hcju/isabelleps huggingface dataset:
replacefacesAt_nth: fixes k :: "nat"   and ns :: "nat list"   and F :: "face list list"   and oldf :: "face"   and newfs :: "face list" assumes "k \<in> set ns"   and "k < |F|"   and "oldf \<notin> set newfs"   and "distinct (F ! k)"   and "distinct newfs"   and "oldf \<in> set (F ! k) \<longrightarrow> set newfs \<inter> set (F ! k) \<subseteq> {oldf}" shows "replacefacesAt ns oldf newfs F ! k = replace oldf newfs (F ! k)"
inject: fixes x21 :: "'b"   and x22 :: "'b llist"   and y21 :: "'b"   and y22 :: "'b llist" shows "(x21 % x22 = y21 % y22) = (x21 = y21 \<and> x22 = y22)"
closest_pair_bf_c0: fixes ps :: "(int \<times> int) list"   and c\<^sub>0 :: "int \<times> int"   and c\<^sub>1 :: "int \<times> int" assumes "1 < length ps"   and "(c\<^sub>0, c\<^sub>1) = closest_pair_bf ps" shows "c\<^sub>0 \<in> set ps"
strict_implies_not_eq: fixes b :: "'d DBMEntry"   and a :: "'d DBMEntry" assumes "b \<prec> a" shows "a \<noteq> b"
backward_finite_path_root_def: fixes r :: "'a"   and x :: "'a" shows "backward_finite_path_root r x \<equiv> connected_root r x \<and> is_inj x \<and> is_p_fun x \<and> point r"
subst_vsubst_trans: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y1 :: "'var"   and y :: "'var" assumes "good X"   and "good Y"   and "fresh ys y1 X" shows "X #[y1 // y]_ys #[Y / y1]_ys = X #[Y / y]_ys"
hyps: shows "length (Gram_Schmidt xs) = length xs \<and> span (set (Gram_Schmidt xs)) = span (set xs) \<and> pairwise orthogonal (set (Gram_Schmidt xs))"
mult_mult_trm_ord_trans: shows "trans (mult (mult trm_ord))"
index_less_def: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd" shows "index_less index key \<equiv> \<forall>x n mi ma. key x \<in> {mi..ma} \<longrightarrow> 0 < n \<longrightarrow> index key x n mi ma < n"
affine_hyperplane: fixes a :: "'a"   and b :: "real" shows "affine {x. a \<bullet> x = b}"
w_2: shows "w = geodesic_segment_param Gyz y t"
asIMOD_igSubstAbsIPresIGWlsAbs: fixes SEM :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'b) semDom" shows "igSubstAbsIPresIGWlsAbs (asIMOD SEM)"
ideal_def: fixes I :: "'c set"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c" shows "ideal I R addition multiplication zero unit \<equiv> subgroup_of_additive_group_of_ring I R addition multiplication zero unit \<and> ideal_axioms I R multiplication"
le: shows "(pp \<le> x + x') = (int_of_integer pp \<le> int_of_integer x + int_of_integer x')"
right_exact_surjec: fixes M :: "('i, 'a) Module"   and N :: "('j, 'a) Module"   and f :: "'i \<Rightarrow> 'j"   and p :: "'j \<Rightarrow> 'k"   and e :: "'k" assumes "R module M"   and "R module N"   and "f \<in> mHom R M N"   and "p \<in> mHom R N (Zm R e)"   and "exact3 R M f N p (Zm R e)" shows "surjec\<^bsub>M,N\<^esub> f"
lc0: fixes a :: "'c \<Rightarrow> 'a" assumes "a \<in> A \<rightarrow> carrier R"   and "lincomb a A = \<zero>\<^bsub>M\<^esub>" shows "\<forall>v\<in>A. a v = \<zero>"
soundness_diagonalize_JNF': fixes bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a"   and A :: "'a mat"   and m :: "nat"   and n :: "nat"   and P :: "'a mat"   and S :: "'a mat"   and Q :: "'a mat" assumes "is_bezout_ext bezout"   and "A \<in> carrier_mat m n"   and "diagonalize_JNF A bezout = (P, S, Q)" shows "P \<in> carrier_mat m m \<and> Q \<in> carrier_mat n n \<and> S \<in> carrier_mat m n \<and> invertible_mat P \<and> invertible_mat Q \<and> isDiagonal_mat S \<and> S = P * A * Q"
