hcju/isabelleps huggingface dataset:
oghoare_Guard: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and F :: "'f set"   and P :: "('e, 'd, 'f) ann"   and f :: "'f"   and g :: "'e set"   and d :: "('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set" assumes "\<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P (Guard f g d) Q, A" shows "\<exists>P' r. P = AnnRec r P' \<and> \<Gamma>, \<Theta> \<turnstile>\<^bsub>/F\<^esub> P' d Q, A \<and> r \<inter> g \<subseteq> pre P' \<and> (r \<inter> - g \<noteq> {} \<longrightarrow> f \<in> F)"
Left_Coset_memE: fixes a :: "'c"   and composition :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and x :: "'d"   and H :: "'e set" assumes "a \<in> coset_notation.Left_Coset composition x H" obtains h :: "'e" where "h \<in> H" and "a = composition x h"
perron_frobenius_axioms_def: fixes A :: "((real, 'a) Finite_Cartesian_Product.vec, 'a) Finite_Cartesian_Product.vec" shows "perron_frobenius_axioms A \<equiv> fixed_mat.irreducible A"
subprob_space_bind: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> 'd measure"   and N :: "'d measure" assumes "subprob_space M"   and "f \<in> M \<rightarrow>\<^sub>M subprob_algebra N" shows "subprob_space (M \<bind> f)"
plus_vec_def: fixes v\<^sub>1 :: "'b Matrix.vec"   and v\<^sub>2 :: "'b Matrix.vec" shows "v\<^sub>1 + v\<^sub>2 \<equiv> Matrix.vec (dim_vec v\<^sub>2) (\<lambda>i. v\<^sub>1 $v i + v\<^sub>2 $v i)"
0: shows "lookup (dematricize rmodes A ds) (weave rmodes (digit_encode (nths (dims (dematricize rmodes A ds)) rmodes) r) (digit_encode (nths (dims (dematricize rmodes A ds)) (- rmodes)) c)) = A $$ (r, c)"
Arr_comp: fixes f :: "('o, 'a) arr"   and g :: "('o, 'a) arr" assumes "arr f"   and "arr g"   and "Dom g = Cod f" shows "COMP g f \<noteq> Null \<and> Dom (COMP g f) \<in> Obj \<and> Cod (COMP g f) \<in> Obj \<and> Map (COMP g f) \<in> Hom (Dom (COMP g f)) (Cod (COMP g f))"
dropWhile_transfer: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "((A ===> (=)) ===> list_all2 A ===> list_all2 A) dropWhile dropWhile"
dynProcModifyReturn_sound: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set"   and P :: "'e set"   and init :: "'e \<Rightarrow> 'e"   and p :: "'e \<Rightarrow> 'd"   and return' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and c :: "'e \<Rightarrow> 'e \<Rightarrow> ('e, 'd, 'f) com"   and Q :: "'e set"   and A :: "'e set"   and Modif :: "'e \<Rightarrow> 'e set"   and ModifAbr :: "'e \<Rightarrow> 'e set"   and return :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and n :: "nat" assumes "\<And>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P dynCall init p return' c Q,A"   and "\<forall>s\<in>P. \<forall>\<sigma> n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/UNIV\<^esub> {\<sigma>} Call (p s) (Modif \<sigma>),(ModifAbr \<sigma>)"   and "\<forall>s t. t \<in> Modif (init s) \<longrightarrow> return' s t = return s t"   and "\<forall>s t. t \<in> ModifAbr (init s) \<longrightarrow> return' s t = return s t" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P dynCall init p return c Q,A"
