hcju/isabelleps huggingface dataset:
im_sheaf_def: fixes S :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and \<FF> :: "'c set \<Rightarrow> 'd set"   and \<rho> :: "'c set \<Rightarrow> 'c set \<Rightarrow> 'd \<Rightarrow> 'd"   and b :: "'d"   and add_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and mult_str :: "'c set \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> 'd"   and zero_str :: "'c set \<Rightarrow> 'd"   and one_str :: "'c set \<Rightarrow> 'd"   and S' :: "'e set"   and is_open' :: "'e set \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'e"   and V :: "'e set" assumes "Comm_Ring.im_sheaf S is_open \<FF> \<rho> b add_str mult_str zero_str one_str S' is_open' f" shows "im_sheaf.im_sheaf S \<FF> f V \<equiv> \<FF> (f \<^sup>\<inverse> S V)"
mod_eq_0D: fixes m :: "nat"   and d :: "nat" assumes "m mod d = 0" shows "\<exists>q. m = d * q"
modular_1_var: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x ; y \<cdot> z \<le> x ; (y \<cdot> x\<^sup>\<smile> ; z)"
composite_simps_7: shows "cod (f \<star> g \<star> h) = cod f \<star> cod g \<star> cod h"
applicableConflict_def: fixes state :: "State" shows "applicableConflict state \<equiv> \<exists>state'. appliedConflict state state'"
etp_axioms: shows "etp I local.domain local.range F F\<^sub>i\<^sub>n\<^sub>v"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b environment"   and b :: "'c environment" assumes "rel_environment R a b" obtains x :: "char list \<Rightarrow> 'b option" and y :: "char list \<Rightarrow> 'c option" where "a = Env x" and "b = Env y" and "rel_fun (=) (rel_option R) x y"   | "a = Malformed" and "b = Malformed"
coprime_hom: fixes hom :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y' :: "'b" assumes "comm_monoid_mult_isom hom" shows "coprime (hom x) y' = coprime x (inv hom y')"
Pair_1: shows "local_iwf t (\<Gamma> \<turnstile> c)"
reachable_props_1: fixes \<phi>' :: "'val"   and g :: "'g"   and \<phi> :: "'val" assumes "\<phi>' \<in> reachable g \<phi>" shows "(phiArg g)\<^sup>*\<^sup>* \<phi> \<phi>'"
cospoly_minus: fixes n :: "nat"   and x :: "real" shows "cospoly n (- x) = cospoly n x"
path_component_of_aux: fixes X :: "'a topology"   and x :: "'a"   and y :: "'a" assumes "path_component_of X x y" shows "path_component_of (subtopology X (path_component_of_set X x)) x y"
Rep_token_run_af: fixes x :: "nat"   and n :: "nat" shows "Rep (token_run x n) \<equiv>\<^sub>P af\<^sub>G \<phi> (w [x \<rightarrow> n])"
list_nA_lt_refl: fixes xs :: "nat list list" shows "xs \<in> list n A \<longrightarrow> xs [\<sqsubseteq>\<^bsub>r\<^esub>] xs"
Diff_partition: fixes A :: "'b set"   and B :: "'b set" assumes "order.greater_eq B A" shows "A \<union> (B - A) = B"
his_le: fixes x :: "nat \<times> 'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and nid :: "'nid" assumes "x \<in> his t n nid" shows "fst x \<le> n"
vertices_graph: fixes g :: "graph" shows "vertices g = [0..<countVertices g]"
thetaZOSeqD_ZOretr: shows "thetaZOSeqD \<subseteq> ZOretr (thetaZOSeqD \<union> ZObis)"
uints_num: fixes n :: "nat" shows "uints n = {i. 0 \<le> i \<and> i < 2 ^ n}"
Bonk_Schramm_extension_unfolded_wo_props_1: shows "well_order Bonk_Schramm_extension_unfolded_wo"
poly_altdef: fixes p :: "'c poly"   and x :: "'c" shows "poly p x = (\<Sum>i\<le>degree p. coeff p i * x ^ i)"
