hcju/isabelleps huggingface dataset:
K_gamma_hom: fixes k :: "nat"   and n :: "nat" assumes "k \<le> n" shows "\<forall>j\<le>n. \<gamma>\<^bsub>k j\<^esub> \<in> Zset"
AR_imp_absolute_retract: fixes S :: "'b set"   and S' :: "'c set"   and U :: "'c set" assumes "AR S"   and "S homeomorphic S'"   and "closedin (top_of_set U) S'" shows "S' retract_of U"
indRelLT_to_TRel: fixes TP :: "'procT"   and TQ :: "'procT"   and TRel :: "('procT \<times> 'procT) set" assumes "TargetTerm TP \<R>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> TargetTerm TQ" shows "(TP, TQ) \<in> TRel"
seq_comp_ref_union_closed: fixes P :: "'c option process"   and Q :: "'c option process" assumes "weakly_sequential P"   and "ref_union_closed P"   and "ref_union_closed Q" shows "ref_union_closed (P ; Q)"
polytope_def: fixes P :: "'a vec set" shows "polytope P = (\<exists>V\<subseteq>carrier_vec n. finite V \<and> P = convex_hull V)"
assms: shows "set ps \<subseteq> set bs \<times> (set gs \<union> set bs)"
nat_inc_closed: fixes n :: "nat" shows "[n] \<cdot> \<one> \<in> carrier Zp"
assms: shows "\<forall>a b. \<exists>a1 b1 d. a = a1 * d \<and> b = b1 * d \<and> ideal_generated {a1, b1} = ideal_generated {1::'a}"
rref_row_rank: fixes A :: "(('a, 'b) vec, 'c) vec" assumes "reduced_row_echelon_form A" shows "row_rank A = card {column (LEAST n. A $ i $ n \<noteq> (0::'a)) A |i. row i A \<noteq> 0}"
his_determ_base: fixes n :: "nat"   and nid' :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and nid :: "'nid" assumes "(n, nid') \<in> his t n nid" shows "nid' = nid"
filterlim_cong: fixes F1 :: "'c filter"   and F1' :: "'c filter"   and F2 :: "'d filter"   and F2' :: "'d filter"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "F1 = F1'"   and "F2 = F2'"   and "\<forall>\<^sub>F x in F2. f x = g x" shows "filterlim f F1 F2 = filterlim g F1' F2'"
lfp_eqI: fixes F :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "mono F"   and "F x = x"   and "\<And>z. F z = z \<Longrightarrow> x \<le> z" shows "lfp F = x"
lat_dist1: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x + y \<sqinter> z = (x + y) \<sqinter> (x + z)"
f1: shows "4 \<le> rk {A, A', C, a, B, P}"
surjective: fixes r :: "('c, 'd) graph_scheme" shows "r = \<lparr>nodes = nodes r, edges = edges r, \<dots> = graph.more r\<rparr>"
sorted_sublist_wrt_entire: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list" assumes "sorted_sublist_wrt R xs 0 (length xs - 1)" shows "sorted_wrt R xs"
simps_6: fixes n :: "nat"   and f :: "recf" shows "encode (Mn n f) = triple_encode 5 n (encode f)"
stmt_inj_term_simp: fixes c :: "stmt" shows "\<langle>c\<rangle> = In1r c"
mk_and_atoms: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "atoms_ltln (mk_and \<phi> \<psi>) \<subseteq> atoms_ltln \<phi> \<union> atoms_ltln \<psi>"
vertices_cases: fixes v :: "'form \<times> nat list" assumes "v |\<in>| vertices" obtains c :: "'form" where "ca__ \<in> set conclusions" and "v = (ca__, [])"   | c :: "'form" and "is" :: "nat list" where "c \<in> set conclusions" and "is \<in> it_paths (it' c)" and "v = (c, 0 # is)"
finite_intersecting: fixes x :: "nat" shows "finite (intersecting (V x))"
openin_Union: fixes K :: "'a set set"   and U :: "'a topology" assumes "\<And>S. S \<in> K \<Longrightarrow> openin U S" shows "openin U (\<Union> K)"
