hcju/isabelleps huggingface dataset:
monad_altc_def: fixes return :: "'a \<Rightarrow> 'b"   and bind :: "'b \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b"   and altc :: "'c cset \<Rightarrow> ('c \<Rightarrow> 'b) \<Rightarrow> 'b" shows "monad_altc return bind altc \<equiv> monad return bind \<and> monad_altc_axioms bind altc"
tauLaw1StatImpRight: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Rel :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and Rel' :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<Psi> \<rhd> P \<lessapprox><Rel> Q"   and "\<And>\<Psi> P Q R. \<lbrakk>(\<Psi>, P, Q) \<in> Rel; \<Psi> \<rhd> Q \<sim> R\<rbrakk> \<Longrightarrow> (\<Psi>, P, R) \<in> Rel'" shows "\<Psi> \<rhd> P \<lessapprox><Rel'> \<tau>.Q"
Csum_def: fixes r :: "('a \<times> 'a) set"   and rs :: "'a \<Rightarrow> ('b \<times> 'b) set" shows "BNF_Cardinal_Arithmetic.Csum r rs \<equiv> |SIGMA i:Field r. Field (rs i)|"
toplevel_summands_ACI_norm_deriv: fixes as :: "'b"   and r :: "'b rexp" shows "(\<Union>a\<in>toplevel_summands r. toplevel_summands \<guillemotleft>deriv as \<guillemotleft>a\<guillemotright>\<guillemotright>) = toplevel_summands \<guillemotleft>deriv as \<guillemotleft>r\<guillemotright>\<guillemotright>"
compact_eq_Heine_Borel: fixes "open" :: "'c set \<Rightarrow> bool"   and S :: "'c set" assumes "class.topological_space open" shows "topological_space.compact open S = (\<forall>C. (\<forall>c\<in>C. open c) \<and> S \<subseteq> \<Union> C \<longrightarrow> (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union> D))"
aligned_add_aligned: fixes x :: "'b word"   and n :: "nat"   and y :: "'b word"   and m :: "nat" assumes "is_aligned x n"   and "is_aligned y m"   and "m \<le> n" shows "is_aligned (x + y) m"
induct: fixes P :: "'a list \<Rightarrow> bool"   and a0 :: "'a list" assumes "\<And>x y xs. P (y # xs) \<Longrightarrow> P (x # y # xs)"   and "P []"   and "\<And>v. P [v]" shows "P a0"
extends_subst_trans: fixes \<sigma> :: "'a \<Rightarrow> 'b option"   and \<tau> :: "'a \<Rightarrow> 'b option"   and \<rho> :: "'a \<Rightarrow> 'b option" assumes "extends_subst \<sigma> \<tau>"   and "extends_subst \<tau> \<rho>" shows "extends_subst \<sigma> \<rho>"
hide_un_D1: fixes P :: "'b process"   and A :: "'b set"   and B :: "'b set" shows "D (P \ (A \<union> B)) \<subseteq> D ((P \ A) \ B)"
that: assumes "prec_spec h A C"   and "prec_spec h B D" shows "thesis"
a_2: shows "f \<in> \<Theta>(\<lambda>x. real x powr p * (1 + integral {a..real x} (\<lambda>u. u powr p * ln u powr - 1 / u powr (p + 1))))"
wf_rcv_append''': fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and t :: "('d, 'c) Term.term" assumes "wf\<^sub>s\<^sub>t V S"   and "fv t \<subseteq> wfrestrictedvars\<^sub>s\<^sub>t S \<union> V" shows "wf\<^sub>s\<^sub>t V (S @ [receive\<langle>t\<rangle>\<^sub>s\<^sub>t])"
gamma_i_component_smooth: shows "\<forall>x\<in>{0..1}. (\<lambda>x. \<gamma> x \<bullet> i) differentiable at x"
mndet_unit: fixes a :: "'b"   and P :: "'b \<Rightarrow> 'b process" shows "mndet {a} P = (a \<rightarrow> P a)"
Un_insert_right: fixes A :: "'a set"   and a :: "'a"   and B :: "'a set" shows "A \<union> (a \<triangleright> B) = a \<triangleright> A \<union> B"
span_INT: fixes f :: "'c \<Rightarrow> 'b set"   and A :: "'c set" shows "span (\<Inter>a\<in>A. span (f a)) = (\<Inter>a\<in>A. span (f a))"
