hcju/isabelleps huggingface dataset:
Rf_limit_Sup: fixes Ns :: "'a literal multiset set llist" assumes "chain (\<triangleright>) Ns" shows "Rf (Liminf_llist Ns) = Rf (Sup_llist Ns)"
r_dummy_prim: fixes f :: "recf"   and a :: "nat"   and n :: "nat" assumes "prim_recfn a f" shows "prim_recfn (a + n) (r_dummy n f)"
empty_rep: fixes q :: "('a, 'b) BinomialHeap" shows "(q = BinomialHeap_loc.empty) = (Rep_BinomialHeap q = [])"
bounded_lattice_top_ow_axioms: shows "bounded_lattice_top_ow U\<^sub>2 (\<sqinter>\<^sub>o\<^sub>w\<^sub>.\<^sub>2) (\<le>\<^sub>o\<^sub>w\<^sub>.\<^sub>2) (<\<^sub>o\<^sub>w\<^sub>.\<^sub>2) (\<squnion>\<^sub>o\<^sub>w\<^sub>.\<^sub>2) top\<^sub>2"
ThrowRed: fixes e :: "char list exp"   and s :: "heap \<times> sfields \<times> sheap"   and b :: "bool"   and e' :: "char list exp"   and s' :: "heap \<times> sfields \<times> sheap"   and b' :: "bool"   and P :: "(char list list \<times> char list exp) prog" assumes "P \<turnstile> \<langle>e,s,b\<rangle> \<rightarrow> \<langle>e',s',b'\<rangle>" shows "P \<turnstile> \<langle>throw e,s,b\<rangle> \<rightarrow> \<langle>throw e',s',b'\<rangle>"
SD_efficient_no_pareto_loser: fixes p :: "'alt pmf" assumes "SD_efficient R p"   and "p \<in> lotteries" shows "set_pmf p \<inter> pareto_losers R = {}"
assms_11: shows "InvariantNoDecisionsWhenConflict (getF state) (getM state) (currentLevel (getM state))"
sorted_remdups_sorted: fixes xs :: "'a list" assumes "sorted xs" shows "sorted (remdups_sorted xs)"
Gcd_dvd: fixes a :: "'a"   and A :: "'a set" assumes "a \<in> A" shows "Gcd A dvd a"
new_character_data_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_character_data_ptr :: "(_) character_data_ptr"   and ptr :: "(_) element_ptr"   and getter :: "('n, 'l, 'm, 'p, 'r option) RElement_scheme \<Rightarrow> 'v" assumes "h \<turnstile> new_character_data \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
wf: shows "P \<turnstile> (E, ws) \<surd>"
length_last_fusion_code: fixes g :: "('a, 'b) generator"   and s :: "'b" shows "length_last_fusion g s = (if list.has_next g s then let (x, s') = list.next g s in fold_fusion g (\<lambda>x (n, uu_). (n + 1, x)) s' (1, x) else (0, undefined))"
Aexp_pre_simps_7: fixes scrut :: "exp"   and e1 :: "exp"   and e2 :: "exp"   and a :: "Arity" shows "Aexp_syn' a (scrut ? e1 : e2) = Aexp_syn' 0 scrut \<squnion> Aexp_syn' a e1 \<squnion> Aexp_syn' a e2"
addSyncEmpty: fixes sa :: "'a event list"   and A :: "'a event set"   and u :: "'a event list"   and y1 :: "'a event" shows "sa setinterleaves (([], u), A) \<and> y1 \<in> A \<longrightarrow> sa @ [y1] setinterleaves (([y1], u @ [y1]), A)"
subst_set_def: fixes \<sigma> :: "'b \<Rightarrow> ('a, 'b) Term.term"   and E :: "(('a, 'b) Term.term \<times> ('a, 'b) Term.term) set" shows "subst_set \<sigma> E = (\<lambda>e. (fst e \<cdot> \<sigma>, snd e \<cdot> \<sigma>)) ` E"
AndE1: fixes \<Gamma> :: "'b formula set"   and F :: "'b formula"   and G :: "'b formula" assumes "\<Gamma> \<turnstile> F \<^bold>\<and> G" shows "\<Gamma> \<turnstile> F"
finite_UN: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and M :: "'c measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> A i \<in> fmeasurable M" shows "\<Union> (A ` I) \<in> fmeasurable M"
finite_nat_iff_bounded: fixes S :: "nat set" shows "finite S = (\<exists>k. S \<subseteq> {..<k})"
