hcju/isabelleps huggingface dataset:
at_set_avoiding: fixes Xs :: "atom set"   and c :: "'a" assumes "finite Xs"   and "finite (supp c)" obtains p :: "perm" where "(p \<bullet> Xs) \<sharp>* c" and "supp p = Xs \<union> p \<bullet> Xs"
smc_Arr_vempty_if_Obj_vempty: assumes "\<AA>\<lparr>Obj\<rparr> = []\<^sub>\<circ>" shows "\<AA>\<lparr>Arr\<rparr> = []\<^sub>\<circ>"
bernoulli'_1: shows "bernoulli' (Suc 0) = 1 / 2"
fls_mult_right_inverse_base_factor: fixes f :: "'a fls"   and y :: "'a" assumes "fls_nth f (fls_subdegree f) * y = (1::'a)" shows "f * fls_right_inverse (fls_base_factor f) y = fls_X_intpow (fls_subdegree f)"
subST_2: shows "T \<subseteq> topspace (Euclidean_space n)"
is_float_zero_def: shows "is_float_zero \<equiv> map_fun real_of_float id ((=) 0)"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>) (\<cdot>) (\<lambda>f. T (\<I>, f)) local.map \<ll>.map"
ket_vec_col: fixes v :: "complex vec" shows "col |v\<rangle> 0 = v"
axioms_1: fixes f :: "'c \<Rightarrow> 'd"   and I :: "'c set"   and R :: "('c, 'e) ring_scheme" assumes "weak_ring_morphism f I R" shows "ideal I R"
K_fresh_not_KeyCryptKey: fixes DK :: "nat"   and K :: "nat"   and evs :: "event list" assumes "\<forall>C. DK \<noteq> priEK C"   and "Key K \<notin> used evs" shows "\<not> KeyCryptKey DK K evs"
vsv_vrange_vempty: assumes "\<R>\<^sub>\<circ> (\<AA>\<lparr>Dom\<rparr>) = []\<^sub>\<circ>" shows "\<AA>\<lparr>Dom\<rparr> = []\<^sub>\<circ>"
c1: shows " Lam [x].P' \<longrightarrow>\<^sub>1 M3"
degree_root_unity: fixes n :: "nat" shows "degree (root_unity n) = n"
vdomain_atD: fixes a :: "V" assumes "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> (F\<lparr>ArrVal\<rparr>)" shows "\<exists>b\<in>elts (\<R>\<^sub>\<circ> (F\<lparr>ArrVal\<rparr>)). F\<lparr>ArrVal\<rparr>\<lparr>a\<rparr> = b"
ls_def: shows "ls = lambda_system \<Omega> (Pow \<Omega>) (outer_measure M f)"
sublist_as_proj_eq_as_1: fixes x :: "('d, 'e) fmap"   and vs :: "'d set"   and s :: "('d, 'e) fmap" assumes "x \<subseteq>\<^sub>f fmrestrict_set vs s" shows "x \<subseteq>\<^sub>f s"
f_abs_summable_on: shows "f abs_summable_on {1..}"
prems: shows "LCons z zs \<sqsubseteq> xs"
lim_real_of_ereal: fixes f :: "'a \<Rightarrow> ereal"   and x :: "real"   and net :: "'a filter" assumes "(f \<longlongrightarrow> ereal x) net" shows "((\<lambda>x. real_of_ereal (f x)) \<longlongrightarrow> x) net"
sin_pi2_plus_odd_kpi: fixes k :: "int" assumes "odd k" shows "sin (pi / 2 + real_of_int k * pi) = - 1"
those_xs: shows "Result.those (map (compile_env_entry \<A> \<O>) (Finca_to_list F1)) = Ok xs"
Imp_1: fixes \<Gamma> :: "'a formula multiset"   and \<Delta> :: "'a formula multiset" shows "(F1, F1, \<Gamma> \<Rightarrow> \<Delta> \<longrightarrow> F1, \<Gamma> \<Rightarrow> \<Delta>) \<and> (\<Gamma> \<Rightarrow> F1, F1, \<Delta> \<longrightarrow> \<Gamma> \<Rightarrow> F1, \<Delta>)"
AbstrLevels_S11opt_A72: fixes i :: "AbstrLevelsID" assumes "sS11opt \<in> AbstrLevel i" shows "sA72 \<notin> AbstrLevel i"