AllowedActs_total_eq: fixes init :: "'c set"   and acts :: "('c \<times> 'c) set set"   and allowed :: "('c \<times> 'c) set set" shows "AllowedActs (mk_total_program (init, acts, allowed)) = insert Id allowed"
assms_2: shows "u \<cdot> p = p \<cdot> u"
vimage2p_def: fixes f :: "'h \<Rightarrow> 'k"   and g :: "'i \<Rightarrow> 'l"   and R :: "'k \<Rightarrow> 'l \<Rightarrow> 'j" shows "BNF_Def.vimage2p f g R = (\<lambda>x y. R (f x) (g y))"
isomorphic_homology_imp_path_connectedness: fixes X :: "'a topology"   and Y :: "'b topology" assumes "homology_group 0 X \<cong> homology_group 0 Y" shows "path_connected_space X = path_connected_space Y"
image_chamber_system: shows "ChamberComplex.chamber_system (f \<turnstile> X) = f \<turnstile> folding.\<C>"
vec_set_def: fixes v :: "'a vec" shows "set\<^sub>v v = ($) v ` {..<dim_vec v}"
supp_def: shows "supp = set_pmf p \<union> set_pmf q'"
cong_to_1_nat: fixes a :: "nat"   and n :: "nat" assumes "[a = 1] (mod n)" shows "n dvd a - 1"
rel_distinct_6: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x1 :: "'b" shows "\<not> rel_xstate R1 R2 Stuck (Normal x1)"
\<omega>: shows "LeftDerives1 (\<alpha> @ \<delta>) i r \<omega> \<and> ix < length (snd r) \<and> snd r ! ix = (\<beta> @ \<delta>') ! j \<and> LeftDerivationFix \<omega> (i + ix) D j (\<beta> @ \<delta>')"
no_head_hd: fixes u :: "'a list"   and b :: "'a list" assumes "u \<in> no_head \<C> b"   and "u \<noteq> \<epsilon>" shows "hd (Dec \<C> u) \<noteq> b"
ivl_disj_un_3: fixes l :: "'a"   and u :: "'a" assumes "l < u" shows "{l} \<union> {l<..<u} = {l..<u}"
bisimilar_reflp: shows "reflp (\<sim>\<cdot>)"
hyps_3: assumes "A \<subseteq> carrier V"   and "a \<in> T ` A \<rightarrow> carrier K" shows "W.module.lincomb a (T ` A) = T (V.module.lincomb (a \<circ> T) A)"
res_li3: fixes x :: "'a"   and y :: "'a" shows "((x \<leftarrow> y) \<cdot> y \<leftarrow> y) \<cdot> y = (x \<leftarrow> y) \<cdot> y"
main_step_aux2_R: fixes qs :: "real poly list" assumes "check_all_const_deg qs = False" shows "set (find_consistent_signs_R qs) = consistent_sign_vectors_R qs UNIV"
E_def: shows "E \<equiv> {(u, v). c (u, v) \<noteq> (0::'a)}"
rel_funD: fixes A :: "'b \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'f"   and x :: "'b"   and y :: "'d" assumes "rel_fun A B f g"   and "A x y" shows "B (f x) (g y)"
reachable_closed_dfs: fixes succs :: "'c \<Rightarrow> 'c list"   and is_node :: "'c \<Rightarrow> bool"   and invariant :: "'d \<Rightarrow> bool"   and ins :: "'c \<Rightarrow> 'd \<Rightarrow> 'd"   and memb :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and empt :: "'d"   and xs :: "'c list" assumes "DFS succs is_node invariant ins memb empt"   and "list_all is_node xs" shows "DFS.reachable succs (set xs) \<subseteq> DFS.set_of is_node memb (gen_dfs succs ins memb empt xs)"
base: fixes data :: "nat \<times> nat \<times> 'd"   and D :: "((('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) \<times> ('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) set"   and bs :: "(('t \<Rightarrow>\<^sub>0 'b) \<times> nat \<times> 'c) list" shows "P data D bs [] (gs @ bs, D)"
coeff_mult: fixes p :: "'b poly"   and q :: "'b poly"   and n :: "nat" shows "coeff (p * q) n = (\<Sum>i\<le>n. coeff p i * coeff q (n - i))"