frontier_of_closures: fixes X :: "'b topology"   and S :: "'b set" shows "X frontier_of S = X closure_of S \<inter> X closure_of (topspace X - S)"
continuous_on_polymonial_function: fixes f :: "'a \<Rightarrow> 'b"   and S :: "'a set" assumes "polynomial_function f" shows "continuous_on S f"
epm_3: fixes e :: "'a"   and w :: "'a" assumes "e \<le> w"   and "injective w" shows "e = w \<sqinter> top * e"
simps_62: fixes x81 :: "pat"   and x82 :: "trm"   and x83 :: "trm"   and x6 :: "(char list \<times> trm) list" shows "(LET x81 = x82 IN x83) \<noteq> Rcd x6"
Agent_EphK: fixes X :: "agent"   and X' :: "ephkey" shows "Messages.Agent X \<noteq> EphK X'"
Gcd_eqI: fixes one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and gcd :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and lcm :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and Gcd :: "'c set \<Rightarrow> 'c"   and Lcm :: "'c set \<Rightarrow> 'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit_factor :: "'c \<Rightarrow> 'c"   and normalize :: "'c \<Rightarrow> 'c"   and a :: "'c"   and A :: "'c set" assumes "class.semiring_Gcd one zero times gcd lcm Gcd Lcm divide plus minus unit_factor normalize"   and "normalize a = a"   and "\<And>b. b \<in> A \<Longrightarrow> dvd.dvd times a b"   and "\<And>c. (\<And>b. b \<in> A \<Longrightarrow> dvd.dvd times c b) \<Longrightarrow> dvd.dvd times c a" shows "Gcd A = a"
intros_1: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" shows "r\<^sup>*\<^sup>* a a"
Gauss_Jordan_in_ij_preserves_previous_elements': fixes k :: "nat"   and A :: "(('b, 'c) vec, 'a) vec"   and j :: "'c"   and n :: "'a"   and i :: "'a" assumes "\<forall>n. is_zero_row_upt_k n k A"   and "mod_type_class.to_nat j < k"   and "A $ n $ mod_type_class.from_nat k \<noteq> (0::'b)" shows "Gauss_Jordan_in_ij A (0::'a) (mod_type_class.from_nat k) $ i $ j = A $ i $ j"
continuous_map_pairwise: fixes Z :: "'c topology"   and X :: "'d topology"   and Y :: "'e topology"   and f :: "'c \<Rightarrow> 'd \<times> 'e" shows "continuous_map Z (prod_topology X Y) f = (continuous_map Z X (fst \<circ> f) \<and> continuous_map Z Y (snd \<circ> f))"
eq: shows "err A = insert Err {x. \<exists>y\<in>A. x = OK y}"
graph_succ_it_axioms: shows "graph_succ_it \<alpha> invar succ_list_it"
dom_assoc: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b local.term set"   and b :: "'b local.term set"   and c :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.ide (free_monoidal_category.comp C) a"   and "partial_magma.ide (free_monoidal_category.comp C) b"   and "partial_magma.ide (free_monoidal_category.comp C) c" shows "partial_magma.dom (free_monoidal_category.comp C) (free_monoidal_category.assoc\<^sub>F\<^sub>M\<^sub>C C a b c) = free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C a b) c"
vwalk_def: fixes p :: "'a list"   and G :: "('a, 'b) pre_digraph" shows "vwalk p G \<equiv> set p \<subseteq> verts G \<and> set (vwalk_arcs p) \<subseteq> arcs_ends G \<and> p \<noteq> []"
WT_gpv_pauses: fixes xs :: "'a list"   and \<I> :: "('a, 'b) \<I>" assumes "set xs \<subseteq> outs_\<I> \<I>" shows "\<I> \<turnstile>g pauses xs \<surd>"