match_rel: fixes t :: "pterm"   and u :: "sterm"   and p :: "Term_Class.term" assumes "pterm_to_sterm t = u" shows "rel_option srelated'.P_env (match p t) (match p u)"
set_1: fixes x1 :: "'f option" shows "results (Resumption.resumption.Done x1) = set_option x1"
cSUP_le_iff: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b multiset"   and u :: "'b multiset" assumes "A \<noteq> {}"   and "subset_mset.bdd_above (f ` A)" shows "(Sup (f ` A) \<subseteq># u) = (\<forall>x\<in>A. f x \<subseteq># u)"
induct: fixes P :: "game \<Rightarrow> game \<Rightarrow> bool"   and a0 :: "game"   and a1 :: "game" assumes "\<And>G H. \<lbrakk>\<And>x. zin x (left_options G) \<Longrightarrow> P x H; \<And>x. zin x (left_options H) \<Longrightarrow> P G x; \<And>x. zin x (right_options G) \<Longrightarrow> P x H; \<And>x. zin x (right_options H) \<Longrightarrow> P G x\<rbrakk> \<Longrightarrow> P G H" shows "P a0 a1"
neg_bounds_compare_7: fixes c :: "'a"   and b :: "'a option" assumes "\<not> c <\<^sub>l\<^sub>b b" shows "c \<ge>\<^sub>l\<^sub>b b"
IH: assumes "p \<in> setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t (A \<cdot>\<^sub>l\<^sub>s\<^sub>s\<^sub>t \<delta>)" shows "\<exists>q\<in>setops\<^sub>l\<^sub>s\<^sub>s\<^sub>t A. fst p = fst q \<and> (\<exists>X\<subseteq>bvars\<^sub>l\<^sub>s\<^sub>s\<^sub>t A. snd p = snd q \<cdot>\<^sub>p rm_vars X \<delta>)"
induct: fixes x1 :: "'form \<times> nat list"   and x2 :: "'form \<times> nat list"   and x3 :: "((('form \<times> nat list) \<times> ('form, 'var) out_port) \<times> ('form \<times> nat list) \<times> ('form, 'var) antecedent) list"   and P :: "'form \<times> nat list \<Rightarrow> 'form \<times> nat list \<Rightarrow> ((('form \<times> nat list) \<times> ('form, 'var) out_port) \<times> ('form \<times> nat list) \<times> ('form, 'var) antecedent) list \<Rightarrow> bool" assumes "path x1 x2 x3"   and "\<And>v. P v v []"   and "\<And>e v' pth. \<lbrakk>e \<in> edges; path (edge_end e) v' pth; P (edge_end e) v' pth\<rbrakk> \<Longrightarrow> P (edge_begin e) v' (e # pth)" shows "P x1 x2 x3"
image_map_indets_ideal: fixes f :: "'c \<Rightarrow> 'd"   and F :: "(('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'e) set" assumes "inj f" shows "map_indets f ` ideal F = ideal (map_indets f ` F) \<inter> P[range f]"
set_empty: fixes xs :: "'g list" shows "(set xs = {}) = (xs = [])"
eventually_at_right_real: fixes a :: "real"   and b :: "real" assumes "a < b" shows "\<forall>\<^sub>F x in at_right a. x \<in> {a<..<b}"
intros: fixes M :: "hmultiset"   and e :: "nat" assumes "\<forall>m\<in>#hmsetmset M. e \<le> local.decode 0 m" shows "local.aligned\<^sub>d e M"
all_security_requirements_fulfilled_eq_rule: fixes G :: "'a list_graph" assumes "wf_list_graph G" shows "TopoS_Composition_Theory_impl.all_security_requirements_fulfilled (TopoS_Composition_Theory_impl.get_IFS (get_impl M)) G = TopoS_Composition_Theory.all_security_requirements_fulfilled (TopoS_Composition_Theory.get_IFS (get_spec M)) (list_graph_to_graph G)"
Cons_1: fixes kv :: "'a \<times> 'b" assumes "sorted_wrt lt (map fst xs)" shows "sorted_wrt lt (map fst (update_by_pair kv xs))"
fst_quot_of_fract_0_imp: fixes x :: "'a fract" assumes "fst (quot_of_fract x) = (0::'a)" shows "snd (quot_of_fract x) = (1::'a)"