algebraic_int_imp_algebraic: fixes x :: "'b" assumes "algebraic_int x" shows "algebraic x"
Gromov_converging_at_boundaryI: fixes a :: "'c"   and u :: "nat \<Rightarrow> 'c" assumes "\<And>M. \<exists>N. \<forall>n\<ge>N. \<forall>m\<ge>N. M \<le> Gromov_product_at a (u m) (u n)" shows "Gromov_converging_at_boundary u"
IH_2: fixes ds :: "bool list" assumes "anybranch T\<^sub>2 (\<lambda>p. red (ds @ p))" shows "anyinternal (cutoff red ds T\<^sub>2) (\<lambda>p. \<not> red (ds @ p))"
MkArr_Map: fixes f :: "(bool, unit) arr" assumes "arr f" shows "MkArr (Dom f) (Cod f) (Map f) = f"
conf_clinit_def: fixes P :: "jvm_prog"   and sh :: "sheap"   and frs :: "frame list" shows "conf_clinit P sh frs \<equiv> distinct_clinit frs \<and> (\<forall>C\<in>set (clinit_classes frs). is_class P C \<and> (\<exists>sfs. sh C = \<lfloor>(sfs, Processing)\<rfloor>))"
distr_PiM_reindex: fixes K :: "'d set"   and M :: "'d \<Rightarrow> 'e measure"   and f :: "'f \<Rightarrow> 'd"   and I :: "'f set" assumes "\<And>i. i \<in> K \<Longrightarrow> prob_space (M i)"   and "inj_on f I"   and "f \<in> I \<rightarrow> K" shows "distr (Pi\<^sub>M K M) (Pi\<^sub>M I (\<lambda>i. M (f i))) (\<lambda>\<omega>. \<lambda>n\<in>I. \<omega> (f n)) = Pi\<^sub>M I (\<lambda>i. M (f i))"
share_append_Ghost\<^sub>s\<^sub>b: fixes sb :: "'b memref list"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and \<S> :: "nat \<Rightarrow> bool option" assumes "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b sb = {}" shows "share (sb @ [Ghost\<^sub>s\<^sub>b A L R W]) \<S> = share sb \<S> \<oplus>\<^bsub>W\<^esub> R \<ominus>\<^bsub>A\<^esub> L"
abs_eq: fixes xb :: "'a"   and xa :: "'b"   and x :: "'a \<Rightarrow> 'b option" shows "Mapping.update xb xa (mapping.Mapping x) = mapping.Mapping (x(xb \<mapsto> xa))"
two: fixes xs :: "('b com \<times> 'b set \<times> ('b \<times> 'b) set \<times> ('b \<times> 'b) set \<times> 'b set) list"   and rely :: "('b \<times> 'b) set"   and pre :: "'b set"   and clist :: "('b com option \<times> 'b) list list"   and x :: "('b com option list \<times> 'b) list"   and s :: "'b" assumes "\<forall>i<length xs. rely \<union> (\<Union>j\<in>{j. j < length xs \<and> j \<noteq> i}. Guar (xs ! j)) \<subseteq> Rely (xs ! i)"   and "pre \<subseteq> (\<Inter>i\<in>{i. i < length xs}. Pre (xs ! i))"   and "\<forall>i<length xs. \<Turnstile> Com (xs ! i) sat [Pre (xs ! i), Rely (xs ! i), Guar (xs ! i), Post (xs ! i)]"   and "length xs = length clist"   and "x \<in> par_cp (ParallelCom xs) s"   and "x \<in> par_assum (pre, rely)"   and "\<forall>i<length clist. clist ! i \<in> cp (Some (Com (xs ! i))) s"   and "x \<propto> clist" shows "\<forall>j i. i < length clist \<and> Suc j < length x \<longrightarrow> clist ! i ! j -c\<rightarrow> clist ! i ! Suc j \<longrightarrow> (snd (clist ! i ! j), snd (clist ! i ! Suc j)) \<in> Guar (xs ! i)"
int_F: fixes f :: "'a \<Rightarrow> 'b" assumes "f \<in> F" shows "f integrable_on cbox a b"
positive_trace: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "positive A" shows "0 \<le> trace A"
simps_1: fixes T :: "'b trm"   and u :: "'b trm"   and S :: "'b trm" shows "replace_subterm T [] u S = (S = u)"
diff: fixes t :: "'b tree"   and n :: "nat" assumes "braun t"   and "size t \<in> {n, n + 1}" shows "diff t n = size t - n"
trans_O_subset: fixes r :: "('b \<times> 'b) set" assumes "trans r" shows "r O r \<subseteq> r"
