hcju/isabelleps huggingface dataset:
funct_OT_12_def: fixes input\<^sub>1 :: "'g \<times> 'g"   and \<sigma> :: "bool" shows "funct_OT_12 input\<^sub>1 \<sigma> = return_spmf ((), if \<sigma> then snd input\<^sub>1 else fst input\<^sub>1)"
monom_mult_dist_left: fixes c :: "'d"   and d :: "'d"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'd" shows "monom_mult (c + d) t p = monom_mult c t p + monom_mult d t p"
compass_reflect_self_inverse_1: shows "rep_equator_reflect ** rep_equator_reflect = mat 1"
Sard_lemma1: fixes P :: "(real, 'a) vec set"   and m :: "real"   and e :: "real"   and w :: "(real, 'a) vec" assumes "dim P < CARD('a)"   and "0 \<le> m"   and "0 \<le> e" obtains S :: "(real, 'a) vec set" where "S \<in> lmeasurable" and "{z. norm (z - w) \<le> m \<and> (\<exists>t\<in>P. norm (z - w - t) \<le> e)} \<subseteq> S" and "Sigma_Algebra.measure lebesgue S \<le> 2 * e * (2 * m) ^ (CARD('a) - 1)"
ee_wfactorsI: fixes a :: "'a"   and b :: "'a"   and as :: "'a list"   and bs :: "'a list" assumes "a \<sim> b"   and "wfactors G as a"   and "wfactors G bs b"   and "a \<in> carrier G"   and "b \<in> carrier G"   and "set as \<subseteq> carrier G"   and "set bs \<subseteq> carrier G" shows "essentially_equal G as bs"
Nats_not_HNatInfinite_iff: fixes x :: "nat star" shows "(x \<in> \<nat>) = (x \<notin> HNatInfinite)"
pf: fixes b :: "'b" assumes "b \<in> Basis" shows "real_polynomial_function (pf b) \<and> (\<forall>x\<in>S. \<bar>f x \<bullet> b - pf b x\<bar> < e / real DIM('b))"
word_bl_Rep': fixes x :: "'a word" shows "length (to_bl x) = LENGTH('a)"
c: fixes s :: "complex" assumes "0 < Re s" shows "cmod (stirling_integral m s) \<le> c / Re s ^ m"
cylinder_prob: fixes p :: "real"   and A :: "nat set set"   and n :: "nat"   and B :: "nat set set" assumes "edge_space p"   and "A \<subseteq> edge_space.S_edges n"   and "B \<subseteq> edge_space.S_edges n"   and "A \<inter> B = {}" shows "Sigma_Algebra.measure (edge_space.P n p) (cylinder (edge_space.S_edges n) A B) = p ^ card A * (1 - p) ^ card B"
euler_mascheroni_LIMSEQ: shows "(\<lambda>n. harm n - ln (real n)) \<longlonglongrightarrow> euler_mascheroni"
outstanding_non_volatile_writes_unshared: fixes i :: "nat"   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" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "non_volatile_writes_unshared \<S>\<^sub>s\<^sub>b sb"
disj_1: shows "h\<^sub>0 \<bottom> h"
space_P: shows "space (sigma (space (Pi\<^sub>F {I} M)) P) = (\<Pi>' i\<in>I. space (M i))"
reduced_homology_group_nsphere_step: fixes p :: "int"   and n :: "nat" shows "\<exists>f\<in>Group.iso (reduced_homology_group p (nsphere n)) (reduced_homology_group (1 + p) (nsphere (Suc n))). \<forall>c\<in>carrier (reduced_homology_group p (nsphere n)). hom_induced (1 + p) (nsphere (Suc n)) {} (nsphere (Suc n)) {} (\<lambda>x i. if i = 0 then - x i else x i) (f c) = f (hom_induced p (nsphere n) {} (nsphere n) {} (\<lambda>x i. if i = 0 then - x i else x i) c)"
assms: shows "u \<cdot> v \<noteq> v \<cdot> u"