polyfunI: fixes N :: "nat set"   and f :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" assumes "\<And>P. (\<And>p. \<lbrakk>vars p \<subseteq> N; \<And>x. insertion x p = f x\<rbrakk> \<Longrightarrow> P) \<Longrightarrow> P" shows "polyfun N f"
prefix_of_pmap_\<Gamma>: fixes \<pi> :: "'b prefix"   and \<sigma> :: "'b trace"   and f :: "'b set \<Rightarrow> 'c set" assumes "prefix_of \<pi> \<sigma>" shows "prefix_of (pmap_\<Gamma> f \<pi>) (map_\<Gamma> f \<sigma>)"
derivg: fixes x :: "real" assumes "x \<in> {a..b}" shows "(g has_real_derivative g' x) (at x)"
assms_3: shows "c \<equiv> lead_coeff g ^ (Suc (degree f) - degree g)"
bl_word_rotr_dt: fixes n :: "nat"   and w :: "'b word" shows "to_bl (word_rotr n w) = drop (LENGTH('b) - n mod LENGTH('b)) (to_bl w) @ take (LENGTH('b) - n mod LENGTH('b)) (to_bl w)"
Ajj: shows "A $$ (j, j) = (0::'a)"
LeftDerivationLadder_cut_prefix: fixes \<delta> :: "symbol list"   and \<alpha> :: "symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and L :: "(nat \<times> nat \<times> nat) list"   and \<gamma> :: "symbol list" assumes "LeftDerivationLadder (\<delta> @ \<alpha>) D L \<gamma>"   and "is_word \<delta>"   and "length \<delta> \<le> ladder_i L 0" shows "\<exists>D' L' \<gamma>'. \<gamma> = \<delta> @ \<gamma>' \<and> LeftDerivationLadder \<alpha> D' L' \<gamma>' \<and> D' = derivation_shift D (length \<delta>) 0 \<and> length L' = length L \<and> ladder_i L' 0 + length \<delta> = ladder_i L 0 \<and> ladder_last_j L' + length \<delta> = ladder_last_j L"
gen_dfs_fe_inv_def: fixes E :: "('b \<times> 'b) set"   and U :: "'c"   and S :: "'b set"   and V0 :: "'b set"   and u0 :: "'b"   and it :: "'b set"   and V :: "'b set"   and brk :: "bool" shows "gen_dfs_fe_inv E U S V0 u0 it V brk \<equiv> (\<not> brk \<longrightarrow> E `` (V - S) \<subseteq> V) \<and> E `` {u0} - it \<subseteq> V \<and> V0 \<subseteq> V \<and> V \<subseteq> V0 \<union> (E - UNIV \<times> S)\<^sup>* `` (E `` {u0} - it - S)"
preserves_arr: fixes f :: "('a, 'a set) Maps.arr arrow_of_spans_data" assumes "Span.arr f" shows "Span.arr (Span.trg f)"
bind_lunit: fixes x :: "'f"   and f :: "'f \<Rightarrow> 'e option" shows "Some x \<bind> f = f x"
prod_carr_mem: fixes M :: "('f, 'a, 'g) Module_scheme"   and N :: "('h, 'a, 'g) Module_scheme"   and m :: "'f"   and n :: "'h" assumes "R module M"   and "R module N"   and "m \<in> carrier M"   and "n \<in> carrier N" shows "(m, n) \<in> M \<times>\<^sub>c N"
terminals_drop: fixes n :: "nat"   and p :: "(symbol \<times> character list) list" shows "terminals (drop n p) = drop n (terminals p)"
less_asym': fixes D :: "('c, 'd) Order_scheme"   and a :: "'c"   and b :: "'c" assumes "Order D"   and "a \<in> carrier D"   and "b \<in> carrier D"   and "a \<prec>\<^bsub>D\<^esub> b" shows "\<not> b \<prec>\<^bsub>D\<^esub> a"
enab_n1: shows "\<turnstile> Enabled \<langle>n1\<rangle>_vars = ($pc1 = #a \<longrightarrow> #0 < $sem)"