hmac_trans_1_4_skr_extr_fake: fixes X :: "msg"   and K :: "msg"   and s' :: "l2_state"   and s :: "l2_state"   and Ra :: "fid_t"   and A :: "agent"   and B :: "agent"   and Rb :: "fid_t"   and Ni :: "msg"   and gnx :: "msg"   and R :: "fid_t"   and KK :: "msg"   and M :: "msg"   and MM :: "chan" assumes "hmac X K \<in> parts (extr (bad s') (ik s') (chan s'))"   and "K \<notin> synth (analz (extr (bad s) (ik s) (chan s)))"   and "s \<in> l2_inv2"   and "(s, s') \<in> l2_step1 Ra A B \<union> l2_step4 Rb A B Ni gnx \<union> l2_skr R KK \<union> l2_dy_fake_msg M \<union> l2_dy_fake_chan MM" shows "hmac X K \<in> parts (extr (bad s) (ik s) (chan s))"
assms: shows "iterate t step ([(f, Partial_Recursive.list_decode x, [])], None) = ([], Some v)"
be: shows "(b, e) \<in> (A \<union> B) ^^ n"
countable_support: shows "countable {x. Sigma_Algebra.measure M {x} \<noteq> 0}"
sets_Collect_conj: fixes M :: "'c measure"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool" assumes "{x \<in> space M. P x} \<in> sets M"   and "{x \<in> space M. Q x} \<in> sets M" shows "{x \<in> space M. Q x \<and> P x} \<in> sets M"
return_only_one_intra_edge: fixes a :: "'edge"   and Q' :: "('var \<Rightarrow> 'val option) \<times> 'ret \<Rightarrow> bool"   and p :: "'pname"   and f' :: "('var \<Rightarrow> 'val option) \<Rightarrow> ('var \<Rightarrow> 'val option) \<Rightarrow> 'var \<Rightarrow> 'val option" assumes "valid_edge a"   and "kind a = Q'\<hookleftarrow>\<^bsub>p\<^esub>f'" shows "\<exists>!a'. valid_edge a' \<and> targetnode a' = targetnode a \<and> intra_kind (kind a')"
f_eq_one: fixes j :: "nat"   and y :: "'a" assumes "j \<in> N"   and "y \<in> U j" shows "f j y = 1"
get_child_nodes_is_strongly_dom_component_safe: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and c :: "(_) object_ptr list"   and ptr' :: "(_) object_ptr"   and children :: "(_) node_ptr list"   and child :: "(_) node_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_dom_component ptr \<rightarrow>\<^sub>r c"   and "h \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"   and "child \<in> set children" shows "(cast\<^sub>n\<^sub>o\<^sub>d\<^sub>e\<^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 child \<in> set c) = (ptr' \<in> set c)"
abs_eq: shows "top = Abs_lifted_distrib_lattice_top (\<lambda>f. top)"
wave_eq_web: fixes f :: "'v \<times> 'v \<Rightarrow> ennreal" shows "wave \<Gamma> f = wave \<Gamma>' f"
assms_1: shows "s \<in> carrier R\<^bsup>\<omega>\<^esup>"
preserves_ide: fixes a :: "'a" assumes "F.A.ide a" shows "F.B.ide (F a)"
drop_drop: fixes n :: "nat"   and m :: "nat"   and xs :: "'d list" shows "drop n (drop m xs) = drop (n + m) xs"
sep_conj_commuteI: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and sep_disj :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and P :: "'c \<Rightarrow> bool"   and Q :: "'c \<Rightarrow> bool"   and h :: "'c" assumes "class.sep_algebra plus zero sep_disj"   and "sep_algebra.sep_conj plus sep_disj P Q h" shows "sep_algebra.sep_conj plus sep_disj Q P h"
lim_ln_over_power: fixes s :: "real" assumes "0 < s" shows "(\<lambda>n. ln (real n) / real n powr s) \<longlonglongrightarrow> 0"