map_add_subsumed1: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option" assumes "f \<subseteq>\<^sub>m g" shows "f ++ g = g"
graph_path_def: fixes \<alpha>e :: "'f \<Rightarrow> ('g \<times> 'h \<times> 'g) set"   and \<alpha>n :: "'f \<Rightarrow> 'g list"   and invar :: "'f \<Rightarrow> bool"   and inEdges' :: "'f \<Rightarrow> 'g \<Rightarrow> ('g \<times> 'h) list" shows "graph_path \<alpha>e \<alpha>n invar inEdges' \<equiv> graph (\<lambda>g. \<lparr>nodes = set (\<alpha>n g), edges = \<alpha>e g\<rparr>) invar \<and> graph_nodes_it (\<lambda>g. \<lparr>nodes = set (\<alpha>n g), edges = \<alpha>e g\<rparr>) invar (\<lambda>g. foldri (\<alpha>n g)) \<and> graph_pred_it (\<lambda>g. \<lparr>nodes = set (\<alpha>n g), edges = \<alpha>e g\<rparr>) invar (\<lambda>g n. foldri (inEdges' g n))"
iptables_bigstep_undecided_to_undecided_in_doubt_allow_approx: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and \<beta> :: "'c \<Rightarrow> 'd \<Rightarrow> ternaryvalue"   and rs :: "'c rule list"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and p :: "'d" assumes "matcher_agree_on_exact_matches \<gamma> \<beta>"   and "good_ruleset rs"   and "\<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow> Undecided" shows "(\<beta>, in_doubt_allow),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Undecided \<or> (\<beta>, in_doubt_allow),p\<turnstile> \<langle>rs, Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Decision FinalAllow"
SoundCall: fixes A :: "(Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool"   and G :: "((Var \<Rightarrow> Val) \<Rightarrow> (Var \<Rightarrow> Val) \<Rightarrow> bool) set"   and n :: "nat" assumes "\<And>n.  \<Turnstile>\<^sub>n ({A} \<union> G)   \<Longrightarrow>  \<Turnstile>\<^sub>n body : A "   and " \<Turnstile>\<^sub>n G  " shows " \<Turnstile>\<^sub>n Call : A "
assms_13: shows "set vs \<inter> \<Union> (FvarsT ` set ts) = {}"
ss423: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "is_map x" shows "(y ; x \<le> z) = (y \<le> z ; x\<^sup>\<smile>)"
bezw_inverse: fixes e :: "nat"   and N :: "nat" assumes "gcd e N = 1" shows "[nat (int e) * nat (fst (bezw e N) mod int N) = 1] (mod nat (int N))"
assms_2: fixes h :: "heap"   and as :: "nat set"   and h' :: "heap" assumes "P (h, as)"   and "relH as h h'"   and "in_range (h', as)" shows "P (h', as)"
wf_induct: fixes r :: "('c \<times> 'c) set"   and P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "wf r"   and "\<And>x. \<forall>y. (y, x) \<in> r \<longrightarrow> P y \<Longrightarrow> P x" shows "P a"
assms_1: shows "R \<subseteq> 1\<^sub>\<sigma>"
that: fixes U :: "'a set" assumes "open U"   and "B \<subseteq> U"   and "emeasure lborel (U - B) \<le> ennreal e" shows "thesis"
simps_1: fixes S :: "'c ltl set"   and \<phi> :: "'c ltl"   and \<psi> :: "'c ltl" shows "eval\<^sub>G S (\<phi> and \<psi>) = eval\<^sub>G S \<phi> and eval\<^sub>G S \<psi>"
single_LeadsTo_I: fixes A :: "'c set"   and F :: "'c program"   and B :: "'c set" assumes "\<And>s. s \<in> A \<Longrightarrow> F \<in> {s} \<longmapsto>w B" shows "F \<in> A \<longmapsto>w B"
to_rat: fixes x :: "'a" shows "to_rat x = (if x \<in> \<rat> then THE y. x = of_rat y else 0)"
rreq_rrep_sn_def: fixes m :: "msg" shows "rreq_rrep_sn m \<equiv> case m of Rreq x xa xb xc xd xe osnc xf xg \<Rightarrow> 1 \<le> osnc | Rrep x xa dsnc xb xc \<Rightarrow> 1 \<le> dsnc | _ \<Rightarrow> True"
