hcju/isabelleps huggingface dataset:
get_component_def: shows "get_component \<equiv> l_get_component\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_component get_root_node to_tree_order"
skip: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and t :: "state" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[], t\<rangle> \<Rightarrow> t"
map_simp: fixes f :: "'c local.term set \<times> 'c local.term set \<times> 'c local.term set" assumes "\<F>C.CCC.arr f" shows "\<F>C.\<alpha>' f = \<F>C.inv (\<F>C.\<alpha>\<^sub>F\<^sub>M\<^sub>C (\<F>C.CCC.cod f)) \<cdot> \<F>C.T.ToCT f"
F_Suc: fixes m :: "nat" assumes "0 < m" shows "F (Suc m) = F m + real (2 * m) / real ((m + 1) * (m + 2))"
bind_is_OK_E2: fixes h :: "'l"   and f :: "('l, 'm, 'o) prog"   and g :: "'o \<Rightarrow> ('l, 'm, 'n) prog"   and x :: "'o" assumes "h \<turnstile> ok Heap_Error_Monad.bind f g"   and "h \<turnstile> f \<rightarrow>\<^sub>r x" obtains h' :: "'l" where "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> ok g x"
induct: fixes x1 :: "pi"   and x2 :: "pi"   and Rel :: "(pi \<times> pi) set"   and P :: "pi \<Rightarrow> pi \<Rightarrow> bool" assumes "(x1, x2) \<in> bangRel Rel"   and "\<And>P Q. (P, Q) \<in> Rel \<Longrightarrow> P (!P) (!Q)"   and "\<And>R T P Q. \<lbrakk>(R, T) \<in> Rel; (P, Q) \<in> bangRel Rel; P P Q\<rbrakk> \<Longrightarrow> P (R \<parallel> P) (T \<parallel> Q)"   and "\<And>P Q a. \<lbrakk>(P, Q) \<in> bangRel Rel; P P Q\<rbrakk> \<Longrightarrow> P (<\<nu>a>P) (<\<nu>a>Q)" shows "P x1 x2"
rhseq: shows "construct_lhs_vector p qs signs = vec_of_list (map (\<lambda>s. rat_of_int (int (card {x. poly p x = 0 \<and> consistent_sign_vec_copr qs x = s}))) signs)"
to_fun_one: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a" assumes "UP_cring R"   and "a \<in> carrier R" shows "UP_cring.to_fun R \<one>\<^bsub>UP R\<^esub> a = \<one>\<^bsub>R\<^esub>"
appReturn: fixes P :: "'a prog"   and pc :: "nat"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and s :: "ty list \<times> ty err list" shows "app\<^sub>i (Return, P, pc, mxs, T\<^sub>r, s) = (\<exists>T ST LT. s = (T # ST, LT) \<and> subtype P T T\<^sub>r)"
fmeasurableD2: fixes A :: "'c set"   and M :: "'c measure" assumes "A \<in> fmeasurable M" shows "emeasure M A \<noteq> top"
approx_mult2: fixes a :: "'a star"   and b :: "'a star"   and c :: "'a star" assumes "a \<approx> b"   and "c \<in> HFinite" shows "c * a \<approx> c * b"
relation_of_Seq: fixes P :: "('c, 'd) action"   and Q :: "('c, 'd) action" shows "action.relation_of (P `;` Q) = (action.relation_of P ;; action.relation_of Q)"
set_insert: fixes x :: "'a"   and A :: "'a set" assumes "x \<in> A" obtains B :: "'a set" where "A = x \<triangleright> B" and "x \<notin> B"
wt_elim_cases_4: fixes E :: "env"   and dt :: "loc \<Rightarrow> ty option"   and C :: "qtname"   and T :: "tys" assumes "E,dt\<Turnstile>In1l (NewC C)\<Colon>T" obtains "T = Inl (Class C)" and "is_acc_class (prg E) (pkg E) C"