the_inv_into_map_impl_def: fixes V :: "'d set"   and f :: "'d \<Rightarrow> 'c" shows "the_inv_into_map_impl V f = FOREACH V (\<lambda>x m. RETURN (m(f x \<mapsto> x))) Map.empty"
fresh_subst_if: fixes j :: "atom"   and i :: "name"   and x :: "trm"   and t :: "trm" shows "j \<sharp> subst i x t = (atom i \<sharp> t \<and> j \<sharp> t \<or> j \<sharp> x \<and> (j \<sharp> t \<or> j = atom i))"
open_closed: fixes "open" :: "'a set \<Rightarrow> bool"   and S :: "'a set" assumes "class.topological_space open" shows "open S = topological_space.closed open (- S)"
r_ifeq_else_diverg_recfn: shows "recfn 3 r_ifeq_else_diverg"
arcs_ends_conv: fixes G :: "('b, 'c) pre_digraph" shows "arcs_ends G = (\<lambda>e. (tail G e, head G e)) ` arcs G"
iFROM_sub: fixes k :: "nat"   and n :: "nat" shows "k \<ominus> [n\<dots>] = (if n \<le> k then [\<dots>k - n] else {})"
abs_sum_np_ineq: fixes C :: "nat set"   and f :: "nat \<Rightarrow> real" assumes "C \<subseteq> PR" shows "\<bar>sum f {..<np}\<bar> \<le> (\<Sum>l\<in>C. \<bar>f l\<bar>) + (\<Sum>l\<in>{..<np} - C. \<bar>f l\<bar>)"
xs_dec: shows "a @ [xs ! i] @ b @ [xs ! j] @ c = xs"
gr: fixes x :: "bool" shows "(if x then map_pmf Inl Da else map_pmf Inr Db) \<bind> case_sum (\<lambda>e. map_pmf Inl (fa e)) (\<lambda>e. map_pmf Inr (fb e)) = (if x then map_pmf Inl Da' else map_pmf Inr Db')"
preserves_iso: fixes f :: "'a" assumes "S.iso f" shows "S.iso (S.src f)"
dcd_1: shows "((\<sigma>, k), \<sigma>', k') \<in> cp"
preserves_cod: fixes f :: "'b" assumes "arr f" shows "cod (\<rr> f) = local.map (cod f)"
has_bochner_integral_iff: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and x :: "'c" shows "has_bochner_integral M f x = (integrable M f \<and> integral\<^sup>L M f = x)"
vector_space_prod_axioms: shows "vector_space_prod (*\<^sub>C) (*\<^sub>C)"
Lim_transform: fixes g :: "'c \<Rightarrow> 'd"   and a :: "'d"   and F :: "'c filter"   and f :: "'c \<Rightarrow> 'd" assumes "(g \<longlongrightarrow> a) F"   and "((\<lambda>x. f x - g x) \<longlongrightarrow> (0::'d)) F" shows "(f \<longlongrightarrow> a) F"
inject: fixes x11 :: "name"   and x12 :: "name"   and y11 :: "name"   and y12 :: "name" shows "(x11[x12] = y11[y12]) = (x11 = y11 \<and> x12 = y12)"
exhaust: fixes y :: "part_witness" obtains "y = a"   | "y = b"   | "y = c"
subspace_neg: fixes S :: "'b set"   and x :: "'b" assumes "subspace S"   and "x \<in> S" shows "- x \<in> S"
component_is_vector: fixes e :: "'a"   and v :: "'a" shows "vector (choose_component e v)"
list_update_append1: fixes i :: "nat"   and xs :: "'b list"   and ys :: "'b list"   and x :: "'b" assumes "i < length xs" shows "(xs @ ys)[i := x] = xs[i := x] @ ys"
l9_8_2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and Q :: "'a"   and A :: "'a"   and C :: "'a"   and B :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.TS Bet P Q A C"   and "Tarski_neutral_dimensionless.OS Bet P Q A B" shows "Tarski_neutral_dimensionless.TS Bet P Q B C"
extremum: fixes a :: "'a" shows "a \<le> top_class.top"
pop_cons: fixes j :: "nat"   and js :: "nat list"   and v :: "int" shows "local.pop (j # js) v = (if v \<le> xs ! j then local.pop js v else j # js)"
noDA: fixes xs :: "('b, 'c) Combinators list"   and s :: "('b, 'c) Combinators" assumes "noDenyAll xs"   and "s \<in> set xs" shows "\<not> member DenyAll s"
