hcju/isabelleps huggingface dataset:
intersection_of_empty: fixes P :: "'b set set \<Rightarrow> bool"   and Q :: "'b set \<Rightarrow> bool" assumes "P {}" shows "(P intersection_of Q) UNIV"
adds_pp_trans: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and s :: "'b"   and t :: "'b"   and v :: "'a" assumes "term_powerprod pair_of_term term_of_pair"   and "s adds t"   and "term_powerprod.adds_pp pair_of_term t v" shows "term_powerprod.adds_pp pair_of_term s v"
list_all_map: fixes f :: "'b \<Rightarrow> bool"   and g :: "'c \<Rightarrow> 'b"   and l :: "'c list" shows "list_all f (map g l) = list_all (f \<circ> g) l"
takeWhile_dropWhile_id: fixes P :: "'f \<Rightarrow> bool"   and xs :: "'f list" shows "takeWhile P xs @ dropWhile P xs = xs"
index_f_def: shows "MN.index_f \<equiv> SOME h. bij_betw h (UNIV // eq_app_right MN.language) (hfset (ord_of MN.min_states))"
rangeE: fixes b :: "'c"   and f :: "'d \<Rightarrow> 'c" assumes "b \<in> range f" obtains x :: "'d" where "b = f x"
microstep_callD: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and pn :: "'a"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "(call(pn)) \<leadsto>\<^bsub>\<Gamma>\<^esub> p" shows "p = \<Gamma> pn"
hyps: assumes "tail_invar bq"   and "\<forall>t\<in>set bq. rank t < rank t'"   and "tree_invar t'" shows "tail_invar (bq @ [t'])"
Cons: fixes xs :: "'a list"   and ys :: "'a list"   and z :: "'a" assumes "xs <~~> ys" shows "z # xs <~~> z # ys"
i_set_MOD: fixes r :: "nat"   and m :: "nat" shows "[ r, mod m ] \<in> i_set"
simps_2: fixes uu :: "('b \<times> nat) tree"   and uv :: "'b \<times> nat" shows "sngl \<langle>uu, uv, \<langle>\<rangle>\<rangle> = True"
orthogonal_complement_def: fixes S :: "'b set" shows "orthogonal_complement S = {x |x. \<forall>y\<in>S. is_orthogonal x y}"
f_def: shows "f = foldl Gauss_Jordan_column_k_det_P (1::'a, 0, A) [0..<Suc k]"
s_hyp_1: shows "S \<subseteq> set (cols (conjugate A))"
progress: fixes v :: "val"   and e :: "exp"   and \<rho> :: "(nat \<times> val) list" assumes "v \<in> E e \<rho>"   and "\<rho> = []"   and "FV e = {}" shows "is_val e \<or> (\<exists>e'. e \<longrightarrow> e')"
hyps: fixes ys :: "'b list" assumes "mset ys = mset xs"   and "\<And>k. k \<in> set ys \<Longrightarrow> filter (\<lambda>x. f k = f x) ys = filter (\<lambda>x. f k = f x) xs"   and "sorted (map f ys)" shows "sort_key f xs = ys"
real_vector_space_on_axioms: shows "real_vector_space_on S1"
div_apart: fixes x :: "nat"   and y :: "nat" assumes "is_even x"   and "is_even y" shows "(x + y) div 2 = x div 2 + y div 2"
intro: fixes U :: "'a set"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "semiring_ow U plus times"   and "ab_group_add_ow U plus zero minus uminus" shows "ring_ow U plus zero minus uminus times"
self_append_conv: fixes xs :: "'f list"   and ys :: "'f list" shows "(xs = xs @ ys) = (ys = [])"
CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P_has_all_failures_Un: fixes P :: "'a process" shows "CHAOS\<^sub>S\<^sub>K\<^sub>I\<^sub>P UNIV \<sqsubseteq>\<^sub>F P"
elims_3: fixes x :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list"   and xa :: "acom"   and xb :: "'a \<Rightarrow> nat"   and xc :: "char list \<Rightarrow> int" assumes "\<not> preList x xa xb xc" obtains xd :: "'a" and e :: "(char list \<Rightarrow> int) \<Rightarrow> nat" and xs :: "('a \<times> ((char list \<Rightarrow> int) \<Rightarrow> nat)) list" and C :: "acom" and l :: "'a \<Rightarrow> nat" and s :: "char list \<Rightarrow> int" where "x = (xd, e) # xs" and "xa = C" and "xb = l" and "xc = s" and "\<not> (l xd = preT C e s \<and> preList xs C l s)"
R1: fixes \<Psi>' :: "'b"   and Q :: "('a, 'b, 'c) psi" shows "(\<Psi>', Q, Q) \<in> Rel"
simps_1: fixes f :: "'b \<Rightarrow> 'a" shows "psubstts f [] = []"
assms_3: assumes "rho zs z = None" shows "freshAbs zs z A"
invariant3p_ts_inv: fixes final :: "'a \<Rightarrow> bool"   and r :: "('d, 'b, 'a, 'c, 'e, 'f) semantics"   and P :: "'g \<Rightarrow> 'b \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> bool"   and convert_RA :: "'d \<Rightarrow>f nat \<Rightarrow> 'f list" assumes "lifting_inv final r P" shows "invariant3p (multithreaded_base.redT final r convert_RA) {s. \<exists>I. ts_inv P I (thr s) (shr s)}"
F_to_idx_impl_def: fixes F :: "'e set set" shows "F_to_idx_impl F \<equiv> SPEC (\<lambda>Flist. distinct Flist \<and> set Flist = F) \<bind> (\<lambda>Flist. let num_acc = length Flist in mk_acc_impl Flist \<bind> (\<lambda>acc. RETURN (num_acc, acc)))"
TC: fixes a :: "V" shows "TC a = a \<squnion> \<Squnion> (TC ` elts a)"
not_less_zero: fixes n :: "'d multiset" shows "\<not> n \<subset># {#}"
elims_3: fixes x :: "'a list"   and xa :: "('a \<times> 'a) list" assumes "\<not> wf_list_graph_impl x xa" obtains V :: "'a list" and v1 :: "'a" and v2 :: "'a" and Es :: "('a \<times> 'a) list" where "x = V" and "xa = (v1, v2) # Es" and "\<not> (v1 \<in> set V \<and> v2 \<in> set V \<and> wf_list_graph_impl V Es)"
pmf_of_set_def: fixes S :: "'b set" assumes "S \<noteq> {}"   and "finite S" shows "pmf_of_set S \<equiv> embed_pmf (\<lambda>x. indicat_real S x / real (card S))"
kcomp_klift: fixes f :: "'b \<Rightarrow> 'd set"   and g :: "'d \<Rightarrow> 'c set" shows "f \<circ>\<^sub>K g = bd\<^sub>\<F> g \<circ> f"
emeasure: fixes A :: "'a set"   and B :: "'b set" assumes "A \<in> sets M1"   and "B \<in> sets M2" shows "emeasure M1 A * emeasure M2 B = emeasure M (A \<times> B)"
infsetsum_Times': fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and A :: "'c set"   and B :: "'d set" assumes "(\<lambda>(x, y). f x y) abs_summable_on A \<times> B" shows "(\<Sum>\<^sub>ax\<in>A. infsetsum (f x) B) = (\<Sum>\<^sub>a(x, y)\<in>A \<times> B. f x y)"
ex_unit_disc_fix_mapping_u_to_zero: fixes u :: "complex_homo" assumes "u \<in> unit_disc" shows "\<exists>M. unit_disc_fix M \<and> moebius_pt M u = 0\<^sub>h"
