hcju/isabelleps huggingface dataset:
strong_inducts: fixes P :: "'a \<Rightarrow> pi \<Rightarrow> bool"   and z :: "'a"   and pi :: "pi" assumes "\<And>z. P z \<zero>"   and "\<And>name1 name2 pi z. (\<And>z. P z pi) \<Longrightarrow> P z (name1{name2}.pi)"   and "\<And>pi z. (\<And>z. P z pi) \<Longrightarrow> P z (\<tau>.pi)"   and "\<And>name1 name2 pi z. \<lbrakk>name2 \<sharp> z; name2 \<sharp> name1; \<And>z. P z pi\<rbrakk> \<Longrightarrow> P z (name1<name2>.pi)"   and "\<And>name1 name2 pi z. (\<And>z. P z pi) \<Longrightarrow> P z ([name1\<frown>name2]pi)"   and "\<And>name1 name2 pi z. (\<And>z. P z pi) \<Longrightarrow> P z ([name1\<noteq>name2]pi)"   and "\<And>pi1 pi2 z. \<lbrakk>\<And>z. P z pi1; \<And>z. P z pi2\<rbrakk> \<Longrightarrow> P z (pi1 \<oplus> pi2)"   and "\<And>pi1 pi2 z. \<lbrakk>\<And>z. P z pi1; \<And>z. P z pi2\<rbrakk> \<Longrightarrow> P z (pi1 \<parallel> pi2)"   and "\<And>name pi z. \<lbrakk>name \<sharp> z; \<And>z. P z pi\<rbrakk> \<Longrightarrow> P z (<\<nu>name>pi)"   and "\<And>pi z. (\<And>z. P z pi) \<Longrightarrow> P z (!pi)" shows "P z pi"
B: fixes a :: "'a"   and b :: "'a" assumes "r' a b" shows "(b \<in> A) = (b \<in> A')"
check_eqv_def: fixes as :: "nat list"   and r :: "nat rexp"   and s :: "nat rexp" shows "check_eqv as r s = (set (add_atoms r (add_atoms s [])) \<subseteq> set as \<and> (case closure as ([(norm r, norm s)], []) of None \<Rightarrow> False | Some ([], x) \<Rightarrow> True | Some (a # list, x) \<Rightarrow> False))"
positive_only_if_decomp: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "positive A" shows "\<exists>M\<in>carrier_mat n n. M * adjoint M = A"
subdiv_ate_4: shows "arc_to_ends H vw = (v, w)"
permutes_others: fixes p :: "'b \<Rightarrow> 'b"   and S :: "'b set"   and x :: "'b" assumes "p permutes S"   and "x \<notin> S" shows "p x = x"
dir_image_def: fixes r :: "('d \<times> 'd) set"   and f :: "'d \<Rightarrow> 'c" shows "dir_image r f = {(f a, f b) |a b. (a, b) \<in> r}"
simps_1: fixes P :: "interface" shows "com_cell (Filler P) = Skip"
ld_ld_less2: fixes x :: "real"   and y :: "real" assumes "2 \<le> x"   and "2 \<le> y" shows "1 + log 2 x + log 2 y \<le> 2 * log 2 (x + y - 1)"
orderI: fixes a :: "'d upred"   and b :: "'d upred" assumes "a = (a \<and> b)" shows "b \<sqsubseteq> a"
ideal_condition: fixes I :: "'a set" assumes "I \<subseteq> carrier R"   and "I \<noteq> {}"   and "\<forall>x\<in>I. \<forall>y\<in>I. x \<plusminus> -\<^sub>a y \<in> I"   and "\<forall>r\<in>carrier R. \<forall>x\<in>I. r \<cdot>\<^sub>r x \<in> I" shows "ideal R I"
funcset_eq: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and g :: "'e \<Rightarrow> 'f" assumes "f \<in> extensional A"   and "g \<in> extensional A"   and "\<forall>x\<in>A. f x = g x" shows "f = g"
lincomb_index: fixes i :: "nat"   and X :: "'a vec set"   and a :: "'a vec \<Rightarrow> 'a" assumes "i < nc"   and "X \<subseteq> mat_kernel A" shows "A.lincomb a X $ i = (\<Sum>x\<in>X. a x * x $ i)"
