hcju/isabelleps huggingface dataset:
initReachable_def: fixes cfg :: "('p, 'v, 's) configuration" shows "initReachable cfg \<equiv> \<exists>cfg0. initial cfg0 \<and> reachable cfg0 cfg"
permutation_ssort: fixes l :: "'a list" shows "ssort l <~~> l"
line_open_segment_iff: fixes i :: "real"   and a :: "'b"   and b :: "'b" assumes "i < 1"   and "a \<noteq> b" shows "{{a -- b}\<^bsub>i\<^esub><--<b} = {a -- b}\<^bsub>`{i<..<1}\<^esub>"
infinite_Un: fixes S :: "'d set"   and T :: "'d set" shows "infinite (S \<union> T) = (infinite S \<or> infinite T)"
as_4: shows "e1 \<inter> e2 \<inter> S = {}"
assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" shows "a * b * c = a * (b * c)"
cp_OclOr: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'a state \<times> 'a state" shows "(X or Y) \<tau> = ((\<lambda>_. X \<tau>) or (\<lambda>_. Y \<tau>)) \<tau>"
emb_hsize_neq: fixes t :: "('a, 'b) tm"   and s :: "('a, 'b) tm" assumes "t \<unrhd>\<^sub>e\<^sub>m\<^sub>b s"   and "t \<noteq> s" shows "hsize s < hsize t"
setFresh: fixes x :: "name"   and xvec :: "name list" shows "(x \<notin> set xvec) = x \<sharp> xvec"
C11: fixes x :: "'a"   and y :: "'a" shows "(x + y)\<^sup>\<star> = (x\<^sup>\<star> \<cdot> y)\<^sup>\<star> \<cdot> x\<^sup>\<star>"
INT_insert: fixes B :: "'b \<Rightarrow> 'a set"   and a :: "'b"   and A :: "'b set" shows "\<Inter> (B ` insert a A) = B a \<inter> \<Inter> (B ` A)"
pathstart_reversepath: fixes g :: "real \<Rightarrow> 'b" shows "pathstart (reversepath g) = pathfinish g"
ipresIGWlsAbs_def: fixes hA :: "'x \<Rightarrow> 'y"   and MOD :: "('z, 'aa, 'ab, 'ac, 'ad, 'ae, 'af, 'x, 'ag) model_scheme"   and MOD' :: "('ah, 'ai, 'ab, 'ac, 'aj, 'ak, 'al, 'y, 'am) model_scheme" shows "ipresIGWlsAbs hA MOD MOD' \<equiv> \<forall>us s A. igWlsAbs MOD (us, s) A \<longrightarrow> igWlsAbs MOD' (us, s) (hA A)"
simps_4: fixes m :: "nat" shows "ackloop [m] = m"
dnf_min_set: fixes \<phi> :: "'b ltln" shows "min_dnf \<phi> = min_set (dnf \<phi>)"
assms_4: shows "pa \<in> proj_set (f a) G"
D2_def: fixes r :: "('c \<times> 'c) set"   and d :: "('d \<times> ('c \<times> 'd) list) \<times> ('d \<times> ('c \<times> 'd) list) \<times> ('d \<times> ('c \<times> 'd) list) \<times> 'd \<times> ('c \<times> 'd) list" shows "D2 r d = (let (\<tau>, \<sigma>, \<sigma>', \<tau>') = d in D r (labels \<tau>) (labels \<sigma>) (labels \<sigma>') (labels \<tau>'))"
lincomb_insert2: fixes S :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and v :: "'c" assumes "finite S"   and "S \<subseteq> carrier M"   and "a \<in> insert v S \<rightarrow> carrier R"   and "v \<notin> S"   and "v \<in> carrier M" shows "lincomb a (insert v S) = a v \<odot>\<^bsub>M\<^esub> v \<oplus>\<^bsub>M\<^esub> lincomb a S"
rawpsubstT_compose_freshVar2_aux: fixes r :: "'trm"   and ts :: "'trm list"   and xs :: "'var list"   and us :: "'var list"   and vs :: "'var list" assumes "r \<in> atrm"   and "set ts \<subseteq> atrm"   and "set xs \<subseteq> var"   and "distinct xs"   and "set us \<subseteq> var"   and "distinct us"   and "set us \<inter> FvarsT r = {}"   and "set us \<inter> \<Union> (FvarsT ` set ts) = {}"   and "set us \<inter> set xs = {}"   and "set vs \<subseteq> var"   and "distinct vs"   and "set vs \<inter> FvarsT r = {}"   and "set vs \<inter> \<Union> (FvarsT ` set ts) = {}"   and "set vs \<inter> set xs = {}"   and "length us = length xs"   and "length vs = length xs"   and "length ts = length xs"   and "set us \<inter> set vs = {}" shows "rawpsubstT (rawpsubstT r (zip (map Var us) xs)) (zip ts us) = rawpsubstT (rawpsubstT r (zip (map Var vs) xs)) (zip ts vs)"