older_seniors_monotonic: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and x :: "nat"   and n :: "nat"   and m :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "x \<le> n" shows "semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x (n + m) \<subseteq> semi_mojmir_def.older_seniors \<Sigma> \<delta> q\<^sub>0 w x n"
fbox_demodalisation3: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "(r y \<le> [x| r z) = (x \<cdot> r y \<le> r z \<cdot> x)"
l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes append_child :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> ((_) heap, exception, unit) prog"   and insert_before :: "(_) object_ptr \<Rightarrow> (_) node_ptr \<Rightarrow> (_) node_ptr option \<Rightarrow> ((_) heap, exception, unit) prog" shows "l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M append_child insert_before \<equiv> append_child = l_append_child\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_append_child insert_before"
chamber1_notin_rfold_im: shows "C \<notin> opp_fold_A \<turnstile> A"
l3_17: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and P :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Bet A B C"   and "Bet A' B' C"   and "Bet A P A'" shows "\<exists>Q. Bet P Q C \<and> Bet B Q B'"
prod_cases3: fixes y :: "'b \<times> 'c \<times> 'd" obtains a :: "'b" and b :: "'c" and c :: "'d" where "y = (a, b, c)"
zip_fun_def: fixes f :: "'g \<Rightarrow> 'h"   and g :: "'g \<Rightarrow> 'i" shows "f \<parallel> g \<equiv> \<lambda>x. (f x, g x)"
WhileFalse: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and \<pi> :: "char list \<Rightarrow> com option"   and c :: "com" assumes "\<not> bval b s" shows "\<pi>: (WHILE b DO c, s) \<Rightarrow> s"
basic_monos_6: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool" assumes "\<And>x. P x \<longrightarrow> Q x" shows "Collect P \<subseteq> Collect Q"
liftAll2_mp: fixes phi :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and chi :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and inp :: "'c \<Rightarrow> 'a option"   and inp' :: "'c \<Rightarrow> 'b option" assumes "liftAll2 (\<lambda>v v'. phi v v' \<longrightarrow> chi v v') inp inp'"   and "liftAll2 phi inp inp'" shows "liftAll2 chi inp inp'"
neg_distr_cond_bset_right_unique: fixes C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and D :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and tytok :: "'f itself" assumes "right_unique C" shows "neg_distr_cond_bset C D tytok"
runit_simps_4: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and f :: "'b" assumes "bicategory V H \<a> \<i> src trg"   and "partial_magma.ide V f" shows "partial_magma.dom V (bicategory.runit V H \<a> \<i> src f) = H f (src f)"
Nml_HcompD_7: fixes t :: "'c local.term"   and u :: "'c local.term" assumes "Nml (t \<^bold>\<star> u)" shows "Src t = Trg u"
eq_general: fixes B :: "'d set"   and A :: "'e set"   and h :: "'e \<Rightarrow> 'd"   and \<gamma> :: "'d \<Rightarrow> 'f"   and \<phi> :: "'e \<Rightarrow> 'f" assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>!x. x \<in> A \<and> h x = y"   and "\<And>x. x \<in> A \<Longrightarrow> h x \<in> B \<and> \<gamma> (h x) = \<phi> x" shows "sum \<phi> A = sum \<gamma> B"
fold_A_chamber_images_1: shows "fold_A ` C = C"