point_equations_1: fixes p :: "'a" assumes "is_point p" shows "p ; 1 = p"
set_eq_fold: fixes x :: "'a Word.word"   and xs :: "'a Word.word list" shows "signed.Max (set (x # xs)) = fold signed.max xs x"
box_empty_imp: fixes a :: "'b"   and b :: "'b" assumes "box a b = {}" shows "g (cbox a b) = \<^bold>1"
rebase_dvd_axioms: shows "rebase_dvd TYPE('pq) TYPE('q)"
map_code: fixes f :: "'b \<Rightarrow> 'a"   and xq :: "'b lazy_sequence" shows "Lazy_Sequence.map f xq = Lazy_Sequence (\<lambda>_. map_option (\<lambda>(x, xq'). (f x, Lazy_Sequence.map f xq')) (yield xq))"
not_prime_elem_zero: shows "\<not> prime_elem (0::'a)"
h7: shows "set (characterize_consistent_signs_at_roots_copr p (qs1 @ qs2)) \<subseteq> set (get_signs (snd (combine_systems p (qs1, matrix1, subsets1, signs1) (qs2, matrix2, subsets2, signs2))))"
distinct_3: fixes x1 :: "'d" shows "Normal x1 \<noteq> Stuck"
Cons_nth_drop_Suc: fixes i :: "nat"   and xs :: "'d list" assumes "i < length xs" shows "xs ! i # drop (Suc i) xs = drop i xs"
col__coplanar: 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" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.Col Bet A B C" shows "Tarski_neutral_dimensionless.Coplanar Bet A B C D"
test_bit_wi: fixes x :: "int"   and n :: "nat" shows "word_of_int x !! n = (n < LENGTH('a) \<and> bit x n)"
sg_subset_of_subG: fixes K :: "'a set"   and H :: "'a set" assumes "G \<guillemotright> K "   and "\<natural>K \<guillemotright> H " shows "H \<subseteq> K"
alphaAbs_all_qFresh_imp_qAFresh: fixes xs :: "'d"   and x :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and xs' :: "'d"   and x' :: "'e"   and X' :: "('f, 'g, 'd, 'e, 'h) qTerm" assumes "alphaAbs_all_qFresh xs x X xs' x' X'" shows "alphaAbs_all_qAFresh xs x X xs' x' X'"
deg_diff_by_const: fixes g :: "nat \<Rightarrow> 'a"   and a :: "'a"   and h :: "nat \<Rightarrow> 'a" assumes "g \<in> carrier (UP R)"   and "a \<in> carrier R"   and "h = g \<oplus>\<^bsub>UP R\<^esub> monom (UP R) a 0" shows "deg R g = deg R h"
set_iterator_product_def: fixes it_a :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and it_b :: "'e \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd" shows "set_iterator_product it_a it_b \<equiv> \<lambda>c f. it_a c (\<lambda>a. it_b a c (\<lambda>b. f (a, b)))"
code: fixes p :: "'d \<Rightarrow> bool"   and g21 :: "'d \<Rightarrow> 'c"   and g22 :: "'d \<Rightarrow> 'd"   and a :: "'d" shows "unfold_llist p g21 g22 a = (if p a then LNil else LCons (g21 a) (unfold_llist p g21 g22 (g22 a)))"
inject_6: fixes x61 :: "Arraytype"   and x62 :: "nat"   and y61 :: "Arraytype"   and y62 :: "nat" shows "(arrV x61 x62 = arrV y61 y62) = (x61 = y61 \<and> x62 = y62)"
of_nat_le_ennreal_iff: fixes r :: "real"   and i :: "nat" assumes "0 \<le> r" shows "(of_nat i \<le> ennreal r) = (real i \<le> r)"
AI_def: shows "AI = lpfp\<^sub>c (step' \<top>)"