gauss_jordan_lemma: fixes m :: "nat"   and n :: "nat"   and A :: "nat \<Rightarrow> nat \<Rightarrow> 'b"   and B :: "nat \<Rightarrow> nat \<Rightarrow> 'b" assumes "m \<le> n"   and "unit A m n"   and "gauss_jordan A m = Some B" shows "unit B 0 n \<and> solution A n (\<lambda>j. B j n)"
is_lasso_pre_prpl_of_lasso: fixes G :: "('e, 'f) graph_rec_scheme"   and L :: "'e lasso" assumes "graph G" shows "graph.is_lasso_prpl_pre G (prpl_of_lasso L) = graph.is_lasso_pre G L"
f\<chi>: shows "D.cones_map (\<psi> (a', a) (\<Psi>.FUN a' (\<iota> \<chi>'))) \<chi> = \<chi>'"
marked_insertions_dequeue_Mark: fixes s :: "'h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state"   and p :: "'h process_name"   and r :: "'g"   and fl :: "bool"   and ws :: "('e, 'f, 'g) mem_store_action list"   and m :: "'h"   and tso_writes_inv :: "('h process_name \<Rightarrow> ('e, 'h, 'f, 'g) Model.local_state) \<Rightarrow> bool" assumes "sys_mem_store_buffers p s = mw_Mark r fl # ws"   and "mut_m.marked_insertions m s"   and "tso_writes_inv s"   and "valid_W_inv s" shows "mut_m.marked_insertions m (s(sys := s sys\<lparr>heap := (sys_heap s)(r := map_option (obj_mark_update (\<lambda>_. fl)) (sys_heap s r)), mem_store_buffers := (mem_store_buffers (s sys))(p := ws)\<rparr>))"
design_axioms_def: fixes block_collection :: "'b set multiset" shows "design_axioms block_collection \<equiv> \<forall>bl. bl \<in># block_collection \<longrightarrow> bl \<noteq> {}"
n_ent_d: shows "\<not> I_of N \<TTurnstile>e {D}"
rec_newrgt1_def: shows "rec_newrgt1 = Cn 2 rec_minus [Cn 2 rec_add [recf.id 2 1, Cn 2 (constn 1) [recf.id 2 0]], Cn 2 rec_scan [recf.id 2 1]]"
circ_sup_one_right_unfold: fixes x :: "'a" assumes "reflexive x" shows "x\<^sup>\<circ> * x = x\<^sup>\<circ>"
neq_NilE: fixes l :: "'a list" assumes "l \<noteq> []" obtains x :: "'a" and xs :: "'a list" where "l = x # xs"
exhaustive_above_fusion_def: fixes g :: "('a, 'c) generator"   and a :: "'a"   and s :: "'c" shows "exhaustive_above_fusion g a s = exhaustive_above a (list.unfoldr g s)"
sub_of_denot_equivl: fixes E :: "char list \<Rightarrow> hterm"   and G :: "char list \<Rightarrow> hterm list \<Rightarrow> bool"   and l :: "fterm literal" shows "eval\<^sub>l E HFun G (l \<cdot>\<^sub>l sub_of_denot E) = eval\<^sub>l E HFun G l"
grid_transitive: fixes a :: "(nat \<times> int) list"   and b :: "(nat \<times> int) list"   and ds :: "nat set"   and c :: "(nat \<times> int) list"   and ds' :: "nat set"   and ds'' :: "nat set" assumes "a \<in> grid b ds"   and "b \<in> grid c ds'"   and "ds' \<subseteq> ds''"   and "ds \<subseteq> ds''" shows "a \<in> grid c ds''"
size_Diff_singleton: fixes x :: "'b"   and M :: "'b multiset" assumes "x \<in># M" shows "size (M - {#x#}) = size M - 1"
chain_reparam_chain'_pathimg_subset': fixes one_chain :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and subdiv :: "(int \<times> (real \<Rightarrow> real \<times> real)) set"   and k :: "int"   and \<gamma> :: "real \<Rightarrow> real \<times> real" assumes "chain_reparam_chain' one_chain subdiv"   and "(k, \<gamma>) \<in> subdiv" shows "\<exists>k' \<gamma>'. (k', \<gamma>') \<in> one_chain \<and> path_image \<gamma> \<subseteq> path_image \<gamma>'"