env: shows "E V' = \<lfloor>T\<rfloor>"
path_connected_open_delete: fixes S :: "'a set"   and a :: "'a" assumes "open S"   and "connected S"   and "2 \<le> DIM('a)" shows "path_connected (S - {a})"
pdivides_zero: fixes K :: "'a list set"   and p :: "'a list list" assumes "subring K (K [X])"   and "p \<in> carrier (K [X]\<^bsub>K [X]\<^esub>)" shows "p pdivides\<^bsub>K [X]\<^esub> []"
braun_sift_down: fixes l :: "'b tree"   and a :: "'b"   and r :: "'b tree" assumes "braun \<langle>l, a, r\<rangle>" shows "braun (sift_down l a r)"
pgallery_map: fixes X :: "'b set set"   and Y :: "'c set set"   and f :: "'b \<Rightarrow> 'c"   and Cs :: "'b set list" assumes "ChamberComplexIsomorphism X Y f"   and "SimplicialComplex.pmaxsimpchain X Cs" shows "SimplicialComplex.pmaxsimpchain Y (f \<Turnstile> Cs)"
\<tau>J_fmla: fixes \<psi> :: "'fmla" assumes "\<psi> \<in> fmla"   and "Fvars \<psi> = {inp}" shows "\<tau>J \<psi> \<in> fmla"
constructive_security_obsf_absorb_cong: fixes real_resource :: "('y + 'z, 'aa + 'ab) resource"   and ideal_resource :: "('ac + 'z, 'ad + 'ab) resource"   and sim :: "('y, 'aa, 'ac, 'ad) converter"   and \<I>_real :: "('y, 'aa) \<I>"   and \<I>_ideal :: "('ac, 'ad) \<I>"   and \<I>_common :: "('z, 'ab) \<I>"   and \<A> :: "('ae, 'af) distinguisher_obsf"   and cnv :: "('ae, 'af, 'y + 'z, 'aa + 'ab) converter_obsf"   and adv :: "real"   and \<I> :: "('ae, 'af) \<I>"   and cnv' :: "('ae, 'af, 'y + 'z, 'aa + 'ab) converter_obsf" assumes "constructive_security_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common (absorb \<A> cnv) adv"   and "exception_\<I> \<I>, exception_\<I> (\<I>_real \<oplus>\<^sub>\<I> \<I>_common) \<turnstile>\<^sub>C cnv \<surd>"   and "exception_\<I> \<I>, exception_\<I> (\<I>_real \<oplus>\<^sub>\<I> \<I>_common) \<turnstile>\<^sub>C cnv' \<surd>"   and "exception_\<I> \<I> \<turnstile>g \<A> \<surd>"   and "exception_\<I> \<I>,exception_\<I> (\<I>_real \<oplus>\<^sub>\<I> \<I>_common) \<turnstile>\<^sub>C cnv \<sim> cnv'" shows "constructive_security_obsf real_resource ideal_resource sim \<I>_real \<I>_ideal \<I>_common (absorb \<A> cnv') adv"
simulate_plan_correct: fixes \<pi>s :: "char list list"   and s :: "nat \<Rightarrow> nat option"   and s' :: "nat \<Rightarrow> nat option" shows "(simulate_plan \<pi>s s = Some s') = path_to s \<pi>s s'"
lunit'_simps_2: fixes f :: "'b" assumes "D.ide f" shows "src\<^sub>D \<l>\<^sub>D\<^sup>-\<^sup>1[f] = src\<^sub>D f"
sorted_prefix: fixes xs :: "'b list"   and ys :: "'b list" assumes "xs \<le> ys"   and "sorted ys" shows "sorted xs"
le_dag_antisym: fixes x :: "dag"   and y :: "dag" assumes "x \<le> y"   and "y \<le> x" shows "x = y"
fset_simps_2: fixes x :: "'a"   and xa :: "'a fset" shows "fset (finsert x xa) = insert x (fset xa)"
slp_of_fa_rev: fixes fa :: "floatarith"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list" shows "slp_of_fa_rev fa M slp (length slp) = (let (M, slp') = slp_of_fa fa M (rev slp) in (M, rev slp', length slp'))"