allAllocationsIntersectionSetEquals: fixes a :: "('c \<times> 'd set) set"   and G :: "'d set"   and N :: "'c set" shows "(a \<in> allAllocations N G) = (a \<in> allocationsUniverse \<and> Domain a \<subseteq> N \<and> \<Union> (Range a) = G)"
wf_until_aux_def: fixes \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and R :: "event_data list set"   and args :: "Monitor.args"   and \<phi> :: "Formula.formula"   and \<psi> :: "Formula.formula"   and aux :: "'muaux"   and ne :: "nat" shows "wf_until_aux \<sigma> V R args \<phi> \<psi> aux ne = (fv \<phi> \<subseteq> fv \<psi> \<and> (\<exists>cur auxlist. valid_muaux args cur aux auxlist \<and> cur = (if ne + length auxlist = 0 then 0 else \<tau> \<sigma> (ne + length auxlist - 1)) \<and> wf_until_auxlist \<sigma> V (args_n args) R (args_pos args) \<phi> (args_ivl args) \<psi> auxlist ne))"
SDG_param_out_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"   and ins :: "'var list"   and outs :: "'var list"   and V :: "'var"   and x :: "nat"   and n :: "'node SDG_node"   and n' :: "'node SDG_node" assumes "valid_edge a"   and "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"   and "(p, ins, outs) \<in> set procs"   and "V = outs ! x"   and "x < length outs"   and "n = Formal_out (sourcenode a, x)"   and "n' = Actual_out (targetnode a, x)" shows "n -p:V\<rightarrow>\<^bsub>out\<^esub> n'"
simps_2: fixes S :: "'b set"   and x :: "'b"   and xs :: "'b list" shows "sstart S (x # xs) = (##) x ` sstart S xs"
col0: shows "col A' 0 = vec n (\<lambda>i. if i = 0 then e else 0)"
pairwise_mono: fixes P :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and A :: "'c set"   and Q :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'c set" assumes "pairwise P A"   and "\<And>x y. P x y \<Longrightarrow> Q x y"   and "B \<subseteq> A" shows "pairwise Q B"
U0_as_alpha: fixes P :: "('d, 'f) urel" shows "P ;; U0 = \<lceil>P\<rceil>\<^sub>0"
PiE_over_singleton_iff: fixes a :: "'c"   and B :: "'c \<Rightarrow> 'd set" shows "Pi\<^sub>E {a} B = (\<Union>b\<in>B a. {\<lambda>x\<in>{a}. b})"
mon_def: fixes G :: "('c, 'e) monoid_scheme"   and H :: "('d, 'f) monoid_scheme" shows "mon G H = {f \<in> hom G H. inj_on f (carrier G)}"
partition_on_le_set_elements: fixes A :: "'b set"   and P :: "'b set set" assumes "finite A"   and "partition_on A P" shows "card P \<le> card A"
False: shows "\<not> a' * ((A + B * sqrt C) / D)\<^sup>2 + b' * (A + B * sqrt C) / D + c' < 0"
GB_adds_lt: fixes G :: "(('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c) set"   and f :: "('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "extended_ord.punit.is_Groebner_basis G"   and "f \<in> extended_ord.punit.pmdl G"   and "f \<noteq> 0" obtains g :: "('a option \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" where "g \<in> G" and "g \<noteq> 0" and "extended_ord.punit.adds_term (extended_ord.lpp g) (extended_ord.lpp f)"
assms_4: shows "\<pi>\<^sub>5 (the (rt2 i)) < \<pi>\<^sub>5 (the (rt1 i))"
subo_trans_subcls: 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 Cs :: "char list list"   and C' :: "char list"   and Cs' :: "char list list" assumes "Subobjs P C (Cs @ C' # rev Cs')" shows "\<forall>C''\<in>set Cs'. (C', C'') \<in> (subcls1 P)\<^sup>+"
domintros_11: fixes v :: "exp0"   and va :: "exp0"   and vc :: "exp0" shows "is_value_dom (exp0.If v va vc)"