enc_welldef: fixes w :: "'a list"   and I :: "(nat + nat set) list"   and w' :: "'a list"   and I' :: "(nat + nat set) list"   and \<phi> :: "'a formula" assumes "enc (w, I) = enc (w', I')"   and "wf_formula (length I) \<phi>"   and "wf_interp_for_formula (w, I) \<phi>"   and "wf_interp_for_formula (w', I') \<phi>" shows "((w, I) \<Turnstile> \<phi>) = ((w', I') \<Turnstile> \<phi>)"
rel_flip: fixes R :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c stream"   and b :: "'d stream" shows "stream_all2 R\<inverse>\<inverse> a b = stream_all2 R b a"
eta_cfun: fixes f :: "'b \<rightarrow> 'c" shows "(\<Lambda> x. f\<cdot>x) = f"
distinct_86: fixes name' :: "name"   and pi' :: "pi"   and pi1 :: "pi"   and pi2 :: "pi" shows "<\<nu>name'>pi' \<noteq> pi1 \<parallel> pi2"
wf_invalid_point: fixes x :: "'b"   and b :: "'b set" assumes "x \<notin> \<V>'"   and "b \<in># \<B>'" shows "x \<notin> b"
sup_Un_eq: fixes R :: "'c set"   and S :: "'c set" shows "sup (\<lambda>x. x \<in> R) (\<lambda>x. x \<in> S) = (\<lambda>x. x \<in> R \<union> S)"
assms_1: shows "p_iiface p \<in> set ifs"
sel_5: fixes x51 :: "'b mrexp2"   and x52 :: "'b mrexp2"   and x53 :: "bool"   and x54 :: "bool" shows "fin (Times2 x51 x52 x53 x54) = x53"
prems_2: shows "insert x V \<subseteq> carrier (vector_space_poly.vs {v. [v ^ CARD('a) = v] (mod u) \<and> degree v < degree u})"
measure_pmf_inverse: fixes x :: "'b pmf" shows "Abs_pmf (measure_pmf x) = x"
assms_1: fixes x :: "'a" shows "(g has_field_derivative g' x) (at x)"
UNIV_I: fixes x :: "'ag" shows "x \<in> UNIV"
uniformly_convergent_eq_Cauchy: fixes X :: "'b set"   and f :: "nat \<Rightarrow> 'b \<Rightarrow> 'c" shows "uniformly_convergent_on X f = uniformly_Cauchy_on X f"
instInpP_def: fixes \<phi> :: "'fmla"   and n :: "nat"   and \<tau> :: "'fmla" shows "instInpP \<phi> n \<tau> \<equiv> let zz = Variable (Suc (Suc n)) in exi zz (cnj (subst \<tau> (Var zz) out) (subst \<phi> (Var zz) inp))"
supp_1: fixes var :: "Vars.var" shows "supp (Var var) = supp var"
no_collisionI: fixes S :: "real set" assumes "\<And>t. t \<in> S \<Longrightarrow> ego.s t \<noteq> other.s t" shows "no_collision S"
ll_def: shows "ll = (\<lambda>x. real x * ln (real x))"
that: fixes T\<^sub>1 :: "'c list"   and c\<^sub>3 :: "'c"   and T\<^sub>2 :: "'c list"   and c\<^sub>2 :: "'c" assumes "T = T\<^sub>1 @ c\<^sub>3 # T\<^sub>2"   and "trace c\<^sub>1 T\<^sub>1 c\<^sub>2"   and "\<forall>x\<in>set T\<^sub>1. P x"   and "P c\<^sub>2"   and "c\<^sub>2 \<Rightarrow> c\<^sub>3"   and "\<not> P c\<^sub>3"   and "trace c\<^sub>3 T\<^sub>2 c\<^sub>4" shows "thesis"
hom_sum: fixes f :: "'b \<Rightarrow> real"   and X :: "'b set" shows "of_real (sum f X) = (\<Sum>x\<in>X. of_real (f x))"
termi_cn: fixes f :: "recf"   and xs :: "nat list"   and gs :: "recf list"   and n :: "nat" assumes "terminate f (map (\<lambda>g. rec_exec g xs) gs)"   and "\<forall>g\<in>set gs. terminate g xs"   and "length xs = n" shows "terminate (Cn n f gs) xs"