guard_x: shows "\<forall>\<tau>. \<tau> \<in> U s \<and> \<tau> \<le> t \<longrightarrow> G (X \<tau>)"
env\<^sub>2'_def: shows "env\<^sub>2' = fmmap convert_term env\<^sub>2"
empty_subts: fixes ts :: "fterm list" shows "ts \<cdot>\<^sub>t\<^sub>s \<epsilon> = ts"
path_linear_image_eq: fixes f :: "'a \<Rightarrow> 'b"   and g :: "real \<Rightarrow> 'a" assumes "linear f"   and "inj f" shows "path (f \<circ> g) = path g"
dghm_HomDom: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF>\<lparr>HomDom\<rparr> = \<AA>"
ring_homD: fixes R :: "('c, 'd) ring_scheme"   and S :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e"   and Phi :: "(nat \<Rightarrow> 'c) \<Rightarrow> 'e" assumes "UP_pre_univ_prop R S h"   and "Phi \<in> ring_hom (UP R) S" shows "ring_hom_cring (UP R) S Phi"
add_basis_naive_def: fixes gs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and ns :: "(('e \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'h" shows "add_basis_naive gs bs ns data = bs @ ns"
eqset_imp_iff: fixes A :: "'a set"   and B :: "'a set"   and x :: "'a" assumes "A = B" shows "(x \<in> A) = (x \<in> B)"
order_pderiv: fixes p :: "'a poly"   and a :: "'a" assumes "pderiv p \<noteq> 0"   and "order a p \<noteq> 0" shows "order a p = Suc (order a (pderiv p))"
lActive_less: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and i :: "nat"   and n :: "nat" assumes "active c (t i)"   and "\<forall>n'>n. \<not> active c (t n')" shows "dynamic_component.lActive active c t \<le> n"
elementOfPartitionOfFiniteSetIsFinite: fixes X :: "'c set"   and XX :: "'c set set" assumes "finite X"   and "XX \<in> all_partitions X" shows "finite XX"
l11_25_aux: fixes P :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p" assumes "P InAngle A B C"   and "\<not> Bet A B C"   and "B Out A' A" shows "P InAngle A' B C"
ample_base_def: fixes ex :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and en :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and int :: "'b \<Rightarrow> 'c"   and ind :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and src :: "'b \<Rightarrow> 'b \<Rightarrow> bool" shows "ample_base ex en int ind src \<equiv> transition_system_interpreted_traces ex en int ind \<and> wellfounded_relation src"
that: fixes as :: "'edge list" assumes "parent_node n -as\<rightarrow>\<^sub>\<surd>* parent_node n'" shows "thesis"
gr_zeroI: fixes n :: "'a" assumes "n = (0::'a) \<Longrightarrow> False" shows "(0::'a) < n"
lemma1: fixes K :: "key"   and H :: "msg set"   and X :: "msg" assumes "Key (invKey K) \<in> analz H" shows "analz (insert (Crypt K X) H) \<subseteq> insert (Crypt K X) (analz (insert X H))"
\<I>_common_def: shows "\<I>_common \<equiv> \<lambda>\<eta>. \<I>_uniform (vld \<eta>) UNIV \<oplus>\<^sub>\<I> \<I>_uniform UNIV (insert None (Some ` vld \<eta>))"
extremum: fixes a :: "'b multiset" shows "\<Empt> \<subseteq># a"