Cons_le_Cons: fixes a :: "'b"   and x :: "'b list"   and b :: "'b"   and y :: "'b list" shows "(a # x \<le> b # y) = (a < b \<or> a = b \<and> x \<le> y)"
list_id: shows "[0..<i] = [0..<j] @ [j..<i]"
allow_all_fun_def: fixes f :: "'c \<Rightarrow> 'd" shows "A\<^sub>f f = A\<^sub>p (Some \<circ> f)"
max_fun_diff_commute: fixes f :: "'b \<Rightarrow> 'd"   and g :: "'b \<Rightarrow> 'd" shows "s.max_fun_diff f g = s.max_fun_diff g f"
sub: fixes a :: "'a" assumes "a \<in> R\<setminus>I" shows "a \<in> R"
asprod_n_1: fixes n :: "int" shows "n *\<^sub>a 1 = ant n"
wtA_gA: fixes at :: "('fsym, 'psym) atm" assumes "Ik.wtA at" shows "GE.wtA (gA at)"
image_image: fixes f :: "'l \<Rightarrow> 'k"   and g :: "'m \<Rightarrow> 'l"   and A :: "'m set" shows "f ` g ` A = (\<lambda>x. f (g x)) ` A"
that: fixes v :: "'interpretation stream" assumes "v \<in> R.language"   and "(!!) u \<approx> (!!) v" shows "thesis"
well_def_def: fixes prog :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" shows "well_def prog \<equiv> healthy (wp prog) \<and> nearly_healthy (wlp prog) \<and> wp_under_wlp prog \<and> sub_distrib_pconj prog \<and> sublinear (wp prog) \<and> bd_cts (wp prog)"
R9_a: shows "pmf (sds R9) a = pmf (sds R35) a"
assms: fixes p2 :: "nat \<times> nat" assumes "(p2, p1) \<in> lex_p" shows "c_assoc_have_key (pr_gr (loc_upb (fst p2) (snd p2))) (c_pair (fst p2) (snd p2)) = 0"
pick_card_in_set: fixes i :: "nat"   and I :: "nat set" assumes "i \<in> I" shows "pick I (card {a \<in> I. a < i}) = i"
lossless_response: shows "\<forall>r w e. lossless_spmf (response r w e)"
comp_char: fixes g :: "'c local.term set"   and f :: "'c local.term set" shows "\<F>\<^sub>SC.comp g f = (if \<F>\<^sub>SC.arr f \<and> \<F>\<^sub>SC.arr g \<and> seq g f then g \<cdot> f else null)"
word_less_iff_unsigned: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and a :: "'b word"   and b :: "'b word" assumes "class.linordered_semidom plus minus less_eq less zero times one" shows "(a < b) = less (semiring_1.unsigned one plus zero a) (semiring_1.unsigned one plus zero b)"
d_imp_sd: fixes scf :: "('a \<Rightarrow> ('b \<times> 'b) set) \<Rightarrow> ('b \<times> 'b) set"   and A :: "'b set"   and Is :: "'a set"   and C :: "'a set"   and x :: "'b"   and y :: "'b" assumes "decisive scf A Is C x y" shows "semidecisive scf A Is C x y"
l2_skr_def: fixes R :: "fid_t"   and K :: "msg" shows "l2_skr R K \<equiv> {(s, s'). R \<noteq> test \<and> R \<notin> partners \<and> in_progress (progress s R) xsk \<and> sklvl1.guessed_frame R xsk = Some K \<and> s' = s\<lparr>ik := {K} \<union> ik s\<rparr>}"
simps_1: fixes x1 :: "'g"   and y1 :: "'g" shows "(Node x1 = Node y1) = (x1 = y1)"
eq_Eq_conv: fixes x :: "'a"   and y :: "'a" shows "(cmp x y = Eq) = (x = y)"
Max_wrt_among_total_preorder: fixes A :: "'a set" shows "Max_wrt_among S A = {x \<in> carrier \<inter> A. \<forall>y\<in>carrier \<inter> A. S y x}"