clw_rel_def: fixes A :: "('c \<times> 'd set) set" shows "clw_rel A = lw_rel O \<langle>A\<rangle>set_rel O br \<Union> top"
isomorphic_symmetric: fixes f :: "'a"   and g :: "'a" assumes "L'.C.isomorphic f g" shows "L'.C.isomorphic g f"
invertible_fixpoint_property: fixes T :: "'b set"   and i :: "'b \<Rightarrow> 'c"   and S :: "'c set"   and r :: "'c \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b" assumes "continuous_on T i"   and "i ` T \<subseteq> S"   and "continuous_on S r"   and "r ` S \<subseteq> T"   and "\<And>y. y \<in> T \<Longrightarrow> r (i y) = y"   and "\<And>f. \<lbrakk>continuous_on S f; f ` S \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>x\<in>S. f x = x"   and "continuous_on T g"   and "g ` T \<subseteq> T" obtains y :: "'b" where "y \<in> T" and "g y = y"
reaches_refl: fixes r :: "'a"   and s :: "'b process_name \<Rightarrow> ('c, 'b, 'd, 'a) Model.local_state" shows "(r reaches r) s"
dist_QuantT_def: fixes x :: "'a['b, 'c]"   and y :: "'a['b, 'c]" shows "dist x y = norm (x - y)"
R: fixes s1' :: "'a"   and s2' :: "'b"   and x :: "'e" assumes "X s1' s2'"   and "(x, s1') \<in> set_spmf (exec_gpv oracle1 gpv s1)"   and "(x, s2') \<in> set_spmf (exec_gpv oracle2 gpv s2)" shows "R (x, s1') (x, s2')"
ModelIkMcalc_axioms: shows "ModelIkMcalc wtFsym wtPsym arOf resOf parOf \<Phi> infTp intT intF intP"
that_1: shows "higher_differentiable_on X f (Suc i)"
induct: fixes x :: "'a"   and \<delta> :: "('a, 'b) ta_rule set"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> b_accessible \<delta>"   and "\<And>q l qs. \<lbrakk>q \<rightarrow> l qs \<in> \<delta>; \<And>x. x \<in> set qs \<Longrightarrow> x \<in> b_accessible \<delta>; \<And>x. x \<in> set qs \<Longrightarrow> P x\<rbrakk> \<Longrightarrow> P q" shows "P x"
mult_Mp_1: fixes f :: "int poly"   and g :: "int poly" shows "pl.eq_m (pl.Mp f * g) (f * g)"
class_cons: fixes C :: "char list"   and x :: "'a cdecl"   and P :: "'a prog" assumes "C \<noteq> fst x" shows "class (x # P) C = class P C"
invertible_fst_Gauss_Jordan_PA: fixes A :: "(('b, 'd) vec, 'c) vec" shows "invertible (fst (Gauss_Jordan_PA A))"
mult_delta_right: fixes x :: "'a"   and b :: "bool"   and y :: "'a" shows "x * (if b then y else (0::'a)) = (if b then x * y else (0::'a))"
smult_sum2: fixes m :: "'a"   and f :: "'b \<Rightarrow> 'a poly"   and S :: "'b set" shows "smult m (sum f S) = (\<Sum>i\<in>S. smult m (f i))"
EM_remainder'_conv_Suc: fixes a :: "int"   and b :: "int"   and n :: "nat"   and A :: "real set"   and f :: "real \<Rightarrow> 'a"   and f' :: "real \<Rightarrow> 'a" assumes "a \<le> b"   and "0 < n"   and "finite A"   and "continuous_on {real_of_int a..real_of_int b} f"   and "continuous_on {real_of_int a..real_of_int b} f'"   and "\<And>x. x \<in> {real_of_int a<..<real_of_int b} - A \<Longrightarrow> (f has_vector_derivative f' x) (at x)" shows "EM_remainder' n f (real_of_int a) (real_of_int b) = ((- 1) ^ Suc n * bernoulli (Suc n) / fact (Suc n)) *\<^sub>R (f (real_of_int b) - f (real_of_int a)) + EM_remainder' (Suc n) f' (real_of_int a) (real_of_int b)"