lemma_3: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and M :: "Points"   and N :: "Points"   and P :: "Points"   and R :: "Points" assumes "is_pappus"   and "desargues_config A B C A' B' C' M N P R"   and "triangle_circumscribes_triangle A' B' C' A B C" shows "col M N P"
elims: fixes x :: "aexp"   and xa :: "'a st option"   and y :: "'a" assumes "aval'' x xa = y" obtains e :: "aexp" where "x = ea__" and "xa = None" and "y = \<bottom>"   | e :: "aexp" and S :: "'a st" where "x = e" and "xa = Some S" and "y = aval' e S"
assms_1: shows "\<lbrace>p'\<rbrace> S \<lbrace>q\<rbrace>\<^sub>u"
strategy_attracts_via_no_deadends: fixes v :: "'a"   and A :: "'a set"   and W :: "'a set"   and p :: "Player"   and \<sigma> :: "'a \<Rightarrow> 'a" assumes "v \<in> V"   and "v \<in> A - W"   and "strategy_attracts_via p \<sigma> v A W" shows "\<not> deadend v"
env_restr_below_subset: fixes S :: "'c set"   and S' :: "'c set"   and m1 :: "'c \<Rightarrow> 'd"   and m2 :: "'c \<Rightarrow> 'd" assumes "S \<subseteq> S'"   and "m1 f|` S' \<sqsubseteq> m2 f|` S'" shows "m1 f|` S \<sqsubseteq> m2 f|` S"
nosubst_preserves_ODE_dom: fixes ODE :: "('f + 'e, 'g) ODE" shows "ODE_dom (OsubstFO ODE \<sigma>) = ODE_dom ODE"
cases: fixes a :: "'s Trace"   and jp :: "'a \<Rightarrow> ('obs, 'aAct, 'c) Protocol" assumes "a \<in> jpTraces jp" obtains s :: "'s" where "a = tInit sa__" and "sa__ \<in> set envInit"   | t :: "'s Trace" and eact :: "'eAct" and aact :: "'a \<Rightarrow> 'aAct" and s :: "'s" where "a = t \<leadsto> s" and "t \<in> jpTraces jp" and "eact \<in> set (envAction (tLast t))" and "\<And>a. aact a \<in> set (actJP jp t a)" and "s = envTrans eact aact (tLast t)"
present_alloc_noteq: fixes h :: "heap"   and a :: "'a array"   and xs :: "'b list" assumes "Array.present h a" shows "a =!!= fst (Array.alloc xs h)"
strict_implies_not_eq: fixes a :: "'b"   and b :: "'b" assumes "a < b" shows "a \<noteq> b"
plus_in_intervalE: fixes xy :: "'a"   and X :: "'a interval"   and Y :: "'a interval" assumes "xy \<in>\<^sub>i X + Y" obtains x :: "'a" and y :: "'a" where "xy = x + y" and "x \<in>\<^sub>i X" and "y \<in>\<^sub>i Y"
s1_s2: fixes x :: "nat"   and a :: "nat"   and b :: "nat"   and r :: "nat"   and y :: "nat" assumes "x < q"   and "a < q"   and "b < q"   and "r < q"   and "y < q" shows "((x + a) mod q * b + q - (a * b + q - r) mod q) mod q = (x * y + q - (x * ((y + q - b) mod q) + q - r) mod q) mod q"
is_extensional: fixes f :: "'a" assumes "\<not> F.A.arr f" shows "(\<tau>'\<tau>.map \<circ> F) f = \<tau>.B.null"
state_update_after_is_CSP: fixes Ac :: "('c, 'd) action"   and sc :: "'d \<times> 'd \<Rightarrow> bool" shows "is_CSP_process (action.relation_of Ac ;; R (true \<turnstile> \<lambda>(A, A'). sc (alpha_rp.more A, alpha_rp.more A') \<and> \<not> wait A' \<and> tr A = tr A'))"
assms_7: shows "x2 \<noteq> (0::'a) \<or> y2 \<noteq> (0::'a) \<or> z2 \<noteq> (0::'a)"