some_worst_def: shows "local.some_worst = (SOME x. x \<in> local.degenerate_lotteries \<and> x \<in> local.worst)"
assms_3: shows "less_multiset_ext\<^sub>D\<^sub>M R M N"
tranclp_rtranclp_tranclp: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c"   and c :: "'c" assumes "r\<^sup>+\<^sup>+ a b"   and "r\<^sup>*\<^sup>* b c" shows "r\<^sup>+\<^sup>+ a c"
assigns_r_comp: fixes \<sigma> :: "'b \<Rightarrow> 'b"   and P :: "('b, 'c) urel" shows "\<langle>\<sigma>\<rangle>\<^sub>a ;; P = \<lceil>\<sigma>\<rceil>\<^sub>s \<dagger> P"
env_no_step_s: fixes s :: "'s"   and c :: "'s multiset"   and e :: "'l"   and s' :: "'s"   and c' :: "'s multiset" assumes "((s, c), e, s', c') \<in> gtrs" obtains csp :: "'s multiset" where "c' = csp + c"
vertical_chop1: shows "\<forall>v. \<exists>u w. v=u--w"
WHILE_unfold: fixes b :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c nres"   and s :: "'c" shows "WHILE b f s = (if b s then f s \<bind> WHILE b f else RETURN s)"
cond_spmf_cong: fixes A :: "'h set"   and p :: "'h spmf"   and B :: "'h set" assumes "A \<inter> set_spmf p = B \<inter> set_spmf p" shows "cond_spmf p A = cond_spmf p B"
toplevel_summands_derivs_Times: fixes xs :: "'b list"   and r :: "'b rexp"   and s :: "'b rexp" shows "toplevel_summands (derivs xs (Times r s)) \<subseteq> {Times (derivs xs r) s} \<union> {r'. \<exists>ys zs. r' \<in> toplevel_summands (derivs ys s) \<and> ys \<noteq> [] \<and> zs @ ys = xs}"
rank_Some_time: fixes x :: "nat"   and n :: "nat"   and j :: "nat" assumes "rank x n = Some j" shows "x \<le> n"
f0': fixes y :: "'b"   and x :: "'a" assumes "y \<in> B x - B' x" shows "f (x, y) = (0::'c)"
small_set_vsubset: fixes A :: "V set"   and B :: "V" assumes "small A"   and "A \<subseteq> elts B" shows "ZFC_in_HOL.set A \<subseteq>\<^sub>\<circ> B"
invariant_1D_5: fixes plr :: "int poly \<times> rat \<times> rat" assumes "invariant_1 plr" shows "poly_cond (poly_real_alg_1 plr)"
class_add_xcpt_eff_eq: fixes wf_md :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and P :: "jvm_prog"   and C :: "char list"   and i :: "instr"   and pc :: "nat"   and \<tau> :: "ty list \<times> ty err list"   and xt :: "ex_table"   and cdec :: "jvm_method class" assumes "wf_prog wf_md P"   and "\<not> is_class P C" shows "xcpt_eff i P pc \<tau> xt = xcpt_eff i (class_add P (C, cdec)) pc \<tau> xt"
set_def: fixes s' :: "'b" shows "State_Monad.set s' = State (\<lambda>_. ((), s'))"
minimalize_offending_overapprox_subseteq_input: fixes sinvar :: "'c graph \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"   and ff :: "('c \<times> 'c) list"   and keeps :: "('c \<times> 'c) list"   and G :: "'c graph"   and nP :: "'c \<Rightarrow> 'd" shows "set (SecurityInvariant_withOffendingFlows.minimalize_offending_overapprox sinvar ff keeps G nP) \<subseteq> set ff \<union> set keeps"
a: shows "( \<Gamma> \<Rightarrow>* \<Delta>, n) \<in> derivable R*"
real_add_minus_iff: fixes x :: "real"   and a :: "real" shows "(x + - a = 0) = (x = a)"
finprod_Un_disjoint: fixes A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "finite A"   and "finite B"   and "A \<inter> B = {}"   and "g \<in> A \<rightarrow> carrier G"   and "g \<in> B \<rightarrow> carrier G" shows "finprod G g (A \<union> B) = finprod G g A \<otimes> finprod G g B"