stabiliser_def: fixes G :: "('d, 'e) monoid_scheme"   and action :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and x :: "'f" assumes "orbit_stabiliser G action" shows "orbit_stabiliser.stabiliser G action x = {g \<in> carrier G. action g x = x}"
fst_map2_val_pair_subset: fixes comp :: "'a \<Rightarrow> 'a \<Rightarrow> order"   and xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'c) list"   and g :: "('a \<times> 'b) list \<Rightarrow> ('a \<times> 'd) list"   and h :: "('a \<times> 'c) list \<Rightarrow> ('a \<times> 'd) list"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "comparator comp"   and "comparator.oalist_inv_raw comp xs"   and "comparator.oalist_inv_raw comp ys"   and "comparator.map2_val_compat comp g"   and "comparator.map2_val_compat comp h" shows "fst ` set (comparator.map2_val_pair comp f g h xs ys) \<subseteq> fst ` set xs \<union> fst ` set ys"
enum_in: fixes p :: "nat"   and n :: "nat"   and base :: "nat \<Rightarrow> nat"   and upd :: "nat \<Rightarrow> nat"   and s :: "(nat \<Rightarrow> nat) set"   and i :: "nat" assumes "kuhn_simplex p n base upd s"   and "i \<le> n" shows "kuhn_simplex.enum base upd i \<in> s"
add1: fixes A :: "'c"   and k :: "'c"   and a :: "'c"   and b :: "'c" assumes "A \<equiv> k + a" shows "A + b \<equiv> k + (a + b)"
edges_of_walkE: fixes v :: "'a"   and w :: "'a"   and xs :: "'a list" assumes "(v, w) \<in> edges_of_walk xs" shows "\<exists>xs_pre xs_post. xs = xs_pre @ v # w # xs_post"
polyhedron_halfspace_le: fixes a :: "'b"   and b :: "real" shows "polyhedron {x. a \<bullet> x \<le> b}"
min_less_iff_disj: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "class.linorder less_eq less" shows "less (ord.min less_eq x y) z = (less x z \<or> less y z)"
assms_5: shows "(M, {#fst \<alpha>_step, fst \<beta>_step#}) \<in> mul_eq r"
prob_eq_1: fixes M :: "'c measure"   and A :: "'c set" assumes "prob_space M"   and "A \<in> sets M" shows "(Sigma_Algebra.measure M A = 1) = (AE x in M. x \<in> A)"
is_monoid: fixes G :: "('f, 'g) monoid_scheme" assumes "Group.group G" shows "Group.monoid G"
cogen_def: fixes P :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and cs :: "'e list"   and w :: "'f" shows "cogen P cs w \<equiv> \<exists>c\<in>set cs. P c w"
subgroup_of_subset: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and K :: "'c set" assumes "subgroup H G"   and "Group.group G"   and "H \<subseteq> K"   and "subgroup K G" shows "subgroup H (G\<lparr>carrier := K\<rparr>)"
less_rat_def: fixes x :: "rat"   and y :: "rat" shows "(x < y) = positive (y - x)"
trm_subst_ident: fixes t :: "('b, 'a) Term.term"   and \<theta> :: "'a \<Rightarrow> ('b, 'a) Term.term" assumes "fv t \<inter> subst_domain \<theta> = {}" shows "t \<cdot> \<theta> = t"
bounded: shows "interaction_bounded_by' (\<lambda>_. True) \<A> q"
