hcju/isabelleps huggingface dataset:
smodule_ideal_coeff_def: fixes R :: "('f, 'g) Ring_scheme"   and M :: "('e, 'f, 'h) Module_scheme"   and A :: "'f set" shows "A \<odot>\<^bsub>R\<^esub> M = linear_span R M A (carrier M)"
notin_List_update: fixes a :: "'c"   and as :: "'c list"   and h :: "'c \<Rightarrow> 'c ref"   and y :: "'c ref"   and x :: "'c ref" assumes "a \<notin> set as" shows "List (h(a := y)) x as = List h x as"
assms_3: shows "h\<^sub>0 \<bottom> h\<^sub>1"
prime_divisor_exists: fixes a :: "'a" assumes "a \<noteq> (0::'a)"   and "\<not> is_unit a" shows "\<exists>b. b dvd a \<and> prime b"
iso_sym: fixes G :: "('f, 'g) monoid_scheme"   and H :: "('h, 'i) monoid_scheme" assumes "Group.group G"   and "G \<cong> H" shows "H \<cong> G"
intro: fixes \<Omega> :: "'c set"   and M :: "'c set set" assumes "\<Omega> \<in> M"   and "\<And>A. A \<in> M \<Longrightarrow> \<Omega> - A \<in> M"   and "\<And>A. \<lbrakk>disjoint_family A; range A \<subseteq> M\<rbrakk> \<Longrightarrow> \<Union> (range A) \<in> M" shows "Dynkin_system_axioms \<Omega> M"
eq_iff_1: fixes tm1 :: "tm"   and tm2 :: "tm"   and tm1a :: "tm"   and tm2a :: "tm" shows "(tm1 IN tm2 = tm1a IN tm2a) = (tm1 = tm1a \<and> tm2 = tm2a)"
cases: fixes a1 :: "CL"   and a2 :: "CL" assumes "(a1, a2) \<in> red" obtains t :: "CL" and t' :: "CL" and u :: "CL" where "a1 = ` ta__ ua__" and "a2 = ` t' ua__" and "(ta__, t') \<in> red"   | u :: "CL" and u' :: "CL" and t :: "CL" where "a1 = ` t u" and "a2 = ` t u'" and "(u, u') \<in> red"   | x :: "CL" and y :: "CL" and z :: "CL" where "a1 = ` ` ` S x__ y__ z" and "a2 = ` ` x__ z ` y__ z"   | x :: "CL" and y :: "CL" where "a1 = ` ` K xa__ y" and "a2 = xa__"   | x :: "CL" where "a1 = ` I x" and "a2 = x"
nth_enumerate_eq: fixes m :: "nat"   and xs :: "'f list"   and n :: "nat" assumes "m < length xs" shows "enumerate n xs ! m = (n + m, xs ! m)"
prodinf_split_initial_segment: fixes f :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "convergent_prod f"   and "\<And>i. i < n \<Longrightarrow> f i \<noteq> (0::'a)" shows "prodinf f = (\<Prod>i. f (i + n)) * prod f {..<n}"
succsr_is_node: fixes x :: "bool list"   and y :: "bool list" assumes "(x, y) \<in> (succsr (subset_succs A))\<^sup>*"   and "nfa_is_node A x" shows "nfa_is_node A y"
vec_of_basis_enum_ket: fixes i :: "'b" shows "vec_of_basis_enum (ket i) = unit_vec CARD('b) (enum_idx i)"
inf2D1: fixes A :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'a"   and y :: "'b" assumes "inf A B x y" shows "A x y"
designI: fixes point_set :: "'a set"   and block_collection :: "'a set multiset" assumes "finite_incidence_system point_set block_collection"   and "\<And>b. b \<in># block_collection \<Longrightarrow> b \<noteq> {}"   and "block_collection \<noteq> {#}"   and "point_set \<noteq> {}" shows "design point_set block_collection"
Inf_exp_def: fixes S :: "('a \<Rightarrow> real) set" shows "Inf_exp S = (\<lambda>s. Inf {f s |f. f \<in> S})"
