # xs) (y # ys)) (Some subs) = Some subs'"
fashoda: fixes f :: "real \<Rightarrow> (real, 2) vec"   and g :: "real \<Rightarrow> (real, 2) vec"   and a :: "(real, 2) vec"   and b :: "(real, 2) vec" assumes "path f"   and "path g"   and "path_image f \<subseteq> cbox a b"   and "path_image g \<subseteq> cbox a b"   and "pathstart f $ 1 = a $ 1"   and "pathfinish f $ 1 = b $ 1"   and "pathstart g $ 2 = a $ 2"   and "pathfinish g $ 2 = b $ 2" obtains z :: "(real, 2) vec" where "z \<in> path_image f" and "z \<in> path_image g"
t0_4: shows "t = t0 \<cdot> \<delta> \<cdot> \<theta>"
b_least2_mono: fixes y1 :: "nat"   and y2 :: "nat"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and x :: "nat" assumes "y1 \<le> y2" shows "b_least2 f x y1 \<le> b_least2 f x y2"
max_ipv4_addr_number: shows "max_ipv4_addr = 4294967295"
bounded_solution_set_sas_plus'_def: fixes \<Psi> :: "('c, 'd) sas_plus_problem"   and k :: "nat" shows "bounded_solution_set_sas_plus' \<Psi> k \<equiv> {\<psi>. SAS_Plus_Semantics.is_serial_solution_for_problem \<Psi> \<psi> \<and> length \<psi> = k}"
pderiv_iszero: fixes p :: "'b poly" assumes "pderiv p = 0" shows "\<exists>h. p = [:h:]"
weak_reduction_bisimulation_and_closures_4: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" assumes "weak_reduction_bisimulation Rel Cal" shows "weak_reduction_bisimulation (symcl (Rel\<^sup>=)) Cal"
wf_empty_rel: shows "minimal_element (\<lambda>_ _. False) UNIV"
verticesFrom_is_nextElem_hd: fixes v' :: "nat"   and f :: "face"   and u :: "nat" assumes "v' \<in> \<V> f"   and "distinct (vertices f)"   and "is_nextElem (verticesFrom f v') u v'" shows "u = last (verticesFrom f v')"
assms_1: shows "r = rat_of_int a / rat_of_int b"
ipurge_fail_aux_t_eq_ref: fixes I :: "('c \<times> 'c) set"   and D :: "'b \<Rightarrow> 'c"   and U :: "'c set"   and xs :: "'b list"   and X :: "'b set" shows "snd (ipurge_fail_aux_t I D U xs X) = ipurge_ref_aux I D U xs X"
add: fixes u :: "'a \<Rightarrow> real"   and v :: "'a \<Rightarrow> real" assumes "u \<in> borel_measurable M"   and "\<And>x. 0 \<le> u x"   and "P u"   and "v \<in> borel_measurable M"   and "\<And>x. 0 \<le> v x"   and "\<And>x. x \<in> space M \<Longrightarrow> u x = 0 \<or> v x = 0"   and "P v" shows "P (\<lambda>x. v x + u x)"
map_comp: fixes g :: "'b \<Rightarrow> 'a"   and f :: "'c \<Rightarrow> 'b"   and v :: "'c blindable\<^sub>h" shows "map_blindable\<^sub>h g (map_blindable\<^sub>h f v) = map_blindable\<^sub>h (g \<circ> f) v"
not_eq_order_implies_strict: fixes a :: "'c list"   and b :: "'c list" assumes "a \<noteq> b"   and "prefix b a" shows "strict_prefix b a"
matrix_right_invertible_ker: fixes M :: "((real, 'b) vec, 'a) vec" shows "(\<exists>M'. M ** M' = mat 1) = (\<forall>x. x v* M = 0 \<longrightarrow> x = 0)"
lambda_system_strong_additive: fixes z :: "'a set"   and x :: "'a set"   and y :: "'a set"   and f :: "'a set \<Rightarrow> ennreal" assumes "z \<in> M"   and "x \<inter> y = {}"   and "x \<in> lambda_system \<Omega> M f"   and "y \<in> lambda_system \<Omega> M f" shows "f (z \<inter> (x \<union> y)) = f (z \<inter> x) + f (z \<inter> y)"
lm042: fixes R :: "('b \<times> 'a) set"   and X :: "'b set"   and Y :: "'b set" shows "R `` (X - Y) = (R || X) `` (X - Y)"
Var\<^sub>0_def: fixes n :: "'a" shows "Var\<^sub>0 n \<equiv> monomial (1::'b) (monomial 1 n)"