load_tmps_fresh_prog_step: fixes i :: "nat"   and ts :: "('a \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'a memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option"   and p' :: "'a"   and is' :: "instr list" assumes "i < length ts"   and "ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"   and "\<theta>\<turnstile> p \<rightarrow>\<^sub>p (p', is')"   and "valid ts" shows "load_tmps is' \<inter> dom \<theta> = {}"
bounded_fst: fixes s :: "('b \<times> 'c) set" assumes "bounded s" shows "bounded (fst ` s)"
partial_span': fixes i :: "nat" assumes "i \<le> length (map of_int_hom.vec_hom fs)" shows "gs.span (fs.gs.gso ` {0..<i}) = gs.span ((!) (map of_int_hom.vec_hom fs) ` {0..<i})"
inject_1: fixes x3 :: "'b"   and y3 :: "'b" shows "(Atom x3 = Atom y3) = (x3 = y3)"
that: fixes \<sigma>' :: "nat \<Rightarrow> 'g"   and \<zeta>' :: "'l net_state" assumes "((\<sigma>, \<zeta>), disconnect(i, i'), \<sigma>', \<zeta>') \<in> automaton.trans (opnet onp n)"   and "\<forall>j. j \<notin> net_ips \<zeta> \<longrightarrow> \<sigma>' j = \<sigma> j"   and "netgmap sr s' = netmask (net_tree_ips n) (\<sigma>', \<zeta>')" shows "thesis"
price_structure_borel_measurable: fixes pyf :: "'a \<Rightarrow> real"   and T :: "nat"   and \<pi> :: "real"   and pr :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "price_structure pyf T \<pi> pr" shows "pr T \<in> borel_measurable (F T)"
limit_ordinal_not_oSuc: fixes x :: "ordinal"   and p :: "ordinal" assumes "limit_ordinal x" shows "x \<noteq> oSuc p"
that: fixes a :: "'a"   and b :: "'a" assumes "K = cbox a b"   and "K \<noteq> {}"   and "K \<subseteq> S" shows "thesis"
timesI: fixes a :: "hf"   and A :: "hf"   and b :: "hf"   and B :: "hf" assumes "a \<^bold>\<in> A"   and "b \<^bold>\<in> B" shows "\<langle>a, b\<rangle> \<^bold>\<in> A * B"
bot_in_ss: shows "{#} \<in> Q_of_state (wrp.Liminf_wstate sswSts)"
comp2le_def: fixes cmp :: "'a \<Rightarrow> 'b \<Rightarrow> comp_res"   and a :: "'a"   and b :: "'b" shows "comp2le cmp a b \<equiv> case cmp a b of GREATER \<Rightarrow> False | _ \<Rightarrow> True"
k_def: shows "k \<equiv> LEAST k. f $$ (M + int k) \<noteq> (0::'a)"
mbisimI: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set" assumes "finite (dom (thr s1))"   and "locks s1 = locks s2"   and "wset s1 = wset s2"   and "interrupts s1 = interrupts s2"   and "wset_thread_ok (wset s1) (thr s1)"   and "\<And>t. thr s1 t = None \<Longrightarrow> thr s2 t = None"   and "\<And>t x1 ln. thr s1 t = \<lfloor>(x1, ln)\<rfloor> \<Longrightarrow> \<exists>x2. thr s2 t = \<lfloor>(x2, ln)\<rfloor> \<and> t \<turnstile> (x1, shr s1) \<approx> (x2, shr s2) \<and> (wset s2 t = None \<or> x1 \<approx>w x2)" shows "s1 \<approx>m s2"
finite_imp_inj_to_nat_seg: fixes A :: "'b set" assumes "finite A" shows "\<exists>f n. f ` A = {i. i < n} \<and> inj_on f A"
assms: fixes f :: "'a \<Rightarrow> 'b" assumes "continuous_on S f"   and "finite (f ` S)" shows "f constant_on S"
dim_def: shows "local.dim = (LEAST na. \<exists>A. finite A \<and> card A = na \<and> A \<subseteq> carrier_vec n \<and> local.span A = carrier_vec n)"
less_imp_le: fixes x :: "'a"   and y :: "'a" assumes "x \<in> U"   and "y \<in> U"   and "x <\<^sub>o\<^sub>w y" shows "x \<le>\<^sub>o\<^sub>w y"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) heap \<Rightarrow> bool"   and known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> bool"   and get_child_nodes :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M :: "(_) object_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes get_child_nodes_locs get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M" shows "l_new_shadow_root_get_child_nodes\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf known_ptr get_child_nodes get_child_nodes_locs type_wf\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_child_nodes_locs\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M"