0: assumes "Can t" shows "Can (Inv t) \<and> Dom (Inv t) = Cod t \<and> Cod (Inv t) = Dom t"
delta_hedging_same_cash_flow: fixes der :: "bool stream \<Rightarrow> real"   and matur :: "nat"   and N :: "bool stream measure"   and w :: "bool stream" assumes "der \<in> borel_measurable (G matur)"   and "N = bernoulli_stream ((1 + r - d) / (u - d))" shows "cls_val_process Mkt (delta_hedging N der matur) matur w = der w"
simps_1: fixes f :: "'a \<Rightarrow> 'b"   and vs :: "'b list" shows "remdups_wrt_rev f [] vs = []"
add_mset_add_mset_same_iff: fixes a :: "'b"   and A :: "'b multiset"   and B :: "'b multiset" shows "(add_mset a A = add_mset a B) = (A = B)"
get_edge_simps_1: fixes f :: "'a"   and d :: "'b"   and t :: "'a" shows "getFrom (f, d, t) = f"
extreme_def: fixes X :: "'b set"   and r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and e :: "'b" shows "extreme X r e \<equiv> e \<in> X \<and> (\<forall>x\<in>X. r x e)"
div_mod_decomp: fixes A :: "nat"   and n :: "nat" shows "A = A div n *n n + A mod n"
DirProds_is_comm_group: fixes I :: "'d set"   and G :: "'d \<Rightarrow> ('e, 'f) monoid_scheme" assumes "\<And>i. i \<in> I \<Longrightarrow> comm_group (G i)" shows "comm_group (DirProds G I)"
simps_2: fixes committed :: "nat set"   and justifying_exec :: "('a \<times> ('b, 'a) obs_event action) llist"   and justifying_ws :: "nat \<Rightarrow> nat"   and more :: "'c" shows "justifying_exec \<lparr>committed = committed, justifying_exec = justifying_exec, justifying_ws = justifying_ws, \<dots> = more\<rparr> = justifying_exec"
QuadRes_1_right: fixes p :: "int" shows "QuadRes p 1"
Ds_through: shows "blinfun_apply (Ds (poincare_map {x \<in> S. s x = 0} x)) (f (poincare_map {x \<in> S. s x = 0} x)) \<noteq> 0"
healthy_inf_cont: fixes hcond :: "'a hrel \<Rightarrow> 'a hrel"   and A :: "'a hrel set" assumes "utp_theory_continuous hcond"   and "A \<subseteq> \<lbrakk>hcond\<rbrakk>\<^sub>H"   and "\<not> A = {}" shows "Lattice.inf (utp_order hcond) A = \<Or>A"
sop_inv_def: fixes R :: "('e, 'f) Ring_scheme"   and s :: "'e \<Rightarrow> 'g \<Rightarrow> 'g"   and i :: "'g \<Rightarrow> 'g"   and A :: "'g set" shows "sop_inv R s i A = (\<forall>r\<in>carrier R. \<forall>x\<in>A. r \<^sub>s\<cdot> (\<^sub>i- x) = (-\<^sub>a\<^bsub>R\<^esub> r) \<^sub>s\<cdot> x)"
assms: fixes x :: "Vars.var"   and e :: "exp"   and a :: "'a" assumes "(x, e) \<in> set \<Gamma>" shows "supp (t a e) \<subseteq> supp e"
ker_im_iff: fixes A :: "'a set"   and f :: "'a \<Rightarrow> 'b" shows "(A \<noteq> {} \<and> A \<subseteq> ker f) = (f ` A = 0)"
bijI: fixes f :: "'a \<Rightarrow> 'b" assumes "inj f"   and "surj f" shows "bij f"
H_reach_conv: fixes x :: "'a"   and y :: "'a" shows "(x \<rightarrow>\<^sup>*\<^bsub>H\<^esub> y) = (proj_verts_H x \<rightarrow>\<^sup>*\<^bsub>G\<^esub> proj_verts_H y)"
base_input_def: fixes m :: "real mat convnet"   and "is" :: "nat list" shows "base_input m is = map2 unit\<^sub>v (input_sizes m) is"
mem_dual_space: fixes E :: "'c \<Rightarrow> real"   and S :: "'c set" shows "(E \<in> dual_space S) = (linear_fun_on S E \<and> extensional0 S E)"
factorial_domain_def: fixes R :: "('c, 'd) ring_scheme" shows "factorial_domain R \<equiv> domain R \<and> factorial_monoid (Ring_Divisibility.mult_of R)"
