hcju/isabelleps huggingface dataset:
uniX: shows "S \<union> insert e X \<union> Y = {0..<m}"
gcd_list_greatest: fixes bs :: "'a list"   and a :: "'a" assumes "\<And>b. b \<in> set bs \<Longrightarrow> a dvd b" shows "a dvd gcd_list bs"
num_params_split_by_prec_le_1: fixes x :: "float poly"   and K :: "nat"   and ord :: "nat" assumes "num_params x \<le> K" shows "num_params (fst (split_by_prec ord x)) \<le> K"
get_e: shows "\<forall>p\<in>s. 0 < get_e p \<and> avoid p (get_e p) \<and> (p \<in> pz \<longrightarrow> cont ff p (get_e p))"
axioms_1: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b" assumes "ChamberComplexAutomorphism X f" shows "ChamberComplexIsomorphism X X f"
split_list: shows "split_half ts = (ls, (sub, sep) # rs)"
new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t_get\<^sub>N\<^sub>o\<^sub>d\<^sub>e: fixes h :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and h' :: "(_) heap"   and ptr :: "(_) node_ptr" assumes "new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h = (new_document_ptr, h')" shows "get ptr h = get ptr h'"
arr_cod: fixes f :: "'a1" assumes "A1.arr f" shows "A1.arr (A1.cod f)"
cases: fixes a1 :: "'b list"   and a2 :: "'b list" assumes "a1 \<approx>2 a2" obtains a :: "'b" and b :: "'b" and xs :: "'b list" where "a1 = a__ # b # xs__" and "a2 = b # a__ # xs__"   | "a1 = []" and "a2 = []"   | xs :: "'b list" and ys :: "'b list" where "a1 = ys__" and "a2 = xsa__" and "xsa__ \<approx>2 ys__"   | a :: "'b" and xs :: "'b list" where "a1 = aa__ # aa__ # xsb__" and "a2 = aa__ # xsb__"   | xs :: "'b list" and ys :: "'b list" and a :: "'b" where "a1 = a # xs" and "a2 = a # ys" and "xs \<approx>2 ys"   | xs1 :: "'b list" and xs2 :: "'b list" and xs3 :: "'b list" where "a1 = xs1" and "a2 = xs3" and "xs1 \<approx>2 xs2" and "xs2 \<approx>2 xs3"
ipresIGSwapAbs_def: fixes hA :: "'h \<Rightarrow> 'i"   and MOD :: "('j, 'k, 'l, 'm, 'n, 'o, 'p, 'h, 'q) model_scheme"   and MOD' :: "('r, 's, 'l, 't, 'u, 'o, 'v, 'i, 'w) model_scheme" shows "ipresIGSwapAbs hA MOD MOD' \<equiv> \<forall>zs z1 z2 us s A. igWlsAbs MOD (us, s) A \<longrightarrow> hA (igSwapAbs MOD zs z1 z2 A) = igSwapAbs MOD' zs z1 z2 (hA A)"
assms_2: shows "alphaAbs_ex_equal_or_qFresh xs x X xs' x' X'"
low_in_def: fixes \<tau> :: "trace" shows "low_in \<tau> \<equiv> the (initial \<tau> ''lowIn'')"
hor_vert_finite_1: shows "finite (two_chain_vertical_boundary two_chain)"
pairs_substI: fixes \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and M :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" assumes "subst_domain \<theta> \<inter> \<Union> (fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r ` M) = {}" shows "M \<cdot>\<^sub>p\<^sub>s\<^sub>e\<^sub>t \<theta> = M"
word_rbl_mul_induct: assumes "length (rev (to_bl x)) \<le> size y" shows "rbl_mul (rev (to_bl x)) (rev (to_bl y)) = take (length (rev (to_bl x))) (rev (to_bl (of_bl (rev (rev (to_bl x))) * y)))"