absolutely_integrable_integrable_bound: fixes S :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> real" assumes "\<And>x. x \<in> S \<Longrightarrow> norm (f x) \<le> g x"   and "f integrable_on S"   and "g integrable_on S" shows "f absolutely_integrable_on S"
a1: shows "\<exists>a. a \<sharp> (h, h a)"
assms_1: fixes v :: "'a vec" assumes "v \<in> set s" shows "dim_vec v = n"
simps_4: fixes t :: "dbtm"   and u :: "dbtm" shows "quot_dbtm (DBEats t u) = HPair (HTuple 1) (HPair (quot_dbtm t) (quot_dbtm u))"
simps_2: fixes xs :: "'h list"   and opt :: "('f, 'g, 'h) env option"   and e :: "('f, 'g, 'h) env" shows "??.Nested_Environment.update_option xs opt (Some e) = (if xs = [] then opt else Some (update xs opt e))"
conga_trans: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p"   and A'' :: "'p"   and B'' :: "'p"   and C'' :: "'p" assumes "A B C CongA A' B' C'"   and "A' B' C' CongA A'' B'' C''" shows "A B C CongA A'' B'' C''"
card_of_empty4: fixes A :: "'e set" shows "(|{}| <o |A|) = (A \<noteq> {})"
assms: shows "cupcake_match_result cenv v0 pes err_v = Rval (e, p, env')"
prime_elem_m_def: fixes f :: "int poly" shows "prime_elem_m f \<equiv> Mp f \<noteq> Mp 0 \<and> \<not> f dvdm 1 \<and> (\<forall>g h. f dvdm g * h \<longrightarrow> f dvdm g \<or> f dvdm h)"
new_tv_Int_free_tv_empty_scheme: fixes n :: "nat"   and sch :: "type_scheme" assumes "new_tv n sch" shows "{x. \<exists>y. x = n + y} \<inter> free_tv sch = {}"
eq_Min_iff: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set"   and m :: "'b" assumes "class.linorder less_eq less"   and "finite A"   and "A \<noteq> {}" shows "(m = linorder.Min less_eq A) = (m \<in> A \<and> (\<forall>a\<in>A. less_eq m a))"
fst_dominant_term_map_ms: fixes f :: "'a \<Rightarrow> 'b"   and basis :: "(real \<Rightarrow> real) list"   and basis' :: "(real \<Rightarrow> real) list"   and C :: "'a ms" assumes "is_lifting f basis basis'" shows "fst (dominant_term (map_ms f C)) = fst (dominant_term C)"
split_int_nb: fixes t :: "MIR.num" assumes "numbound0 t" shows "numbound0 (fst (split_int t)) \<and> numbound0 (snd (split_int t))"
cet_pair: fixes \<Gamma> :: "nat \<Rightarrow> pdf_type"   and e1 :: "cexpr"   and t1 :: "pdf_type"   and e2 :: "cexpr"   and t2 :: "pdf_type" assumes "\<Gamma> \<turnstile>\<^sub>c e1 : t1"   and "\<Gamma> \<turnstile>\<^sub>c e2 : t2" shows "\<Gamma> \<turnstile>\<^sub>c <e1, e2>\<^sub>c : PRODUCT t1 t2"
l_get_parent_wf_axioms: shows "l_get_parent_wf ShadowRootClass.type_wf ShadowRootClass.known_ptr ShadowRootClass.known_ptrs heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M Shadow_DOM.parent_child_rel Shadow_DOM.get_child_nodes Shadow_DOM.get_parent"
lcm_eq_0_iff: fixes a :: "'a"   and b :: "'a" shows "(lcm a b = (0::'a)) = (a = (0::'a) \<or> b = (0::'a))"
bounded_linear_sub: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "bounded_linear f"   and "bounded_linear g" shows "bounded_linear (\<lambda>x. f x - g x)"
zip_Cons_Cons: fixes x :: "'c"   and xs :: "'c list"   and y :: "'d"   and ys :: "'d list" shows "zip (x # xs) (y # ys) = (x, y) # zip xs ys"
code: fixes out :: "'e"   and c :: "('d, 'e, 'f) rpv" shows "Generative_Probabilistic_Value.Pause out c = GPV (return_spmf (IO out c))"