TerminationLessAfterExhaustiveUnitPropagate: fixes state :: "State"   and F0 :: "Literal list list"   and Vbl :: "nat set" assumes "exhaustiveUnitPropagate_dom state"   and "InvariantUniq (getM state)"   and "InvariantConsistent (getM state)"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchCharacterization (getF state) (getWatch1 state) (getWatch2 state) (getM state)"   and "InvariantConflictFlagCharacterization (getConflictFlag state) (getF state) (getM state)"   and "InvariantQCharacterization (getConflictFlag state) (getQ state) (getF state) (getM state)"   and "InvariantUniqQ (getQ state)"   and "InvariantVarsM (getM state) F0 Vbl"   and "InvariantVarsQ (getQ state) F0 Vbl"   and "InvariantVarsF (getF state) F0 Vbl"   and "finite Vbl"   and "getSATFlag state = UNDEF" shows "let state' = exhaustiveUnitPropagate state in state' = state \<or> (state', state) \<in> terminationLessState1 (vars F0 \<union> Vbl)"
supply_wfp: fixes p :: "prog"   and us :: "nat list" assumes "wfp p"   and "length us = length (para p)" shows "\<exists>\<E>\<in>envs p. supply p us = Some \<E>"
Erdos_Rado_sunflower_same_card: fixes F :: "'b set set"   and k :: "nat"   and r :: "nat" assumes "\<forall>A\<in>F. finite A \<and> card A = k"   and "(r - 1) ^ k * fact k < card F" shows "\<exists>S\<subseteq>F. sunflower S \<and> card S = r \<and> {} \<notin> S"
iso_tuple_UNIV_I: fixes x :: "'d" shows "x \<in> UNIV \<equiv> True"
homogeneousD: fixes p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and s :: "'b \<Rightarrow>\<^sub>0 nat"   and t :: "'b \<Rightarrow>\<^sub>0 nat" assumes "homogeneous p"   and "s \<in> keys p"   and "t \<in> keys p" shows "deg_pm s = deg_pm t"
ab_diff_conv_add_uminus: fixes a :: "'b"   and b :: "'b" shows "a - b = a + - b"
interval_of_zero: shows "interval_of (0::'b) = 0"
character_data_ptr_kinds_def: fixes heap :: "(_) heap" shows "character_data_ptr_kinds heap = the |`| cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>c\<^sub>h\<^sub>a\<^sub>r\<^sub>a\<^sub>c\<^sub>t\<^sub>e\<^sub>r\<^sub>_\<^sub>d\<^sub>a\<^sub>t\<^sub>a\<^sub>_\<^sub>p\<^sub>t\<^sub>r |`| ffilter is_character_data_ptr_kind\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^sub>_\<^sub>p\<^sub>t\<^sub>r (node_ptr_kinds heap)"
card_eq_0_iff: fixes A :: "'b set" shows "(card A = 0) = (A = {} \<or> infinite A)"
atLeast_def: fixes l :: "'b word" shows "signed.atLeast l \<equiv> {x. l \<le>s x}"
nat_cut_le_Max_le: fixes I :: "nat set"   and t :: "nat" assumes "I \<down>\<le> t \<noteq> {}" shows "Max (I \<down>\<le> t) \<le> t"
concat_bit_eq_iff: fixes n :: "nat"   and k :: "int"   and l :: "int"   and r :: "int"   and s :: "int" shows "(concat_bit n k l = concat_bit n r s) = (take_bit n k = take_bit n r \<and> l = s)"
Least_in_nonempty_set_imp_ex: fixes A :: "'a set"   and P :: "'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b" assumes "A \<noteq> {}"   and "P (LEAST y. \<exists>x\<in>A. y = f x)" shows "\<exists>x\<in>A. P (f x)"
elims: fixes x :: "'c list"   and y :: "('c list \<times> 'c list) set" assumes "splitset_fun x = y" obtains "x = []" and "y = {([], [])}"   | xa :: "'c" and xs :: "'c list" where "x = xa # xs" and "y = insert ([], xa # xs) (apfst ((#) xa) ` splitset_fun xs)"
expand_set: fixes g :: "'d \<Rightarrow> 'a" shows "G g = F.F g {a. g a \<noteq> \<^bold>1}"