arc_expanded: fixes x :: "'a" shows "(bijective (x * top) \<and> bijective (x[[top]])) = (coreflexive (x * top * x\<^sup>T) \<and> coreflexive ((x[[top]]) * x) \<and> times_top_class.total (top * x))"
carrier_local_ring_at_def: shows "carrier_local_ring_at \<equiv> local.carrier_quotient_ring"
chain_closure: fixes \<phi> :: "'a"   and C :: "'a set set" assumes "\<not> \<tturnstile> \<phi>"   and "subset.chain {B. \<not> B \<turnstile> \<phi>} C" shows "\<not> \<Union> C \<turnstile> \<phi>"
Upper_cong: fixes L :: "('a, 'b) gorder_scheme"   and A :: "'a set"   and A' :: "'a set" assumes "weak_partial_order L"   and "A \<subseteq> carrier L"   and "A' \<subseteq> carrier L"   and "A {.=}\<^bsub>L\<^esub> A'" shows "Upper L A = Upper L A'"
partition3_def: fixes x :: "'a"   and xs :: "'a list" shows "partition3 x xs = (filter (\<lambda>y. y < x) xs, filter (\<lambda>y. y = x) xs, filter ((<) x) xs)"
Soundness: fixes \<alpha> :: "symbol list" assumes "finished_item \<alpha> \<in> \<II>" shows "\<exists>p. pvalid_with p (finished_item \<alpha>) 0 [] \<and> p \<in> ll"
inverse_floatarith_def: fixes f :: "floatarith" shows "inverse f = Inverse f"
orientable_5_implies_1: fixes x :: "'a" assumes "local.comp_inf.symmetric (x \<squnion> id x) \<longrightarrow> bounded_distrib_allegory.loop_orientable top (\<sqinter>) (\<le>) (\<squnion>) id (x \<squnion> id x)" shows "bounded_distrib_allegory.loop_super_orientable top (\<sqinter>) (\<le>) (\<squnion>) id x"
VClock: shows "\<forall>p t i. correct p t \<and> te p i \<le> t \<and> t < te p (i + 1) \<longrightarrow> VC p t = IC p i t"
split_set_1: fixes ts :: "('a btree \<times> 'a) list"   and z :: "'a"   and ls :: "('a btree \<times> 'a) list"   and a :: "'a btree"   and b :: "'a"   and rs :: "('a btree \<times> 'a) list" assumes "split ts z = (ls, (a, b) # rs)" shows "(a, b) \<in> set ts"
trcl_one_elem: fixes c :: "'a"   and e :: "'b"   and c' :: "'a"   and t :: "('a \<times> 'b \<times> 'a) set" assumes "(c, e, c') \<in> t" shows "(c, [e], c') \<in> trcl t"
lens_plus_subcong: fixes Y\<^sub>1 :: "'d \<Longrightarrow> 'e"   and Y\<^sub>2 :: "'f \<Longrightarrow> 'e"   and X\<^sub>1 :: "'g \<Longrightarrow> 'e"   and X\<^sub>2 :: "'h \<Longrightarrow> 'e" assumes "Y\<^sub>1 \<bowtie> Y\<^sub>2"   and "X\<^sub>1 \<subseteq>\<^sub>L Y\<^sub>1"   and "X\<^sub>2 \<subseteq>\<^sub>L Y\<^sub>2" shows "X\<^sub>1 +\<^sub>L X\<^sub>2 \<subseteq>\<^sub>L Y\<^sub>1 +\<^sub>L Y\<^sub>2"
labeling_consistent: fixes a :: "'b"   and b :: "'a" assumes "(a, b) \<in> R\<inverse>" shows "b.L a = a.L b"
map_poly_pdivmod: fixes hom :: "'b \<Rightarrow> 'c"   and p :: "'b poly"   and q :: "'b poly" assumes "field_hom hom" shows "map_prod (map_poly hom) (map_poly hom) (p div q, p mod q) = (map_poly hom p div map_poly hom q, map_poly hom p mod map_poly hom q)"
lower_asymptotic_density_infinite_interval_3: shows "lower_asymptotic_density UNIV = 1"