listrel_trans: fixes r :: "('g \<times> 'g) set" assumes "trans r" shows "trans (listrel r)"
bre_cond_def: fixes Qi :: "'d set" shows "bre_cond Qi \<equiv> {(Q, W). W \<noteq> {} \<and> Qi \<inter> Q = {}}"
assms: shows "(\<lambda>n. u n / real n) \<longlonglongrightarrow> l"
has_integral_integrable_integral: fixes f :: "'c \<Rightarrow> 'd"   and i :: "'d"   and s :: "'c set" shows "(f has_integral i) s = (f integrable_on s \<and> integral s f = i)"
simps_2: shows "ImZ \<i>\<^sub>\<int> = 1"
a_\<beta>: fixes p :: "fm"   and a :: "real"   and bs :: "real list"   and l :: "int"   and x :: "int" assumes "iszlfm p (a # bs)"   and "d_\<beta> p l"   and "0 < l" shows "iszlfm (a_\<beta> p l) (a # bs) \<and> d_\<beta> (a_\<beta> p l) 1 \<and> Ifm (real_of_int (l * x) # bs) (a_\<beta> p l) = Ifm (real_of_int x # bs) p"
dghm_singleton_ArrMap_vdomain: fixes j :: "V"   and \<CC> :: "V" shows "\<D>\<^sub>\<circ> (dghm_singleton j \<CC>\<lparr>ArrMap\<rparr>) = \<CC>\<lparr>Arr\<rparr>"
1: fixes \<mu>\<nu> :: "'c \<times> 'c" assumes "C.VV.arr \<mu>\<nu>" shows "D.VV.arr (FF \<mu>\<nu>)"
mkM_simps_2: fixes t :: "'s Trace"   and t' :: "'s Trace"   and T :: "'s Trace set"   and a :: "'a" assumes "(t, t') \<in> relations (mkM T) a" shows "jview a t = jview a t'"
quat_norm_pow_2: fixes q :: "quat" shows "(quat_of_real (norm q))\<^sup>2 = q * Quaternions.cnj q"
tensor_preserves_iso: fixes f :: "'a"   and g :: "'a" assumes "local.iso f"   and "local.iso g" shows "local.iso (f \<otimes>\<^sub>1 g)"
subst_add_disjnt: fixes t :: "'a"   and env\<^sub>1 :: "(name, 'a) fmap"   and env\<^sub>2 :: "(name, 'a) fmap" assumes "fdisjnt (frees t) (fmdom env\<^sub>1)" shows "subst t (env\<^sub>1 ++\<^sub>f env\<^sub>2) = subst t env\<^sub>2"
better_LImp_intro: fixes z :: "name"   and N :: "trm"   and y :: "name"   and P :: "trm"   and b :: "coname"   and a :: "coname"   and M :: "trm"   and x :: "name"   and c :: "coname" assumes "z \<sharp> (N, [y].P)"   and "b \<sharp> [a].M"   and "a \<sharp> N" shows "Cut <b>.ImpR x.<a>.M b z.ImpL <c>.N y.P z \<longrightarrow>\<^sub>l Cut <a>.Cut <c>.N x.M y.P"
simps_3: fixes xs :: "'f"   and x :: "'g"   and delta' :: "'h"   and inp' :: "'d \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qTerm option"   and binp' :: "'e \<Rightarrow> ('d, 'e, 'f, 'g, 'h) qAbs option" shows "(qVar xs x #= qOp delta' inp' binp') = False"
incseq: shows "incseq (\<lambda>i. {g (l i)<..<g (u i)})"
t_instance_s: shows "is_wt_instance_of_cond \<Gamma> t s"
sum_cong_aux: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "\<And>x. x \<in> A \<Longrightarrow> f x = g x" shows "sum f A = sum g A"
in_set_lookup_inserts_tries: fixes v :: "'c"   and key :: "'c \<Rightarrow> 'd list"   and vs :: "'c list"   and t :: "('d, 'c list) trie" shows "(v \<in> set (lookup_tries (inserts_tries key vs t) (key v))) = (v \<in> set vs \<union> set (lookup_tries t (key v)))"
poly_gcd_0_iff: fixes p :: "'b poly"   and q :: "'b poly"   and x :: "'b" shows "(poly (gcd p q) x = (0::'b)) = (poly p x = (0::'b) \<and> poly q x = (0::'b))"
pwalk_simps_1: fixes v :: "'a" assumes "v = s \<or> dist v = \<infinity> \<or> v \<notin> verts G" shows "pwalk v = []"
keys_subset_Keys: fixes f :: "'b \<Rightarrow>\<^sub>0 'c"   and F :: "('b \<Rightarrow>\<^sub>0 'c) set" assumes "f \<in> F" shows "keys f \<subseteq> Keys F"