toplevel_summands_lderivs_Star_nonempty: fixes xs :: "'a list"   and r :: "'b rexp" assumes "xs \<noteq> []" shows "toplevel_summands (lderivs xs (Star r)) \<subseteq> {Times (lderivs ys r) (Star r) |ys. \<exists>zs. ys \<noteq> [] \<and> zs @ ys = xs}"
boundeffect_cor: fixes V :: "variable set"   and \<alpha> :: "game"   and \<omega> :: "variable \<Rightarrow> real"   and I :: "interp"   and X :: "(variable \<Rightarrow> real) set" assumes "V \<inter> BVG \<alpha> = {}" shows "(\<omega> \<in> game_sem I \<alpha> X) = (\<omega> \<in> game_sem I \<alpha> (selectlike X \<omega> V))"
ha: fixes x :: "real" shows "a = insertion (nth_default 0 (xs' @ x # xs)) A"
wfr: shows "\<turnstile> \<rho>,\<eta> : \<Gamma>"
of_int_neg_numeral: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and k :: "num" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.of_int one plus zero minus (- numeral k) = uminus (numeral.numeral one plus k)"
\<gamma>: shows "\<gamma> < \<omega> \<up> \<delta>"
vector_scalar_mult_def: fixes c :: "'a"   and x :: "('a, 'b) vec" shows "c *s x = (\<chi>i. c * x $ i)"
f1: shows "memory_write asi address (st_byte_mask instr address) (st_data0 instr curr_win rd address s) s \<noteq> None"
observable_from_rel: fixes rel :: "('a \<times> ('b \<times> 'c) \<times> 'a) set"   and q0 :: "'a" assumes "ASC_Example.observable_rel rel" shows "observable (from_rel rel q0)"
zero_tag_const_def: shows "0 \<equiv> \<tau>\<^sub>c\<^sub>s\<^sub>t (0::'b)"
assms_5: shows "\<forall>h\<in>HmodG. f ((1::'f) \<delta>\<delta> h) = f' ((1::'f) \<delta>\<delta> h)"
Suc: shows "(pderiv ^^ n) (p * q) = (\<Sum>k\<le>n. Polynomial.smult (of_nat (n choose k)) ((pderiv ^^ k) p * (pderiv ^^ (n - k)) q))"
has_derivative_transform_within: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'c \<Rightarrow> 'd"   and x :: "'c"   and s :: "'c set"   and d :: "real"   and g :: "'c \<Rightarrow> 'd" assumes "(f has_derivative f') (at x within s)"   and "0 < d"   and "x \<in> s"   and "\<And>x'. \<lbrakk>x' \<in> s; dist x' x < d\<rbrakk> \<Longrightarrow> f x' = g x'" shows "(g has_derivative f') (at x within s)"
semiring_norm_171: fixes v :: "num"   and w :: "num"   and y :: "'a" shows "numeral v * (- numeral w * y) = - numeral (v * w) * y"
sorted_hd_le_last: fixes xs :: "'a list" assumes "sorted xs"   and "xs \<noteq> []" shows "hd xs \<le> last xs"
countable_insert_eq: fixes x :: "'d"   and A :: "'d set" shows "countable (insert x A) = countable A"
Bk_not_tape_start: fixes list :: "cell list"   and m :: "nat"   and lm :: "nat list"   and ys :: "cell list" shows "(Bk # list = <m # lm> @ ys) = False"
cod_closed: fixes f :: "'a" assumes "right a f" shows "right a (cod f)"
card_of_Un_diff_infinite: fixes A :: "'b set"   and B :: "'b set" assumes "infinite A"   and "|B| <o |A|" shows "|A - B| =o |A|"
unitaryI: fixes P :: "'a \<Rightarrow> real" assumes "sound P"   and "bounded_by 1 P" shows "unitary P"
exists_exactly_one_following_state: fixes channel :: "nat \<Rightarrow> ('a \<times> 'a) option"   and trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and send :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and recv :: "nat \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> bool"   and ev :: "('a, 'b, 'c) event"   and c :: "('a, 'b, 'c) configuration" assumes "distributed_system channel"   and "distributed_system.can_occur channel trans send recv ev c" shows "\<exists>!c'. distributed_system.next channel trans send recv c ev c'"