add_true: fixes b :: "assn"   and p :: "'a Heap"   and a :: "'a \<Rightarrow> assn" assumes "<b> p <a>\<^sub>t" shows "<b * true> p <a>\<^sub>t"
map_add_assoc: fixes m1 :: "'c \<Rightarrow> 'd option"   and m2 :: "'c \<Rightarrow> 'd option"   and m3 :: "'c \<Rightarrow> 'd option" shows "m1 ++ (m2 ++ m3) = m1 ++ m2 ++ m3"
nths_nth: fixes n :: "nat"   and A :: "nat set"   and xs :: "'a list" assumes "n \<in> A"   and "n < length xs" shows "nths xs A ! card {i. i < n \<and> i \<in> A} = xs ! n"
sin_z_over_z_series': fixes z :: "'a" assumes "z \<noteq> (0::'a)" shows "(\<lambda>n. sin_coeff (n + 1) *\<^sub>R z ^ n) sums (sin z / z)"
less_half_sum: fixes a :: "'a"   and b :: "'a" assumes "a < b" shows "a < (a + b) / ((1::'a) + (1::'a))"
Out: fixes generat :: "('b, 'a, ('b, 'a, 'c) rpv) generat"   and gpv :: "('b, 'a, 'c) gpv" assumes "generat \<in> set_spmf (the_gpv gpv)"   and "x \<in> generat_outs generat" shows "P gpv"
self_codomain_iff_ide: fixes C :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "partial_magma C" shows "(a \<in> partial_magma.codomains C a) = partial_magma.ide C a"
Fun_param_in_subterms: fixes x :: "('c, 'd) Term.term"   and X :: "('c, 'd) Term.term list"   and f :: "'c" assumes "x \<in> set X" shows "x \<sqsubseteq> Fun f X"
1: shows "lfp (\<lambda>x. fst (F (x, lfp (\<lambda>y. snd (F (x, y)))))) \<le> x"
test0_conv_test1: fixes Fs\<^sub>2 :: "'a list list"   and m :: "'b \<Rightarrow> 'a option"   and Fs\<^sub>1 :: "'b list list" assumes "[] \<notin> set Fs\<^sub>2" shows "pr_iso_test1 m Fs\<^sub>1 Fs\<^sub>2 = pr_iso_test0 m Fs\<^sub>1 Fs\<^sub>2"
g_iso: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "is_conjugation f g" shows "x \<le> y \<longrightarrow> g x \<le> g y"
spmf_spmf_of_pmf: fixes p :: "'b pmf"   and x :: "'b" shows "spmf (spmf_of_pmf p) x = pmf p x"
rep_eq: fixes x :: "('a, 'b) IEEE.float" shows "sign x = (case Rep_float x of (s, uu_, uua_) \<Rightarrow> unat s)"
g2: fixes x :: "'a discr \<rightarrow> 'b" shows "cont (G x)"
converse_rtrancl_induct: fixes a :: "'c"   and b :: "'c"   and r :: "('c \<times> 'c) set"   and P :: "'c \<Rightarrow> bool" assumes "(a, b) \<in> r\<^sup>*"   and "P b"   and "\<And>y z. (y, z) \<in> r \<and> (z, b) \<in> r\<^sup>* \<and> P z \<Longrightarrow> P y" shows "P a"
vcard_aa: shows "2\<^sub>\<nat> = vcard [a, a]\<^sub>\<circ>"
seq_empty_all: fixes s :: "nat \<Rightarrow> 'b"   and i :: "nat"   and j :: "nat" assumes "emptyseq s" shows "s i = s j"
liftM_type_1: fixes \<Gamma> :: "nat \<Rightarrow> Syntax.type"   and \<Delta> :: "nat \<Rightarrow> Syntax.type"   and t :: "trm"   and T :: "Syntax.type"   and U :: "Syntax.type" assumes "\<Gamma> , \<Delta> \<turnstile>\<^sub>T t : T" shows "\<forall>k. \<Gamma> , \<Delta>\<langle>k:U\<rangle> \<turnstile>\<^sub>T liftM_trm t k : T"
newton_interpolation_poly_int: fixes xs_ys :: "(int \<times> int) list" assumes "distinct (map fst xs_ys)" shows "newton_interpolation_poly_int xs_ys = (let rxs_ys = map (\<lambda>(x, y). (rat_of_int x, rat_of_int y)) xs_ys; rp = newton_interpolation_poly rxs_ys in if \<forall>x\<in>set (coeffs rp). is_int_rat x then Some (map_poly int_of_rat rp) else None)"