axioms_3: fixes V\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and H\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<a>\<^sub>B :: "'r \<Rightarrow> 'r \<Rightarrow> 'r \<Rightarrow> 'r"   and \<i>\<^sub>B :: "'r \<Rightarrow> 'r"   and src\<^sub>B :: "'r \<Rightarrow> 'r"   and trg\<^sub>B :: "'r \<Rightarrow> 'r"   and V\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and H\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's"   and \<a>\<^sub>C :: "'s \<Rightarrow> 's \<Rightarrow> 's \<Rightarrow> 's"   and \<i>\<^sub>C :: "'s \<Rightarrow> 's"   and src\<^sub>C :: "'s \<Rightarrow> 's"   and trg\<^sub>C :: "'s \<Rightarrow> 's"   and V\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't"   and H\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't"   and \<a>\<^sub>D :: "'t \<Rightarrow> 't \<Rightarrow> 't \<Rightarrow> 't"   and \<i>\<^sub>D :: "'t \<Rightarrow> 't"   and src\<^sub>D :: "'t \<Rightarrow> 't"   and trg\<^sub>D :: "'t \<Rightarrow> 't"   and F :: "'r \<Rightarrow> 's"   and \<Phi>\<^sub>F :: "'r \<times> 'r \<Rightarrow> 's"   and G :: "'s \<Rightarrow> 't"   and \<Phi>\<^sub>G :: "'s \<times> 's \<Rightarrow> 't" assumes "composite_pseudofunctor V\<^sub>B H\<^sub>B \<a>\<^sub>B \<i>\<^sub>B src\<^sub>B trg\<^sub>B V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>\<^sub>F G \<Phi>\<^sub>G" shows "bicategory V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D"
parPres: fixes P :: "ccs"   and Rel :: "(ccs \<times> ccs) set"   and Q :: "ccs"   and Rel' :: "(ccs \<times> ccs) set"   and R :: "ccs" assumes "P \<leadsto>[Rel] Q"   and "(P, Q) \<in> Rel"   and "\<And>S T U. (S, T) \<in> Rel \<Longrightarrow> (S \<parallel> U, T \<parallel> U) \<in> Rel'" shows "P \<parallel> R \<leadsto>[Rel'] Q \<parallel> R"
left_idem: fixes a :: "'e"   and b :: "'e" shows "inf a (inf a b) = inf a b"
project_inject_cancel: fixes project :: "'b \<Rightarrow> 'c"   and inject :: "'c \<Rightarrow> 'b"   and x :: "'c" assumes "project_inject project inject" shows "project (inject x) = x"
lin: fixes x :: "'a"   and y :: "'a" shows "R x y \<or> R y x"
map_pair: fixes f :: "'d \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'c"   and A :: "'d pmf"   and B :: "'e pmf" shows "map_pmf (\<lambda>(a, b). (f a, g b)) (pair_pmf A B) = pair_pmf (map_pmf f A) (map_pmf g B)"
dn_def: shows "dn \<equiv> d (n + 1) powr 2 ^ (n + 1)"
not_prime_imp_ex_prod_nat: fixes m :: "nat" assumes "1 < m"   and "\<not> prime m" shows "\<exists>n k. m = n * k \<and> 1 < n \<and> n < m \<and> 1 < k \<and> k < m"
length_swap: fixes i :: "nat"   and xs :: "'b list" shows "length (swap i xs) = length xs"
P1_real_ddh1_\<sigma>_true: fixes \<sigma> :: "bool"   and M :: "'a \<times> 'a"   and z :: "'f"   and \<A> :: "('a \<times> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'f \<Rightarrow> (('a \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<Rightarrow> 'g \<Rightarrow> ((('a \<times> 'a) \<times> 'a \<times> 'a) \<times> 'g) spmf) \<times> ('g \<Rightarrow> 'e spmf)"   and D :: "'e \<times> 'a \<Rightarrow> bool spmf" assumes "\<sigma> = True" shows "P1_real_model M \<sigma> z \<A> \<bind> D = local.ddh.DDH1 (P1_DDH_mal_adv_\<sigma>_true M z \<A> D)"
to_stream_to_omega: fixes s :: "'b stream" shows "to_stream (to_omega s) = s"
fimage_fconst_on_ne: fixes B :: "V"   and A :: "V"   and c :: "V" assumes "B \<inter>\<^sub>\<circ> A \<noteq> []\<^sub>\<circ>" shows "fconst_on A c `\<^sub>\<bullet> B = set {c}"
hyps_2: fixes q :: "ref" assumes "Dag q (rep \<propto> low) (rep \<propto> high) rrt"   and "\<forall>pt\<in>set_of rrt. rep pt = pt" shows "Dag q (rep \<propto> lowa) (rep \<propto> higha) rrt"