col_transpose: fixes i :: "nat"   and A :: "'b mat" assumes "i < dim_row A" shows "col A\<^sup>T i = row A i"
mono_maps_chain_to_chain: fixes f :: "'b set \<Rightarrow> 'c set"   and C :: "nat \<Rightarrow> 'b set" assumes "mono f"   and "chain C" shows "chain (f \<circ> C)"
valid_path_in_V: fixes P :: "'a llist" assumes "valid_path P" shows "lset P \<subseteq> V"
openin_topology\<^sub>N_D: fixes N :: "'a quasinorm"   and U :: "'a set"   and x :: "'a" assumes "openin (topology\<^sub>N N) U"   and "x \<in> U" shows "\<exists>e>0. \<forall>y. eNorm N (y - x) < e \<longrightarrow> y \<in> U"
delete_root_cases: fixes x :: "'b tree" obtains n :: "'b" and r :: "'b tree" and h :: "nat" where "x = MKT na__ ET r ha__"   | n :: "'b" and v :: "'b" and va :: "'b tree" and vb :: "'b tree" and vc :: "nat" and h :: "nat" where "x = MKT nb__ (MKT v__ va__ vb__ vc__) ET hb__"   | n :: "'b" and v :: "'b" and va :: "'b tree" and vb :: "'b tree" and vc :: "nat" and vd :: "'b" and ve :: "'b tree" and vf :: "'b tree" and vg :: "nat" and h :: "nat" where "x = MKT n (MKT v va vb vc) (MKT vd ve vf vg) h"   | "x = ET"
assms: shows "equivalentFormulae (F1 @ val2form v @ F2) F'"
simps_1: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and u :: "nat"   and v :: "nat" shows "Graph.isPath c u [] v = (u = v)"
basic_subtype_x_ObjectType: fixes \<tau> :: "'a basic_type"   and \<C> :: "'a" assumes "\<tau> \<sqsubset>\<^sub>B \<langle>\<C>\<rangle>\<^sub>\<T>" obtains "\<tau> = OclVoid"   | \<C>' :: "'a" where "\<tau> = \<langle>\<C>'\<rangle>\<^sub>\<T>" and "\<C>' < \<C>"
lm023: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N1 :: "'c set"   and N2 :: "'c set" assumes "a \<in> allAllocations N1 G" shows "a \<in> allAllocations (N1 \<union> N2) G"
u_Abs_def: shows "u_Abs \<equiv> apply_sign Abs_freeletter"
OclAnd_false2: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(X and false) = false"
simps_1: fixes wits :: "'a list"   and w :: "'a vec" shows "adjuster_wit wits w [] = (wits, 0\<^sub>v n)"
distinct_21: fixes x2 :: "('c, 'd) Term.term"   and x71 :: "'d list"   and x72 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and x73 :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list" shows "receive\<langle>x2\<rangle> \<noteq> \<forall>x71\<langle>\<or>\<noteq>: x72 \<or>\<notin>: x73\<rangle>"
exec_instr: fixes ta :: "('addr, 'thread_id, 'heap) jvm_thread_action"   and xcp :: "'addr option"   and h' :: "'heap"   and stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and ins :: "'addr instr list"   and pc :: "nat"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and t :: "'thread_id"   and h :: "'heap"   and stk :: "'addr val list"   and loc :: "'addr val list"   and ci :: "('addr, 'heap) check_instr"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list" assumes "(ta, xcp, h', [(stk', loc', undefined, undefined, pc')]) \<in> exec_instr (ins ! pc) P t h stk loc undefined undefined pc []"   and "pc < length ins"   and "ci_app ci (ins ! pc) P h stk loc undefined undefined pc []" shows "exec_meth ci P ins xt t h (stk, loc, pc, None) ta h' (stk', loc', pc', xcp)"