indecomposable_ordertype_ge: fixes \<alpha> :: "V"   and A :: "V set"   and B :: "V set" assumes "indecomposable \<alpha>"   and "\<alpha> \<le> tp A"   and "small A"   and "small B" shows "\<alpha> \<le> tp B \<or> \<alpha> \<le> tp (A - B)"
abs_ln_one_plus_x_minus_x_bound_nonneg: fixes x :: "real" assumes "0 \<le> x"   and "x \<le> 1" shows "\<bar>ln (1 + x) - x\<bar> \<le> x\<^sup>2"
associative: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<in> f ` A"   and "b \<in> f ` A"   and "c \<in> f ` A" shows "addB (addB a b) c = addB a (addB b c)"
rel_refl: fixes Ra :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "'g list" assumes "\<And>x. Ra x x" shows "list_all2 Ra x x"
T_splay_simps_5: fixes x :: "'a"   and b :: "'a"   and a :: "'a"   and A :: "'a tree"   and B :: "'a tree"   and CD :: "'a tree" assumes "x < b"   and "a < x" shows "T_splay x \<langle>\<langle>A, a, B\<rangle>, b, CD\<rangle> = (if B = \<langle>\<rangle> then 1 else T_splay x B + 1)"
edges_are_in_V_1: fixes v :: "'a"   and w :: "'a" assumes "v \<rightarrow>\<^bsub>G'\<^esub> w" shows "v \<in> V\<^bsub>G'\<^esub>"
submatrix_hma_def: fixes A :: "(('b, 'e) Finite_Cartesian_Product.vec, 'f) Finite_Cartesian_Product.vec"   and I :: "nat set"   and J :: "nat set" shows "submatrix_hma A I J = (\<chi>a b. A $h Bij_Nat.from_nat (pick I (Bij_Nat.to_nat a)) $h Bij_Nat.from_nat (pick J (Bij_Nat.to_nat b)))"
of_nat_0_le_iff: fixes one :: "'g"   and times :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and plus :: "'g \<Rightarrow> 'g \<Rightarrow> 'g"   and zero :: "'g"   and n :: "nat" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq zero (semiring_1.of_nat one plus zero n)"
int_pow_inv: fixes x :: "(((nat \<Rightarrow> real) \<Rightarrow> nat \<Rightarrow> real) \<Rightarrow>\<^sub>0 int) set"   and i :: "int" assumes "x \<in> carrier (relative_homology_group (int n) (nsphere n) (equator n))" shows "inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> i = inv\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> (x [^]\<^bsub>relative_homology_group (int n) (nsphere n) (equator n)\<^esub> i)"
next_free_and: fixes \<phi> :: "'b pltl"   and \<psi> :: "'b pltl" shows "next_free (\<phi> and\<^sub>p \<psi>) = (next_free \<phi> \<and> next_free \<psi>)"
transpose_combine_matrix: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and A :: "'c matrix"   and B :: "'d matrix" assumes "f (0::'c) (0::'d) = (0::'b)" shows "transpose_matrix (combine_matrix f A B) = combine_matrix f (transpose_matrix A) (transpose_matrix B)"
interaction_bounded_by_start: fixes "consider" :: "'d \<Rightarrow> bool"   and gpv :: "('e, 'd, 'f) gpv"   and n :: "enat"   and m :: "enat" assumes "interaction_bounded_by consider gpv n"   and "n \<le> m" shows "interaction_bounded_by consider gpv m"
wordinterval_intersection_def: fixes r1 :: "'b wordinterval"   and r2 :: "'b wordinterval" shows "wordinterval_intersection r1 r2 \<equiv> wordinterval_compress (??.WordInterval.wordinterval_intersection' r1 r2)"