notrkis0: shows "\<forall>i\<in>{0..k}. i \<noteq> k mod p \<longrightarrow> int (n mod p choose i) * (1 + fps_X ^ p) ^ (n div p) $ (k - i) = 0"
lm064: fixes R :: "('a \<times> 'b) set" assumes "runiq R" shows "sum (toFunction R) (Domain R) = sum snd R"
cases: fixes a1 :: "'node"   and a2 :: "'edge list"   and a3 :: "'node" assumes "a1 -a2\<rightarrow>* a3" obtains n :: "'node" where "a1 = na__" and "a2 = []" and "a3 = na__" and "valid_node na__"   | n'' :: "'node" and as :: "'edge list" and n' :: "'node" and a :: "'edge" and n :: "'node" where "a1 = n" and "a2 = a # as" and "a3 = n'" and "n'' -as\<rightarrow>* n'" and "valid_edge a" and "sourcenode a = n" and "targetnode a = n''"
c_ipurge_aux_append: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and U :: "'f set"   and xs :: "'e list"   and x :: "'e" shows "c_ipurge_aux I D U (xs @ [x]) = (if D x \<in> c_sources_aux I D U [x] then c_ipurge_aux I D (insert (D x) U) xs @ [x] else c_ipurge_aux I D U xs)"
intros_13: fixes c :: "i32"   and testop :: "testop" shows "\<lparr>[$C ConstInt32 c, $Testop T_i32 testop]\<rparr> \<leadsto> \<lparr>[$C ConstInt32 (wasm_bool (app_testop_i testop c))]\<rparr>"
measures_lesseq: fixes f :: "'b \<Rightarrow> nat"   and x :: "'b"   and y :: "'b"   and fs :: "('b \<Rightarrow> nat) list" assumes "f x \<le> f y"   and "(x, y) \<in> measures fs" shows "(x, y) \<in> measures (f # fs)"
prj_u_def: shows "PRJ('e\<^sub>\<bottom>) = liftprj oo u_prj"
fls_shift_def: fixes n :: "int"   and f :: "'a fls" shows "fls_shift n f \<equiv> Abs_fls (\<lambda>k. fls_nth f (k + n))"
assms_3: shows "t \<in> {0..Gromov_product_at x y z}"
is_class_Thread: fixes wf_md :: "'a prog \<Rightarrow> String.literal \<Rightarrow> 'a mdecl \<Rightarrow> bool"   and P :: "'a prog" assumes "wf_prog wf_md P" shows "is_class P Thread"
l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_scdom_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M get_scdom_component is_strongly_scdom_component_safe is_weakly_scdom_component_safe get_owner_document get_disconnected_nodes to_tree_order"
loose_bvar_decr_unchanged: fixes t :: "Core.term"   and k :: "nat" assumes "\<not> loose_bvar t k" shows "decr k t = t"
case_prod_conv: fixes f :: "'d \<Rightarrow> 'f \<Rightarrow> 'c"   and a :: "'d"   and b :: "'f" shows "(case (a, b) of (c, d) \<Rightarrow> f c d) = f a b"
trless: fixes i :: "nat"   and j :: "nat" assumes "i \<noteq> j" shows "transition_idx s T {i, j} < n"
preferred_greater_my_U: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<in> local.\<P>"   and "q \<in> local.\<P>"   and "mix_pmf (local.my_U p) local.some_best local.some_worst \<succeq>[\<R>] mix_pmf (local.my_U q) local.some_best local.some_worst \<and> (mix_pmf (local.my_U q) local.some_best local.some_worst, mix_pmf (local.my_U p) local.some_best local.some_worst) \<notin> \<R>" shows "local.my_U q < local.my_U p"
subst_syntactic_and_axioms_def: fixes P :: "'b \<Rightarrow> bool" shows "subst_syntactic_and_axioms P \<equiv> \<forall>t env. P t \<longrightarrow> fmpred (\<lambda>_. P) env \<longrightarrow> P (subst t env)"