inf2: fixes B :: "'c"   and k :: "'c"   and b :: "'c"   and a :: "'c" assumes "B \<equiv> inf k b" shows "inf a B \<equiv> inf k (inf a b)"
all_in_list_insort: fixes xs :: "('b, 'c) Combinators list"   and l :: "'b set list"   and x :: "('b, 'c) Combinators" assumes "all_in_list xs l"   and "singleCombinators (x # xs)"   and "in_list x l" shows "all_in_list (FWNormalisationCore.insort x xs l) l"
scale_right_distrib: fixes a :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and x :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and y :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" shows "local.punit.mult_scalar a (x + y) = local.punit.mult_scalar a x + local.punit.mult_scalar a y"
X_not_zero: assumes "\<one> \<noteq> \<zero>" shows "X_poly R \<noteq> \<zero>\<^bsub>P\<^esub>"
sorted_iff_nth_Suc: fixes xs :: "'a list" shows "linorder.sorted (\<lambda>x y. y \<le> x) xs = (\<forall>i. Suc i < length xs \<longrightarrow> xs ! Suc i \<le> xs ! i)"
cases: fixes a1 :: "bool"   and a2 :: "char list list"   and a3 :: "v sem_env"   and a4 :: "'b state"   and a5 :: "dec list"   and a6 :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_decs a1 a2 a3 a4 a5 a6" obtains ck :: "bool" and mn :: "char list list" and env :: "v sem_env" and s :: "'b state" where "a1 = ck__" and "a2 = mn__" and "a3 = enva__" and "a4 = s" and "a5 = []" and "a6 = (s, Rval (make_sem_env nsEmpty nsEmpty))"   | ck :: "bool" and mn :: "char list list" and s1 :: "'b state" and s2 :: "'b state" and env :: "v sem_env" and d :: "dec" and ds :: "dec list" and e :: "v error_result" where "a1 = cka__" and "a2 = mna__" and "a3 = envb__" and "a4 = s1__" and "a5 = d__ # ds__" and "a6 = (s2__, Rerr e)" and "evaluate_dec cka__ mna__ envb__ s1__ d__ (s2__, Rerr e)"   | ck :: "bool" and mn :: "char list list" and s1 :: "'b state" and s2 :: "'b state" and s3 :: "'b state" and env :: "v sem_env" and d :: "dec" and ds :: "dec list" and new_env :: "v sem_env" and r :: "(v sem_env, v) result" where "a1 = ck" and "a2 = mn" and "a3 = env" and "a4 = s1" and "a5 = d # ds" and "a6 = (s3, combine_dec_result new_env r)" and "evaluate_dec ck mn env s1 d (s2, Rval new_env) \<and> evaluate_decs ck mn (extend_dec_env new_env env) s2 ds (s3, r)"
compose_delete_twist: fixes k :: "'d"   and xs :: "('d \<times> 'f) list"   and ys :: "('f \<times> 'e) list" shows "compose (delete k xs) ys = delete k (compose xs ys)"
variation_cases_1: fixes x :: "real"   and y :: "real" assumes "0 < x"   and "0 < y" shows "variation x y = 0"
equiv_not_sym: fixes x :: "'a"   and y :: "'a" assumes "\<not> x \<sim> y" shows "\<not> y \<sim> x"
Image_fold: fixes R :: "('b \<times> 'c) set"   and S :: "'b set" assumes "finite R" shows "R `` S = Finite_Set.fold (\<lambda>(x, y) A. if x \<in> S then insert y A else A) {} R"
is_denormal_def: fixes a :: "('a, 'b) IEEE.float" shows "is_denormal a = (IEEE.exponent a = 0 \<and> fraction a \<noteq> 0)"
lgrid_eq: fixes pa :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list" assumes "pa \<in> grid b {d}" shows "lgrid b {d} lm \<inter> (grid pa {d} - {pa}) = lgrid pa {d} lm - {pa}"
rel_simps_76: fixes n :: "num" shows "(0::'a) \<noteq> numeral n"
l5_6: 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 D :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a"   and D' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Le Bet Cong A B C D"   and "Cong A B A' B'"   and "Cong C D C' D'" shows "Tarski_neutral_dimensionless.Le Bet Cong A' B' C' D'"