rtr_nomac_eq: fixes ifl :: "interface list"   and pi :: "'a Simple_Packet.simple_packet_ext"   and mlf :: "'a word \<Rightarrow> 48 word option"   and rt :: "'a routing_rule list"   and fw :: "'a simple_rule list" assumes "iface_packet_check ifl pi \<noteq> None"   and "mlf (fromMaybe (p_dst pi) (next_hop (routing_table_semantics rt (p_dst pi)))) \<noteq> None" shows "\<exists>x. map_option (p_l2dst_update (\<lambda>_. x)) (simple_linux_router_nol12 rt fw pi) = simple_linux_router rt fw mlf ifl pi"
subcls1I: fixes P :: "'b prog"   and C :: "char list"   and D :: "char list"   and rest :: "fdecl list \<times> 'b mdecl list" assumes "class P C = \<lfloor>(D, rest)\<rfloor>"   and "C \<noteq> Object" shows "P \<turnstile> C \<prec>\<^sup>1 D"
qfree_and: fixes \<phi>\<^sub>1 :: "'a fm"   and \<phi>\<^sub>2 :: "'a fm" assumes "qfree \<phi>\<^sub>1"   and "qfree \<phi>\<^sub>2" shows "qfree (and \<phi>\<^sub>1 \<phi>\<^sub>2)"
assms: shows "\<exists>!i. n \<le> i \<and> \<parallel>c\<parallel>\<^bsub>t i\<^esub>"
INTER_eq: fixes B :: "'b \<Rightarrow> 'a set"   and A :: "'b set" shows "\<Inter> (B ` A) = {y. \<forall>x\<in>A. y \<in> B x}"
Proc_CFG_SeqFirst: fixes c\<^sub>1 :: "cmd"   and n :: "label"   and et :: "p_edge_kind"   and n' :: "label"   and c\<^sub>2 :: "cmd" assumes "c\<^sub>1 \<turnstile> n -et\<rightarrow>\<^sub>p n'"   and "n' \<noteq> Exit" shows "c\<^sub>1;; c\<^sub>2 \<turnstile> n -et\<rightarrow>\<^sub>p n'"
RedCASSucceed: fixes h :: "'heap"   and a :: "'addr"   and D :: "String.literal"   and F :: "String.literal"   and v :: "'addr val"   and v' :: "'addr val"   and h' :: "'heap"   and extTA :: "('addr, 'thread_id, 'heap) external_thread_action \<Rightarrow> ('addr, 'thread_id, 'b, 'heap) Jinja_thread_action"   and P :: "'addr J_prog"   and t :: "'thread_id"   and l :: "String.literal \<Rightarrow> 'addr val option" assumes "heap_read h a (CField D F) v"   and "heap_write h a (CField D F) v' h'" shows "extTA,P,t \<turnstile> \<langle>addr a\<bullet>compareAndSwap(D\<bullet>F, Val v, Val v'),(h, l)\<rangle> -\<lbrace>ReadMem a (CField D F) v, WriteMem a (CField D F) v'\<rbrace>\<rightarrow> \<langle>true,(h', l)\<rangle>"
preal_exists_bound: fixes A :: "rat set" assumes "Dedekind_Real.cut A" shows "\<exists>x>0. x \<notin> A"
QCong_def: fixes l :: "'p \<Rightarrow> 'p \<Rightarrow> bool" shows "QCong l \<equiv> \<exists>A B. \<forall>X Y. Cong A B X Y = l X Y"
ex_sup: fixes x :: "'c"   and y :: "'c" assumes "Lattice.class.lattice TYPE('c)" shows "\<exists>sup. is_sup x y sup"
simps_2: fixes f :: "'b" shows "pderiv_coeffs_code f [] = []"
injective_def: fixes f :: "'b \<Rightarrow> 'c" shows "injective f \<equiv> \<forall>x y. f x = f y \<longrightarrow> x = y"
is_extensional: fixes f :: "'a" assumes "\<not> arr f" shows "cod \<iota> \<otimes> f = null"
eq_nodes_dfa_trans: fixes M :: "nat bdd list \<times> bool list"   and v :: "nat"   and p :: "nat"   and q :: "nat"   and bs :: "bool list" assumes "eq_nodes M v p q"   and "is_alph v bs" shows "eq_nodes M v (dfa_trans M p bs) (dfa_trans M q bs)"
