hcju/isabelleps huggingface dataset:
store_sub2_low_equal_sub1: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and addr :: "word32"   and instr :: "sparc_operation \<times> inst_operand list"   and curr_win :: "'b word"   and rd :: "word5"   and y :: "'b sparc_state"   and ya :: "'b sparc_state" assumes "low_equal s1 s2 \<and> memory_write 10 addr (st_byte_mask instr addr) (st_data0 instr curr_win rd addr s2) s1 = Some y \<and> memory_write 10 addr (st_byte_mask instr addr) (st_data0 instr curr_win rd addr s2) s2 = Some ya" shows "low_equal (y\<lparr>traps := insert data_access_exception (traps y)\<rparr>) (ya\<lparr>traps := insert data_access_exception (traps ya)\<rparr>)"
parent_child_rel_child_in_heap: fixes h :: "(_) heap"   and parent :: "(_) object_ptr"   and child_ptr :: "(_) object_ptr" assumes "heap_is_wellformed\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M h"   and "ShadowRootClass.type_wf h"   and "ShadowRootClass.known_ptr parent"   and "(parent, child_ptr) \<in> Shadow_DOM.parent_child_rel h" shows "child_ptr |\<in>| object_ptr_kinds h"
set_times_intro: fixes a :: "'a"   and C :: "'a set"   and b :: "'a"   and D :: "'a set" assumes "a \<in> C"   and "b \<in> D" shows "a * b \<in> C * D"
simps_59: fixes x61 :: "label"   and x62 :: "expr"   and x63 :: "stmt"   and x31 :: "jump"   and x32 :: "stmt" shows "x61\<bullet> While(x62) x63 \<noteq> x31\<bullet> x32"
H_assign_var: fixes P :: "(char list \<Rightarrow> 'a) \<Rightarrow> bool"   and Q :: "(char list \<Rightarrow> 'a) \<Rightarrow> bool"   and v :: "char list"   and e :: "(char list \<Rightarrow> 'a) \<Rightarrow> 'a" assumes "\<forall>s. P s \<longrightarrow> Q (s(v := e s))" shows "rel_kat.H \<lceil>P\<rceil> (v ::= e) \<lceil>Q\<rceil>"
neg: shows "\<not> (\<exists>D\<in>decomps\<^sub>e\<^sub>s\<^sub>t A N \<I>. \<forall>t. A \<turnstile> t \<longrightarrow> A \<union> ik\<^sub>e\<^sub>s\<^sub>t D \<turnstile>\<^sub>c t)"
atMost_def: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and u :: "'c" shows "ord.atMost less_eq u \<equiv> {x. less_eq x u}"
Pair_inject: fixes a :: "'g"   and b :: "'h"   and a' :: "'g"   and b' :: "'h" assumes "(a, b) = (a', b')" obtains "a = a'" and "b = b'"
is_local_def: fixes U :: "'a set" shows "stx.is_local U \<equiv> local_ring stx.carrier_stalk stx.add_stalk stx.mult_stalk (stx.zero_stalk U) (stx.one_stalk U)"
A: fixes n :: "nat" shows "2 * \<Sum> {1..n} = n * Suc n"
times_left_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and z :: "'c"   and y :: "'c"   and x :: "'c" assumes "SN_Orders.class.ordered_semiring_0 plus zero times less_eq"   and "less_eq zero z"   and "less_eq y x" shows "less_eq (times y z) (times x z)"
take_is_prefix: fixes n :: "nat"   and xs :: "'b list" shows "prefix (xs \<down> n) xs"
loc1_a_s_2: shows "results_in a s \<le> b"
set_sorted_list_of_set: fixes less_eq :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and less :: "'k \<Rightarrow> 'k \<Rightarrow> bool"   and A :: "'k set" assumes "class.linorder less_eq less"   and "finite A" shows "set (linorder.sorted_list_of_set less_eq A) = A"
PO_refines_implies_Range_init: fixes R :: "('a \<times> 'b) set"   and Ta :: "('a, 'c) TS_scheme"   and Tc :: "('b, 'd) TS_scheme" assumes "PO_refines R Ta Tc" shows "init Tc \<subseteq> Range R"