buffer_defs_3: fixes ic :: "state \<Rightarrow> 'b"   and q :: "state \<Rightarrow> 'b list"   and oc :: "state \<Rightarrow> 'b" shows "Deq ic q oc \<equiv> #0 < length<$q> \<and> oc$ = hd<$q> \<and> q$ = tl<$q> \<and> ic$ = $ic"
nprv_dsjIR: fixes F :: "'fmla set"   and \<psi> :: "'fmla"   and \<phi> :: "'fmla" assumes "nprv F \<psi>"   and "F \<subseteq> fmla"   and "finite F"   and "\<phi> \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv F (dsj \<phi> \<psi>)"
qo_on_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a set" shows "qo_on P A = (reflp_on P A \<and> transp_on P A)"
steps_z_beta_complete: fixes X :: "'d set"   and v :: "'d \<Rightarrow> nat"   and n :: "nat"   and not_in_X :: "'d"   and A :: "('e \<times> ('d, real) cconstraint \<times> 'f \<times> 'd list \<times> 'e) set \<times> ('e \<Rightarrow> ('d, real) cconstraint)"   and l :: "'e"   and Z :: "('d \<Rightarrow> real) set"   and l' :: "'e"   and Z' :: "('d \<Rightarrow> real) set"   and k :: "'d \<Rightarrow> nat" assumes "Regions X v n not_in_X"   and "A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>* \<langle>l', Z'\<rangle>"   and "valid_abstraction A X (\<lambda>x. real (k x))"   and "Z \<subseteq> Regions.V X" shows "\<exists>Z''. Regions.steps_z_beta X k v n not_in_X A l Z l' Z'' \<and> Z' \<subseteq> Z''"
concurrent_expunge_delete_independent_technical: fixes i :: "'a"   and "is" :: "'a set"   and xs :: "('a \<times> ('a, 'b) operation) event list"   and j :: "nat"   and e :: "'b"   and mo :: "'a"   and ir :: "'a" assumes "i \<in> is"   and "xs prefix of j"   and "(i, Expunge e mo i) \<in> set (node_deliver_messages xs)"   and "(ir, Delete is e) \<in> set (node_deliver_messages xs)" shows "hb (i, Expunge e mo i) (ir, Delete is e)"
ereal_divide: fixes r :: "real"   and p :: "real" shows "ereal r / ereal p = (if p = 0 then ereal r * \<infinity> else ereal (r / p))"
le: fixes i :: "nat"   and j :: "nat" assumes "i \<le> n"   and "j \<le> n"   and "i \<noteq> j" shows "M i j \<le> M1 i j"
valid_sops_stmt: fixes ts :: "((stmt \<times> nat) \<times> instr list \<times> (nat \<Rightarrow> 'a option) \<times> (stmt \<times> nat) memref list \<times> 'b \<times> 'c) list"   and i :: "nat"   and s :: "stmt"   and t :: "nat"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> 'a option"   and sb :: "(stmt \<times> nat) memref list"   and \<D> :: "'b"   and \<O> :: "'c" assumes "separated_tmps ts"   and "i < length ts"   and "ts ! i = ((s, t), is, \<theta>, sb, \<D>, \<O>)" shows "valid_sops_stmt t s"
indhom_eq: fixes A :: "('c, 'd) Ring_scheme"   and R :: "('e, 'f) Ring_scheme"   and f :: "'c \<Rightarrow> 'e" assumes "Ring A"   and "Ring R"   and "f \<in> rHom A R" shows "f\<degree>\<^bsub>A,rimg A R f\<^esub> = f\<degree>\<^bsub>A,R\<^esub>"
isSubobjs_subclsR: 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 C :: "char list"   and Cs :: "char list list"   and D :: "char list"   and D' :: "char list"   and Cs' :: "char list list" assumes "is_subobj P (C, Cs @ [D, D'] @ Cs')" shows "P \<turnstile> D \<prec>\<^sub>R D'"
bind_eq_None_conv: fixes a :: "'g option"   and f :: "'g \<Rightarrow> 'f option" shows "(a \<bind> f = None) = (a = None \<or> f (the a) = None)"
g2: shows "g2 \<in> carrier G \<and> \<phi> g2 y = z"
vint_fpair_in_fproduct_vint: fixes a :: "int"   and b :: "int" shows "[a\<^sub>\<int>, b\<^sub>\<int>]\<^sub>\<circ> \<in>\<^sub>\<circ> \<int>\<^sub>\<circ> ^\<^sub>\<times> 2\<^sub>\<nat>"