cfiov: shows "f\<inverse> O ov \<subseteq> ov"
arr_cod: fixes f :: "('o, 'a) arr" assumes "arr f" shows "arr (cod f)"
inv_all'_def: fixes p :: "'a mat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> bool"   and A :: "'a mat" shows "local.inv_all' p A \<equiv> \<forall>i j. i < dim_row A \<longrightarrow> j < dim_row A \<longrightarrow> p A i j"
p: fixes y :: "xml" shows "monotone sum_bot.le_fun sum_bot_ord (p1 y)"
IfTrue: fixes eval :: "exp \<Rightarrow> (loc \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> (loc \<Rightarrow> nat) \<Rightarrow> bool"   and e :: "exp"   and s :: "loc \<Rightarrow> nat"   and s' :: "loc \<Rightarrow> nat"   and c0 :: "com"   and s1 :: "loc \<Rightarrow> nat"   and c1 :: "com" assumes "eval e s 0 s'"   and "execp eval c0 s' s1" shows "execp eval (IF e THEN c0 ELSE c1) s s1"
intros_4: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int"   and c\<^sub>1 :: "com"   and t :: "char list \<Rightarrow> int"   and c\<^sub>2 :: "com" assumes "bval b s"   and "(c\<^sub>1, s) \<Rightarrow> t" shows "(IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<Rightarrow> t"
index_mat_1: fixes i :: "nat"   and nr :: "nat"   and j :: "nat"   and nc :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'b" assumes "i < nr"   and "j < nc" shows "mat nr nc f $$ (i, j) = f (i, j)"
map_def: fixes f :: "'t" shows "S'.map f = (if S'.arr f then f else S'.null)"
ray_to_rel_frontier: fixes S :: "'c set"   and a :: "'c"   and l :: "'c" assumes "bounded S"   and "a \<in> rel_interior S"   and "a + l \<in> affine hull S"   and "l \<noteq> (0::'c)" obtains d :: "real" where "0 < d" and "a + d *\<^sub>R l \<in> rel_frontier S" and "\<And>e. \<lbrakk>0 \<le> e; e < d\<rbrakk> \<Longrightarrow> a + e *\<^sub>R l \<in> rel_interior S"
has_products_def: fixes I :: "'c set" shows "has_products I \<equiv> I \<noteq> UNIV \<and> (\<forall>J D. discrete_diagram J (\<cdot>) D \<and> Collect (partial_magma.arr J) = I \<longrightarrow> (\<exists>a. has_as_product J D a))"
t_stop_nonneg: shows "0 \<le> other.t_stop"
intro: fixes impl :: "('d, 'c, 'e) subseqs_foldr_impl"   and invariant :: "'c \<Rightarrow> 'd list \<Rightarrow> nat \<Rightarrow> 'e \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<And>base elements n out state. subseqs_foldr impl base elements n = (out, state) \<Longrightarrow> invariant base elements n state \<and> set out = subseqs_foldr_impl.S f base elements n"   and "\<And>state out state' base elements n. \<lbrakk>next_subseqs_foldr impl state = (out, state'); invariant base elements n state\<rbrakk> \<Longrightarrow> invariant base elements (Suc n) state' \<and> set out = subseqs_foldr_impl.S f base elements (Suc n)" shows "correct_subseqs_foldr_impl f impl invariant"