digraph_isomorphism_inj_on_arcs: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "digraph_isomorphism hom" shows "inj_on (iso_arcs hom) (arcs G)"
long_suf: fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>s v"   and "\<^bold>|v\<^bold>| \<le> \<^bold>|u\<^bold>|" shows "u = v"
Conj_eqvt: fixes p :: "perm"   and A :: "fm"   and B :: "fm" shows "p \<bullet> A AND B = (p \<bullet> A) AND (p \<bullet> B)"
observableE: fixes ob :: "'a \<Rightarrow> 'b"   and P :: "'a set"   and s :: "'a"   and s' :: "'a" assumes "observable ob P"   and "ob s = ob s'"   and "s' \<in> P" shows "s \<in> P"
set_quicksort_acc: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and ac :: "'b list"   and xs :: "'b list" shows "set (ord.quicksort_acc less ac xs) = set ac \<union> set xs"
match_ex_table_SomeD: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'a) list) list"   and C :: "char list"   and pc :: "nat"   and xt :: "(nat \<times> nat \<times> char list \<times> nat \<times> nat) list"   and pc' :: "nat"   and d' :: "nat" assumes "match_ex_table P C pc xt = \<lfloor>(pc', d')\<rfloor>" shows "\<exists>(f, t, D, h, d)\<in>set xt. matches_ex_entry P C pc (f, t, D, h, d) \<and> h = pc' \<and> d = d'"
finite_measure_Union': fixes A :: "'a set"   and B :: "'a set" assumes "A \<in> events"   and "B \<in> events" shows "prob (A \<union> B) = prob A + prob (B - A)"
"2_1": shows "loc_imps_fw c loc1 loc2 M xs"
simps_3: fixes dims :: "nat list"   and S1 :: "com"   and S2 :: "com"   and \<rho> :: "complex mat" shows "state_sig.denote dims (S1;; S2) \<rho> = state_sig.denote dims S2 (state_sig.denote dims S1 \<rho>)"
AnalBinds_not_there: fixes x :: "Vars.var"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and exp :: "exp \<Rightarrow> 'b \<rightarrow> 'a"   and ae :: "Vars.var \<Rightarrow> 'b\<^sub>\<bottom>" assumes "x \<notin> domA \<Gamma>" shows "(ExpAnalysis.AnalBinds exp \<Gamma>\<cdot>ae) x = \<bottom>"
fimage_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"
sum_associative: fixes X :: "'b set"   and f :: "'b \<Rightarrow> nat"   and c :: "nat" assumes "finite X" shows "(\<Sum>x\<in>X. f x + c) = sum f X + card X * c"
lookup_uminus_fun: fixes s :: "'a \<Rightarrow>\<^sub>0 'b" shows "lookup (- s) = - lookup s"
kronecker_factorization_main_def: fixes df :: "int \<Rightarrow> int list"   and dp :: "int \<Rightarrow> int list"   and bnd :: "nat"   and p :: "int poly" shows "kronecker_factorization_main df dp bnd p \<equiv> if degree p \<le> 1 then None else let p = primitive_part p; js = kronecker_samples bnd; cjs = map (\<lambda>j. (poly p j, j)) js in case map_of cjs 0 of None \<Rightarrow> let djs = map (\<lambda>(v, j). map (Pair j) (if j = 0 then dp v else df v)) cjs in map_option the (find_map_filter newton_interpolation_poly_int (\<lambda>go. case go of None \<Rightarrow> False | Some g \<Rightarrow> dvd_int_poly_non_0 g p \<and> 1 \<le> degree g) (concat_lists djs)) | Some j \<Rightarrow> Some [:- j, 1:]"