simps_5: fixes M :: "('e, 'f, 'g) KripkeStructure"   and w :: "'g"   and as :: "'e list"   and \<phi> :: "('e, 'f) Kform" shows "M, w \<Turnstile> \<^bold>C\<^bsub>as\<^esub> \<phi> = (\<forall>w'\<in>(\<Union> (relations M ` set as))\<^sup>+ `` {w}. M, w' \<Turnstile> \<phi>)"
iTILL_not_empty: fixes n :: "nat" shows "[\<dots>n] \<noteq> {}"
currentLevelPrefixToLevel: fixes level :: "nat"   and M :: "('a \<times> bool) list" shows "currentLevel (prefixToLevel level M) \<le> level"
closed_map_def: fixes X1 :: "'a topology"   and X2 :: "'b topology"   and f :: "'a \<Rightarrow> 'b" shows "closed_map X1 X2 f \<equiv> \<forall>U. closedin X1 U \<longrightarrow> closedin X2 (f ` U)"
Red1ThrowNull: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>throw null,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer),s\<rangle>"
IH: fixes s :: "'a state" assumes "is_cupcake_all_env env_"   and "is_cupcake_exp e" shows "\<exists>s'. Big_Step_Unclocked_Single.evaluate env_ s e (s', Rerr err_)"
distance0_def: shows "distance0 = v\<^sub>e * \<delta> - v\<^sub>o * \<delta> - a\<^sub>o * \<delta>\<^sup>2 / 2"
ap_pure_dlist: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'e dnelist" shows "single f \<diamondop> x = dnelist.map f x"
totatives_less: fixes x :: "nat"   and n :: "nat" assumes "x \<in> totatives n"   and "1 < n" shows "x < n"
word_le_not_less: fixes b :: "'b word"   and a :: "'b word" shows "(b \<le> a) = (\<not> a < b)"
strong_induct: fixes P :: "'a \<Rightarrow> dbfm \<Rightarrow> bool"   and c :: "'a"   and dbfm :: "dbfm" assumes "\<And>x1 x2 c. P c (DBMem x1 x2)"   and "\<And>x1 x2 c. P c (DBEq x1 x2)"   and "\<And>x1 x2 c. \<lbrakk>\<And>c. P c x1; \<And>c. P c x2\<rbrakk> \<Longrightarrow> P c (DBDisj x1 x2)"   and "\<And>x c. (\<And>c. P c x) \<Longrightarrow> P c (DBNeg x)"   and "\<And>x c. (\<And>c. P c x) \<Longrightarrow> P c (DBEx x)" shows "P c dbfm"
rule_sub_lem_1_c: fixes v :: "i"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>"   and \<Theta> :: "\<o>" assumes "[\<^bold>\<box>(\<psi> \<^bold>\<equiv> \<chi>) in v]" shows "[\<Theta> \<^bold>\<rightarrow> \<psi> \<^bold>\<equiv> (\<Theta> \<^bold>\<rightarrow> \<chi>) in v]"
neg_literal_notin_imp_true_cls: fixes A :: "'a"   and C :: "'a literal multiset"   and I :: "'a set" assumes "Neg A \<in># C"   and "A \<notin> I" shows "I \<Turnstile> C"
G_def: shows "G = {g \<in> borel_measurable M. \<forall>A\<in>sets M. \<integral>\<^sup>+ x. g x * indicator A x \<partial>M \<le> emeasure N A}"
pringsheim_landau: fixes f :: "complex fds"   and c :: "real"   and A :: "complex set"   and g :: "complex \<Rightarrow> complex" assumes "nonneg_dirichlet_series f"   and "abs_conv_abscissa f = ereal c"   and "\<And>s. \<lbrakk>s \<in> A; c < Re s\<rbrakk> \<Longrightarrow> g s = eval_fds f s"   and "g holomorphic_on A"   and "open A"   and "complex_of_real c \<in> A" shows "False"
ctvalid_def: fixes C :: "(('a \<Rightarrow> state \<Rightarrow> bool) \<times> com \<times> ('a \<Rightarrow> state \<Rightarrow> bool)) set"   and P :: "'a \<Rightarrow> state \<Rightarrow> bool"   and c :: "com"   and Q :: "'a \<Rightarrow> state \<Rightarrow> bool" shows "(C \<Turnstile>\<^sub>t {P} c {Q}) = ((\<forall>(P', c', Q')\<in>C. \<Turnstile>\<^sub>t {P'} c' {Q'}) \<longrightarrow> \<Turnstile>\<^sub>t {P} c {Q})"