commutes_with_composition: fixes \<eta> :: "'c \<Rightarrow> 'd"   and M :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and M' :: "'d set"   and composition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and unit' :: "'d"   and x :: "'c"   and y :: "'c" assumes "monoid_homomorphism \<eta> M composition unit M' composition' unit'"   and "x \<in> M"   and "y \<in> M" shows "\<eta> (composition x y) = composition' (\<eta> x) (\<eta> y)"
measurable_PiM_component_rev: fixes i :: "'b"   and I :: "'b set"   and f :: "'c \<Rightarrow> 'd"   and M :: "'b \<Rightarrow> 'c measure"   and N :: "'d measure" assumes "i \<in> I"   and "f \<in> M i \<rightarrow>\<^sub>M N" shows "(\<lambda>x. f (x i)) \<in> Pi\<^sub>M I M \<rightarrow>\<^sub>M N"
gcd_dvdI2: fixes b :: "'a"   and c :: "'a"   and a :: "'a" assumes "b dvd c" shows "gcd a b dvd c"
cod_trg_eq_dom_trg: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<mu> :: "'b arrow_of_spans_data" assumes "arrow_of_spans C \<mu>" shows "partial_magma.cod C (Leg1 (Cod \<mu>)) = partial_magma.cod C (Leg1 (Dom \<mu>))"
Nset_injTr1: fixes n :: "nat"   and f :: "nat \<Rightarrow> nat" assumes "\<forall>l\<le>Suc n. f l \<le> Suc n"   and "inj_on f {i. i \<le> Suc n}"   and "f (Suc n) = Suc n" shows "inj_on f {i. i \<le> n}"
ii: shows "\<forall>s. pre C (\<lambda>s. I s + 1) s \<le> I s + \<up> (bval b s) \<and> Q s \<le> I s + \<up> (\<not> bval b s)"
segmentD_orbit: fixes x :: "'b"   and f :: "'b \<Rightarrow> 'b"   and y :: "'b"   and z :: "'b" assumes "x \<in> segment f y z" shows "x \<in> orbit f y"
InstrElimNext: fixes M :: "Var list \<times> (int \<times> Instr) list \<times> int"   and l :: "int"   and s :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and n :: "nat"   and ll :: "int"   and t :: "Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list"   and I :: "Instr"   and c :: "Val"   and x :: "Var"   and f :: "Val \<Rightarrow> Val \<Rightarrow> Val"   and g :: "Val \<Rightarrow> Val"   and d :: "Class"   and F :: "Field" assumes "(M, l, s, n, ll, t) \<in> Step"   and "get_ins M l = Some I"   and "I = const c \<or> I = dup \<or> I = pop \<or> I = swap \<or> I = load x \<or> I = store x \<or> I = binop f \<or> I = unop g \<or> I = new d \<or> I = getfield d F \<or> I = putfield d F \<or> I = checkcast d" shows "ll = l + 1"
Radj_eq_iff: fixes vid1 :: "'a"   and vid2 :: "'a"   and vid3 :: "'a"   and fid1 :: "'b"   and fid2 :: "'b"   and fid3 :: "'b"   and pid1 :: "'c"   and pid2 :: "'c"   and pid3 :: "'c"   and pid4 :: "'c"   and a :: "(real, 'a) vec \<times> (real, 'a) vec"   and b :: "(real, 'a) vec \<times> (real, 'a) vec"   and x :: "'a"   and y :: "'a" assumes "ids vid1 vid2 vid3 fid1 fid2 fid3 pid1 pid2 pid3 pid4" shows "(a = b) = (ids.Radj x y a = ids.Radj x y b)"
good_item_simps_70: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and xsa :: "'varSort"   and xa :: "'var" assumes "good X"   and "good Y"   and "fresh xsa xa Y" shows "Y #[X / xa]_xsa = Y"
igWlsDisj_def: fixes MOD :: "('a, 'b, 'c, 'd, 'e, 'f, 'g, 'h, 'i) model_scheme" shows "igWlsDisj MOD \<equiv> \<forall>s s' X. igWls MOD s X \<and> igWls MOD s' X \<longrightarrow> s = s'"
iso_class_def: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category C" shows "category.iso_class C f \<equiv> {f'. category.isomorphic C f f'}"