integrable_on_def: fixes f :: "'d \<Rightarrow> 'e"   and i :: "'d set" shows "(f integrable_on i) = (\<exists>y. (f has_integral y) i)"
induct: fixes x1 :: "nat \<Rightarrow> classDef option"   and x2 :: "nat"   and x3 :: "nat"   and x4 :: "nat list"   and x5 :: "exp"   and P :: "(nat \<Rightarrow> classDef option) \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat list \<Rightarrow> exp \<Rightarrow> bool" assumes "mbody(x1,x2,x3) = x4 . x5"   and "\<And>CT C CDef m mDef xs e. \<lbrakk>CT C = Some CDef; lookup (cMethods CDef) (\<lambda>md. mName md = m) = Some mDef; varDefs_names (mParams mDef) = xs; mBody mDef = e\<rbrakk> \<Longrightarrow> P CT m C xs e"   and "\<And>CT C CDef m D xs e. \<lbrakk>CT C = Some CDef; lookup (cMethods CDef) (\<lambda>md. mName md = m) = None; cSuper CDef = D; mbody(CT,m,D) = xs . e; P CT m D xs e\<rbrakk> \<Longrightarrow> P CT m C xs e" shows "P x1 x2 x3 x4 x5"
ennreal_spmf_map: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d spmf"   and x :: "'c" shows "ennreal (spmf (map_spmf f p) x) = emeasure (measure_spmf p) (f -` {x})"
simps_5: fixes c :: "'c"   and ux :: "'d" shows "collect_clks (GE c ux) = {c}"
DomainI: fixes a :: "'a"   and b :: "'b"   and r :: "('a \<times> 'b) set" assumes "(a, b) \<in> r" shows "a \<in> Domain r"
adjacent_int_decomp: fixes x :: "'b set"   and y :: "'b set" assumes "x \<sim> y"   and "x \<noteq> y" shows "\<exists>v. v \<notin> y \<and> x = insert v (x \<inter> y)"
sorted_wrt_concat_map: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and h :: "'b \<Rightarrow> 'a"   and xs :: "'b list"   and f :: "'b \<Rightarrow> 'b list" assumes "sorted_wrt P (map h xs)"   and "\<And>x. x \<in> set xs \<Longrightarrow> sorted_wrt P (map h (f x))"   and "\<And>x y u v. \<lbrakk>P (h x) (h y); x \<in> set xs; y \<in> set xs; u \<in> set (f x); v \<in> set (f y)\<rbrakk> \<Longrightarrow> P (h u) (h v)" shows "sorted_wrt P (concat (map (map h \<circ> f) xs))"
per23_preserves_bet: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "Bet A B C"   and "A \<noteq> B'"   and "A \<noteq> C'"   and "Col A B' C'"   and "Per A B' B"   and "Per A C' C" shows "Bet A B' C'"
bdd_above_Icc: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b" assumes "class.preorder less_eq less" shows "preorder.bdd_above less_eq (ord.atLeastAtMost less_eq a b)"
from_nat_into: fixes A :: "'d set"   and n :: "nat" assumes "A \<noteq> {}" shows "from_nat_into A n \<in> A"
case_cong: fixes option :: "'c option"   and option' :: "'c option"   and f1 :: "'d"   and g1 :: "'d"   and f2 :: "'c \<Rightarrow> 'd"   and g2 :: "'c \<Rightarrow> 'd" assumes "option = option'"   and "option' = None \<Longrightarrow> f1 = g1"   and "\<And>x2. option' = Some x2 \<Longrightarrow> f2 x2 = g2 x2" shows "(case option of None \<Rightarrow> f1 | Some x2 \<Rightarrow> f2 x2) = (case option' of None \<Rightarrow> g1 | Some x2 \<Rightarrow> g2 x2)"
hyps_3: assumes "\<And>xsa ys. xs = xsa @ ys \<Longrightarrow> set xsa \<inter> set ys = {}" shows "distinct xs"
lprefix_lnullD: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "xs \<sqsubseteq> ys"   and "lnull ys" shows "lnull xs"
submonoid_axioms: shows "submonoid \<ww> carrier_local_ring_at add_local_ring_at zero_local_ring_at"