DynCom: fixes P :: "'e set"   and \<Gamma> :: "'f \<Rightarrow> ('e, 'f, 'g) com option"   and \<Theta> :: "('e set \<times> 'f \<times> 'e set \<times> 'e set) set"   and F :: "'g set"   and c :: "'e \<Rightarrow> ('e, 'f, 'g) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<forall>s\<in>P. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P c s Q,A" shows "\<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> P DynCom c Q,A"
simps_1: fixes ExcessAtPG :: "nat \<Rightarrow> nat" shows "ExcessTable_cont ExcessAtPG [] = []"
prob_reachable_le: fixes m :: "nat"   and n :: "nat"   and x :: "'s"   and y :: "'s"   and w :: "'s" assumes "m \<le> n" shows "p x y m * p y w (n - m) \<le> p x w n"
nb\<^sub>1: shows "\<forall>op'\<in>set ((\<phi> \<Psi> )\<^sub>\<O>). \<exists>op\<in>set (\<Psi>\<^sub>\<O>\<^sub>+). op' = \<phi>\<^sub>O \<Psi> op"
rep_eq: fixes x :: "fbint"   and xa :: "fbint" shows "local.dist x xa = N (int_of_fbint x - int_of_fbint xa)"
assms_2: shows "closedin (top_of_set (f ` S)) T"
swapOpt: fixes x :: "'a"   and y :: "'a"   and \<sigma> :: "'a list" shows "T\<^sub>p_opt [x, y] \<sigma> \<le> 1 + T\<^sub>p_opt [y, x] \<sigma>"
module_pair_axioms: shows "module_pair (*\<^sub>C) (*\<^sub>C)"
the_elem_eq: fixes x :: "'a" shows "the_elem {x} = x"
length_merge: fixes xs :: "'c list"   and ys :: "'c list" shows "length (merge xs ys) = length xs + length ys"
thunks_Cons: fixes x :: "Vars.var"   and e :: "exp"   and \<Gamma> :: "(Vars.var \<times> exp) list" shows "thunks ((x, e) # \<Gamma>) = (if isVal e then thunks \<Gamma> - {x} else insert x (thunks \<Gamma>))"
arctan_aform_err_def: fixes p :: "nat"   and X :: "(real \<times> real pdevs) \<times> real" shows "arctan_aform_err p X = (let l = Inf_aform_err p X; u = Sup_aform_err p X in min_range_mono p floatarith.Arctan (\<lambda>x. 1 / (floatarith.Num 1 + x * x)) l u X)"
intm: shows "integrable M (\<lambda>x. fm x * g x)"
hom_one: fixes h :: "'f \<Rightarrow> 'g"   and G :: "('f, 'h) monoid_scheme"   and H :: "('g, 'i) monoid_scheme" assumes "h \<in> hom G H"   and "Group.group G"   and "Group.group H" shows "h \<one>\<^bsub>G\<^esub> = \<one>\<^bsub>H\<^esub>"
is_empty_alist_of: fixes q :: "('c, 'd) pq" assumes "is_empty q" shows "alist_of q = []"
ideduct_reduce: fixes arity :: "'a \<Rightarrow> nat"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and public :: "'a \<Rightarrow> bool"   and M :: "('a, 'b) Term.term set"   and M' :: "('a, 'b) Term.term set"   and t :: "('a, 'b) Term.term" assumes "intruder_model arity Ana"   and "intruder_model.intruder_deduct arity public Ana (M \<union> M') t"   and "\<And>t'. t' \<in> M' \<Longrightarrow> intruder_model.intruder_deduct arity public Ana M t'" shows "intruder_model.intruder_deduct arity public Ana M t"