word_eqI: fixes u :: "'b word"   and v :: "'b word" assumes "\<And>n. n < size u \<longrightarrow> u !! n = v !! n" shows "u = v"
convexD: fixes s :: "'a set"   and x :: "'a"   and y :: "'a"   and u :: "real"   and v :: "real" assumes "convex s"   and "x \<in> s"   and "y \<in> s"   and "0 \<le> u"   and "0 \<le> v"   and "u + v = 1" shows "u *\<^sub>R x + v *\<^sub>R y \<in> s"
rel_fun_Grp: fixes h :: "'g \<Rightarrow> 'e"   and A :: "'f set"   and g :: "'f \<Rightarrow> 'h" shows "rel_fun (BNF_Def.Grp UNIV h)\<inverse>\<inverse> (BNF_Def.Grp A g) = BNF_Def.Grp {f. f ` range h \<subseteq> A} (map_fun h g)"
create_opt_sol_satisfies: fixes A :: "rat mat"   and b :: "rat vec"   and c :: "rat vec"   and X :: "(nat, rat) mapping" assumes "create_optimal_solutions A b c = Inr X" shows "\<langle>X\<rangle> \<Turnstile>\<^sub>c\<^sub>s set (x_times_c_geq_y_times_b c b # mat_leqb_eqc A b c @ from_index_geq0_vector (dim_vec c) (0\<^sub>v (dim_vec b)))"
f_def: shows "f \<equiv> \<lambda>x. - 1 - C1 * sqrt (x - a)"
segment_segment: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier D"   and "b \<in> carrier D" shows "(segment (Iod D (segment D b)) a = segment D a) = (segment D a \<subseteq> segment D b)"
assert_demonic_prop: fixes p :: "'e \<Rightarrow> bool"   and r :: "'e \<Rightarrow> 'd \<Rightarrow> bool" shows "{. p .} \<circ> [: r :] = {. p .} \<circ> [: (\<lambda>x y. p x) \<sqinter> r :]"
eq_fold: fixes A :: "'a set"   and x :: "'a" assumes "finite A" shows "\<Squnion>\<^sub>f\<^sub>i\<^sub>n insert x A = Finite_Set.fold sup x A"
fj: assumes "((\<lambda>x. f ((1 - x) *\<^sub>R c + x *\<^sub>R b) * (b - c)) has_integral j) {0..1}" shows "((\<lambda>x. f ((1 - x) *\<^sub>R a + x *\<^sub>R b) * (b - a)) has_integral j) {k..1}"
two_comparisons_into_case_order_8: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and x :: "'a"   and y :: "'a"   and Q :: "'b"   and P :: "'b"   and R :: "'b" assumes "comparator comp" shows "(if lt_of_comp comp x y then Q else if y = x then P else R) = (case comp x y of Eq \<Rightarrow> P | Lt \<Rightarrow> Q | Gt \<Rightarrow> R)"
part_partGen: fixes P :: "'a set set" shows "part (\<Union> P) (partGen P)"
comp_min_basis_adds: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and xs :: "('t \<Rightarrow>\<^sub>0 'c) list" assumes "p \<in> set xs"   and "p \<noteq> 0" obtains q :: "'t \<Rightarrow>\<^sub>0 'c" where "q \<in> set (comp_min_basis xs)" and "lt q adds\<^sub>t lt p"
assms_1: shows "\<forall>t\<in>M. \<exists>s\<in>N. \<exists>\<delta>. t = s \<cdot> \<delta> \<and> wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t \<delta> \<and> wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range \<delta>)"
mono_atLeast: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and B :: "'b" assumes "class.order less_eq less" shows "order.mono_set less_eq (ord.atLeast less_eq B)"
intros_1: fixes p :: "('b, 'a) fm" assumes "tautology p" shows "\<turnstile> p"
hom1_2: shows "homotopic_with (\<lambda>x. True) Y Y (f1 \<circ> g1) id"
toplevel_summands_finite: fixes r :: "'c rexp" shows "finite (toplevel_summands r)"
assms_4: shows "\<nexists>c A'. c \<in> carrier_vec n \<and> A' \<subseteq> A \<and> Suc (card A') = dim_span (insert b A) \<and> (\<forall>a\<in>A'. c \<bullet> a = (0::'a)) \<and> lin_indpt A' \<and> (\<forall>a\<^sub>i\<in>A. (0::'a) \<le> c \<bullet> a\<^sub>i) \<and> c \<bullet> b < (0::'a)"