borel_measurable_subalgebra: fixes N :: "'c measure"   and M :: "'c measure"   and f :: "'c \<Rightarrow> 'd" assumes "sets N \<subseteq> sets M"   and "space N = space M"   and "f \<in> borel_measurable N" shows "f \<in> borel_measurable M"
handy_if_lemma: fixes P :: "bool"   and A :: "'b"   and B :: "'b" shows "((if P then Some A else None) = Some B) = (P \<and> A = B)"
size_Suc_Diff1: fixes x :: "'c"   and M :: "'c multiset" assumes "x \<in># M" shows "Suc (size (M - {#x#})) = size M"
simps_2: fixes j :: "nat"   and xs :: "nat list"   and i :: "nat"   and bezout :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<times> 'b \<times> 'b \<times> 'b \<times> 'b"   and P :: "'b mat"   and A :: "'b mat"   and Q :: "'b mat" shows "diagonal_to_Smith_i_PQ_JNF (j # xs) i bezout (P, A, Q) = (if A $$ (i, i) dvd A $$ (j, j) then diagonal_to_Smith_i_PQ_JNF xs i bezout (P, A, Q) else let (p, q, u, v, d) = bezout (A $$ (i, i)) (A $$ (j, j)); A' = diagonal_step_JNF A i j d v; (P', Q') = diagonal_step_PQ_JNF A i j bezout in diagonal_to_Smith_i_PQ_JNF xs i bezout (P' * P, A', Q * Q'))"
max_fun_diff_in: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "f \<noteq> g"   and "f \<in> rs.FINFUNC"   and "g \<in> rs.FINFUNC" shows "s.max_fun_diff f g \<in> {a \<in> Field s. f a \<noteq> g a}"
2: shows "is_set_to_sorted_list ordR Rk Rs tsl"
simps_1: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and f :: "'a" assumes "monoidal_language C" shows "monoidal_language.Inv C \<^bold>\<langle>f\<^bold>\<rangle> = \<^bold>\<langle>category.inv C f\<^bold>\<rangle>"
pred_set: fixes P :: "'c \<Rightarrow> bool" shows "llist_all P = (\<lambda>x. Ball (lset x) P)"
lm065: fixes f :: "('a \<times> 'b) set"   and X :: "'a set" assumes "runiq (f || X)" shows "sum (toFunction (f || X)) (X \<inter> Domain f) = sum snd (f || X)"
asc_soundness: fixes M1 :: "('d, 'e, 'f) FSM"   and M2 :: "('d, 'e, 'f) FSM"   and T :: "'d list set"   and \<Omega> :: "('d, 'e) ATC set" assumes "OFSM M1"   and "OFSM M2" shows "M1 \<preceq> M2 \<longrightarrow> M1 \<preceq>\<lbrakk>T.\<Omega>\<rbrakk> M2"
el_at_j: shows "mod_arr j \<in># arr_mset arr m r\<^sub>0"
is_free_free: fixes name :: "name" shows "is_free (Free name)"
unfold: shows "COPY' = (left`?`x \<rightarrow> right`!`x \<rightarrow> COPY')"
0: fixes ps :: "(int \<times> int) list" assumes "ps \<in> {ps. distinct ps \<and> sorted_fst ps}" shows "real (time (closest_pair_rec_tm ps)) \<le> (closest_pair_recurrence \<circ> length) ps"
lc_in_carrier_2: shows "combine Cs' Us \<in> carrier R"
fin_succs: fixes G :: "nat graph_rec" assumes "cfg_doms G" shows "\<forall>v\<in>set (g_V G). finite (cfg_doms.succs G v)"
ordermap_type: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "small A" shows "ordermap A r \<in> A \<rightarrow> elts (ordertype A r)"
augment: fixes E1 :: "'edge set"   and E2 :: "'edge set" assumes "forest E1"   and "forest E2"   and "card E2 < card E1" shows "\<exists>e\<in>E1 - E2. forest (insert e E2)"