elims: fixes x :: "'a wordinterval"   and xa :: "'a wordinterval list"   and y :: "'a wordinterval list" assumes "partIps x xa = y" obtains uu_ :: "'a wordinterval" where "x = uu_" and "xa = []" and "y = []"   | s :: "'a wordinterval" and t :: "'a wordinterval" and ts :: "'a wordinterval list" where "x = s" and "xa = t # ts" and "y = (if wordinterval_empty s then t # ts else if wordinterval_empty (wordinterval_intersection s t) then t # partIps s ts else if wordinterval_empty (wordinterval_setminus t s) then t # partIps (wordinterval_setminus s t) ts else wordinterval_intersection t s # wordinterval_setminus t s # partIps (wordinterval_setminus s t) ts)"
lemma_st_part_major2: fixes x :: "real star"   and t :: "real star" assumes "x \<in> HFinite"   and "isLub \<real> {s \<in> \<real>. s < x} t" shows "\<forall>r\<in>\<real>. 0 < r \<longrightarrow> \<bar>x - t\<bar> < r"
letter_less_def: fixes R :: "('a \<times> 'a) set" shows "letter_less R = {(a, b). (snd a, snd b) \<in> R}"
succs_mono: fixes p :: "'a prog \<Rightarrow> char list \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog"   and i :: "instr"   and mxs :: "nat"   and mpc :: "nat"   and rT :: "ty"   and \<tau>' :: "ty list \<times> ty err list"   and \<tau> :: "ty list \<times> ty err list"   and pc :: "nat" assumes "wf_prog p P"   and "app\<^sub>i (i, P, mxs, mpc, rT, \<tau>')"   and "P \<turnstile> \<tau> \<le>\<^sub>i \<tau>'" shows "set (succs i \<tau> pc) \<subseteq> set (succs i \<tau>' pc)"
abs_split: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" shows "P \<bar>a\<bar> = (((0::'b) \<le> a \<longrightarrow> P a) \<and> (a < (0::'b) \<longrightarrow> P (- a)))"
analz_Un_EncSet: fixes K :: "msg set"   and G :: "msg set"   and H :: "msg set" assumes "K \<subseteq> range LtK"   and "Enc_keys_clean (G \<union> H)" shows "analz (EncSet G K \<union> H) \<subseteq> EncSet G K \<union> analz (G \<union> H)"
verticesFrom_congs: fixes v :: "nat"   and f :: "face" assumes "v \<in> \<V> f" shows "vertices f \<cong> verticesFrom f v"
submatrix_def: fixes A :: "'b mat"   and I :: "nat set"   and J :: "nat set" shows "submatrix A I J = Matrix.mat (card {i. i < dim_row A \<and> i \<in> I}) (card {j. j < dim_col A \<and> j \<in> J}) (\<lambda>(i, j). A $$ (pick I i, pick J j))"
join_pmf_def: fixes M :: "'d pmf pmf" shows "join_pmf M = M \<bind> (\<lambda>x. x)"
rcset_to_rcset: fixes A :: "'f set" assumes "countable A" shows "rcset (the_inv rcset A) = A"
next_symbol_starts_item_\<beta>: fixes x :: "item"   and t :: "symbol" assumes "wellformed_item x"   and "next_symbol x = Some t" shows "\<exists>\<delta>. item_\<beta> x = t # \<delta>"
valid_lists_downset_aux: fixes xs :: "Vars.var list"   and S :: "Vars.var set"   and CoCalls :: "CoCalls" assumes "xs \<in> valid_lists S CoCalls" shows "butlast xs \<in> valid_lists S CoCalls"
mono: fixes a :: "'c"   and c :: "'c"   and b :: "'c"   and d :: "'c" assumes "a \<le> c"   and "b \<le> d" shows "a \<sqinter> b \<le> c \<sqinter> d"
PB_QB: fixes a :: "'b" assumes "QB a" shows "PB a"