cases: fixes \<I> :: "('d, 'e) \<I>"   and a :: "('f, 'd, 'e) gpv" assumes "finite_gpv \<I> a" obtains gpv :: "('f, 'd, 'e) gpv" where "a = gpv" and "\<And>out c input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> finite_gpv \<I> (c input)"
mult_right_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z * x \<le> z * y"
anti_deduction: fixes H :: "fmla set"   and A :: "fmla"   and B :: "fmla" assumes "H \<turnstile> A IMP B" shows "insert A H \<turnstile> B"
resource_of_parallel_oracle: fixes oracle1 :: "'e \<Rightarrow> 'a \<Rightarrow> ('c \<times> 'e) spmf"   and oracle2 :: "'f \<Rightarrow> 'b \<Rightarrow> ('d \<times> 'f) spmf"   and s1 :: "'e"   and s2 :: "'f" shows "RES (oracle1 \<ddagger>\<^sub>O oracle2) (s1, s2) = RES oracle1 s1 \<parallel> RES oracle2 s2"
walk_edges_vertices_1: fixes G :: "('a, 'b) Graph_scheme"   and v :: "'a"   and w :: "'a"   and xs :: "'a list" assumes "Digraph G"   and "(v, w) \<in> Digraph.edges_of_walk xs" shows "v \<in> set xs"
assms_4: shows "bl' \<in># remove1_mset bl B"
coeff_pcompose_monom: fixes j :: "nat"   and n :: "nat"   and f :: "'a poly"   and i :: "nat" assumes "j < n" shows "poly.coeff (f \<circ>\<^sub>p Polynomial.monom (1::'a) n) (n * i + j) = (if j = 0 then poly.coeff f i else (0::'a))"
simps_9: fixes jmps :: "jump set"   and c1 :: "stmt"   and C :: "qtname"   and vn :: "vname"   and c2 :: "stmt" shows "jumpNestingOkS jmps (Try c1 Catch(C vn) c2) = (jumpNestingOkS jmps c1 \<and> jumpNestingOkS jmps c2)"
runProc_gState_progress_rel: fixes prog :: "program"   and g :: "'a gState_scheme"   and p :: "pState"   and name :: "String.literal"   and args :: "expr list" assumes "program_inv prog"   and "gState_inv prog g"   and "pState_inv prog p"   and "cl_inv (g, p)" shows "(g, fst (runProc name args prog g p)) \<in> gState_progress_rel prog"
atLeast1_atMost_eq_remove0: fixes n :: "nat" shows "{Suc 0..n} = {..n} - {0}"
suc: fixes X :: "'a set set"   and A :: "'a set set" assumes "X \<in> subset.suc_Union_closed A" shows "subset.suc A X \<in> subset.suc_Union_closed A"
assms_1: shows "vars F0 \<subseteq> Vbl"
m: shows "Monotonic (\<lambda>X. P ;; X \<triangleleft> b \<triangleright>\<^sub>r II)"
bind_nonempty': fixes f :: "'c \<Rightarrow> 'd measure"   and M :: "'c measure"   and N :: "'d measure"   and x :: "'c" assumes "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N"   and "x \<in> space M" shows "M \<bind> f = join (distr M (subprob_algebra N) f)"
pelims: fixes x :: "'a literal"   and y :: "'a literal" assumes "x\<^sup>c = y"   and "complement_dom x" obtains P :: "char list" and ts :: "'a list" where "x = Pos P__ ts__" and "y = Neg P__ ts__" and "complement_dom (Pos P__ ts__)"   | P :: "char list" and ts :: "'a list" where "x = Neg P ts" and "y = Pos P ts" and "complement_dom (Neg P ts)"
map_Val_nthrow_neq: fixes vs :: "val list"   and vs' :: "val list"   and ex' :: "'a exp"   and es' :: "'a exp list" assumes "map Val vs = map Val vs' @ throw ex' # es'" shows "False"
S_def: shows "local.S = {v. norm1 v = 1 \<and> (\<forall>i. 0 \<le> v $h i) \<and> (\<forall>i. HMA_Connect.spectral_radius A * v $h i \<le> (local.B *v v) $h i)}"