wary_\<rho>s: shows "wary (subst \<rho> s)"
keys_frag_extend: fixes f :: "'d \<Rightarrow> 'c \<Rightarrow>\<^sub>0 int"   and c :: "'d \<Rightarrow>\<^sub>0 int" shows "Poly_Mapping.keys (frag_extend f c) \<subseteq> (\<Union>x\<in>Poly_Mapping.keys c. Poly_Mapping.keys (f x))"
Gromov_extension_isometry_boundary_to_boundary: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c Gromov_completion" assumes "isometry_on UNIV f"   and "x \<in> Gromov_boundary" shows "Gromov_extension f x \<in> Gromov_boundary"
simps_2: fixes uu :: "'a"   and uv :: "bool" shows "restrict Trueif uu uv = Trueif"
erf_remainder_integral_0_conv_erfc: fixes x :: "real" assumes "0 < x" shows "erf_remainder_integral 0 x = sqrt pi / 2 * erfc x"
inj_on_compose: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and g :: "'d \<Rightarrow> 'e" assumes "bij_betw f A B"   and "inj_on g B" shows "inj_on (compose A g f) A"
map_def: fixes f :: "'j" shows "Fun_fg.map f = (if J.arr f then Fun_g.map (J.cod f) \<cdot> Fun_f.map f else C.null)"
assoc: fixes a :: "'a DBMEntry"   and b :: "'a DBMEntry"   and c :: "'a DBMEntry" shows "dbm_add (dbm_add a b) c = dbm_add a (dbm_add b c)"
assms_3: fixes x :: "'a"   and y :: "'a"   and ys :: "'a list" assumes "xs = x # y # ys" shows "P (x # y # ys)"
lookup_lcs_pp: fixes s :: "('c, 'b) pp"   and t :: "('c, 'b) pp"   and x :: "'c" shows "lookup_pp (lcs s t) x = max (lookup_pp s x) (lookup_pp t x)"
shiftr_le_0: fixes w :: "'b word"   and n :: "nat" assumes "unat w < 2 ^ n" shows "w >> n = 0"
LeastMethodDef_def: fixes P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and C :: "char list"   and M :: "char list"   and mthd :: "ty list \<times> ty \<times> char list list \<times> expr"   and Cs :: "char list list" shows "P \<turnstile> C has least M = mthd via Cs \<equiv> (Cs, mthd) \<in> MethodDefs P C M \<and> (\<forall>(Cs', mthd')\<in>MethodDefs P C M. P,C \<turnstile> Cs \<sqsubseteq> Cs')"
poly_deg_homogenize_le: fixes x :: "'c"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd" shows "poly_deg (homogenize x p) \<le> poly_deg p"
bounded_clinear_right: fixes a :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "bounded_clinear (cblinfun_apply a)"
option_Max_by_def: fixes cmp_f :: "'a \<Rightarrow> 'b"   and A :: "'a set" shows "option_Max_by cmp_f A \<equiv> if A = {} then None else Some (Max_by cmp_f A)"
right_unit: fixes a :: "'a set" assumes "a \<in> contained.Class ` G" shows "a [\<cdot>] contained.Class \<one> = a"
seq_comp_secure_aux_2: fixes I :: "('c \<times> 'c) set"   and D :: "'d option \<Rightarrow> 'c"   and P :: "'d option process"   and Q :: "'d option process"   and ws :: "'d option list"   and Z :: "'d option set"   and xs :: "'d option list"   and zs :: "'d option list"   and y :: "'d option" assumes "secure_termination I D"   and "ref_union_closed P"   and "sequential P"   and "secure P I D"   and "secure Q I D"   and "(ws, Z) \<in> seq_comp_failures P Q"   and "ws = xs @ zs"   and "(xs @ [y], {}) \<in> seq_comp_failures P Q" shows "(xs @ y # ipurge_tr I D (D y) zs, ipurge_ref I D (D y) zs Z) \<in> seq_comp_failures P Q"
i_take_drop: fixes m :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> 'b" shows "f \<Down> n \<up> m = f \<Up> m \<Down> (n - m)"