pvalid_is_derivation_terminals_item_\<beta>: fixes p :: "(symbol \<times> character list) list"   and x :: "item" assumes "pvalid p x" shows "\<exists>\<delta>. is_derivation (terminals p @ item_\<beta> x @ \<delta>)"
deutsch_transform_coeff_1: shows "U\<^sub>f $$ (0, 1) = of_nat (f 0)"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>C) (\<cdot>\<^sub>C) (F \<circ> G) (F \<circ> G) (F \<circ> G)"
cone_list_def: fixes Vs :: "'a vec list" shows "cone_list Vs = {b. \<exists>c. nonneg_lincomb_list c Vs b}"
conf_AddrI: fixes s :: "st"   and a :: "loc"   and obj :: "obj"   and G :: "prog"   and T :: "ty" assumes "heap s a = Some obj"   and "G\<turnstile>obj_ty obj\<preceq>T" shows "G,s\<turnstile>Addr a\<Colon>\<preceq>T"
zmult_zminus_right: fixes w :: "int"   and z :: "int" shows "w * - z = - (w * z)"
simps_8: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case Two of One \<Rightarrow> f1 | Two \<Rightarrow> f2 | Three \<Rightarrow> f3) = f2"
Spectrum_def: fixes A :: "(('b, 'c) Finite_Cartesian_Product.vec, 'c) Finite_Cartesian_Product.vec" shows "Spectrum A = Collect (eigen_value A)"
a_comm: fixes G :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "abelian_monoid G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "x \<oplus>\<^bsub>G\<^esub> y = y \<oplus>\<^bsub>G\<^esub> x"
length_map_upt: fixes f :: "nat \<Rightarrow> 'b"   and a :: "nat"   and b :: "nat" shows "length (map f [a..<b]) = b - a"
inline1_def: fixes callee :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g, 'h, 'i) gpv" shows "inline1 callee \<equiv> curry (spmf.fixp_fun (\<lambda>inline1 (gpv, s). the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). curry inline1 (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv))))))"
ct_eq_flip: shows "ct_eq\<inverse>\<inverse> = ct_eq"
pwd_owner_iii_def: fixes n :: "nat" shows "pwd_owner_iii n \<equiv> SOME x. pred_owner_iii n (pwd_spy_ii n) x"
some_elem_def: fixes s :: "'a set" shows "some_elem s = (SOME x. x \<in> s)"
attach_shadow_root_def: fixes known_ptr :: "(_) object_ptr \<Rightarrow> bool"   and set_shadow_root :: "(_) element_ptr \<Rightarrow> (_) shadow_root_ptr option \<Rightarrow> ((_) heap, exception, unit) prog"   and set_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and set_mode :: "(_) shadow_root_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, unit) prog"   and set_mode_locs :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and attach_shadow_root :: "(_) element_ptr \<Rightarrow> shadow_root_mode \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr) prog"   and type_wf :: "(_) heap \<Rightarrow> bool"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and get_shadow_root :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, (_) shadow_root_ptr option) prog"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and element_ptr :: "(_) element_ptr"   and shadow_root_mode :: "shadow_root_mode" assumes "l_attach_shadow_root\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M known_ptr set_shadow_root set_shadow_root_locs set_mode set_mode_locs attach_shadow_root type_wf get_tag_name get_tag_name_locs get_shadow_root get_shadow_root_locs" shows "attach_shadow_root element_ptr shadow_root_mode = Heap_Error_Monad.bind (get_tag_name element_ptr) (\<lambda>tag. Heap_Error_Monad.bind (if tag \<notin> safe_shadow_root_element_types then error HierarchyRequestError else return ()) (\<lambda>_. Heap_Error_Monad.bind (get_shadow_root element_ptr) (\<lambda>prev_shadow_root. Heap_Error_Monad.bind (if prev_shadow_root \<noteq> None then error HierarchyRequestError else return ()) (\<lambda>_. Heap_Error_Monad.bind new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M (\<lambda>new_shadow_root_ptr. Heap_Error_Monad.bind (set_mode new_shadow_root_ptr shadow_root_mode) (\<lambda>_. Heap_Error_Monad.bind (set_shadow_root element_ptr (Some new_shadow_root_ptr)) (\<lambda>_. return new_shadow_root_ptr)))))))"
reflpI: fixes r :: "'g \<Rightarrow> 'g \<Rightarrow> bool" assumes "\<And>x. r x x" shows "reflp r"
f2_le_f2'e: shows "f2 \<le> f2' * e"
image_iff: fixes z :: "'a"   and f :: "'c \<Rightarrow> 'a"   and A :: "'c set" shows "(z \<in> f ` A) = (\<exists>x\<in>A. z = f x)"
cmp_simps'_1: fixes \<mu> :: "'b"   and \<nu> :: "'b" assumes "B.arr \<mu>"   and "B.arr \<nu>"   and "src\<^sub>B \<mu> = trg\<^sub>B \<nu>" shows "C.arr (\<Phi>\<^sub>H (\<mu>, \<nu>))"
lift_preserves_beta': fixes r :: "Core.term"   and s :: "Core.term"   and i :: "nat" assumes "r \<rightarrow>\<^sub>\<beta>\<^sup>* s" shows "lift r i \<rightarrow>\<^sub>\<beta>\<^sup>* lift s i"
fresh_name'_def: shows "fresh_name' \<equiv> map_fun ((`) literal.explode) literal.Abs_literal fresh_name"
graph_empty_e: fixes v :: "'b set" shows "graph (LG {} v)"
a1: assumes "(a, b) \<in>\<^sub>p set qs" shows "thesis"
"1_3": shows "j \<le> f i \<or> nr \<le> i"
Lambert_W_eq_minus_exp_minus1: shows "Lambert_W (- exp (- 1)) = - 1"
intro: fixes A :: "'c set"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "antisymmetric A less_eq"   and "transitive A less_eq" shows "near_ordered_set A less_eq"
irr: fixes ga :: "int poly"   and h :: "int poly" assumes "degree_m ga = 0"   and "degree_m h = 0"   and "[:0, 1:] = Mp (ga * h)" shows "False"
cases: fixes x :: "'a bro" obtains a :: "'a" where "x = L2 a"   | t1 :: "'a bro" and a1 :: "'a" and t2 :: "'a bro" and a2 :: "'a" and t3 :: "'a bro" where "x = N3 t1 a1 t2 a2 t3"   | "x = N0"   | v :: "'a bro" where "x = N1 v__"   | v :: "'a bro" and va :: "'a" and vb :: "'a bro" where "x = N2 v va vb"
filtermap_at_top_linear_eq: fixes c :: "'c"   and b :: "'c" assumes "c \<noteq> (0::'c)" shows "filtermap (\<lambda>x. x * c + b) at_top = (if (0::'c) < c then at_top else at_bot)"
special_coset_inj: fixes a :: "'w" shows "inj_on ((+o) a \<circ> genby) (Pow S)"