nsum_memr: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "\<forall>i\<le>n. f i \<in> carrier R" shows "\<forall>l\<le>n. \<Sigma>\<^sub>e R f l \<in> carrier R"
hol_pal_lem3: fixes S :: "complex set"   and a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex"   and k :: "real"   and f :: "complex \<Rightarrow> complex" assumes "convex S"   and "open S"   and "a \<in> S"   and "b \<in> S"   and "c \<in> S"   and "d \<noteq> 0"   and "d \<bullet> a \<le> k"   and "f holomorphic_on {z \<in> S. d \<bullet> z < k}"   and "f holomorphic_on {z \<in> S. k < d \<bullet> z}"   and "continuous_on S f" shows "contour_integral (linepath a b) f + contour_integral (linepath b c) f + contour_integral (linepath c a) f = 0"
RInitThrowReds: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and a :: "nat"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog"   and C :: "char list"   and sfs :: "sfields"   and i :: "init_state"   and sh'' :: "sheap"   and e\<^sub>0 :: "char list exp" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow>* \<langle>Throw a,(h', l', sh'),b'\<rangle>"   and "sh' C = \<lfloor>(sfs, i)\<rfloor>"   and "sh'' = sh'(C \<mapsto> (sfs, Error))" shows "P \<turnstile> \<langle>RI (C,e) ; [] \<leftarrow> e\<^sub>0,s,b\<rangle> \<rightarrow>* \<langle>Throw a,(h', l', sh''),b'\<rangle>"
reachable_conv: fixes u :: "'a"   and v :: "'a" shows "(u \<rightarrow>\<^sup>*\<^bsub>with_proj (list_digraph G)\<^esub> v) = ((u, v) \<in> Restr ((parcs (list_digraph G))\<^sup>*) (pverts (list_digraph G)))"
t_def: shows "t = lcs (lp p) (lp q) - lp q"
identity_arrow_of_spans_axioms_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" shows "identity_arrow_of_spans_axioms C \<mu> \<equiv> partial_magma.ide C (Chn \<mu>)"
that: fixes g :: "real \<Rightarrow> real" assumes "g square_integrable S"   and "\<And>i. i \<in> I \<Longrightarrow> orthonormal_coeff S w (\<lambda>x. f x - g x) i = 0"   and "(\<lambda>n. l2norm S (\<lambda>x. (\<Sum>i\<in>I \<inter> {..n}. orthonormal_coeff S w f i * w i x) - g x)) \<longlonglongrightarrow> 0" shows "thesis"
homeomorphic_unicoherent_eq: fixes S :: "'c set"   and T :: "'d set" assumes "S homeomorphic T" shows "unicoherent S = unicoherent T"
cand_def: fixes P :: "bool"   and Q :: "bool" shows "cand P Q = (if P then Q else False)"
sxalloc_cases: fixes G :: "prog"   and s :: "state"   and s' :: "state" assumes "G\<turnstile>s \<midarrow>sxalloc\<rightarrow> s'" obtains sa :: "st" where "s = Norm sa__" and "s' = Norm sa__"   | j :: "jump" and sa :: "st" where "s = (Some (Jump j), saa__)" and "s' = (Some (Jump j), saa__)"   | e :: "error" and sa :: "st" where "s = (Some (Error e), sab__)" and "s' = (Some (Error e), sab__)"   | a :: "loc" and sa :: "st" where "s = (Some (Xcpt (Loc a__)), sa)" and "s' = (Some (Xcpt (Loc a__)), sa)"   | s0 :: "st" and xn :: "xname" and a :: "loc" and x :: "abopt" and s1 :: "st" where "s = (Some (Xcpt (Std xn)), s0)" and "s' = (Some (Xcpt (Loc a)), s1)" and "G\<turnstile>Norm s0 \<midarrow>halloc CInst (SXcpt xn)\<succ>a\<rightarrow> (x, s1)"
inorder_balR: fixes l :: "('c \<times> nat) tree"   and a :: "'c"   and r :: "('c \<times> nat) tree" shows "Tree2.inorder (balR l a r) = Tree2.inorder l @ a # Tree2.inorder r"