induct: fixes x :: "('b com option \<times> 'b) list"   and P :: "('b com option \<times> 'b) list \<Rightarrow> bool" assumes "x \<in> cptn_mod"   and "\<And>Pa s. P [(Pa, s)]"   and "\<And>Pa t xs s. \<lbrakk>(Pa, t) # xs \<in> cptn_mod; P ((Pa, t) # xs)\<rbrakk> \<Longrightarrow> P ((Pa, s) # (Pa, t) # xs)"   and "\<And>Pa s t xs. \<lbrakk>(Some Pa, s) -c\<rightarrow> (None, t); (None, t) # xs \<in> cptn_mod; P ((None, t) # xs)\<rbrakk> \<Longrightarrow> P ((Some Pa, s) # (None, t) # xs)"   and "\<And>P0 s ys b P1. \<lbrakk>(Some P0, s) # ys \<in> cptn_mod; P ((Some P0, s) # ys); s \<in> b\<rbrakk> \<Longrightarrow> P ((Some (Cond b P0 P1), s) # (Some P0, s) # ys)"   and "\<And>P1 s ys b P0. \<lbrakk>(Some P1, s) # ys \<in> cptn_mod; P ((Some P1, s) # ys); s \<notin> b\<rbrakk> \<Longrightarrow> P ((Some (Cond b P0 P1), s) # (Some P1, s) # ys)"   and "\<And>P0 s xs zs P1. \<lbrakk>(Some P0, s) # xs \<in> cptn_mod; P ((Some P0, s) # xs); zs = map (lift P1) xs\<rbrakk> \<Longrightarrow> P ((Some (Seq P0 P1), s) # zs)"   and "\<And>P0 s xs P1 ys zs. \<lbrakk>(Some P0, s) # xs \<in> cptn_mod; P ((Some P0, s) # xs); fst (last ((Some P0, s) # xs)) = None; (Some P1, snd (last ((Some P0, s) # xs))) # ys \<in> cptn_mod; P ((Some P1, snd (last ((Some P0, s) # xs))) # ys); zs = map (lift P1) xs @ ys\<rbrakk> \<Longrightarrow> P ((Some (Seq P0 P1), s) # zs)"   and "\<And>Pa s xs b zs. \<lbrakk>(Some Pa, s) # xs \<in> cptn_mod; P ((Some Pa, s) # xs); s \<in> b; zs = map (lift (While b Pa)) xs\<rbrakk> \<Longrightarrow> P ((Some (While b Pa), s) # (Some (Seq Pa (While b Pa)), s) # zs)"   and "\<And>Pa s xs b zs ys. \<lbrakk>(Some Pa, s) # xs \<in> cptn_mod; P ((Some Pa, s) # xs); fst (last ((Some Pa, s) # xs)) = None; s \<in> b; zs = map (lift (While b Pa)) xs @ ys; (Some (While b Pa), snd (last ((Some Pa, s) # xs))) # ys \<in> cptn_mod; P ((Some (While b Pa), snd (last ((Some Pa, s) # xs))) # ys)\<rbrakk> \<Longrightarrow> P ((Some (While b Pa), s) # (Some (Seq Pa (While b Pa)), s) # zs)" shows "P x"
assms_5: shows "A, n \<turnstile> ((\<^bold>@ i p) # (\<^bold>\<diamond> Nom i) # ps, a) # branch"
invertible_right_inverse: fixes u :: "'a" assumes "invertible u"   and "u \<in> G" shows "u \<cdot> inverse u = \<one>"
simps_2: fixes v :: "'a tree"   and va :: "'a"   and vb :: "'a tree" shows "mrbst_union \<langle>v, va, vb\<rangle> \<langle>\<rangle> = return_pmf \<langle>v, va, vb\<rangle>"
eq: shows "except (MP_oalist xs) {fst ct} = MP_oalist (OAlist_tl_ntm xs)"
evC: fixes K :: "'a set" assumes "compact K"   and "K \<subseteq> T" shows "\<forall>\<^sub>F i in sequentially. K \<subseteq> C i"
open_vec_def: fixes S :: "('c, 'd) vec set" shows "open S = (\<forall>x\<in>S. \<exists>A. (\<forall>i. open (A i) \<and> x $ i \<in> A i) \<and> (\<forall>y. (\<forall>i. y $ i \<in> A i) \<longrightarrow> y \<in> S))"
LPHIL0\<^sub>n\<^sub>o\<^sub>r\<^sub>m_def: shows "LPHIL0\<^sub>n\<^sub>o\<^sub>r\<^sub>m = P\<^sub>n\<^sub>o\<^sub>r\<^sub>m\<lbrakk>Tr\<^sub>l\<^sub>p,Up\<^sub>l\<^sub>p\<rbrakk>"
list_allD: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and x :: "'a" assumes "list_all P xs"   and "x \<in> set xs" shows "P x"
ord_strict_higher: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and q :: "'t \<Rightarrow>\<^sub>0 'c" shows "(p \<prec>\<^sub>p q) = (\<exists>v. lookup p v = (0::'c) \<and> lookup q v \<noteq> (0::'c) \<and> higher p v = higher q v)"
negation_free_and: fixes \<phi> :: "atom fm"   and \<psi> :: "atom fm" assumes "negation_free \<phi>"   and "negation_free \<psi>" shows "negation_free (and \<phi> \<psi>)"