simps_6: fixes v :: "'c"   and vd :: "'c"   and ve :: "'c Tree"   and vf :: "'c Tree"   and va :: "'c"   and vb :: "'c Tree"   and vc :: "'c Tree" shows "is_heap (T v (T vd ve vf) (T va vb vc)) = (val (T va vb vc) \<le> v \<and> is_heap (T va vb vc) \<and> val (T vd ve vf) \<le> v \<and> is_heap (T vd ve vf))"
set_2: fixes x21 :: "'e"   and x22 :: "'e list" shows "set (x21 # x22) = insert x21 (set x22)"
invc_upd_1: fixes t :: "(('c \<times> 'd) \<times> color) tree"   and x :: "'c"   and y :: "'d" assumes "invc t"   and "color t = Black" shows "invc (upd x y t)"
wf_invalid_point: fixes x :: "'a"   and b :: "'a set" assumes "x \<notin> \<V>"   and "b \<in># \<B>" shows "x \<notin> b"
mset_eq_permutation: fixes xs :: "'b list"   and ys :: "'b list" assumes "mset xs = mset ys" obtains p :: "nat \<Rightarrow> nat" where "p permutes {..<length ys}" and "permute_list p ys = xs"
exhaust: fixes y :: "('g, 'h, 'i) generat" obtains x1 :: "'g" where "y = Pure x1"   | x21 :: "'h" and x22 :: "'i" where "y = IO x21 x22"
that: fixes op :: "('a, 'b) sas_plus_operator" assumes "op \<in> set \<psi>"   and "ops = [\<phi>\<^sub>O \<Psi> op]" shows "thesis"
subdegree_add_ge: fixes f :: "'a fps"   and g :: "'a fps" assumes "f \<noteq> - g" shows "min (subdegree f) (subdegree g) \<le> subdegree (f + g)"
maxim_singleton: fixes x :: "'b" assumes "x \<in> Field s" shows "s.maxim {x} = x"
min_gallery_betw_chamber_distance: fixes C :: "'a set"   and Cs :: "'a set list"   and D :: "'a set" assumes "domain.min_gallery (C # Cs @ [D])" shows "domain.chamber_distance C D = Suc (length Cs)"
cod_simp_2: shows "partial_magma.cod (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} {False, True} (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) quiver.One = quiver.One"
join: fixes a :: "'a\<^sub>m"   and b :: "'a\<^sub>m"   and ab :: "'a\<^sub>m" shows "(m a b = Some ab) = (bo a ab \<and> bo b ab \<and> (\<forall>u. bo a u \<longrightarrow> bo b u \<longrightarrow> bo ab u))"
label_1: fixes i :: "'a"   and x :: "'a" assumes "i \<in> Basis" shows "label x i \<le> 1"
ex_limitE: fixes J :: "'ag \<Rightarrow> 'ag \<Rightarrow> 'ag"   and C :: "'ah \<Rightarrow> 'ah \<Rightarrow> 'ah"   and D :: "'ag \<Rightarrow> 'ah" assumes "diagram J C D"   and "\<exists>a \<chi>. limit_cone J C D a \<chi>" obtains a :: "'ah" and \<chi> :: "'ag \<Rightarrow> 'ah" where "limit_cone J C D a \<chi>"
c: fixes z :: "complex" assumes "s0 < ereal (Re z)" shows "eval_fds (fds_ln l f) z - g z = c"
simps_1: fixes x1 :: "'g"   and y1 :: "'g" shows "(Inl x1 = Inl y1) = (x1 = y1)"
take_rews_2: fixes n :: "nat"   and discr :: "bool discr" shows "\<psi>\<^sup>E\<^bsub>Suc n\<^esub>\<cdot>(CB\<cdot>discr) = CB\<cdot>discr"
Compl_eq_Diff_UNIV: fixes A :: "'c set" shows "- A = UNIV - A"
ex_mono: fixes A' :: "assigned" assumes "Env\<turnstile> B' \<guillemotright>t\<guillemotright> A'"   and "nrm A \<subseteq> nrm A'"   and "\<And>l. brk A l \<subseteq> brk A' l" shows "P"
evar_gext_f: fixes G :: "prog"   and s1 :: "st"   and e :: "Term.var"   and vf :: "vvar"   and s2 :: "state"   and s :: "st"   and v :: "val"   and x :: "abopt" assumes "G\<turnstile>Norm s1 \<midarrow>e=\<succ>vf\<rightarrow> s2" shows "s\<le>|snd (assign (snd vf) v (x, s))"
heapify_correct: fixes n :: "nat"   and xs :: "'b list"   and t :: "'b tree"   and ys :: "'b list" assumes "n \<le> length xs"   and "heapify n xs = (t, ys)" shows "size t = n \<and> heap t \<and> braun t \<and> mset_tree t = mset (take n xs)"