homotopy_equivalent_space_def: fixes X :: "'c topology"   and Y :: "'d topology" shows "X homotopy_equivalent_space Y \<equiv> \<exists>f g. continuous_map X Y f \<and> continuous_map Y X g \<and> homotopic_with (\<lambda>x. True) X X (g \<circ> f) id \<and> homotopic_with (\<lambda>x. True) Y Y (f \<circ> g) id"
sel_2: fixes x21 :: "'b"   and x22 :: "'b llist" shows "ltl (LCons x21 x22) = x22"
pickT_pred_pickT_unique: fixes cl :: "('test, 'atom, 'choice) cmd list"   and P :: "nat \<Rightarrow> nat set set"   and II :: "nat set"   and n :: "nat" assumes "\<And>n. n < length cl \<Longrightarrow> part {..<brn (cl ! n)} (P n) \<and> {} \<notin> P n"   and "pickT_pred cl P II n" shows "n = pickT cl P II"
init_s: fixes t :: "real" assumes "t \<le> 0" shows "ego.s t = s\<^sub>e"
map_filter_def: fixes f :: "'c \<Rightarrow> 'b option"   and xs :: "'c list" shows "List.map_filter f xs = map (the \<circ> f) (filter (\<lambda>x. f x \<noteq> None) xs)"
cod_simp: fixes f :: "'d" assumes "D'.arr f" shows "D'.cod f = D.cod f"
Int_Union_pairwise_disjoint: fixes \<A> :: "'b set set"   and \<B> :: "'b set set" assumes "disjoint (\<A> \<union> \<B>)" shows "\<Union> \<A> \<inter> \<Union> \<B> = \<Union> (\<A> \<inter> \<B>)"
pfinite_gpv_sub_gpvs: fixes \<I> :: "('h, 'i) \<I>"   and gpv :: "('j, 'h, 'i) gpv"   and gpv' :: "('j, 'h, 'i) gpv" assumes "pfinite_gpv \<I> gpv"   and "gpv' \<in> sub_gpvs \<I> gpv"   and "\<I> \<turnstile>g gpv \<surd>" shows "pfinite_gpv \<I> gpv'"
less_term_alt: shows "(<) = in_rel (leq_term - Id)"
map_1: fixes f1 :: "'d \<Rightarrow> 'b"   and f2 :: "'e \<Rightarrow> 'c"   and x1 :: "'d" shows "map_result f1 f2 (Rval x1) = Rval (f1 x1)"
not_choice_simps_6: fixes l :: "'d"   and fips :: "'a \<Rightarrow> nat set"   and fmsg :: "'a \<Rightarrow> 'b"   and p :: "('a, 'b, 'c, 'd) seqp" shows "not_choice ({l}groupcast(fips, fmsg) . p)"
MGFn_free_wt: fixes G :: "prog"   and t :: "Term.term"   and A :: "state triple set"   and n :: "nat" assumes "(\<exists>T L C. \<lparr>prg = G, cls = C, lcl = L\<rparr>\<turnstile>t\<Colon>T) \<longrightarrow> G,A\<turnstile>{=:n} t\<succ> {G\<rightarrow>}" shows "G,A\<turnstile>{=:n} t\<succ> {G\<rightarrow>}"
w_in_hom_1: shows "\<guillemotleft>\<mu>.chine : src (tab\<^sub>0 (local.dom \<mu>)) \<rightarrow> src (tab\<^sub>0 (cod \<mu>))\<guillemotright>"
subst_dom_elim: fixes s :: "'a \<Rightarrow> ('b, 'a) Term.term"   and t :: "('b, 'a) Term.term" assumes "subst_domain s \<inter> range_vars s = {}" shows "fv (t \<cdot> s) \<inter> subst_domain s = {}"
on_tripleD_2: fixes l1 :: "'e"   and v1 :: "'f"   and v2 :: "'f"   and l2 :: "'e"   and v3 :: "'g"   and v4 :: "'g"   and R :: "('f \<times> 'g) set" assumes "((l1, v1, v2), l2, v3, v4) \<in> on_triple R" shows "(v1, v3) \<in> R"
iterates_Suc: fixes body :: "bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and G :: "'a \<Rightarrow> bool"   and i :: "nat" shows "iterates body G (Suc i) = wp (body ;; Embed (iterates body G i) \<^bsub>\<guillemotleft> G \<guillemotright>\<^esub>\<oplus> Skip)"
elims: fixes x :: "frame"   and y :: "char list list" assumes "frame_clinit_classes x = y" obtains stk :: "val list" and loc :: "val list" and C :: "char list" and M :: "char list" and pc :: "nat" and ics :: "init_call_status" where "x = (stk, loc, C, M, pc, ics)" and "y = (if M = clinit then [C] else []) @ ics_classes ics"