lhd_cons_code: fixes g :: "'d \<Rightarrow> ('c, 'd) step"   and s :: "'d" shows "lhd_cons g s = (case g s of Skip s' \<Rightarrow> lhd_cons g s' | Yield x xa \<Rightarrow> x)"
linorder_cases: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "class.linorder less_eq less" obtains "less x y"   | "x = y"   | "less y x"
mrSet: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and r :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "applied_rule_rev Cp x p = \<lfloor>r\<rfloor>" shows "r \<in> set p"
continuous_at_within_powr: fixes a :: "'a"   and s :: "'a set"   and f :: "'a \<Rightarrow> real"   and g :: "'a \<Rightarrow> real" assumes "continuous (at a within s) f"   and "continuous (at a within s) g"   and "f a \<noteq> 0" shows "continuous (at a within s) (\<lambda>x. f x powr g x)"
interval_integral_substitution_integrable: fixes a :: "ereal"   and b :: "ereal"   and g :: "real \<Rightarrow> real"   and g' :: "real \<Rightarrow> real"   and f :: "real \<Rightarrow> 'a"   and A :: "ereal"   and B :: "ereal" assumes "a < b"   and "\<And>x. \<lbrakk>a < ereal x; ereal x < b\<rbrakk> \<Longrightarrow> (g has_real_derivative g' x) (at x)"   and "\<And>x. \<lbrakk>a < ereal x; ereal x < b\<rbrakk> \<Longrightarrow> isCont f (g x)"   and "\<And>x. \<lbrakk>a < ereal x; ereal x < b\<rbrakk> \<Longrightarrow> isCont g' x"   and "\<And>x. \<lbrakk>a \<le> ereal x; ereal x \<le> b\<rbrakk> \<Longrightarrow> 0 \<le> g' x"   and "((ereal \<circ> g \<circ> real_of_ereal) \<longlongrightarrow> A) (at_right a)"   and "((ereal \<circ> g \<circ> real_of_ereal) \<longlongrightarrow> B) (at_left b)"   and "set_integrable lborel (einterval a b) (\<lambda>x. g' x *\<^sub>R f (g x))"   and "set_integrable lborel (einterval A B) f" shows "interval_lebesgue_integral lborel A B f = LBINT x=a..b. g' x *\<^sub>R f (g x)"
closed_2: shows "has_all_wt_instances_of \<Gamma> (\<Union> ((set \<circ> fst \<circ> Ana) ` set M)) (set M)"
quote_all_insert: fixes p :: "perm"   and i :: "name"   and V :: "name set" shows "quote_all p (insert i V) = insert (QuoteP (Var i) (Var (p \<bullet> i))) (quote_all p V)"
simps_1: fixes carrier :: "'b set"   and mult :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and add :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and more :: "'c" shows "\<zero>\<^bsub>\<lparr>carrier = carrier, monoid.mult = mult, one = one, zero = zero, add = add, \<dots> = more\<rparr>\<^esub> = zero"
product_part_in_second_factor: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "O x y"   and "P z (x \<otimes> y)" shows "P z y"
coprime_n_idealsTr3: fixes n :: "nat"   and J :: "nat \<Rightarrow> 'a set" shows "(\<forall>k\<le>Suc n. ideal R (J k)) \<and> (\<forall>i\<le>Suc n. \<forall>j\<le>Suc n. i \<noteq> j \<longrightarrow> coprime_ideals R (J i) (J j)) \<longrightarrow> coprime_ideals R (i\<Pi>\<^bsub>R,n\<^esub> J) (J (Suc n))"
POR_lf_rep_def: fixes R :: "(Three \<Rightarrow> ValD) admS dual \<times> (Three \<Rightarrow> ValD) admS" shows "POR_lf_rep R \<equiv> POR_base_lf_rep R \<union> fn_lf_rep R"
assms_1: shows "has_least_prefixpoint (\<lambda>x. Rf x y)"