assms_2: shows "A \<subseteq>\<^sub>\<circ> B"
subst_sends_dom_to_img: fixes v :: "'c"   and \<theta> :: "'c \<Rightarrow> ('d, 'c) Term.term" assumes "v \<in> subst_domain \<theta>" shows "fv (Var v \<cdot> \<theta>) \<subseteq> range_vars \<theta>"
row: fixes A :: "((real, 'n) vec, 'n) vec"   and m :: "'n"   and n :: "'n"   and c :: "real" assumes "P A"   and "m \<noteq> n" shows "P (\<chi>i. if i = m then row m A + c *\<^sub>R row n A else row i A)"
Entry_predecessor_of_High: fixes a :: "'edge" assumes "valid_edge a"   and "targetnode a = (_High_)" shows "sourcenode a = (_Entry_)"
bij_from_nat: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs" shows "bij_betw (mod_type.from_nat Abs) {0..<CARD('b)} UNIV"
upd_nested_keys: fixes m :: "('a, ('b, 'c) mapping) mapping"   and d :: "'c"   and f :: "'c \<Rightarrow> 'c"   and X :: "('a \<times> 'b) set" shows "Mapping.keys (upd_nested m d f X) = Mapping.keys m \<union> fst ` X"
status_aux: fixes i :: "nat"   and \<sigma> :: "'a status"   and f :: "'a"   and ss :: "'b list" assumes "i \<in> set (status \<sigma> (f, length ss))" shows "ss ! i \<in> set ss"
l_neq_1_exp_neq_0: fixes G :: "('c, 'd) cyclic_group_scheme"   and l :: "'c"   and t :: "nat" assumes "cyclic_group G"   and "l \<in> carrier G"   and "l \<noteq> \<one>\<^bsub>G\<^esub>"   and "l = \<^bold>g\<^bsub>G\<^esub> [^]\<^bsub>G\<^esub> t" shows "t \<noteq> 0"
indep_iff_rank_of: fixes X :: "'a set" assumes "X \<subseteq> \<E>" shows "indep_in \<E> X = (\<E>.rank_of X = card X)"
obind_NF_wp: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and g :: "'c \<Rightarrow> 'd \<Rightarrow> 'e option"   and Q :: "'e \<Rightarrow> 'd \<Rightarrow> bool"   and P :: "'d \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c option" assumes "\<And>r. ovalidNF (R r) (g r) Q"   and "ovalidNF P f R" shows "ovalidNF P (f |>> g) Q"
BINDING_preserved_2: fixes x :: "name"   and M :: "trm"   and B :: "ty"   and M' :: "trm" assumes "x:M \<in> BINDINGn B (\<parallel><B>\<parallel>)"   and "M \<longrightarrow>\<^sub>a* M'" shows "x:M' \<in> BINDINGn B (\<parallel><B>\<parallel>)"
u_simps_3: shows "trg (tab\<^sub>0 g \<star> Tfg.p\<^sub>0) = src (f \<star> g)"
prob_space: shows "X.prob (space (distr M MX X)) = 1"
sup_bot_right: fixes x :: "'d upred" shows "(x \<or> false) = x"
igSubstAbsIPresIGWlsAbsSTR_def: fixes MOD :: "('h, 'i, 'varSort, 'sort, 'j, 'k, 'l, 'm, 'n) model_scheme" shows "igSubstAbsIPresIGWlsAbsSTR MOD \<equiv> \<forall>ys Y y us s A. igWlsAbs MOD (us, s) (igSubstAbs MOD ys Y y A) = (igWls MOD (asSort ys) Y \<and> igWlsAbs MOD (us, s) A)"
eventually_const: fixes F :: "'b filter"   and P :: "bool" assumes "F \<noteq> \<bottom>" shows "(\<forall>\<^sub>F x in F. P) = P"
def_body: fixes x :: "'b" assumes "x \<noteq> \<bottom>"   and "x \<noteq> null" shows "g x \<noteq> \<bottom> \<and> g x \<noteq> null"
nested_prop_atoms_finite: fixes \<phi> :: "'b ltln" shows "finite (nested_prop_atoms \<phi>)"
strip_eq_Seq: fixes c :: "'c acom"   and c1 :: "com"   and c2 :: "com" shows "(strip c = c1;; c2) = (\<exists>d1 d2. c = d1;; d2 \<and> strip d1 = c1 \<and> strip d2 = c2)"
assms: fixes x :: "'a"   and y :: "'b"   and y' :: "'b" assumes "(x, y) \<in> R"   and "(x, y') \<in> R" shows "y = y'"