intros_1: fixes x :: "'k"   and \<Gamma> :: "('k, 'l) web_scheme"   and f :: "'k \<times> 'k \<Rightarrow> ennreal" assumes "x \<in> A \<Gamma>" shows "x \<in> SAT \<Gamma> f"
avl_insert: fixes t :: "('b \<times> bal) tree"   and x :: "'b" assumes "avl t" shows "avl (AVL_Bal2_Set.insert x t)"
Int_lower2: fixes A :: "'d set"   and B :: "'d set" shows "A \<inter> B \<subseteq> B"
proj_ord_def: fixes cmp :: "(('a, 'b) pp \<times> nat) nat_term_order" shows "proj_ord cmp = Abs_nat_term_order (proj_comp cmp)"
map_fst_enumerate: fixes n :: "nat"   and xs :: "'d list" shows "map fst (List.enumerate n xs) = [n..<n + length xs]"
conv_dist_sup: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and conv :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "class.bounded_distrib_allegory one times inf less_eq less sup bot top conv" shows "conv (sup x y) = sup (conv x) (conv y)"
eq_LConsD: fixes xs :: "'a llist"   and y :: "'a"   and ys :: "'a llist" assumes "xs = LCons y ys" shows "xs \<noteq> LNil \<and> lhd xs = y \<and> ltl xs = ys"
wp: shows "\<I> \<turnstile>g GPV p \<surd>"
comp_inv_arr': fixes f :: "'b" assumes "\<phi>.B.iso f" shows "\<phi>.B.inv f \<cdot>\<^sub>B f = \<phi>.B.dom f"
spair_sigs_alt_spp: fixes p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" shows "spair_sigs p q = spair_sigs_spp (spp_of p) (spp_of q)"
assms_1: shows "basis_in \<E> B\<^sub>1"
rel_induct: fixes R1 :: "'n \<Rightarrow> 'o \<Rightarrow> bool"   and x :: "('n, 'p, 'q, 'r, 's, 't) T"   and y :: "('o, 'p, 'q, 'r, 's, 't) T"   and Q :: "('n, 'p, 'q, 'r, 's, 't) T \<Rightarrow> ('o, 'p, 'q, 'r, 's, 't) T \<Rightarrow> bool" assumes "rell_T R1 x y"   and "\<And>a b. rell_G Q R1 a b \<Longrightarrow> Q (C_T a) (C_T b)" shows "Q x y"
quotientI2: fixes x :: "'a"   and A :: "'a set"   and X :: "'a set"   and r :: "('a \<times> 'a) set" assumes "x \<in> A"   and "X = r `` {x}" shows "X \<in> A // r"
comm_semiring_axioms_def: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c" shows "class.comm_semiring_axioms plus times \<equiv> \<forall>a b c. times (plus a b) c = plus (times a c) (times b c)"
diamond_cube_gen_def: shows "diamond_cube_gen \<equiv> \<lambda>(x, y). (d * x_coord x, (2 * y - 1) * (d * diamond_y_gen (x_coord x)))"
inv_closed: fixes x :: "'a" assumes "x \<in> carrier G" shows "f (inv\<^bsub>G\<^esub> x) \<in> carrier C"
open_inside: fixes S :: "'a set" assumes "closed S" shows "open (inside S)"
s_id_iff: fixes a :: "'a"   and A :: "'a set" shows "((a, A) \<in> 1\<^sub>\<sigma>) = (A = {a})"
plus_period: fixes x :: "'b" shows "sin (x + (2::'b) * of_real pi) = sin x"
h4: shows "satisfy_equation p (qs1 @ qs2) (get_subsets (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2)))) (get_signs (snd (combine_systems p (qs1, calculate_data p qs1) (qs2, calculate_data p qs2))))"
subtopology_restrict: fixes X :: "'c topology"   and S :: "'c set" shows "subtopology X (topspace X \<inter> S) = subtopology X S"
prefixes_length: fixes t :: "nat"   and j :: "nat"   and b :: "nat" assumes "local.prefixes t j \<down>= b" shows "t < e_length b"
Vagree_def: fixes \<nu> :: "variable \<Rightarrow> real"   and \<nu>' :: "variable \<Rightarrow> real"   and V :: "variable set" shows "Vagree \<nu> \<nu>' V \<equiv> \<forall>i. i \<in> V \<longrightarrow> \<nu> i = \<nu>' i"
cnf_disj: fixes R :: "'b literal list" shows "cnf (disj_of_clause R) = {set R}"
select_convs_3: fixes mReturn :: "nat"   and mName :: "nat"   and mParams :: "varDef list"   and mBody :: "exp"   and more :: "'a" shows "mParams \<lparr>mReturn = mReturn, mName = mName, mParams = mParams, mBody = mBody, \<dots> = more\<rparr> = mParams"