expectation_rqs_cost: fixes n :: "nat" shows "measure_pmf.expectation (rqs_cost n) real = rqs_cost_exp n"
f_def: fixes v :: "('a, 'm) vec" shows "f v = (\<Sum>j\<in>UNIV. A $ j $ (THE k. v = X $ k) *s Y $ j)"
comp_right_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z * x \<le> z * y"
least_image: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e" assumes "A \<noteq> {}"   and "\<And>k l. \<lbrakk>k \<in> A; l \<in> A; k \<le> l\<rbrakk> \<Longrightarrow> f k \<le> f l" shows "least (f ` A) = f (least A)"
hyps: fixes P :: "(('a, 'c) vec, 'c) vec"   and A :: "(('a, 'b) vec, 'c) vec" shows "snd (row_add_iterate_PA (P, A) n i j) = row_add_iterate A n i j"
New1ArrayRed: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and T :: "ty" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>newA T\<lfloor>e\<rceil>,s\<rangle> -ta\<rightarrow> \<langle>newA T\<lfloor>e'\<rceil>,s'\<rangle>"
bezout_matrix_not_zero: fixes bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd"   and a :: "'e"   and b :: "'e"   and A :: "(('d, 'f) vec, 'e) vec"   and j :: "'f" assumes "is_bezout_ext bezout"   and "a \<noteq> b"   and "A $ a $ j \<noteq> (0::'d)" shows "(bezout_matrix A a b j bezout ** A) $ a $ j \<noteq> (0::'d)"
word_unat_power: fixes n :: "nat" shows "2 ^ n = word_of_nat (2 ^ n)"
PSQ_D: shows "(P', S, Q') = diagonal_to_Smith_PQ D bezout"
small: shows "\<parallel>fs ! 0\<parallel>\<^sup>2 \<le> \<alpha> ^ i * \<parallel>gso i\<parallel>\<^sup>2"
rep_eq: fixes S :: "'c set" assumes "S \<noteq> {}"   and "finite S" shows "pmf (local.pmf_of_set S) = (\<lambda>x. indicat_real S x / real (card S))"
of_exps_def: fixes a :: "nat"   and b :: "nat"   and i :: "nat" shows "Term_Order.of_exps a b i = (THE u. rep_nat_term u = (pp_of_fun (\<lambda>x. if x = 0 then a else if x = 1 then b else 0), if \<exists>v. snd (rep_nat_term v) = i then i else 0))"
Gcd_factorial_def: fixes A :: "'a set" shows "Gcd_factorial A = (if A \<subseteq> {0::'a} then 0::'a else normalize (\<Prod>\<^sub># (Inf (prime_factorization ` (A - {0::'a})))))"
to_Bonk_Schramm_extension_def: fixes x :: "'c" shows "to_Bonk_Schramm_extension x = abs_Bonk_Schramm_extension (basepoint x)"
denotational_is_big_step: fixes s :: "char list \<Rightarrow> int"   and t :: "char list \<Rightarrow> int"   and c :: "com" shows "((s, t) \<in> D c) = (c, s) \<Rightarrow> t"
comm_neutral: fixes a :: "'a" shows "a \<sqinter> top = a"
sup_ge1: fixes x :: "'a"   and y :: "'a" shows "x \<le> x \<squnion> y"
gen_subalgebra_sig_sets: fixes G :: "'c set set"   and M :: "'c measure" assumes "G \<subseteq> Pow (space M)" shows "sets (gen_subalgebra M G) = sigma_sets (space M) G"
not_present_alloc: fixes h :: "heap"   and xs :: "'b list" shows "\<not> Array.present h (fst (Array.alloc xs h))"