RID\<^sub>LI_3: fixes s :: "('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('a \<Rightarrow> ('b, 'a, 'c) val option) \<times> ('b, 'a, 'c) expr"   and \<sigma> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and \<tau> :: "'a \<Rightarrow> ('b, 'a, 'c) val option"   and e :: "('b, 'a, 'c) expr"   and r :: "'b" assumes "s = (\<sigma>, \<tau>, e)"   and "r \<in> RID\<^sub>E e" shows "r \<in> RID\<^sub>L s"
edom_ae2ce: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "NoCardinalityAnalysis Aexp Aheap" shows "edom (ae2ce ae) = edom ae"
bres_galois: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<cdot> y \<le> z) = (y \<le> x \<rightarrow> z)"
1: fixes n :: "nat"   and m :: "int" assumes "0 < n" shows "\<exists>k. plusi (exp (\<i> * (of_int m * complex_of_real pi) - complex_of_real (ln (real n + sqrt ((real n)\<^sup>2 - 1))))) = of_int k * 2"
mk_alt_consistency_def: fixes C :: "('c, 'd) form set set" shows "mk_alt_consistency C = {S. \<exists>f. psubst f ` S \<in> C}"
fixp_induct: fixes P :: "(('b, 'c2, 'r2) gpv \<Rightarrow> 's2 \<Rightarrow> 's1 \<Rightarrow> ('b \<times> 's2 \<times> 's1 + 'c \<times> ('r1 \<times> 's1, 'c, 'r) rpv \<times> ('r2 \<times> 's2, 'c1, 'r1) rpv \<times> ('b, 'c2, 'r2) rpv) spmf) \<Rightarrow> bool" assumes "spmf.admissible (\<lambda>inline2. P (curry (curry inline2)))"   and "P (\<lambda>inline2 gpv s2. return_pmf None)"   and "\<And>s1. P s1 \<Longrightarrow> P (\<lambda>a aa aaa. the_gpv a \<bind> (\<lambda>a. case a of Pure x \<Rightarrow> return_spmf (Inl (x, aa, aaa)) | IO out rpv \<Rightarrow> inline1 callee1 (callee2 aa out) aaa \<bind> (\<lambda>a. case a of Inl (b, c) \<Rightarrow> (case b of (r2, s2) \<Rightarrow> s1 (rpv r2) s2) c | Inr (x, rpv'', rpv') \<Rightarrow> return_spmf (Inr (x, rpv'', rpv', rpv)))))" shows "P local.inline2"
zero_change_imp_all_effects_submap: fixes as :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) list"   and vs :: "'a set"   and s :: "('a, 'b) fmap"   and b :: "('a, 'b) fmap \<times> ('a, 'b) fmap"   and s' :: "('a, 'b) fmap" assumes "vars_change as vs s = []"   and "sat_precond_as s as"   and "ListMem b as"   and "fmrestrict_set vs s = fmrestrict_set vs s'" shows "fmrestrict_set vs (snd b) \<subseteq>\<^sub>f fmrestrict_set vs s'"
float_divr_nonpos_pos_upper_bound: fixes x :: "float"   and y :: "float"   and prec :: "nat" assumes "real_of_float x \<le> 0"   and "0 \<le> real_of_float y" shows "real_of_float (float_divr prec x y) \<le> 0"
AA2: shows "Esigma (3\<^sup>2 * 7 * 13 * 107) = Esigma (3\<^sup>2 * 7 * 13) * Esigma 107"
quasi_isometry_betweenD_5: fixes lambda :: "real"   and C :: "real"   and X :: "'b set"   and Y :: "'c set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and y :: "'b" assumes "lambda C -quasi_isometry_between X Y f"   and "x \<in> X"   and "y \<in> X" shows "1 / lambda * dist x y - C \<le> dist (f x) (f y)"