dens: shows "has_parametrized_subprob_density (state_measure V' \<Gamma>) (\<lambda>\<rho>. dens_ctxt_measure \<Y> \<rho> \<bind> (\<lambda>\<sigma>. expr_sem \<sigma> e)) (stock_measure t) f"
hext_refl: fixes h :: "heap" shows "h \<unlhd> h"
mydiv_mono: fixes a :: "nat"   and b :: "nat"   and k :: "nat" assumes "a \<le> b" shows "mydiv a k \<le> mydiv b k"
quot_set_iff: fixes a :: "'a"   and A :: "'a set"   and x :: "'a" shows "(a \<in> A \<div> x) = (x * a \<in> A)"
s\<^sub>1_r: shows "s\<^sub>1 r = Some (\<sigma>, \<tau>, \<E> [Rfork e])"
shadow_root_ptr_shadow_root_ptr_cast: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "is_shadow_root_ptr_kind (cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r shadow_root_ptr)"
standard_rules_mono: fixes C\<^sub>1 :: "'b set"   and C\<^sub>2 :: "'b set"   and L\<^sub>1 :: "'b Standard_Constant set"   and L\<^sub>2 :: "'b Standard_Constant set" assumes "C\<^sub>1 \<subseteq> C\<^sub>2"   and "L\<^sub>1 \<subseteq> L\<^sub>2" shows "standard_rules C\<^sub>1 L\<^sub>1 \<subseteq> standard_rules C\<^sub>2 L\<^sub>2"
sum_strict_mono_ex1: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "finite A"   and "\<forall>x\<in>A. f x \<le> g x"   and "\<exists>a\<in>A. f a < g a" shows "sum f A < sum g A"
OUT: fixes x :: "'v"   and y :: "'v" assumes "edge \<Delta> x y" shows "edg (Vertex x) (Edge x y)"
bound_inside: fixes z :: "complex" assumes "z \<in> A" shows "cmod (H z) \<le> C"
Domainp_sum_with: fixes t :: "'d set"   and A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and r :: "'d \<Rightarrow> 'd"   and p :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and p' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'d"   and z' :: "'e" assumes "\<And>x. x \<in> t \<Longrightarrow> Domainp A (r x)"   and "t \<subseteq> Collect (Domainp A)"   and "rel_fun A (rel_fun A A) p p'"   and "A z z'" shows "Domainp A (sum_with p z r t)"
InvariantsClAfterExplainUIP: fixes state :: "State"   and F0 :: "Literal list list" assumes "applyExplainUIP_dom state"   and "InvariantUniq (getM state)"   and "InvariantCFalse (getConflictFlag state) (getM state) (getC state)"   and "InvariantCEntailed (getConflictFlag state) F0 (getC state)"   and "InvariantClCharacterization (getCl state) (getC state) (getM state)"   and "InvariantCnCharacterization (getCn state) (getC state) (getM state)"   and "InvariantClCurrentLevel (getCl state) (getM state)"   and "InvariantUniqC (getC state)"   and "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))"   and "InvariantEquivalentZL (getF state) (getM state) F0"   and "getConflictFlag state"   and "0 < currentLevel (getM state)" shows "let state' = applyExplainUIP state in InvariantCFalse (getConflictFlag state') (getM state') (getC state') \<and> InvariantCEntailed (getConflictFlag state') F0 (getC state') \<and> InvariantClCharacterization (getCl state') (getC state') (getM state') \<and> InvariantCnCharacterization (getCn state') (getC state') (getM state') \<and> InvariantClCurrentLevel (getCl state') (getM state') \<and> InvariantUniqC (getC state')"
segment_construction: shows "\<forall>a b c q. \<exists>x. Bet q a x \<and> Cong a x b c"
swapEnv_idEnv: fixes xs :: "'varSort"   and x :: "'var"   and y :: "'var" shows "idEnv &[x \<and> y]_xs = idEnv"