new_document_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_document_ptr :: "(_) document_ptr"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'o, 'r option) RElement_scheme \<Rightarrow> 'w" assumes "h \<turnstile> new_document \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_document \<rightarrow>\<^sub>r new_document_ptr" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
xor_zero_right: fixes x :: "'b" shows "x XOR (0::'b) = x"
uniform_limit1: shows "uniform_limit {z. Re z \<in> {x..y}} (\<lambda>a z. complex_interval_lebesgue_integral lborel a (ereal (exp (- 1))) (f z)) (\<lambda>z. complex_interval_lebesgue_integral lborel 0 (ereal (exp (- 1))) (f z)) (at_right 0)"
None_in_opt: fixes A :: "'a set" shows "None \<in> opt A"
le_multiset_plus_left_nonempty: fixes M :: "'b multiset"   and N :: "'b multiset" assumes "M \<noteq> {#}" shows "N < M + N"
finy: shows "finite {i \<in> I. y i \<noteq> \<one>\<^bsub>A i\<^esub>}"
finite_vertex_set: shows "finite V\<^bsub>H_x\<^esub>"
bot_less1: fixes X :: "'d process" shows "\<bottom> \<le> X"
AccDef: fixes i :: "AbstrLevelsID"   and C :: "CSet" shows "Acc i C = DAcc i C \<union> \<Union> (Acc i ` DAcc i C)"
giapath_if_slim_giapath: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" assumes "pre_digraph.gen_iapath (with_proj slim) (verts3 (with_proj G)) u p v" shows "gen_iapath (verts3 (with_proj G)) u p v"
leg0_in_hom_2: shows "\<guillemotleft>r\<^sub>0 : r\<^sub>0 \<Rightarrow> r\<^sub>0\<guillemotright>"
induct: fixes x :: "stack"   and P :: "stack \<Rightarrow> bool" assumes "SSN x"   and "\<And>k. \<lbrakk>\<And>k'.  k \<mapsto> k'  \<Longrightarrow> SSN k'; \<And>k'.  k \<mapsto> k'  \<Longrightarrow> P k'\<rbrakk> \<Longrightarrow> P k" shows "P x"
o_subst: fixes \<pi>1 :: "Preliminaries.var \<Rightarrow> 'c trm"   and \<pi>2 :: "Preliminaries.var \<Rightarrow> 'c trm" shows "subst \<pi>1 \<circ> subst \<pi>2 = subst (subst \<pi>1 \<circ> \<pi>2)"
Max_wrt_among_total_preorder: fixes A :: "'alt set" shows "Max_wrt_among (of_weak_ranking (x # xs)) A = {xa \<in> \<Union> (set (x # xs)) \<inter> A. \<forall>y\<in>\<Union> (set (x # xs)) \<inter> A. of_weak_ranking (x # xs) y xa}"
times_monomial_monomial: fixes t :: "'b"   and c :: "'c"   and v :: "'b"   and d :: "'c" shows "monomial c t * monomial d v = monomial (c * d) (t + v)"
h_eq_p_gen: fixes y :: "real"   and x :: "real" assumes "y \<in> T"   and "open_segment x y \<inter> T = {}" shows "h x = p y"
elim_land: fixes P :: "'b \<Rightarrow> bool"   and Q :: "'b \<Rightarrow> bool"   and s :: "'b" assumes "(P land Q) s" obtains "P s" and "Q s"
fmdomE: fixes x :: "'e"   and m :: "'e \<rightharpoonup>\<^sub>f 'f" assumes "x |\<in>| fmdom m" obtains y :: "'f" where "fmlookup m x = Some y"
u_1: shows "u \<sqsubseteq>\<^sub>s\<^sub>e\<^sub>t M"
elims: fixes x :: "int list"   and y :: "int list" assumes "take_incr x = y" obtains "x = []" and "y = []"   | xa :: "int" where "x = [xa__]" and "y = [xa__]"   | xa :: "int" and ya :: "int" and xs :: "int list" where "x = xa # ya # xs" and "y = (if xa < ya then xa # take_incr (ya # xs) else [xa])"