reachable_def: fixes s\<^sub>0 :: "'b"   and lstep :: "'d \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool" shows "system.reachable s\<^sub>0 lstep \<equiv> (\<lambda>s s'. \<exists>l. lstep l s s')\<^sup>*\<^sup>* s\<^sub>0"
cases: fixes x :: "nat \<times> nat" obtains i :: "nat" and j :: "nat" where "x = (i, j)"
step_awalk_to_apath: fixes u :: "'a"   and p :: "'b list"   and v :: "'a"   and q :: "'b list"   and r :: "'b list"   and s :: "'b list" assumes "awalk u p v"   and "awalk_cyc_decomp p = (q, r, s)"   and "\<not> distinct (awalk_verts u p)" shows "awalk_to_apath p = awalk_to_apath (q @ s)"
2: shows "exec_plan s (as_proj as vs) \<in> valid_states PROB"
push_bit_eq_0_iff: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and division_segment :: "'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" assumes "class.unique_euclidean_semiring_with_bit_shifts divide plus minus zero times one modulo euclidean_size division_segment bit push_bit drop_bit take_bit" shows "(push_bit n a = zero) = (a = zero)"
assms: shows "Cond enc = (\<exists>Rel. (\<forall>S. (SourceTerm S, TargetTerm (\<lbrakk>S\<rbrakk>)) \<in> Rel) \<and> Pred Rel)"
that: fixes \<D> :: "'a set set" assumes "countable \<D>"   and "\<Union> \<D> \<subseteq> cbox a b"   and "\<And>K. K \<in> \<D> \<Longrightarrow> interior K \<noteq> {} \<and> (\<exists>c d. K = cbox c d)"   and "pairwise (\<lambda>A B. interior A \<inter> interior B = {}) \<D>"   and "\<And>K. K \<in> \<D> \<Longrightarrow> \<exists>x\<in>S \<inter> K. K \<subseteq> g x"   and "\<And>u v. cbox u v \<in> \<D> \<Longrightarrow> \<exists>n. \<forall>i\<in>Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2 ^ n"   and "S \<subseteq> \<Union> \<D>" shows "thesis"
p1_def: fixes x :: "real"   and y :: "real" shows "p1 x y = - (21 / 34) - 69 * x / 38 + 19 * x\<^sup>2 / 15 - 9 * x ^ 3 / 28 - 6 * x ^ 4 / 43 + 14 * y / 29 + 31 * x * y / 21 + 182 * x\<^sup>2 * y / 47 - 35 * x ^ 3 * y / 16 - 3 * y\<^sup>2 / 17 - 2 * x * y\<^sup>2 / 9 - 31 * x\<^sup>2 * y\<^sup>2 / 20 + y ^ 3 / 102 + x * y ^ 3 / 59"
differentiable_eqI: fixes g :: "'a \<Rightarrow> 'b"   and x :: "'a"   and X :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "g differentiable at x"   and "\<And>x. x \<in> X \<Longrightarrow> f x = g x"   and "x \<in> X"   and "open X" shows "f differentiable at x"
sumZero: fixes P :: "pi" shows "P \<oplus> \<zero> \<sim>\<^sub>e P"
frontier_minimal_separating_closed_pointwise: fixes S :: "'a set"   and a :: "'a"   and b :: "'a" assumes "closed S"   and "a \<notin> S"   and "\<not> connected_component (- S) a b"   and "\<And>T. \<lbrakk>closed T; T \<subset> S\<rbrakk> \<Longrightarrow> connected_component (- T) a b" shows "frontier (connected_component_set (- S) a) = S"
dg_def: shows "dg_Rel \<alpha> = [dg_Rel \<alpha>\<lparr>Obj\<rparr>, dg_Rel \<alpha>\<lparr>Arr\<rparr>, dg_Rel \<alpha>\<lparr>Dom\<rparr>, dg_Rel \<alpha>\<lparr>Cod\<rparr>]\<^sub>\<circ>"
Ln_of_nat: fixes n :: "nat" assumes "0 < n" shows "Ln (of_nat n) = complex_of_real (ln (real n))"
w_def_1: shows "s \<unrhd>\<^sub>e\<^sub>m\<^sub>b w"