OV_inv1_obsv_round: fixes r :: "nat"   and S :: "process set"   and v :: "val"   and d_f :: "process \<Rightarrow> val option"   and Ob :: "process set" shows "{OV_inv1} obsv_round r S v d_f Ob {> OV_inv1}"
sappend_def: fixes Al :: "'b list set"   and Bl :: "'b list set" shows "Al @@ Bl = {al @ bl |al bl. al \<in> Al \<and> bl \<in> Bl}"
mismatch_incopm: fixes u :: "'b list"   and v :: "'b list"   and x :: "'b"   and y :: "'b" assumes "\<^bold>|u\<^bold>| = \<^bold>|v\<^bold>|"   and "x \<noteq> y" shows "\<not> u \<cdot> [x] \<bowtie> v \<cdot> [y]"
l_set_tag_name_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_set_tag_name_get_child_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.type_wf set_tag_name set_tag_name_locs DocumentClass.known_ptr Core_DOM_Functions.get_child_nodes Core_DOM_Functions.get_child_nodes_locs"
assms_2: shows "x \<sharp> \<langle>A\<^sub>F, \<Psi>\<^sub>F\<rangle>"
inorder_join: fixes l :: "'b tree"   and r :: "'b tree" shows "inorder (join l r) = inorder l @ inorder r"
finsubset: shows "finite ({x. f x \<noteq> c} - {i, b})"
while_rule: fixes P :: "'b \<Rightarrow> bool"   and s :: "'b"   and b :: "'b \<Rightarrow> bool"   and c :: "'b \<Rightarrow> 'b"   and Q :: "'b \<Rightarrow> bool"   and r :: "('b \<times> 'b) set" assumes "P s"   and "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> P (c s)"   and "\<And>s. \<lbrakk>P s; \<not> b s\<rbrakk> \<Longrightarrow> Q s"   and "wf r"   and "\<And>s. \<lbrakk>P s; b s\<rbrakk> \<Longrightarrow> (c s, s) \<in> r" shows "Q (while b c s)"
set_return_spmf: fixes x :: "'d" shows "set_spmf (return_spmf x) = {x}"
neq_le_trans: fixes a :: "'d"   and b :: "'d" assumes "a \<noteq> b"   and "b \<le> a" shows "b < a"
assms_3: shows "\<exists>u. ordinal_utility (\<P> outcomes) \<R> (\<lambda>x. measure_pmf.expectation x u)"
if_eval_elim: fixes e :: "'BExp"   and c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and annos :: "'Var ModeUpd list"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>Stmt.If e c\<^sub>1 c\<^sub>2 \<otimes> annos, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "(c' = c\<^sub>1 \<and> ev\<^sub>B mem e \<or> c' = c\<^sub>2 \<and> \<not> ev\<^sub>B mem e) \<and> mds' = mds \<oplus> annos \<and> mem' = mem"
REP_in_hhom_1: fixes F :: "('a, 'a set) arr"   and A :: "('a, 'a set) arr"   and B :: "('a, 'a set) arr" assumes "\<guillemotleft>F : A \<rightarrow> B\<guillemotright>" shows "\<guillemotleft>REP F : src (REP A) \<rightarrow>\<^sub>B src (REP B)\<guillemotright>"
effect_lengthE: fixes a :: "'b array"   and h :: "heap"   and h' :: "heap"   and r :: "nat" assumes "effect (Array.len a) h h' r" obtains "r = Array.length h' a" and "h' = h"
Rep_downset_cases: fixes y :: "'a set" assumes "y \<in> range \<down>" obtains x :: "'a downset" where "y = Rep_downset x"
rel_set_pos_distr_iff: fixes A :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and A' :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "rel_set_pos_distr_cond A A' = True"
id_left: fixes f :: "'a set_arrow" assumes "f \<in> Ar\<^bsub>BB\<^esub>" shows "Id\<^bsub>BB\<^esub> (Cod\<^bsub>BB\<^esub> f) \<bullet>\<^bsub>BB\<^esub> f = f"
rel_eq: shows "rel_optionT' (=) (=) = (=)"
nn_integral_cong: fixes M :: "'b measure"   and u :: "'b \<Rightarrow> ennreal"   and v :: "'b \<Rightarrow> ennreal" assumes "\<And>x. x \<in> space M \<Longrightarrow> u x = v x" shows "integral\<^sup>N M u = integral\<^sup>N M v"