fcd_ge: fixes x :: "complex" assumes "x \<in> interior S \<and> x \<in> interior {x. k \<le> d \<bullet> x}" shows "f field_differentiable at x"
allUses_def: fixes u :: "'g \<Rightarrow> 'node \<Rightarrow> 'val set"   and p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g"   and n :: "'node" shows "uninst.allUses u p g n \<equiv> u g n \<union> uninst.phiUses p g n"
integrable_diff: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "integrable M f"   and "integrable M g" shows "integrable M (\<lambda>x. f x - g x)"
star_ref: fixes x :: "'a" shows "(1::'a) \<le> x\<^sup>\<star>"
sets_M: shows "sets M = sets (stream_space (count_space UNIV))"
prj_chine_2: shows "THfgh.p\<^sub>1 \<star> THfgh.chine \<cong> THfgh.\<omega>.chine \<star> THfgh.p\<^sub>1"
finvimage_mono: fixes r' :: "V"   and r :: "V"   and A' :: "V"   and A :: "V" assumes "r' \<subseteq>\<^sub>\<circ> r"   and "A' \<subseteq>\<^sub>\<circ> A" shows "r' -`\<^sub>\<bullet> A' \<subseteq>\<^sub>\<circ> r -`\<^sub>\<bullet> A"
sources_idem: fixes I :: "('c \<times> 'c) set"   and D :: "'d \<Rightarrow> 'c"   and u :: "'c"   and xs :: "'d list" shows "sources I D u (ipurge_tr_rev I D u xs) = sources I D u xs"
poly_lift_hom_add: fixes S :: "('e, 'f) ring_scheme"   and \<phi> :: "'a \<Rightarrow> 'e"   and p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "cring S"   and "\<phi> \<in> ring_hom R S"   and "p \<in> carrier (UP R)"   and "q \<in> carrier (UP R)" shows "poly_lift_hom R S \<phi> (p \<oplus>\<^bsub>UP R\<^esub> q) = poly_lift_hom R S \<phi> p \<oplus>\<^bsub>UP S\<^esub> poly_lift_hom R S \<phi> q"
remove_strictly_subsumed_clauses_in_Q: fixes C :: "'a literal list"   and N :: "('a literal list \<times> nat) list"   and P :: "('a literal list \<times> nat) list"   and Q :: "('a literal list \<times> nat) list"   and Q' :: "('a literal list \<times> nat) list"   and n :: "nat" assumes "C \<in> fst ` set N" shows "wstate_of_dstate (N, P, Q @ Q', n) \<leadsto>\<^sub>w\<^sup>* wstate_of_dstate (N, P, Q @ filter (Not \<circ> strictly_subsume [C] \<circ> fst) Q', n)"
division_decomp: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a dvd b * c" shows "\<exists>b' c'. a = b' * c' \<and> b' dvd b \<and> c' dvd c"
ereal_divide_right_mono: fixes x :: "ereal"   and y :: "ereal"   and z :: "ereal" assumes "x \<le> y"   and "0 < z" shows "x / z \<le> y / z"
lc_Abs: fixes T :: "Unary_PCF.type"   and X :: "(char list \<times> Unary_PCF.type) fset"   and e :: "expr" assumes "\<forall>x. (x, T) |\<notin>| X \<longrightarrow> lc (open0_Var (x, T) e)" shows "lc (\<Lambda>\<langle>T\<rangle> e)"
in_null_sets_measure_pmfI: fixes A :: "'b set"   and p :: "'b pmf" assumes "A \<inter> set_pmf p = {}" shows "A \<in> null_sets (measure_pmf p)"