sel_10: fixes x1 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x2 :: "('b, 'a) state \<Rightarrow> (nat, 'b \<times> 'a) mapping"   and x3 :: "('b, 'a) state \<Rightarrow> (nat, 'b \<times> 'a) mapping"   and x4 :: "('b, 'a) state \<Rightarrow> nat \<Rightarrow> 'a option"   and x5 :: "('b, 'a) state \<Rightarrow> nat \<Rightarrow> 'a option"   and x6 :: "('b, 'a) state \<Rightarrow> nat \<Rightarrow> 'b"   and x7 :: "('b, 'a) state \<Rightarrow> nat \<Rightarrow> 'b"   and x8 :: "((nat, 'b \<times> 'a) mapping \<Rightarrow> (nat, 'b \<times> 'a) mapping) \<Rightarrow> ('b, 'a) state \<Rightarrow> ('b, 'a) state"   and x9 :: "nat \<Rightarrow> 'a \<Rightarrow> 'a atom"   and x10 :: "nat \<Rightarrow> 'a \<Rightarrow> 'a atom"   and x11 :: "rat \<Rightarrow> rat \<Rightarrow> bool" shows "Direction.GE (Direction x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11) = x10"
td_ext_ubin: shows "td_ext uint word_of_int (uints LENGTH('a)) (take_bit LENGTH('a))"
wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t_decomp_rm\<^sub>e\<^sub>s\<^sub>t_subset: fixes A :: "('fun, 'var) extstrand_step list" shows "wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t (decomp_rm\<^sub>e\<^sub>s\<^sub>t A) \<subseteq> wfrestrictedvars\<^sub>e\<^sub>s\<^sub>t A"
fun_upds_apply: fixes m :: "'b \<Rightarrow> 'a option"   and xs :: "'b list"   and ys :: "'a list"   and x :: "'b" shows "(m(xs [\<mapsto>] ys)) x = (let xs' = take (length ys) xs in if x \<in> set xs' then \<lfloor>ys ! last_index xs' x\<rfloor> else m x)"
eq1: fixes a :: "nat"   and b :: "nat"   and i :: "nat" shows "snd (Term_Order.of_exps a b i) = i"
sub_block_mem: fixes p :: "('b, 'c) fm"   and block :: "('b, 'c) fm list \<times> 'c"   and f :: "'c \<Rightarrow> 'd" assumes "p on block" shows "sub f p on sub_block f block"
f1: fixes x :: "'a" assumes "norm x = 1" shows "norm (f x) = 1"
assms_3: shows "\<not> \<not> (\<exists>w\<in>V\<^bsub>subgame V'\<^esub>. v \<rightarrow>\<^bsub>subgame V'\<^esub> w)"
lens_equiv_def: fixes X :: "'e \<Longrightarrow> 'f"   and Y :: "'g \<Longrightarrow> 'f" shows "X \<approx>\<^sub>L Y = (X \<subseteq>\<^sub>L Y \<and> Y \<subseteq>\<^sub>L X)"
upd_None_map_le: fixes f :: "'d \<Rightarrow> 'e option"   and x :: "'d" shows "f(x := None) \<subseteq>\<^sub>m f"
integral_holo: fixes a :: "real"   and b :: "real"   and A :: "complex set" assumes "0 < min a b"   and "max a b < 1" shows "(\<lambda>z. complex_interval_lebesgue_integral lborel (ereal a) (ereal b) (f z)) holomorphic_on A"
mid_le_2: fixes i :: "float interval" shows "mid i \<le> upper i"
exhaust: fixes y :: "intv" obtains x1 :: "nat" where "y = Const x1"   | x2 :: "nat" where "y = Intv x2"   | x3 :: "nat" where "y = Greater x3"
rreq_dip_in_vD_dip_eq_ip: fixes i :: "nat" shows "paodv i \<TTurnstile> onl \<Gamma>\<^sub>A\<^sub>O\<^sub>D\<^sub>V (\<lambda>(\<xi>, l). (l \<in> {PRreq-:16..PRreq-:18} \<longrightarrow> dip \<xi> \<in> vD (rt \<xi>)) \<and> (l \<in> {PRreq-:5, PRreq-:6} \<longrightarrow> dip \<xi> = ip \<xi>) \<and> (l \<in> {PRreq-:15..PRreq-:18} \<longrightarrow> dip \<xi> \<noteq> ip \<xi>))"