basis_reduction_add_rows_cost_def: fixes upw :: "bool"   and i :: "nat"   and state :: "(int vec list \<times> int vec list) \<times> int iarray iarray \<times> int iarray" shows "basis_reduction_add_rows_cost upw i state = (if upw then basis_reduction_add_rows_loop_cost state i i (small_fs_state state) else (state, 0))"
maddux_141: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.relation_algebra minus uminus inf less_eq less sup bot top composition converse unit" shows "(inf (composition x y) z = bot) = (inf (composition (converse x) z) y = bot)"
2: shows "\<forall>y. (y, s) \<in> R \<longrightarrow> (\<exists>X. wls y X)"
uminus_float: fixes x :: "real" assumes "x \<in> float" shows "- x \<in> float"
enat_1_iff_2: fixes x :: "nat" shows "(1 = enat x) = (x = 1)"
continuous_map_axioms: shows "continuous_map S is_open S' is_open' f"
IH: assumes "list_constr xa (length qs)" shows "(prod_list (map ((!) sign) xa) = 1) = (0 < poly (prod_list (retrieve_polys qs xa)) x)"
Constrains_weaken_R: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set"   and B' :: "'b set" assumes "F \<in> A Co A'"   and "A' \<subseteq> B'" shows "F \<in> A Co B'"
CI_def: fixes J :: "nat set" shows "CI J = distr (C 0 (up_to J) (\<lambda>x. undefined)) (Pi\<^sub>M J M) (\<lambda>f. restrict f J)"
sel_2: fixes x :: "complex"   and y :: "complex" shows "Im (x + y) = Im x + Im y"
bipartite_web_of_sel_7: fixes \<Gamma> :: "('a, 'b) web_scheme" shows "B (bipartite_web_of \<Gamma>) = Inr ` (- A \<Gamma>)"
that: fixes s :: "nat \<Rightarrow> 'a" assumes "(f \<circ> s) \<longlonglongrightarrow> l"   and "\<And>i. s i \<in> T"   and "\<And>C. \<lbrakk>compact C; C \<subseteq> T\<rbrakk> \<Longrightarrow> \<forall>\<^sub>F i in sequentially. s i \<notin> C" shows "thesis"
receive_right: fixes \<sigma> :: "nat \<Rightarrow> 'a"   and m :: "'c"   and \<sigma>' :: "nat \<Rightarrow> 'a" assumes "P (\<sigma>, receive m, \<sigma>')" shows "P (\<sigma>, \<tau>, \<sigma>')"
sub: shows "{x. \<not> (f x = x \<and> g x = x)} \<subseteq> h ` S"
homotopic_with_compose_continuous_map_right: fixes p :: "('b \<Rightarrow> 'c) \<Rightarrow> bool"   and X2 :: "'b topology"   and X3 :: "'c topology"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and X1 :: "'d topology"   and h :: "'d \<Rightarrow> 'b"   and q :: "('d \<Rightarrow> 'c) \<Rightarrow> bool" assumes "homotopic_with p X2 X3 f g"   and "continuous_map X1 X2 h"   and "\<And>j. p j \<Longrightarrow> q (j \<circ> h)" shows "homotopic_with q X1 X3 (f \<circ> h) (g \<circ> h)"
\<iota>_in_hom: shows "\<guillemotleft>\<i>[src \<mu>] : src \<mu> \<star> src \<mu> \<Rightarrow> src \<mu>\<guillemotright>"
lateEarlySim: fixes P :: "pi"   and Rel :: "(pi \<times> pi) set"   and Q :: "pi" assumes "P \<leadsto>\<^sub>l[Rel] Q" shows "P \<leadsto>\<^sub>e[Rel] Q"
not_eq_order_implies_strict: fixes a :: "nat list"   and b :: "nat list" assumes "a \<noteq> b"   and "b \<le>\<^sub>v a" shows "b <\<^sub>v a"
DoS_eq_\<F>\<^sub>SC: shows "DoS.map = \<F>\<^sub>SC.map"
kuhn_combinatorial: fixes p :: "nat"   and n :: "nat"   and lab :: "(nat \<Rightarrow> nat) \<Rightarrow> nat \<Rightarrow> nat" assumes "0 < p"   and "\<forall>x j. (\<forall>j. x j \<le> p) \<and> j < n \<and> x j = 0 \<longrightarrow> lab x j = 0"   and "\<forall>x j. (\<forall>j. x j \<le> p) \<and> j < n \<and> x j = p \<longrightarrow> lab x j = 1" shows "odd (card {s. ksimplex p n s \<and> (reduced n \<circ> lab) ` s = {..n}})"