abs_Gromov_completion_in_Gromov_boundary: fixes u :: "nat \<Rightarrow> 'a" assumes "Gromov_converging_at_boundary u" shows "abs_Gromov_completion u \<in> Gromov_boundary"
set_tag_name_writes: fixes element_ptr :: "(_) element_ptr"   and tag :: "char list"   and h :: "(_) heap"   and h' :: "(_) heap" shows "writes (set_tag_name_locs element_ptr) (set_tag_name element_ptr tag) h h'"
assms_4: shows "InvariantEquivalentZL (getF state) (getM state) F0"
disj_idem: fixes P :: "'b \<Rightarrow> bool" shows "(P \<or> P) = P"
length_pos_if_in_set: fixes x :: "'e"   and xs :: "'e list" assumes "x \<in> set xs" shows "0 < length xs"
valid_Tree\<^sub>\<alpha>_Act\<^sub>\<alpha>: fixes satisfies :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and transition :: "'b \<Rightarrow> ('d, 'b) residual \<Rightarrow> bool"   and P :: "'b"   and \<alpha> :: "'d"   and t\<^sub>\<alpha> :: "('e, 'c, 'd) Tree\<^sub>\<alpha>" assumes "nominal_ts satisfies transition" shows "nominal_ts.valid_Tree\<^sub>\<alpha> satisfies transition P (Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha>) = (\<exists>\<alpha>'' t\<^sub>\<alpha>' P'. Act\<^sub>\<alpha> \<alpha> t\<^sub>\<alpha> = Act\<^sub>\<alpha> \<alpha>'' t\<^sub>\<alpha>' \<and> transition P \<langle>\<alpha>'',P'\<rangle> \<and> nominal_ts.valid_Tree\<^sub>\<alpha> satisfies transition P' t\<^sub>\<alpha>')"
mult_divide_mult_cancel_left_if: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and c :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide (times c a) (times c b) = (if c = zero then zero else divide a b)"
state_to_wm_def: fixes s :: "predicate \<times> object list \<Rightarrow> bool" shows "state_to_wm s = {Atom (predAtm p xs) |p xs. s (p, xs)}"
all_imp_eqI: fixes P :: "'b \<Rightarrow> bool"   and P' :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and Q' :: "'b \<Rightarrow> bool" assumes "P = P'"   and "\<And>x. P x \<Longrightarrow> Q x = Q' x" shows "(\<forall>x. P x \<longrightarrow> Q x) = (\<forall>x. P' x \<longrightarrow> Q' x)"
span_minimal: fixes S :: "'b set"   and T :: "'b set" assumes "S \<subseteq> T"   and "local.subspace T" shows "local.span S \<subseteq> T"
prems: fixes i :: "nat" assumes "i < count_weights s (Pool m1 m2)" shows "w1 i = w2 i"
add_less_same_cancel2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "class.ordered_ab_semigroup_monoid_add_imp_le plus zero minus less_eq less" shows "less (plus a b) b = less a zero"
enum_def: fixes enum :: "'a list"   and enum_all :: "('a \<Rightarrow> bool) \<Rightarrow> bool"   and enum_ex :: "('a \<Rightarrow> bool) \<Rightarrow> bool" shows "class.enum enum enum_all enum_ex \<equiv> (UNIV = set enum \<and> distinct enum) \<and> (\<forall>P. enum_all P = Ball UNIV P) \<and> (\<forall>P. enum_ex P = Bex UNIV P)"
norm: fixes k :: "nat" shows "cmod (\<chi> k) = (if coprime k n then 1 else 0)"
l_def: shows "l = Min (insert 0 {x. poly p x = 0}) - 1"
dd_subs_n: shows "set_mset DD \<subseteq> N"
winding_number_part_circlepath_pos_less: fixes s :: "real"   and t :: "real"   and w :: "complex"   and z :: "complex"   and r :: "real" assumes "s < t"   and "cmod (w - z) < r" shows "0 < Re (winding_number (part_circlepath z r s t) w)"
open_translation: fixes S :: "'b set"   and a :: "'b" assumes "open S" shows "open ((+) a ` S)"