block_size_t: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and u_block_size :: "int"   and grouping :: "int"   and min_index :: "int" assumes "t_covering_design point_set block_collection u_block_size grouping min_index" shows "grouping \<le> u_block_size"
pos_less_minus_divideC_eq: fixes c :: "complex"   and a :: "'a"   and b :: "'a" assumes "0 < c" shows "(a < - (b /\<^sub>C c)) = (c *\<^sub>C a < - b)"
squarefree_primorial': fixes x :: "nat" shows "squarefree (primorial' x)"
in_arcs_imp_in_arcs_ends: fixes G :: "('c, 'd) pre_digraph"   and e :: "'d" assumes "wf_digraph G"   and "e \<in> arcs G" shows "tail G e \<rightarrow>\<^bsub>G\<^esub> head G e"
content_dvd_coeffs: fixes c :: "'a"   and p :: "'a poly" assumes "c \<in> set (coeffs p)" shows "content p dvd c"
simplify_pathE: fixes g :: "'b ugraph"   and u :: "'b"   and p :: "('b \<times> 'b) list"   and v :: "'b" assumes "path g u p v" obtains p' :: "('b \<times> 'b) list" where "path g u p' v" and "simple p'" and "set p' \<subseteq> set p"
simps_4: fixes p :: "dir list"   and dir :: "dir"   and t :: "('c, 'd) tm"   and s :: "('c, 'd) tm" shows "emb_step_at (dir.Right # p) dir (App t s) = App t (emb_step_at p dir s)"
induct: fixes r :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and x1 :: "'g"   and x2 :: "'h list"   and x3 :: "'g"   and P :: "'g \<Rightarrow> 'h list \<Rightarrow> 'g \<Rightarrow> bool" assumes "rtrancl3p r x1 x2 x3"   and "\<And>a. P a [] a"   and "\<And>a bs a' b a''. \<lbrakk>rtrancl3p r a bs a'; P a bs a'; r a' b a''\<rbrakk> \<Longrightarrow> P a (bs @ [b]) a''" shows "P x1 x2 x3"
digit_to_char_simps_1: shows "digit_to_char 0 = CHR ''0''"
atrm_imp_trm: fixes t :: "'trm" assumes "t \<in> atrm" shows "t \<in> trm"
is_gcd_connect: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and x :: "'a" assumes "class.idom times one plus zero minus uminus"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "x gcdof\<^bsub>\<lparr>carrier = UNIV - {zero}, monoid.mult = times, one = one\<rparr>\<^esub> a b = dvd.is_gcd times x a b"
\<theta>': shows "\<guillemotleft>\<theta>' : p\<^sub>0 \<star> w' \<Rightarrow> u\<guillemotright>"
intros_2: fixes M :: "'b measure"   and N :: "'b measure" assumes "space M = space N"   and "sets M \<subset> sets N" shows "M \<le> N"
inv_in_hom: fixes f :: "('a S.E.term, 'a) S.arr"   and a :: "('a S.E.term, 'a) S.arr"   and b :: "('a S.E.term, 'a) S.arr" assumes "S.iso f"   and "\<guillemotleft>f : a \<Rightarrow>\<^sub>S b\<guillemotright>" shows "\<guillemotleft>S.inv f : b \<Rightarrow>\<^sub>S a\<guillemotright>"
nprv_eql_symE01: fixes t1 :: "'trm"   and t2 :: "'trm"   and F :: "'fmla set"   and \<psi> :: "'fmla" assumes "eql t1 t2 \<in> F"   and "\<psi> = eql t2 t1 \<or> \<psi> \<in> F"   and "F \<subseteq> fmla"   and "finite F"   and "\<psi> \<in> fmla"   and "t1 \<in> trm"   and "t2 \<in> trm" shows "nprv F \<psi>"
ereal_minus_minus_image: fixes S :: "ereal set" shows "uminus ` uminus ` S = S"