simps_2: fixes x :: "'c"   and y :: "'d"   and t1 :: "(('c \<times> 'd) \<times> nat) tree"   and a :: "'c"   and b :: "'d"   and lv :: "nat"   and t2 :: "(('c \<times> 'd) \<times> nat) tree" shows "update x y \<langle>t1, ((a, b), lv), t2\<rangle> = (case cmp x a of LT \<Rightarrow> split (skew \<langle>update x y t1, ((a, b), lv), t2\<rangle>) | EQ \<Rightarrow> \<langle>t1, ((x, y), lv), t2\<rangle> | GT \<Rightarrow> split (skew \<langle>t1, ((a, b), lv), update x y t2\<rangle>))"
simps_1: fixes valuation :: "Literal list" shows "clauseFalse [] valuation = True"
times_interval_tendsto: fixes X :: "nat \<Rightarrow> 'a interval"   and x :: "'a"   and Y :: "nat \<Rightarrow> 'a interval"   and y :: "'a" assumes "X \<longlonglongrightarrow>\<^sub>i x"   and "Y \<longlonglongrightarrow>\<^sub>i y" shows "(\<lambda>i. X i * Y i) \<longlonglongrightarrow>\<^sub>i x * y"
get_ET_def: fixes psr :: "word32" shows "get_ET psr \<equiv> ucast (psr AND 32 >> 5)"
cases: fixes a1 :: "'b \<Rightarrow> ('c, 'd) trm"   and a2 :: "('c + 'b, 'd) trm" assumes "TadmitFO a1 a2" obtains \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm" and \<theta> :: "('c + 'b, 'd) trm" where "a1 = \<sigma>''__" and "a2 = Differential \<theta>" and "TadmitFFO \<sigma>''__ \<theta>" and "NTUadmit \<sigma>''__ \<theta> UNIV" and "dfree (TsubstFO \<theta> \<sigma>''__)"   | \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm" and args :: "'d \<Rightarrow> ('c + 'b, 'd) trm" and f :: "'c + 'b" where "a1 = \<sigma>'''__" and "a2 = $f f args" and "\<And>i. TadmitFO \<sigma>'''__ (args i)"   | \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm" and \<theta>1 :: "('c + 'b, 'd) trm" and \<theta>2 :: "('c + 'b, 'd) trm" where "a1 = \<sigma>''''__" and "a2 = Plus \<theta>1__ \<theta>2__" and "TadmitFO \<sigma>''''__ \<theta>1__" and "TadmitFO \<sigma>''''__ \<theta>2__"   | \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm" and \<theta>1 :: "('c + 'b, 'd) trm" and \<theta>2 :: "('c + 'b, 'd) trm" where "a1 = \<sigma>'''''__" and "a2 = Times \<theta>1 \<theta>2" and "TadmitFO \<sigma>'''''__ \<theta>1" and "TadmitFO \<sigma>'''''__ \<theta>2"   | \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm" and x :: "'d" where "a1 = \<sigma>''''''__" and "a2 = $' x__"   | \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm" and x :: "'d" where "a1 = \<sigma>'''''''__" and "a2 = Var x"   | \<sigma> :: "'b \<Rightarrow> ('c, 'd) trm" and r :: "real" where "a1 = \<sigma>" and "a2 = Const r"
rep_eq: fixes x :: "'a dense"   and xa :: "'a dense" shows "(x \<le> xa) = (Rep_dense x \<le> Rep_dense xa)"
additive_increasing: fixes f :: "'a set \<Rightarrow> ennreal" assumes "positive M f"   and "Measure_Space.additive M f" shows "increasing M f"
SYSTEM'_def: shows "SYSTEM' \<equiv> (SEND' \<lbrakk>SYN\<rbrakk> REC') \ SYN"
shadow_root_ptr_shadow_root_ptr_cast: fixes shadow_root_ptr :: "(_) shadow_root_ptr" shows "is_shadow_root_ptr_kind (cast shadow_root_ptr)"
pheap_del_min: fixes h :: "'b heap" assumes "pheap h" shows "pheap (del_min h)"