c1_on_open_def: fixes f :: "'c \<Rightarrow> 'c"   and f' :: "'c \<Rightarrow> 'c \<Rightarrow>\<^sub>L 'c"   and X :: "'c set" shows "c1_on_open f f' X \<equiv> open X \<and> (\<forall>x. x \<in> X \<longrightarrow> (f has_derivative blinfun_apply (f' x)) (at x)) \<and> continuous_on X f'"
power_neg_one_If: fixes n :: "nat" shows "(- (1::'b)) ^ n = (if even n then 1::'b else - (1::'b))"
V_mono_Suc: fixes j :: "nat" shows "V j \<subseteq> V (Suc j)"
HomFtorMapLemma1: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d"   and f :: "'e"   and x :: "ZF" assumes "LSCategory C"   and "X \<in> obj\<^bsub>C\<^esub>"   and "f \<in> mor\<^bsub>C\<^esub>"   and "x |\<in>| (Hom\<^bsub>C\<^esub> X dom\<^bsub>C\<^esub> f)" shows "m2z\<^bsub>C\<^esub>z2m\<^bsub>C\<^esub>x ;;\<^bsub>C\<^esub> f |\<in>| (Hom\<^bsub>C\<^esub> X cod\<^bsub>C\<^esub> f)"
B: fixes s :: "trm"   and y :: "name"   and n :: "trm"   and l :: "stack" assumes "t = [s]"   and "[y__]n__\<ggreater>L = [y]n\<ggreater>l"   and "r = n[y::=s] \<star> l" shows "P"
length_l_partition_by_median: fixes ps :: "(real, 'a) vec list"   and l :: "(real, 'a) vec list"   and m :: "real"   and r :: "(real, 'a) vec list"   and k :: "'a" assumes "0 < length ps"   and "\<forall>k. distinct (map (\<lambda>p. p $ k) ps)"   and "(l, m, r) = partition_by_median k ps" shows "length l = (length ps - 1) div 2 + 1"
partial_state2_axioms: shows "partial_state2 vars1 vars2"
rtrancl_trans: fixes x :: "'g"   and y :: "'g"   and r :: "('g \<times> 'g) set"   and z :: "'g" assumes "(x, y) \<in> r\<^sup>*"   and "(y, z) \<in> r\<^sup>*" shows "(x, z) \<in> r\<^sup>*"
card_atLeastLessThan: fixes l :: "nat"   and u :: "nat" shows "card {l..<u} = u - l"
eq: shows "ZFC_in_HOL.succ \<beta> = \<Squnion> A"
abs_conv_abscissa_deriv: fixes f :: "'a fds" shows "abs_conv_abscissa (fds_deriv f) = abs_conv_abscissa f"
acc_run_run: fixes G :: "('b, 'c) gb_graph_rec_scheme"   and r :: "nat \<Rightarrow> 'b" assumes "gb_graph G"   and "gb_graph.is_acc_run G r" shows "graph_defs.is_run G r"
not_eq_newpkt_rrep: fixes hops :: "nat"   and dip :: "nat"   and dsn :: "nat"   and oip :: "nat"   and sip :: "nat" shows "\<not> eq_newpkt (Rrep hops dip dsn oip sip)"
assms_2: shows "\<^bold>|u\<^bold>| < \<^bold>|v\<^bold>|"
cont2contlub: fixes f :: "'b \<Rightarrow> 'c" assumes "cont f" shows "contlub f"
sep_conj_sep_impl_sep_conj2: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and sep_disj :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a \<Rightarrow> bool"   and R :: "'a \<Rightarrow> bool"   and h :: "'a"   and Q :: "'a \<Rightarrow> bool" assumes "class.sep_algebra plus zero sep_disj"   and "sep_algebra.sep_conj plus sep_disj P R h" shows "sep_algebra.sep_conj plus sep_disj P (sep_algebra.sep_impl plus sep_disj Q (sep_algebra.sep_conj plus sep_disj Q R)) h"
buffon_prob_aux: shows "emeasure (buffon l d) {True} = emeasure lborel (buffon_set l d) / ennreal (2 * d * pi)"
reflcl_trancl: fixes r :: "('b \<times> 'b) set" shows "(r\<^sup>+)\<^sup>= = r\<^sup>*"
rPar2: fixes \<alpha> :: "act"   and P' :: "ccs"   and \<C> :: "'a" assumes "!P \<longmapsto>\<alpha> \<prec> P'"   and "\<And>\<C>. Prop \<C> !P \<alpha> P'" shows "Prop \<C> (P \<parallel> !P) \<alpha> (P \<parallel> P')"