f_2_lemma_1: fixes x :: "int" shows "f_2 (f_2 x) = x"
Cons_parallelI2: fixes a :: "'c"   and b :: "'c"   and as :: "'c list"   and bs :: "'c list" assumes "a = b"   and "as \<parallel> bs" shows "a # as \<parallel> b # bs"
Max_Image_distrib: fixes X :: "'a set"   and f1 :: "'a \<Rightarrow> 'b"   and f2 :: "'a \<Rightarrow> 'b" assumes "finite X"   and "X \<noteq> {}" shows "(MAX x\<in>X. max (f1 x) (f2 x)) = max (Max (f1 ` X)) (Max (f2 ` X))"
min_int_poly_unique: fixes p :: "int poly"   and x :: "'b" assumes "p represents x"   and "irreducible p"   and "0 < Polynomial.lead_coeff p" shows "min_int_poly x = p"
AXIOMS_in_CANDs_1: fixes B :: "ty" shows "AXIOMSn B \<subseteq> \<parallel>(B)\<parallel>"
one_closed: shows "\<one>\<^bsub>R\<^esub> \<in> carrier R"
degen_skip_condition: fixes k :: "nat"   and cs :: "('b \<Rightarrow> bool) list"   and w :: "'b stream" assumes "k < length cs"   and "infs (degen cs) (w ||| k ## sscan (count cs) w k)" obtains u :: "'b list" and a :: "'b" and v :: "'b stream" where "w = u @- a ## v" and "fold (count cs) u k = k" and "(cs ! k) a"
assms_3: shows "ipresIGSubstAll h hA MOD MOD'"
xs_bad: shows "bad (\<lambda>ys xs. length ys \<le> n \<and> length xs \<le> n \<and> ext gt ys xs) xs"
hdvd_one_eq_one: fixes x :: "nat star" assumes "is_unit x" shows "x = 1"
MkIde_Dom: fixes a :: "('obj, 'obj) concrete_category.arr" assumes "arr a" shows "MkIde (Dom a) = local.dom a"
trivial_ker_imp_inj: fixes G :: "('b, 'c) monoid_scheme"   and H :: "('d, 'e) monoid_scheme"   and h :: "'b \<Rightarrow> 'd" assumes "group_hom G H h"   and "kernel G H h = {\<one>\<^bsub>G\<^esub>}" shows "inj_on h (carrier G)"
csol_eq: fixes s :: "real" assumes "t - e \<le> s"   and "s \<le> t0" shows "csol t0 x0 (t - e) s = flow t0 x0 s"
assms_3: shows "\<nu> \<star> \<mu> \<noteq> null"
cdiov: shows "d\<inverse> O ov \<subseteq> ov \<union> f\<inverse> \<union> d\<inverse>"
get_tag_name_delete_shadow_root: fixes h :: "(_) heap"   and shadow_root_ptr :: "(_) shadow_root_ptr"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool"   and ptr' :: "(_) element_ptr" assumes "h \<turnstile> delete\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M shadow_root_ptr \<rightarrow>\<^sub>h h'"   and "r \<in> get_tag_name_locs ptr'" shows "r h h'"
lzip_simps_2: fixes xs :: "'c llist" shows "lzip xs LNil = LNil"
step_4_correct: fixes p :: "prog" assumes "spec\<^sub>4 p" shows "spec\<^sub>3 p"
apex_def: shows "Cod_\<rho>.apex \<equiv> C.dom Cod_\<rho>.leg0"
fun_of_def: fixes S :: "('e \<times> 'f) set" shows "fun_of S \<equiv> \<lambda>x y. (x, y) \<in> S"
add_ub: fixes x :: "'a"   and y :: "'a" shows "x \<le> x + y"
mult_1_right: fixes a :: "'a" shows "a * (1::'a) = a"
injects_1: fixes cfun :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue"   and cfun' :: "(C \<rightarrow> CValue) \<rightarrow> C \<rightarrow> CValue" shows "(CFn\<cdot>cfun = CFn\<cdot>cfun') = (cfun = cfun')"
focus_times: fixes X :: "'b set"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c"   and q :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "focus X (p * q) = focus X p * focus X q"