end_and_opposite_are_endpoints_in_S: fixes a :: "proj2"   and b :: "proj2"   and p :: "proj2"   and l :: "proj2_line" assumes "a \<in> hyp2"   and "b \<in> hyp2"   and "p \<in> S"   and "proj2_incident a l"   and "proj2_incident b l"   and "proj2_incident p l" shows "are_endpoints_in_S p (endpoint_in_S p b) a b"
is_sup_maximum: fixes x :: "'c"   and y :: "'c" shows "is_sup x y (maximum x y)"
seqE: fixes \<Gamma> :: "char list \<Rightarrow> 'c rule list option"   and \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and p :: "'d"   and rs\<^sub>1 :: "'c rule list"   and rs\<^sub>2 :: "'c rule list"   and s :: "state"   and t :: "state" assumes "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1 @ rs\<^sub>2, s\<rangle> \<Rightarrow> t" obtains ti :: "state" where "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> ti" and "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>2, ti\<rangle> \<Rightarrow> t" and "no_matching_Goto \<gamma> p rs\<^sub>1"   | "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>rs\<^sub>1, s\<rangle> \<Rightarrow> t" and "\<not> no_matching_Goto \<gamma> p rs\<^sub>1"
mhead_static_simp: fixes m :: "methd" shows "is_static (mhead m) = is_static m"
simps_1: fixes t :: "'b tree23" shows "treeI (TI t) = t"
zcount_zmset_of_nonneg: fixes M :: "'a multiset"   and t :: "'a" shows "0 \<le> zcount (zmset_of M) t"
h_id: fixes p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" assumes "p \<in> P[- X]" shows "poly_eval (\<lambda>x. monomial (a x) 0) (focus X p) = p"
simps_1: fixes k :: "nat"   and ls :: "('a btree \<times> 'a) list"   and sep :: "'a"   and rs :: "('a btree \<times> 'a) list" shows "rebalance_middle_tree k ls Leaf sep rs Leaf = Node (ls @ (Leaf, sep) # rs) Leaf"
f_Exec_snoc: fixes trans_fun :: "'e \<Rightarrow> 'd \<Rightarrow> 'd"   and xs :: "'e list"   and x :: "'e"   and c :: "'d" shows "f_Exec_Comp trans_fun (xs @ [x]) c = trans_fun x (f_Exec_Comp trans_fun xs c)"
lookup_default_bind_pmf_aux: fixes p :: "'d pmf"   and f :: "'d \<Rightarrow> 'c pmf"   and A :: "'d set"   and x :: "'c" shows "Mapping.lookup_default 0 (bind_pmf_aux p f A) x = (if x \<in> (\<Union>y\<in>A. set_pmf (f y)) then measure_pmf.expectation p (\<lambda>y. indicat_real A y * pmf (f y) x) else 0)"
annir: fixes x :: "'a" shows "x \<cdot> (zero_class.zero::'a) = (zero_class.zero::'a)"
mult_eq_if: fixes m :: "nat"   and n :: "nat" shows "m * n = (if m = 0 then 0 else n + (m - 1) * n)"
prime_elem_dvd_mult_iff: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and p :: "'b"   and a :: "'b"   and b :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "comm_semiring_1.prime_elem times one zero p" shows "dvd.dvd times p (times a b) = (dvd.dvd times p a \<or> dvd.dvd times p b)"
unique_half_chamber_system_f: fixes f' :: "'a \<Rightarrow> 'a"   and g' :: "'a \<Rightarrow> 'a" assumes "OpposedThinChamberComplexFoldings X f' g' C0"   and "g' ` C0 = D0" shows "f' \<turnstile> folding_g.\<C> = f \<turnstile> folding_g.\<C>"
coset_mult_inv2: fixes G :: "('c, 'd) monoid_scheme"   and M :: "'c set"   and x :: "'c"   and y :: "'c" assumes "Group.group G"   and "M #>\<^bsub>G\<^esub> x = M #>\<^bsub>G\<^esub> y"   and "x \<in> carrier G"   and "y \<in> carrier G"   and "M \<subseteq> carrier G" shows "M #>\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> y = M"