IH_1: fixes \<sigma> :: "nat \<Rightarrow> val" assumes "\<forall>x\<in>free_vars b. val_type (\<sigma> x) = \<Gamma> x" shows "sets (expr_sem \<sigma> b) = sets (stock_measure BOOL)"
stable_onI: fixes ds :: "'d set"   and X :: "'x set" assumes "individually_rational_on ds X"   and "stable_no_blocking_on ds X" shows "stable_on ds X"
conjugate_zero_iff: fixes conjugate :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.conjugatable_ring conjugate plus zero minus uminus times" shows "(conjugate a = zero) = (a = zero)"
is_regular_add_sheaf_spec: fixes s :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and U :: "'a set set"   and s' :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "is_regular s U"   and "is_regular s' U"   and "U \<subseteq> Spec" shows "is_regular (add_sheaf_spec U s s') U"
echelon_form_of_def: fixes A :: "(('d, 'e) vec, 'f) vec"   and bezout :: "'d \<Rightarrow> 'd \<Rightarrow> 'd \<times> 'd \<times> 'd \<times> 'd \<times> 'd" shows "echelon_form_of A bezout = echelon_form_of_upt_k A (ncols A - 1) bezout"
prems_2: shows "a \<in> T ` insert x A \<rightarrow> carrier K"
lub_is_union: fixes S :: "'d set set" shows "lub S = \<Union> S"
C\<^sub>i\<^sub>i: fixes y :: "i" shows "((\<^bold>\<forall>x. cod y \<cdot> x \<cong> x \<^bold>\<leftarrow> E (cod y \<cdot> x)) \<^bold>\<and> (\<^bold>\<forall>x. x \<cdot> cod y \<cong> x \<^bold>\<leftarrow> E (x \<cdot> cod y))) \<^bold>\<and> cod y \<cdot> y \<cong> y \<^bold>\<leftarrow> E y"
g_2: shows "inj_on g ({a..b} \<union> s)"
big_step_induct: fixes x1a :: "com"   and x1b :: "char list \<Rightarrow> int"   and x2a :: "char list \<Rightarrow> int"   and P :: "com \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int) \<Rightarrow> bool" assumes "(x1a, x1b) \<Rightarrow> x2a"   and "\<And>s. P SKIP s s"   and "\<And>x a s. P (x ::= a) s (s[a/x])"   and "\<And>c\<^sub>1 s\<^sub>1 s\<^sub>2 c\<^sub>2 s\<^sub>3. \<lbrakk>(c\<^sub>1, s\<^sub>1) \<Rightarrow> s\<^sub>2; P c\<^sub>1 s\<^sub>1 s\<^sub>2; (c\<^sub>2, s\<^sub>2) \<Rightarrow> s\<^sub>3; P c\<^sub>2 s\<^sub>2 s\<^sub>3\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) s\<^sub>1 s\<^sub>3"   and "\<And>b s c\<^sub>1 t c\<^sub>2. \<lbrakk>bval b s; (c\<^sub>1, s) \<Rightarrow> t; P c\<^sub>1 s t\<rbrakk> \<Longrightarrow> P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) s t"   and "\<And>b s c\<^sub>2 t c\<^sub>1. \<lbrakk>\<not> bval b s; (c\<^sub>2, s) \<Rightarrow> t; P c\<^sub>2 s t\<rbrakk> \<Longrightarrow> P (IF b THEN c\<^sub>1 ELSE c\<^sub>2) s t"   and "\<And>b s c. \<not> bval b s \<Longrightarrow> P (WHILE b DO c) s s"   and "\<And>b s\<^sub>1 c s\<^sub>2 s\<^sub>3. \<lbrakk>bval b s\<^sub>1; (c, s\<^sub>1) \<Rightarrow> s\<^sub>2; P c s\<^sub>1 s\<^sub>2; (WHILE b DO c, s\<^sub>2) \<Rightarrow> s\<^sub>3; P (WHILE b DO c) s\<^sub>2 s\<^sub>3\<rbrakk> \<Longrightarrow> P (WHILE b DO c) s\<^sub>1 s\<^sub>3" shows "P x1a x1b x2a"