conds: shows "[[x d e]] \<and> path ae a e \<and> path be b e"
simps_7: fixes Extend :: "'a \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'b"   and Length :: "'b \<Rightarrow> nat"   and satisfies0 :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and r :: "'a \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> bool"   and \<AA> :: "'b"   and k :: "'a"   and \<phi> :: "('d, 'a) aformula" shows "Formula_Operations.satisfies_gen Extend Length satisfies0 r \<AA> (FAll k \<phi>) = (\<forall>P. r k P (Length \<AA>) \<longrightarrow> Formula_Operations.satisfies_gen Extend Length satisfies0 r (Extend k 0 \<AA> P) \<phi>)"
del_UnEdge_odd_even: fixes G :: "('a, 'b) graph"   and v :: "'a"   and w :: "'b"   and v' :: "'a" assumes "valid_unMultigraph G"   and "finite (edges G)"   and "finite (nodes G)"   and "(v, w, v') \<in> edges G"   and "odd (degree v G)"   and "even (degree v' G)" shows "num_of_odd_nodes (del_unEdge v w v' G) = num_of_odd_nodes G"
conversep_conversep: fixes r :: "'d \<Rightarrow> 'e \<Rightarrow> bool" shows "r\<inverse>\<inverse>\<inverse>\<inverse> = r"
upper_triangular_snd_QR_decomposition: fixes A :: "((real, 'a) vec, 'b) vec" assumes "rank A = ncols A" shows "upper_triangular (snd (QR_decomposition A))"
chain_lnth_rel: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b llist"   and j :: "nat" assumes "chain R xs"   and "enat (Suc j) < llength xs" shows "R (lnth xs j) (lnth xs (Suc j))"
ladder: shows "LeftDerivationLadder \<alpha> D L \<gamma>"
select_convs_4: fixes E_ES :: "'a set"   and I_ES :: "'a set"   and O_ES :: "'a set"   and Tr_ES :: "'a list set"   and more :: "'b" shows "Tr_ES \<lparr>E_ES = E_ES, I_ES = I_ES, O_ES = O_ES, Tr_ES = Tr_ES, \<dots> = more\<rparr> = Tr_ES"
starfun_inverse2: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'d star" shows "inverse ((*f* f) x) = (*f* (\<lambda>x. inverse (f x))) x"
balance_left_rbt_less: fixes k :: "'a"   and a :: "('a, 'c) rbt"   and b :: "('a, 'c) rbt"   and x :: "'a"   and t :: "'c" assumes "a |\<guillemotleft> k"   and "b |\<guillemotleft> k"   and "x < k" shows "balance_left a x t b |\<guillemotleft> k"
is_monoidal_category: shows "monoidal_category S.comp (\<lambda>\<mu>\<nu>. S.hcomp (fst \<mu>\<nu>) (snd \<mu>\<nu>)) S.\<alpha> \<i>[a]"
connected_ivt_hyperplane: fixes S :: "'a set"   and x :: "'a"   and y :: "'a"   and a :: "'a"   and b :: "real" assumes "connected S"   and "x \<in> S"   and "y \<in> S"   and "a \<bullet> x \<le> b"   and "b \<le> a \<bullet> y" shows "\<exists>z\<in>S. a \<bullet> z = b"
P4: shows "\<forall>x. Dependent x \<longrightarrow> (\<exists>y. Independent y \<and> y explains x)"
thmD7_LeftDerivation: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and N :: "symbol"   and \<gamma> :: "symbol list"   and \<alpha> :: "symbol list" assumes "LeftDerivation [\<SS>] D (N # \<gamma>)"   and "is_nonterminal N"   and "(N, \<alpha>) \<in> \<RR>" shows "init_item (N, \<alpha>) 0 \<in> \<pi> 0 {} Init"
monom_mult_inj: fixes m :: "'c monom"   and m1 :: "'c monom"   and m2 :: "'c monom" assumes "m * m1 = m * m2" shows "m1 = m2"
intro: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "semilattice_order f less_eq less"   and "semilattice_fset f" shows "semilattice_order_fset f less_eq less"