unatSuc2: fixes n :: "'b word" assumes "n + 1 \<noteq> 0" shows "unat (n + 1) = Suc (unat n)"
scale_def: fixes a :: "'a"   and v :: "'b \<times> 'c" shows "local.scale a v = (a *a fst v, a *b snd v)"
sees_methods_Object: fixes P :: "'a prog"   and D :: "char list"   and fs :: "fdecl list"   and ms :: "'a mdecl list"   and Mm :: "char list \<Rightarrow> ((staticb \<times> ty list \<times> ty \<times> 'a) \<times> char list) option" assumes "class P Object = \<lfloor>(D, fs, ms)\<rfloor>"   and "Mm = map_option (\<lambda>m. (m, Object)) \<circ> map_of ms" shows "P \<turnstile> Object sees_methods Mm"
lex_triple_def: shows "Recursive.lex_triple \<equiv> less_than <*lex*> Recursive.lex_pair"
get_ancestors_reads: fixes h :: "(_) heap"   and node_ptr :: "(_) object_ptr"   and h' :: "(_) heap" assumes "Shadow_DOM.heap_is_wellformed h" shows "reads Shadow_DOM.get_ancestors_locs (Shadow_DOM.get_ancestors node_ptr) h h'"
dvd_mult_imp_div: fixes a :: "'a"   and c :: "'a"   and b :: "'a" assumes "a * c dvd b" shows "a dvd b div c"
gbinomial_partial_sum_poly: fixes m :: "nat"   and a :: "'a"   and x :: "'a"   and y :: "'a" shows "(\<Sum>k\<le>m. (of_nat m + a gchoose k) * x ^ k * y ^ (m - k)) = (\<Sum>k\<le>m. (- a gchoose k) * (- x) ^ k * (x + y) ^ (m - k))"
convergent_Cauchy: fixes "open" :: "'c set \<Rightarrow> bool"   and uniformity :: "('c \<times> 'c) filter"   and X :: "nat \<Rightarrow> 'c" assumes "class.uniform_space open uniformity"   and "topological_space.convergent open X" shows "uniform_space.Cauchy uniformity X"
case_distrib: fixes h :: "'c \<Rightarrow> 'b"   and f1 :: "'c"   and f2 :: "'d \<Rightarrow> 'c"   and err :: "'d err" shows "h (case err of Err \<Rightarrow> f1 | OK x \<Rightarrow> f2 x) = (case err of Err \<Rightarrow> h f1 | OK x \<Rightarrow> h (f2 x))"
assms_1: shows "\<forall>\<^sub>F n in sequentially. norm (f n) \<le> norm (g n)"
rcons_correct: fixes t :: "('a, 'b) FingerTree"   and a :: "'a \<times> 'b" shows "FingerTree.toList (FingerTree.rcons t a) = FingerTree.toList t @ [a]"
bilateral_substitutes_on_def: fixes A :: "'x set"   and f :: "'x set \<Rightarrow> 'x set" shows "bilateral_substitutes_on A f = (\<not> (\<exists>B\<subseteq>A. \<exists>a b. {a, b} \<subseteq> A \<and> Xd a \<notin> Xd ` B \<and> Xd b \<notin> Xd ` B \<and> b \<notin> f (B \<union> {b}) \<and> b \<in> f (B \<union> {a, b})))"
assms_2: shows "w \<in> set (awalk_verts u p)"
map_def: fixes f :: "'a" shows "S.map f = (if S.arr f then f else S.null)"
equality: fixes r :: "('c, 'd) Graph_scheme"   and r' :: "('c, 'd) Graph_scheme" assumes "V\<^bsub>r\<^esub> = V\<^bsub>r'\<^esub>"   and "E\<^bsub>r\<^esub> = E\<^bsub>r'\<^esub>"   and "Graph.more r = Graph.more r'" shows "r = r'"
simps_2: fixes t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" shows "SkewBinomialHeapStruc.rank_invar [t] = True"
sample_const: fixes p :: "'p pmf"   and m :: "'m" shows "sample p (\<lambda>_. m) = m"