less_kbo_less: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "less_kbo s t" shows "s < t"
flip_altsublist_chain_G_in_Q: fixes ss :: "'w list"   and xss :: "'w list list"   and ts :: "'w list" assumes "Abs_freelist ss \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"   and "flip_altsublist_chain (ss # xss @ [ts])" shows "Abs_freelist ts \<in> \<langle>\<Union>g\<in>FreeGroup S. lconjby g ` Abs_freeword ` map (charpair S) ` R\<rangle>"
hyps_2: shows "v \<notin> subst_domain \<sigma>1"
graph_homo_inv: fixes G :: "('a, 'b) labeled_graph"   and f :: "'b \<Rightarrow> 'c" assumes "graph G"   and "inj_on f (vertices G)" shows "graph_homomorphism (map_graph_fn G f) G ((on_graph G f)\<inverse>)"
zip_Cons_Cons: fixes x :: "'d"   and xs :: "'d list"   and y :: "'e"   and ys :: "'e list" shows "x # xs || y # ys = (x, y) # (xs || ys)"
nat_mod_as_int: shows "(mod) = (\<lambda>a b. nat (int a mod int b))"
Units_eq: shows "Units (relative_homology_group (int n) (nsphere n) (equator n)) = carrier (relative_homology_group (int n) (nsphere n) (equator n))"
properties_def: fixes lambda :: "nat"   and mu :: "nat" shows "local.properties lambda mu \<equiv> 0 < lambda \<and> inj_on seq {0..<mu + lambda} \<and> (\<forall>i\<ge>mu. \<forall>j. seq (i + j * lambda) = seq i)"
independent_insertI: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and S :: "'b set" assumes "vector_space scale"   and "a \<notin> module.span scale S"   and "\<not> module.dependent scale S" shows "\<not> module.dependent scale (insert a S)"
intros_1: fixes P :: "complex mat" assumes "is_quantum_predicate P" shows "\<turnstile>\<^sub>p {P} SKIP {P}"
wegdamit: fixes A :: "'c list"   and Strat :: "(nat \<times> nat list) list"   and b :: "'c"   and x :: "'c"   and y :: "'c"   and init :: "'c list" assumes "length A < length Strat"   and "b \<notin> {x, y}" shows "ALGxy_det Strat (A @ [b]) init x y = ALGxy_det Strat A init x y"
Derives1_skip_suffix: fixes i :: "nat"   and a :: "symbol list"   and c :: "symbol list"   and r :: "symbol \<times> symbol list"   and b :: "symbol list" assumes "i < length a"   and "Derives1 (a @ c) i r (b @ c)" shows "Derives1 a i r b"
term_variants_pred_iff_in_term_variants: fixes P :: "'c \<Rightarrow> 'c list"   and t :: "('c, 'd) Term.term"   and s :: "('c, 'd) Term.term" shows "term_variants_pred P t s = (s \<in> set (term_variants P t))"
adds_term_canc: fixes t :: "'a"   and u :: "'t"   and v :: "'t" shows "(t \<oplus> u adds\<^sub>t t \<oplus> v) = (u adds\<^sub>t v)"
subspace_span: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and S :: "'b set" assumes "Modules.module scale" shows "module.subspace scale (module.span scale S)"
ltl_const_equiv_implies_ltl_prop_equiv: fixes \<phi> :: "'b ltln"   and \<psi> :: "'b ltln" assumes "\<phi> \<sim>\<^sub>C \<psi>" shows "\<phi> \<sim>\<^sub>P \<psi>"
measurable_envelopeD2: fixes M :: "'b measure"   and A :: "'b set"   and E :: "'b set" assumes "measurable_envelope M A E" shows "emeasure M E = outer_measure_of M A"