elims: fixes x :: "'a list"   and y :: "'a option" assumes "safe_hd x = y" obtains xa :: "'a" and uu_ :: "'a list" where "x = xa # uu_" and "y = Some xa"   | "x = []" and "y = None"
f_simps_2: fixes n :: "nat"   and y :: "'v" shows "f n (SOURCE, \<langle>y\<rangle>) = (if y = b then d_IN (h n) b - d_IN (h 0) b else 0)"
induct: fixes P :: "'b \<Rightarrow> 'b tree \<Rightarrow> bool"   and a0 :: "'b"   and a1 :: "'b tree" assumes "\<And>p. P p \<langle>\<rangle>"   and "\<And>p al a ar. \<lbrakk>\<And>x21 x22 x23. \<lbrakk>a \<le> p; ar = \<langle>x21, x22, x23\<rangle>; x22 \<le> p\<rbrakk> \<Longrightarrow> P p x23; \<And>x21 x22 x23. \<lbrakk>a \<le> p; ar = \<langle>x21, x22, x23\<rangle>; \<not> x22 \<le> p\<rbrakk> \<Longrightarrow> P p x21; \<And>x21 x22 x23. \<lbrakk>\<not> a \<le> p; al = \<langle>x21, x22, x23\<rangle>; x22 \<le> p\<rbrakk> \<Longrightarrow> P p x23; \<And>x21 x22 x23. \<lbrakk>\<not> a \<le> p; al = \<langle>x21, x22, x23\<rangle>; \<not> x22 \<le> p\<rbrakk> \<Longrightarrow> P p x21\<rbrakk> \<Longrightarrow> P p \<langle>al, a, ar\<rangle>" shows "P a0 a1"
A_def: shows "A = (\<lambda>z. contour_integral (linepath 0 z) (\<lambda>t. exp (- t\<^sup>2)))"
of_ocircline_def: shows "of_ocircline \<equiv> map_fun rep_ocircline abs_circline id"
openin_inverse: fixes x :: "'b topology" shows "topology (openin x) = x"
vector_sub_project_orthogonal: fixes b :: "'a"   and x :: "'a" shows "b \<bullet> (x - (b \<bullet> x / (b \<bullet> b)) *\<^sub>R b) = 0"
mult_pCons_right: fixes p :: "'c poly"   and a :: "'c"   and q :: "'c poly" shows "p * pCons a q = smult a p + pCons (0::'c) (p * q)"
bounded_image: fixes b :: "real" shows "bounded (f ` {0..b})"
neq_iff: fixes x :: "'b word"   and y :: "'b word" shows "(x \<noteq> y) = (x <s y \<or> y <s x)"
knows2know_k: fixes A :: "specID"   and m :: "Keys" assumes "knows A [kE m]" shows "know A (kKS m)"
repeat_mset_right: fixes a :: "nat"   and b :: "nat"   and A :: "'b multiset" shows "repeat_mset a (repeat_mset b A) = repeat_mset (a * b) A"
\<chi>: shows "\<guillemotleft>\<chi> : p\<^sub>1 \<star> w \<Rightarrow> k\<^sup>* \<star> h \<star> p\<^sub>0 \<star> w\<guillemotright>"
lfinite_LConsI: fixes xs :: "'b llist"   and x :: "'b" assumes "lfinite xs" shows "lfinite (x % xs)"
bst_delete: fixes t :: "('a \<times> 'b) tree"   and x :: "'a" assumes "Tree2.bst t" shows "Tree2.bst (delete x t)"
inv_inv: fixes x :: "nat" assumes "x \<in> carrier G" shows "inv\<^bsub>G\<^esub> (inv\<^bsub>G\<^esub> x) = x"
finite_set_decidable: fixes X :: "nat set" assumes "finite X" shows "decidable X"
e_prim_cong: fixes e1 :: "exp"   and e1' :: "exp"   and e2 :: "exp"   and e2' :: "exp"   and f :: "nat \<Rightarrow> nat \<Rightarrow> nat" assumes "E e1 = E e1'"   and "E e2 = E e2'" shows "E (EPrim f e1 e2) = E (EPrim f e1' e2')"
polyn_Suc: fixes n :: "nat"   and c :: "nat \<times> (nat \<Rightarrow> 'a)" assumes "Suc n \<le> fst c" shows "polyn_expr R X (Suc n) (Suc n, snd c) = polyn_expr R X n c \<plusminus> snd c (Suc n) \<cdot>\<^sub>r X^\<^bsup>R Suc n\<^esup>"