assms_1: shows "\<forall>z\<in>set zs. \<forall>y\<in>set ys. \<forall>x\<in>set xs. gt z y \<longrightarrow> gt y x \<longrightarrow> gt z x"
a1: shows "\<forall>x\<in>keys (mapping_of p). var \<notin> keys x"
higher_id_iff: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" shows "(higher p v = p) = (p = 0 \<or> v \<prec>\<^sub>t tt p)"
substitution_lemma: fixes S :: "nat \<Rightarrow> typ"   and B :: "nat \<Rightarrow> type_scheme"   and sch :: "type_scheme" shows "$ S (bound_scheme_inst B sch) = bound_scheme_inst ($ S \<circ> B) ($ S sch)"
l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf DocumentClass.known_ptr get_child_nodes get_child_nodes_locs"
ex_un_null: shows "\<exists>!n. \<forall>f. CC.comp n f = n \<and> CC.comp f n = n"
zero_less_one: fixes one :: "'k"   and zero :: "'k"   and less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool" assumes "class.zero_less_one one zero less_eq less" shows "less zero one"
collapse: fixes SkewBinomialTree :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" shows "SkewBinomialHeapStruc.Node (SkewBinomialHeapStruc.val SkewBinomialTree) (SkewBinomialHeapStruc.prio SkewBinomialTree) (SkewBinomialHeapStruc.rank SkewBinomialTree) (SkewBinomialHeapStruc.children SkewBinomialTree) = SkewBinomialTree"
le_add_same_cancel2: fixes plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less_eq a (plus b a) = less_eq zero b"
homeomorphism_moving_points_exists_gen: fixes K :: "'b set"   and x :: "'b \<Rightarrow> 'c"   and S :: "'c set"   and y :: "'b \<Rightarrow> 'c"   and T :: "'c set" assumes "finite K"   and "\<And>i. i \<in> K \<Longrightarrow> x i \<in> S \<and> y i \<in> S"   and "pairwise (\<lambda>i j. x i \<noteq> x j \<and> y i \<noteq> y j) K"   and "2 \<le> aff_dim S"   and "openin (top_of_set (affine hull S)) S"   and "S \<subseteq> T"   and "T \<subseteq> affine hull S"   and "connected S" shows "\<exists>f g. homeomorphism T T f g \<and> (\<forall>i\<in>K. f (x i) = y i) \<and> {x. \<not> (f x = x \<and> g x = x)} \<subseteq> S \<and> bounded {x. \<not> (f x = x \<and> g x = x)}"
rtranclp_mono: fixes r :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "r \<le> s" shows "r\<^sup>*\<^sup>* \<le> s\<^sup>*\<^sup>*"
fun_relI: fixes A :: "('a \<times> 'b) set"   and f :: "'a \<Rightarrow> 'c"   and f' :: "'b \<Rightarrow> 'd"   and B :: "('c \<times> 'd) set" assumes "\<And>a a'. (a, a') \<in> A \<Longrightarrow> (f a, f' a') \<in> B" shows "(f, f') \<in> A \<rightarrow> B"
cmp_interpretation: shows "identity_arrow_of_tabulations_in_maps (\<cdot>) (\<star>) \<a> \<i> src trg (r \<star> s) rs.\<rho>\<sigma>.tab (tab\<^sub>0 s \<star> rs.p\<^sub>0) (tab\<^sub>1 r \<star> rs.p\<^sub>1) (r \<star> s) rs.tab (tab\<^sub>0 (r \<star> s)) (tab\<^sub>1 (r \<star> s)) (r \<star> s)"
unlessE_def: fixes P :: "bool"   and f :: "'c \<Rightarrow> (('d + unit) \<times> 'c) set \<times> bool" shows "unlessE P f \<equiv> if P then returnOk () else f"
eq_expgen: fixes X :: "cmsg"   and Y :: "cmsg" assumes "eq X Y" shows "(\<forall>X'. X = cExp cGen X' \<longrightarrow> (\<exists>Z. Y = cExp cGen Z \<and> eq X' Z)) \<and> (\<forall>Y'. Y = cExp cGen Y' \<longrightarrow> (\<exists>Z. X = cExp cGen Z \<and> eq Y' Z))"