prems_5: shows "\<forall>t1\<le>Suc t. req t1 = []"
lnull_def: fixes llist :: "'b llist" shows "lnull llist = (llist = LNil)"
tree_rank_height: fixes e :: "'a"   and a :: "'b"   and r :: "nat"   and ts :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.tree_invar (SkewBinomialHeapStruc.Node e a r ts)" shows "SkewBinomialHeapStruc.height_tree (SkewBinomialHeapStruc.Node e a r ts) = r"
cases: fixes r :: "'a monoid" obtains carrier :: "'a set" and mult :: "'a \<Rightarrow> 'a \<Rightarrow> 'a" and one :: "'a" where "r = \<lparr>carrier = carrier, monoid.mult = mult, one = one\<rparr>"
ereal_abs_add: fixes a :: "ereal"   and b :: "ereal" shows "\<bar>a + b\<bar> \<le> \<bar>a\<bar> + \<bar>b\<bar>"
incidence_system_isomorphism_def: fixes \<V> :: "'c set"   and \<B> :: "'c set multiset"   and \<V>' :: "'d set"   and \<B>' :: "'d set multiset"   and bij_map :: "'c \<Rightarrow> 'd" shows "incidence_system_isomorphism \<V> \<B> \<V>' \<B>' bij_map \<equiv> incidence_system \<V> \<B> \<and> incidence_system \<V>' \<B>' \<and> incidence_system_isomorphism_axioms \<V> \<B> \<V>' \<B>' bij_map"
trace_parallel_plan_strips_not_nil: fixes I :: "'a \<Rightarrow> bool option"   and \<pi> :: "'a strips_operator list list" shows "trace_parallel_plan_strips I \<pi> \<noteq> []"
all_roots_real_mult: fixes p :: "real poly"   and q :: "real poly" shows "all_roots_real (p * q) = (all_roots_real p \<and> all_roots_real q)"
has_integral_set_lebesgue: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "f absolutely_integrable_on S" shows "(f has_integral set_lebesgue_integral lebesgue S f) S"
A: shows "vid_on (\<BB>\<lparr>Obj\<rparr>)\<lparr>A\<rparr> = A"
is_iso_def: fixes \<phi> :: "'a \<Rightarrow> 'b"   and Fs\<^sub>1 :: "'a list list"   and Fs\<^sub>2 :: "'b list list" shows "is_iso \<phi> Fs\<^sub>1 Fs\<^sub>2 \<equiv> is_Iso \<phi> (set Fs\<^sub>1) (set Fs\<^sub>2)"
onB: shows "on_triple (f_a \<union> f_b) `` edges B = on_triple (Id_on (vertices B) O (f_a \<union> f_b)) `` edges B"
snoc: shows "suffixes xs = rev (prefix (length xs) (\<lambda>i. drop i xs))"
less_than_iff: fixes x :: "nat"   and y :: "nat" shows "((x, y) \<in> less_than) = (x < y)"
f1: shows "Deliver (i, Append i e2) \<notin> set (history k)"
cfs_monom: fixes R :: "('c, 'd) ring_scheme"   and a :: "'c"   and m :: "nat"   and n :: "nat" assumes "UP_ring R"   and "a \<in> carrier R" shows "monom (UP R) a m n = (if m = n then a else \<zero>\<^bsub>R\<^esub>)"
agreeDT: fixes s :: "'d \<Rightarrow> bool"   and env :: "('d, bool) mapping"   and x :: "'d" assumes "agree s env"   and "Mapping.lookup env x = Some True" shows "s x"
proper_part_transitivity: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "MM3"   and "PP x y"   and "PP y z" shows "PP x z"
assms: shows "y \<in> carrier (residue_ring (p ^ k))"
spmf_lub_spmf: fixes Y :: "'h spmf set"   and x :: "'h" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y"   and "Y \<noteq> {}" shows "spmf (lub_spmf Y) x = (\<Squnion>p\<in>Y. spmf p x)"