map_bset_binsert: fixes f :: "'c \<Rightarrow> 'a"   and x :: "'c"   and X :: "'c set['b]" shows "map_bset f (binsert x X) = binsert (f x) (map_bset f X)"
less_rat: fixes b :: "int"   and d :: "int"   and a :: "int"   and c :: "int" assumes "b \<noteq> 0"   and "d \<noteq> 0" shows "(Fract a b < Fract c d) = (a * d * (b * d) < c * b * (b * d))"
length_meld: shows "\<forall>ts v. (\<exists>va u. v=va\<parallel>u \<and> 0 < \<parallel>ext va\<parallel> \<and> 0 < \<parallel>ext u\<parallel>) \<longrightarrow> 0 < \<parallel>ext v\<parallel>"
WellDefinedTy: fixes A :: "'a"   and S :: "('a, 'b) Signature" assumes "A \<in> Ty\<^bsub>S\<^esub>" shows "Sig S \<triangleright> \<turnstile> A Type"
intro: fixes G :: "('c, 'd) pre_digraph" assumes "symmetric G" shows "sym_digraph_axioms G"
inext_nth_less_less_card_conv: fixes I :: "nat set"   and t :: "nat"   and n :: "nat" assumes "I \<down>\<ge> t \<noteq> {}" shows "(I \<rightarrow> n < t) = (n < card (I \<down>< t))"
bind_SBE_def: fixes f :: "'b \<Rightarrow> ('d \<times> 'b) set option"   and g :: "'d \<Rightarrow> 'b \<Rightarrow> ('c \<times> 'b) set option" shows "bind_SBE f g = (\<lambda>\<sigma>. case f \<sigma> of None \<Rightarrow> None | Some S \<Rightarrow> let S' = (\<lambda>(out, \<sigma>'). g out \<sigma>') ` S in if None \<in> S' then None else Some (\<Union> (the ` S')))"
subsetD: fixes x :: "'a" assumes "x \<in> V" shows "x \<in> E"
WTrtInit: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and e :: "char list exp"   and T :: "ty"   and C :: "char list"   and Cs :: "char list list"   and b :: "bool" assumes "P,E,h,sh \<turnstile> e : T"   and "\<forall>C'\<in>set (C # Cs). is_class P C'"   and "\<not> sub_RI e"   and "\<forall>C'\<in>set (tl Cs). \<exists>sfs. sh C' = \<lfloor>(sfs, Processing)\<rfloor>"   and "b \<longrightarrow> (\<forall>C'\<in>set Cs. \<exists>sfs. sh C' = \<lfloor>(sfs, Processing)\<rfloor>)"   and "distinct Cs"   and "supercls_lst P Cs" shows "P,E,h,sh \<turnstile> INIT C (Cs,b) \<leftarrow> e : T"
induct: fixes P :: "nat \<Rightarrow> 'b list \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "'b list" assumes "\<And>n xs. \<lbrakk>\<And>x. \<lbrakk>n \<noteq> 0; x = n div 2\<rbrakk> \<Longrightarrow> P x xs; \<And>x xa xb. \<lbrakk>n \<noteq> 0; x = n div 2\<rbrakk> \<Longrightarrow> P (n - 1 - x) (tl xb)\<rbrakk> \<Longrightarrow> P n xs" shows "P a0 a1"
assms: shows "x \<in> set (unlabel (transaction_strand T))"
Range_iff: fixes a :: "'e"   and r :: "('f \<times> 'e) set" shows "(a \<in> Range r) = (\<exists>y. (y, a) \<in> r)"
flow_has_derivative: fixes t :: "real"   and x0 :: "'a" assumes "t \<in> existence_ivl0 x0" shows "(flow0 x0 has_derivative (\<lambda>i. i *\<^sub>R f (flow0 x0 t))) (at t)"
filterlim_def: fixes f :: "'a \<Rightarrow> 'b"   and F2 :: "'b filter"   and F1 :: "'a filter" shows "filterlim f F2 F1 = (filtermap f F1 \<le> F2)"
neq_NilE: fixes l :: "'g list" assumes "l \<noteq> []" obtains x :: "'g" and xs :: "'g list" where "l = x # xs"
perm_simple_struct_induct: fixes P :: "perm \<Rightarrow> bool"   and p :: "perm" assumes "P 0"   and "\<And>p a b. \<lbrakk>P p; a \<noteq> b; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> P ((a \<rightleftharpoons> b) + p)" shows "P p"