pref_equal: fixes u :: "'b list"   and v :: "'b list"   and w :: "'b list" assumes "u \<le>p v \<cdot> w"   and "\<^bold>|u\<^bold>| = \<^bold>|v\<^bold>|" shows "u = v"
lexlist_ext: fixes xs :: "'b list"   and ys :: "'b list" assumes "Lex xs = Lex ys" shows "xs = ys"
d_IN_lift_Inner: fixes x :: "'v"   and f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "x \<noteq> source \<Delta>" shows "d_IN (lift f) (Inner x) = d_IN f x"
odd_numeral_BitM: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and w :: "num" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (numeral.numeral one plus (Num.BitM w))"
arc_imp_path: fixes g :: "real \<Rightarrow> 'b" assumes "arc g" shows "path g"
dvd_div_iff_mult: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and c :: "'a"   and b :: "'a"   and a :: "'a" assumes "class.algebraic_semidom divide plus minus zero times one"   and "c \<noteq> zero"   and "dvd.dvd times c b" shows "dvd.dvd times a (divide b c) = dvd.dvd times (times a c) b"
star_inductl_var_equiv: fixes x :: "'a"   and y :: "'a" shows "(x\<^sup>\<star> ; y \<le> y) = (x ; y \<le> y)"
fps_nth_Abs_fps: fixes f :: "'d list \<Rightarrow> 'c"   and n :: "'d list" shows "Abs_fps f $ n = f n"
intro: fixes set_of :: "'c \<Rightarrow> 'b set"   and r :: "'b \<Rightarrow> 'c" assumes "\<And>a. finite {b. (a, b) \<in> {(a, b). b \<in> set_of (r a)}\<^sup>*}" shows "relation_subsumption_impl_axioms r set_of"
least_least: fixes k :: "'b"   and A :: "'b set" assumes "k \<in> A" shows "least A \<le> k"
simps: fixes cs :: "('a \<times> constraint) list"   and J :: "'a set"   and s :: "QDelta ns_constraint list \<times> (('a, ('a \<times> QDelta atom) list) mapping \<times> ((nat, QDelta) mapping \<Rightarrow> (nat, QDelta) mapping) \<times> 'a list) \<times> ('a, QDelta) state" shows "invariant_simplex cs J (Simplex_State s) = invariant_simplex' cs J s"
hd_fst: fixes V :: "nat set"   and V' :: "nat set"   and \<Gamma> :: "nat \<Rightarrow> pdf_type"   and \<delta> :: "(nat \<Rightarrow> val) \<Rightarrow> ennreal"   and e :: "expr"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal" assumes "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d e \<Rightarrow> f" shows "(V, V', \<Gamma>, \<delta>) \<turnstile>\<^sub>d Fst $$ e \<Rightarrow> (\<lambda>\<rho> x. \<integral>\<^sup>+ y. f \<rho> <|x, y|> \<partial>stock_measure (the (expr_type \<Gamma> (Snd $$ e))))"
well_analyzed_inv: fixes A :: "('fun, 'var) extstrand_step list"   and t :: "('fun, 'var) Term.term" assumes "well_analyzed (A @ [Decomp t])" shows "t \<in> subterms\<^sub>s\<^sub>e\<^sub>t (ik\<^sub>e\<^sub>s\<^sub>t A \<union> assignment_rhs\<^sub>e\<^sub>s\<^sub>t A) - range Var"
sub_comp: fixes f :: "'g \<Rightarrow> 'f"   and g :: "'h \<Rightarrow> 'g"   and p :: "('e, 'h) fm" shows "sub f (sub g p) = sub (f \<circ> g) p"
block_Suc_3: fixes b :: "nat"   and i :: "nat"   and t :: "nat"   and k :: "nat" assumes "b = block i t"   and "k = gap i t"   and "change_on_neither b k" shows "block i (Suc t) = e_snoc b 0"
Un_empty_left: fixes B :: "'b set" shows "{} \<union> B = B"
apply_cltn2_preserve_incident: fixes p :: "proj2"   and A :: "cltn2"   and l :: "proj2_line" shows "proj2_incident (apply_cltn2 p A) (apply_cltn2_line l A) = proj2_incident p l"
simps_1: fixes T :: "POPLmarkRecord.type" shows "\<parallel>PVar T\<parallel>\<^sub>p = 1"
intro: fixes R :: "('e, 'f) ring_scheme" assumes "cring R"   and "domain_axioms R" shows "domain R"