ExcessNotAtRecList_subset: fixes ps :: "(nat \<times> nat) list"   and g :: "graph" shows "set (ExcessNotAtRecList ps g) \<subseteq> set (map fst ps)"
mult_pos_pos: fixes a :: "'m"   and b :: "'m" assumes "(0::'m) < a"   and "(0::'m) < b" shows "(0::'m) < a * b"
Abs_bset_inverse': fixes A :: "'c set" assumes "|A| <o natLeq +c |UNIV|" shows "set_bset (Abs_bset A) = A"
restrictA_cong: fixes m1 :: "('a \<times> 'b) list"   and V :: "'a set"   and V' :: "'a set"   and m2 :: "('a \<times> 'b) list" assumes "\<And>x. x \<in> domA m1 \<Longrightarrow> (x \<in> V) = (x \<in> V')"   and "m1 = m2" shows "restrictA V m1 = restrictA V' m2"
measurable_restrict_space1: fixes f :: "'a \<Rightarrow> 'b"   and M :: "'a measure"   and N :: "'b measure"   and \<Omega> :: "'a set" assumes "f \<in> M \<rightarrow>\<^sub>M N" shows "f \<in> restrict_space M \<Omega> \<rightarrow>\<^sub>M N"
store_word_mem_privilege: fixes s :: "'b sparc_state"   and addr :: "word32"   and data :: "word32"   and byte_mask :: "word4"   and asi :: "word8"   and s' :: "'b sparc_state" assumes "store_word_mem s addr data byte_mask asi = Some s' \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
Units_inv_closed: fixes x :: "'a" assumes "x \<in> Units G" shows "inv x \<in> carrier G"
a_closed: fixes x :: "'a"   and y :: "'a" shows "apd_d (ad x \<cdot> ad y) = ad x \<cdot> ad y"
treesize_Leaf: fixes T :: "tree" assumes "treesize T = 0" shows "T = Leaf"
preserves_src: fixes x :: "'c" shows "E (src\<^sub>C x) = src (E x)"
case_2: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "nat set \<Rightarrow> 'e \<Rightarrow> 'd"   and f3 :: "nat \<Rightarrow> 'e \<Rightarrow> 'd"   and f4 :: "nat \<Rightarrow> 'd"   and f5 :: "'e \<Rightarrow> 'd"   and f6 :: "nat \<Rightarrow> 'd"   and f7 :: "'e \<Rightarrow> 'd"   and f8 :: "'d"   and x21 :: "nat set"   and x22 :: "'e" shows "(case groupcast x21 x22 of broadcast x \<Rightarrow> f1 x | groupcast x xa \<Rightarrow> f2 x xa | unicast x xa \<Rightarrow> f3 x xa | \<not>unicast x \<Rightarrow> f4 x | send x \<Rightarrow> f5 x | deliver x \<Rightarrow> f6 x | receive x \<Rightarrow> f7 x | \<tau>\<^sub>s \<Rightarrow> f8) = f2 x21 x22"
ENR_neighborhood_retract: fixes U :: "'a set"   and S :: "'a set"   and T :: "'a set" assumes "ENR U"   and "S retract_of T"   and "openin (top_of_set U) T" shows "ENR S"
left_hom_with_unit_axioms: shows "left_hom_with_unit (\<cdot>) (\<star>) \<a> \<iota> b"
conservativeI3: fixes M :: "'b measure"   and T :: "'b \<Rightarrow> 'b" assumes "qmpt M T"   and "\<And>A. \<lbrakk>A \<in> sets M; \<forall>n>0. (T ^^ n) -` A \<inter> A = {}\<rbrakk> \<Longrightarrow> A \<in> null_sets M" shows "conservative M T"
set_ebounds: fixes as :: "atom list" shows "set (ebounds as) = {(r / c, (- 1 / c) *\<^sub>s cs) |r c cs. Eq r (c # cs) \<in> set as \<and> c \<noteq> 0}"