dg_in_Vset: fixes \<alpha> :: "V"   and \<CC> :: "V"   and \<beta> :: "V" assumes "digraph \<alpha> \<CC>"   and "\<Z> \<beta>"   and "\<alpha> \<in>\<^sub>\<circ> \<beta>" shows "\<CC> \<in>\<^sub>\<circ> Vset \<beta>"
img_mapsto: fixes a :: "'s" assumes "ide a" shows "img \<in> hom unity a \<rightarrow> Univ"
inf_sup_distrib2: fixes y :: "'a"   and z :: "'a"   and x :: "'a" shows "y \<squnion> z \<sqinter> x = (y \<sqinter> x) \<squnion> (z \<sqinter> x)"
per_col: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and B :: "'a"   and C :: "'a"   and A :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "B \<noteq> C"   and "Tarski_neutral_dimensionless.Per Bet Cong A B C"   and "Tarski_neutral_dimensionless.Col Bet B C D" shows "Tarski_neutral_dimensionless.Per Bet Cong A B D"
y_2: shows "y \<in> cl (insert x X - {y})"
DIAndSound_lemma: shows "sound (proof_result (proof_take 61 DIAndProof))"
SUP_upper: fixes i :: "'b"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'a" assumes "i \<in> A" shows "\<Sqinter> (f ` A) \<le> f i"
gen_dfs_simps_2: fixes succs :: "'b \<Rightarrow> 'b list"   and ins :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and memb :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and S :: "'a"   and x :: "'b"   and xs :: "'b list" shows "gen_dfs succs ins memb S (x # xs) = (if memb x S then gen_dfs succs ins memb S xs else gen_dfs succs ins memb (ins x S) (succs x @ xs))"
set_lub_spmf: fixes Y :: "'a spmf set" assumes "Complete_Partial_Order.chain (ord_spmf (=)) Y" shows "set_spmf (lub_spmf Y) = \<Union> (set_spmf ` Y)"
mult_zero_right: fixes a :: "'a" shows "a * (0::'a) = (0::'a)"
odd_mirror_traverse_tree: fixes path :: "dir list"   and t :: "'a tree" shows "root (traverse_tree path (odd_mirror t)) = root (traverse_tree (odd_mirror_path path) t)"
simps_1: fixes Q :: "(char list \<Rightarrow> nat) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" shows "Nielson_VCGi.pre SKIP Q = Q"
add_mono: fixes a :: "complex"   and b :: "complex"   and c :: "complex"   and d :: "complex" assumes "a \<le>\<^sub>\<complex> b"   and "c \<le>\<^sub>\<complex> d" shows "a + c \<le>\<^sub>\<complex> b + d"
f': fixes tx :: "real \<times> 'a" assumes "tx \<in> T \<times> X'" shows "((\<lambda>(t, x). f t x) has_derivative f' tx) (at tx within T \<times> X')"
loop: shows "pathfinish \<gamma>2 = pathstart \<gamma>2"
retraction_is_epi: fixes g :: "'a \<times> 'a" assumes "VV.retraction g" shows "VV.epi g"
tdev_slist_sum: fixes xs :: "(nat, 'd) slist" shows "tdev_slist xs = sum (abs \<circ> snd) (set_slist xs)"
has_integral_restrict_open_subinterval: fixes f :: "'a \<Rightarrow> 'b"   and i :: "'b"   and c :: "'a"   and d :: "'a"   and a :: "'a"   and b :: "'a" assumes "(f has_integral i) (cbox c d)"   and "cbox c d \<subseteq> cbox a b" shows "((\<lambda>x. if x \<in> box c d then f x else (0::'b)) has_integral i) (cbox a b)"
split_sels_1: fixes P :: "'f \<Rightarrow> bool"   and f1 :: "'f"   and f2 :: "nat \<Rightarrow> 'f"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = ((nat = 0 \<longrightarrow> P f1) \<and> (nat = Suc (nat.pred nat) \<longrightarrow> P (f2 (nat.pred nat))))"