evaluate_iff_sym_3: fixes eval :: "v sem_env \<Rightarrow> exp0 \<Rightarrow> 'b state \<Rightarrow> 'b state \<times> (v, v) result"   and eval_list :: "v sem_env \<Rightarrow> exp0 list \<Rightarrow> 'b state \<Rightarrow> 'b state \<times> (v list, v) result"   and eval_match :: "v sem_env \<Rightarrow> v \<Rightarrow> (pat \<times> exp0) list \<Rightarrow> v \<Rightarrow> 'b state \<Rightarrow> 'b state \<times> (v, v) result"   and env :: "v sem_env"   and st :: "'b state"   and v :: "v"   and pes :: "(pat \<times> exp0) list"   and v' :: "v"   and r :: "'b state \<times> (v, v) result" assumes "eval eval eval_list eval_match" shows "evaluate_match True env st v pes v' r = (eval_match env v pes v' st = r)"
dense_pp: fixes uminus :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.p_algebra uminus inf less_eq less sup bot top" shows "(uminus x = bot) = (uminus (uminus x) = top)"
gcd_mult_lcm: fixes a :: "'a"   and b :: "'a" shows "gcd a b * lcm a b = normalize a * normalize b"
gen_t_le_gen_alpha_t: fixes n :: "nat"   and A :: "type_scheme list"   and t :: "typ" assumes "new_tv n A" shows "gen A t \<le> gen A ($ (\<lambda>x. TVar (if x \<in> free_tv A then x else n + x)) t)"
apath_def: fixes u :: "'a"   and p :: "('a \<times> 'a) list"   and v :: "'a" shows "G.apath u p v \<equiv> G.awalk u p v \<and> distinct (pawalk_verts u p)"
sset_cycle: fixes xs :: "'d list" assumes "xs \<noteq> []" shows "sset (cycle xs) = set xs"
closed_negations: fixes S :: "'a set" assumes "closed S" shows "closed (uminus ` S)"
asymp_equiv_compose: fixes f :: "'a \<Rightarrow> 'b"   and G :: "'a filter"   and g :: "'a \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'a"   and F :: "'c filter" assumes "f \<sim>[G] g"   and "filterlim h G F" shows "f \<circ> h \<sim>[F] g \<circ> h"
frequently_eq_eventually: fixes P :: "'a \<Rightarrow> bool" shows "frequently P F = eventually P F"
additive_def: fixes f :: "'e \<Rightarrow> 'f" shows "Modules.additive f \<equiv> \<forall>x y. f (x + y) = f x + f y"
order_tendstoI: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and "open" :: "'c set \<Rightarrow> bool"   and y :: "'c"   and f :: "'d \<Rightarrow> 'c"   and F :: "'d filter" assumes "class.order_topology less_eq less open"   and "\<And>a. less a y \<Longrightarrow> \<forall>\<^sub>F x in F. less a (f x)"   and "\<And>a. less y a \<Longrightarrow> \<forall>\<^sub>F x in F. less (f x) a" shows "filterlim f (topological_space.nhds open y) F"
as'_def: shows "as' \<equiv> new_addrs h as h'"
mset_remove1: fixes a :: "'k"   and xs :: "'k list" shows "mset (remove1 a xs) = mset xs - {#a#}"
41: shows "bprv (imp (subst \<phi> \<langle>\<psi>\<rangle> xx) (cnj (subst \<phi> \<langle>\<psi>\<rangle> xx) (SS \<langle>\<chi>\<rangle> \<langle>\<psi>\<rangle>)))"
abc_only_cba_1: fixes \<E> :: "'b set"   and \<P> :: "'b set set"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "MinkowskiBetweenness \<E> \<P> betw"   and "betw a b c" shows "\<not> betw b a c"
const_g: shows "const (\<lambda>\<tau>. g (X \<tau>) (Y \<tau>))"