abs_eq: fixes x :: "64 word" shows "uint_of_uint64 (Abs_uint64 x) = Abs_uint (ucast x)"
simps: fixes ant :: "'a form multiset"   and suc :: "'a form" shows "succ ( ant \<Rightarrow>* suc) = suc"
cases: fixes x :: "'b poly \<times> 'b poly" obtains p :: "'b poly" and q :: "'b poly" where "x = (p, q)"
reject: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile>\<^sub>g \<langle>[Rule m Reject], Undecided\<rangle> \<Rightarrow> Decision FinalDeny"
Im_Ln_less_pi: fixes z :: "complex" assumes "z \<notin> \<real>\<^sub>\<le>\<^sub>0" shows "Im (Ln z) < pi"
rotate_rl: fixes k :: "nat"   and xs :: "'b list" shows "rotate k (rotater k xs) = xs"
dropWhile_cong: fixes l :: "'a list"   and k :: "'a list"   and P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool" assumes "l = k"   and "\<And>x. x \<in> list.set l \<Longrightarrow> P x = Q x" shows "dropWhile P l = dropWhile Q k"
semiring_simprules_8: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<otimes> y \<otimes> z = x \<otimes> (y \<otimes> z)"
init: fixes \<sigma> :: "'a"   and p :: "'b" assumes "(\<sigma>, p) \<in> init A" shows "P \<sigma>"
analz_Un_PairAgentSet: fixes G :: "msg set"   and H :: "msg set" shows "analz (PairSet AgentSet G \<union> H) \<subseteq> PairSet AgentSet G \<union> AgentSet \<union> analz (G \<union> H)"
no_loops: fixes e :: "'b" assumes "e \<in> arcs G" shows "tail G e \<noteq> head G e"
IH: fixes x2a :: "(('fun, 'atom, 'sets) prot_fun, 'a) Term.term" assumes "x2a \<in> set T"   and "wf\<^sub>t\<^sub>r\<^sub>m x2a" shows "finite {a. timpls_transformable_to_pred A x2a a}"
simps_3: shows "eval_ternary_And TernaryFalse TernaryTrue = TernaryFalse"
shp_def: shows "shp \<equiv> snd \<circ> snd"
last_message_append: fixes xs :: "'b message_af list"   and ys :: "'b message_af list" shows "last_message (xs @ ys) = (if last_message ys = NoMsg then last_message xs else last_message ys)"
trimmingD_le: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and e :: "'a \<times> 'a" assumes "trimming \<Gamma> f g" shows "g e \<le> f e"
last_butlast_tl: fixes bs :: "('d \<times> 'e) list"   and x :: "'d \<times> 'e"   and as :: "('d \<times> 'e) list" assumes "butlast bs = x # butlast as"   and "bs \<noteq> []"   and "as \<noteq> []"   and "fst (last as) = fst (last bs)"   and "snd (last as) = snd (last bs)" shows "as = tl bs"
disc_9: fixes x2 :: "('e, 'f) Term.term" shows "stateful_strand_step.is_Receive (receive\<langle>x2\<rangle>)"
bezout_ring_iff_fin_gen_principal_ideal2: shows "(\<forall>I. finitely_generated_ideal I \<longrightarrow> principal_ideal I) = class.bezout_ring (*) (1::'b) (+) (0::'b) (-) uminus"
norm_axioms_def: fixes V :: "'c set"   and norm :: "'c \<Rightarrow> real" shows "norm_axioms V norm \<equiv> \<forall>x. x \<in> V \<longrightarrow> (norm x = 0) = (x = (0::'c))"
min_height_balanced: fixes kdt :: "'b kdt" assumes "balanced kdt" shows "min_height kdt = nat \<lfloor>log 2 (real (KD_Tree.size_kdt kdt))\<rfloor>"
s1_in_S: shows "s1 \<in> S\<^bsub>SES\<^esub>"
has_overlap_lst_def: fixes rects :: "'a rectangle list" shows "has_overlap_lst rects = (let ops = all_ops rects in \<exists>k<length ops. has_overlap_at_k rects k)"
j_def: shows "j = (GREATEST a. \<not> is_zero_row a (Gauss_Jordan A))"