wp_assigns_r: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and r :: "'b upred" shows "\<langle>\<sigma>\<rangle>\<^sub>a wp r = \<sigma> \<dagger> r"
inject: fixes x1 :: "char list"   and x2 :: "hterm list"   and y1 :: "char list"   and y2 :: "hterm list" shows "(HFun x1 x2 = HFun y1 y2) = (x1 = y1 \<and> x2 = y2)"
fst_D: shows "fst (D ! ladder_n L 0) = ladder_i L 1"
monom_eq_0_iff: fixes a :: "'c"   and n :: "nat" shows "(monom a n = 0) = (a = (0::'c))"
SC_WbisT_imp_SC_Wbis: fixes c :: "('test, 'atom) com" assumes "SC_WbisT c" shows "SC_Wbis c"
STCal_step_1: fixes Source :: "'a processCalculus"   and Target :: "'b processCalculus"   and S :: "'a"   and P' :: "('a, 'b) Proc" shows "SourceTerm S \<longmapsto>STCal Source Target P' = (\<exists>S'. P' = SourceTerm S' \<and> S \<longmapsto>Source S')"
fin: fixes K :: "'a set" assumes "K \<in> \<D>" shows "finite {L \<in> \<D>. K \<subseteq> L}"
pcompose_assoc: fixes p :: "'a poly"   and q :: "'a poly"   and r :: "'a poly" shows "p \<circ>\<^sub>p (q \<circ>\<^sub>p r) = p \<circ>\<^sub>p q \<circ>\<^sub>p r"
dbl_dec_def: fixes minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and one :: "'c"   and x :: "'c" assumes "class.neg_numeral minus plus zero uminus" shows "neg_numeral.dbl_dec minus plus one x = minus (plus x x) one"
\<mu>: shows "natural_isomorphism C D G G' \<mu>"
invKey_K: fixes K :: "nat" assumes "K \<in> symKeys" shows "invKey K = K"
exp_scaleR_has_vector_derivative_right: fixes A :: "'a"   and t :: "real"   and T :: "real set" shows "((\<lambda>t. exp (t *\<^sub>R A)) has_vector_derivative exp (t *\<^sub>R A) * A) (at t within T)"
prod_ennreal: fixes A :: "'b set"   and f :: "'b \<Rightarrow> real" assumes "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i" shows "(\<Prod>i\<in>A. ennreal (f i)) = ennreal (prod f A)"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and order :: "order" shows "P (case order of Eq \<Rightarrow> f1 | Lt \<Rightarrow> f2 | Gt \<Rightarrow> f3) = ((order = Eq \<longrightarrow> P f1) \<and> (order = Lt \<longrightarrow> P f2) \<and> (order = Gt \<longrightarrow> P f3))"
cases: fixes a1 :: "'a"   and a2 :: "'a"   and r :: "('a \<times> 'a) set" assumes "(a1, a2) \<in> r\<^sup>+" obtains a :: "'a" and b :: "'a" where "a1 = a__" and "a2 = b__" and "(a__, b__) \<in> r"   | a :: "'a" and b :: "'a" and c :: "'a" where "a1 = a" and "a2 = c" and "(a, b) \<in> r\<^sup>+" and "(b, c) \<in> r"
reachable_dom_acyclic2: fixes n2 :: "nat"   and n1 :: "nat" assumes "reachable n2"   and "strict_dominate n1 n2" shows "\<not> dominate n2 n1"
isomorphic_to_zero_right: fixes smult' :: "('b, 'a) aezfun \<Rightarrow> 'd \<Rightarrow> 'd" assumes "isomorphic smult' 0" shows "V = 0"
h: fixes x :: "'a" shows "h x = (case f x of Terminal_corec a \<Rightarrow> TERMINAL a | Linear_corec b x' \<Rightarrow> LINEAR b (h x') | Branch_corec c xs \<Rightarrow> BRANCH c (\<lambda>d. h (xs d)) | Resumption_corec r \<Rightarrow> Rep_resumption r)"
a: shows "x:M \<in> IMPLEFT (B IMP C) (\<parallel><B>\<parallel>) (\<parallel>(C)\<parallel>)"
deriv_in_subterms_3: fixes p1 :: "('b, 'c, 'd, 'e) seqp"   and p2 :: "('b, 'c, 'd, 'e) seqp"   and p :: "('b, 'c, 'd, 'e) seqp" assumes "p1 \<oplus> p2 \<in> subterms p" shows "p1 \<in> subterms p"