multiset_subset_insert: fixes x :: "'a"   and xs :: "'a multiset" shows "{ps. ps \<subseteq># add_mset x xs} = {ps. ps \<subseteq># xs} \<union> add_mset x ` {ps. ps \<subseteq># xs}"
suff_all: fixes x :: "'b llist"   and A :: "'b set"   and s :: "'b llist" assumes "x \<in> suff A s" shows "x \<in> A\<^sup>\<infinity>"
mrw_value_dom_mono: fixes vs :: "'a \<times> addr_loc \<Rightarrow> ('a val \<times> bool) option"   and P :: "'b prog"   and ob :: "('a, 'c) obs_event action" shows "dom vs \<subseteq> dom (mrw_value P vs ob)"
cinner_commute': fixes y :: "'b"   and x :: "'b" shows "cnj \<langle>y, x\<rangle> = \<langle>x, y\<rangle>"
prems_1: shows "indecomposable (SUP \<xi>\<in>elts \<gamma>. \<xi>)"
real_sqrt_lt_1_iff: fixes x :: "real" shows "(sqrt x < 1) = (x < 1)"
group_left_neutral: fixes a :: "'b perm" shows "1 * a = a"
norm_triangle_le: fixes x :: "'a"   and y :: "'a"   and e :: "real" assumes "norm x + norm y \<le> e" shows "norm (x + y) \<le> e"
Min_Un: fixes A :: "'a Word.word set"   and B :: "'a Word.word set" assumes "finite A"   and "A \<noteq> {}"   and "finite B"   and "B \<noteq> {}" shows "signed.Min (A \<union> B) = signed.min (signed.Min A) (signed.Min B)"
permute_set_eq_image: fixes p :: "perm"   and X :: "'c set" shows "p \<bullet> X = permute p ` X"
le_fin_member': fixes a :: "'item"   and u :: "'item list"   and v :: "'item list" assumes "[a] \<preceq>\<^sub>F u @ v"   and "a \<in> set u" shows "[a] \<preceq>\<^sub>F u"
minus_ivl_def: fixes i1 :: "ivl"   and i2 :: "ivl" shows "minus_ivl i1 i2 = (if is_empty i1 \<or> is_empty i2 then Abs_Int2_ivl.empty else case (i1, i2) of (I l1 h1, I l2 h2) \<Rightarrow> I (l1 - h2) (h1 - l2))"
nrename_NotL_aux: fixes R :: "trm"   and x :: "name"   and y :: "name"   and c :: "coname"   and N :: "trm"   and u :: "name" assumes "R[x\<turnstile>n>y] = NotL <c>.N u" shows "x = u \<and> x = y \<or> x \<noteq> u"
mem_CD_on_Cd: fixes x :: "'a"   and ds :: "'b set"   and X :: "'a set" shows "(x \<in> TruePref_tax.CD_on ds X) = (x \<in> TruePref_tax.Cd (Xd x) X \<and> Xd x \<in> ds)"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b formula"   and b :: "'c formula" assumes "rel_formula R a b" obtains x :: "'b" and y :: "'c" where "a = Atom x" and "b = Atom y" and "R x y"   | "a = \<bottom>" and "b = \<bottom>"   | xa :: "'b formula" and ya :: "'c formula" where "a = \<^bold>\<not> xa" and "b = \<^bold>\<not> ya" and "rel_formula R xa ya"   | x1a :: "'b formula" and x2 :: "'b formula" and y1a :: "'c formula" and y2 :: "'c formula" where "a = x1a \<^bold>\<and> x2" and "b = y1a \<^bold>\<and> y2" and "rel_formula R x1a y1a" and "rel_formula R x2 y2"   | x1b :: "'b formula" and x2a :: "'b formula" and y1b :: "'c formula" and y2a :: "'c formula" where "a = x1b \<^bold>\<or> x2a" and "b = y1b \<^bold>\<or> y2a" and "rel_formula R x1b y1b" and "rel_formula R x2a y2a"   | x1c :: "'b formula" and x2b :: "'b formula" and y1c :: "'c formula" and y2b :: "'c formula" where "a = x1c \<^bold>\<rightarrow> x2b" and "b = y1c \<^bold>\<rightarrow> y2b" and "rel_formula R x1c y1c" and "rel_formula R x2b y2b"
ih: fixes D :: "'a literal list" shows "wstate_of_dstate (N, P @ (D @ D', j) # P', Q, n) \<leadsto>\<^sub>w\<^sup>* wstate_of_dstate (N, P @ (D @ reduce [C] D D', j) # P', Q, n)"