indexed_const_in_carrier: fixes K :: "'a set"   and k :: "'a"   and m :: "'c multiset" assumes "K \<subseteq> carrier R"   and "k \<in> K" shows "indexed_const k m \<in> carrier R"
wf_residual_networkD_3: assumes "\<Delta>''.wf_residual_network"   and "edge \<Delta>'' (source \<Delta>'') (sink \<Delta>'')" shows "False"
s0_le_s_stop: shows "s\<^sub>e \<le> ego.s_stop"
proper: shows "\<exists>a. a \<noteq> (1::'a)"
inv_stereographic_cvec_r3_def: fixes z :: "complex \<times> complex" shows "inv_stereographic_cvec_r3 z = (let (z1, z2) = z in if z2 = 0 then (0, 0, 1) else let z = z1 / z2; X = Re (2 * z / (1 + z * cnj z)); Y = Im (2 * z / (1 + z * cnj z)); Z = ((cmod z)\<^sup>2 - 1) / (1 + (cmod z)\<^sup>2) in (X, Y, Z))"
eq_fun_eq_val: fixes f :: "'e \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f"   and x :: "'e" assumes "f = g" shows "f x = g x"
latin_rect_card_col: fixes s :: "(latin_type \<Rightarrow> nat) set"   and m :: "nat"   and n :: "nat"   and x :: "nat" assumes "latin_rect s m n"   and "x < n" shows "card {e Col |e. e \<in> s \<and> e Num = x} = m"
coeff_mpoly_of_poly: fixes i :: "nat"   and p :: "'a poly"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (mpoly_of_poly i p) m = (poly.coeff p (lookup m i) when keys m \<subseteq> {i})"
walk_edges_decomp_simple: fixes v :: "'a"   and w :: "'a"   and xs :: "'a list" shows "edges_of_walk (v # w # xs) = {(v, w)} \<union> edges_of_walk (w # xs)"
score_from_list_def: fixes P1 :: "nat list list fset"   and P2 :: "nat list list fset"   and e :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset"   and s :: "nat list \<Rightarrow> nat list \<Rightarrow> (nat list \<times> (nat \<times> nat) \<times> transition) fset \<Rightarrow> nat" shows "score_from_list P1 P2 e s = (let pairs = (\<lambda>(l1, l2). zip l1 l2) |`| (P1 |\<times>| P2) in Let ((\<lambda>l. step_score l e s) |`| pairs) fSum)"
iop_i_i: fixes G :: "('c, 'd) Group_scheme"   and a :: "'c" assumes "Group G"   and "a \<in> carrier G" shows "\<rho>\<^bsub>G\<^esub> (\<rho>\<^bsub>G\<^esub> a) = a"
intro: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "EdomArityAnalysis Aexp" shows "CoCallArityEdom Aexp"
lfp_the_while_option: fixes f :: "'b set \<Rightarrow> 'b set"   and C :: "'b set" assumes "mono f"   and "\<And>X. X \<subseteq> C \<Longrightarrow> f X \<subseteq> C"   and "finite C" shows "lfp f = the (while_option (\<lambda>A. f A \<noteq> A) f {})"
h_prop: shows "(cms\<^sub>2 ! k, subst h (snd (mems ! k))) \<leadsto> \<langle>c\<^sub>h, snd (cms\<^sub>1' ! k), mem\<^sub>h\<rangle> \<and> (cms\<^sub>1' ! k, subst h mem\<^sub>1') \<approx> \<langle>c\<^sub>h, snd (cms\<^sub>1' ! k), mem\<^sub>h\<rangle>"
interval_cbox_cart: fixes a :: "(real, 'a) vec"   and b :: "(real, 'a) vec" shows "{a..b} = cbox a b"
cases: fixes x :: "'a word list" obtains "x = []"   | xa :: "'a word" and ys :: "'a word list" where "x = xa # ys"