pop2_def: fixes stk :: "nat list"   and v :: "int" shows "local.pop2 stk v \<equiv> monadic_WHILEIT (\<lambda>_. set stk \<subseteq> {0..<length xs}) (\<lambda>x. case x of [] \<Rightarrow> RETURN False | k # stk \<Rightarrow> ASSERT (k < length xs) \<bind> (\<lambda>_. RETURN (v \<le> xs ! k))) mop_list_tl stk"
Diagonalize_Dom: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "monoidal_language C"   and "monoidal_language.Arr C t" shows "monoidal_language.Diagonalize C (monoidal_language.Dom C t) = monoidal_language.Dom C (monoidal_language.Diagonalize C t)"
assms: shows "fin_join_ti s x (Suc i) = []"
power_le_imp_le_exp: 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 :: "'a"   and m :: "nat"   and n :: "nat" assumes "class.linordered_semidom plus minus less_eq less zero times one"   and "less one a"   and "less_eq (power.power one times a m) (power.power one times a n)" shows "m \<le> n"
simps: fixes C :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'c \<Rightarrow> 'd \<Rightarrow> ('c \<times> 'd) set \<times> bool"   and a1 :: "'c"   and a2 :: "'d" shows "whileLoop_terminates C B a1 a2 = ((\<exists>r s. a1 = r \<and> a2 = s \<and> \<not> C r s) \<or> (\<exists>r s. a1 = r \<and> a2 = s \<and> C r s \<and> (\<forall>(r', s')\<in>fst (B r s). whileLoop_terminates C B r' s')))"
zero_less_iff_neq_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and n :: "'a" assumes "class.canonically_ordered_monoid_add plus zero less_eq less" shows "less zero n = (n \<noteq> zero)"
Image_empty_rtrancl_Image_id: fixes R :: "('b \<times> 'b) set"   and v :: "'b" assumes "R `` {v} = {}" shows "R\<^sup>* `` {v} = {v}"
rep_eq: fixes x :: "'e \<Rightarrow> 'c"   and xa :: "('e, 'd) mapping" shows "mapping.rep (map_keys x xa) = (\<lambda>xb. if x -` {xb} \<noteq> {} then mapping.rep xa (THE k. x -` {xb} = {k}) else None)"
remove_child_removes_first_child: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and node_ptr :: "(_) node_ptr"   and children :: "(_) node_ptr list"   and h' :: "(_) heap" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptrs h"   and "h \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r node_ptr # children"   and "h \<turnstile> Shadow_DOM.remove_child ptr node_ptr \<rightarrow>\<^sub>h h'" shows "h' \<turnstile> Shadow_DOM.get_child_nodes ptr \<rightarrow>\<^sub>r children"
xy_together: shows "\<tau> \<Turnstile> H .allInstances()->includes\<^sub>S\<^sub>e\<^sub>t(x) and H .allInstances()->includes\<^sub>S\<^sub>e\<^sub>t(y) or H .allInstances@pre()->includes\<^sub>S\<^sub>e\<^sub>t(x) and H .allInstances@pre()->includes\<^sub>S\<^sub>e\<^sub>t(y)"
vconn_implies_hpar_1: fixes \<mu> :: "'d"   and f :: "'d"   and g :: "'d" assumes "\<guillemotleft>\<mu> : f \<rightarrow>\<^sub>D g\<guillemotright>" shows "src\<^sub>D \<mu> = src\<^sub>D f"
rm_vars_ident: fixes t :: "('b, 'a) Term.term"   and vs :: "'a set"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "fv t \<inter> vs = {}" shows "t \<cdot> rm_vars vs \<theta> = t \<cdot> \<theta>"
simps_8: fixes v :: "poly"   and va :: "nat" shows "isnpolyh (Pw v va) = (\<lambda>k. False)"
coeff_Var: fixes i :: "nat"   and m :: "nat \<Rightarrow>\<^sub>0 nat" shows "MPoly_Type.coeff (Var i) m = ((1::'a) when m = Poly_Mapping.single i 1)"
divide_complex_def: fixes x :: "complex"   and y :: "complex" shows "x / y = x * inverse y"