case_4: fixes f1 :: "'a"   and f2 :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and f3 :: "'b \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and f4 :: "('b, 'c) Combinators \<Rightarrow> ('b, 'c) Combinators \<Rightarrow> 'a"   and x41 :: "('b, 'c) Combinators"   and x42 :: "('b, 'c) Combinators" shows "(case x41 \<oplus> x42 of DenyAll \<Rightarrow> f1 | DenyAllFromTo x xa \<Rightarrow> f2 x xa | AllowPortFromTo x xa xb \<Rightarrow> f3 x xa xb | x \<oplus> xa \<Rightarrow> f4 x xa) = f4 x41 x42"
assms: shows "(_Exit_) \<in> obs_intra n' \<lfloor>HRB_slice S\<rfloor>\<^bsub>CFG\<^esub>"
dfree_DiffVar_simps: fixes x :: "'b" shows "dfree ($' x) = False"
map_closed: fixes a :: "'b" assumes "a \<in> \<FF> (fst x)" shows "\<rho> (fst x) z' a \<in> \<FF> z'"
strict_implies_order: fixes a :: "'t \<Rightarrow>\<^sub>0 'c"   and b :: "'t \<Rightarrow>\<^sub>0 'c" assumes "a \<prec>\<^sub>p b" shows "a \<preceq>\<^sub>p b"
SN_rel_on_alt_imp_SN_rel_on: fixes R :: "('a \<times> 'a) set"   and S :: "('a \<times> 'a) set"   and T :: "'a set" assumes "SN_rel_on_alt R S T" shows "SN_rel_on R S T"
rotate2_1: fixes x :: "(real, 2) vec" shows "rotate2 x $ 1 = - x $ 2"
vbrelation_axioms: fixes a :: "V"   and b :: "V" shows "vbrelation (set {\<langle>a, b\<rangle>})"
lossless_R1_14: fixes msgs :: "bool \<times> bool \<times> bool \<times> bool"   and C :: "bool \<times> bool" shows "lossless_spmf (R1_14 msgs C)"
apply_cltn2_unique: fixes l :: "proj2_line"   and m :: "proj2_line"   and p :: "proj2"   and q :: "proj2"   and C :: "cltn2" assumes "l \<noteq> m"   and "proj2_incident p l"   and "proj2_incident p m"   and "proj2_incident q (apply_cltn2_line l C)"   and "proj2_incident q (apply_cltn2_line m C)" shows "apply_cltn2 p C = q"
ring_homomorphism_def: fixes \<eta> :: "'c \<Rightarrow> 'd"   and R :: "'c set"   and addition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and multiplication :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and unit :: "'c"   and R' :: "'d set"   and addition' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and multiplication' :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero' :: "'d"   and unit' :: "'d" shows "ring_homomorphism \<eta> R addition multiplication zero unit R' addition' multiplication' zero' unit' \<equiv> (Set_Theory.map \<eta> R R' \<and> ring R addition multiplication zero unit) \<and> ring R' addition' multiplication' zero' unit' \<and> group_homomorphism \<eta> R addition zero R' addition' zero' \<and> monoid_homomorphism \<eta> R multiplication unit R' multiplication' unit'"
additive_def: fixes f :: "'q \<Rightarrow> 'r" shows "additive f \<equiv> \<forall>x y. f (x + y) = f x + f y"
not_zero_suc_j: shows "\<not> is_zero_row (mod_type_class.from_nat (Suc j)) A"
Image_singleton: fixes r :: "('d \<times> 'c) set"   and a :: "'d" shows "r `` {a} = {b. (a, b) \<in> r}"
restr_conv: fixes A :: "'d set"   and al :: "('d \<times> 'c) list"   and k :: "'d" shows "map_of (restrict A al) k = (map_of al |` A) k"
emeasure_eq_measure: fixes M :: "'b pmf"   and A :: "'b set" shows "emeasure (measure_pmf M) A = ennreal (measure_pmf.prob M A)"
C: shows "finite {y. [(x, y)] \<bullet> Tvec \<noteq> Tvec}"
gen_iapath_def: fixes G :: "('b, 'c) pre_digraph"   and V :: "'b set"   and u :: "'b"   and p :: "'c list"   and v :: "'b" shows "pre_digraph.gen_iapath G V u p v \<equiv> u \<in> V \<and> v \<in> V \<and> pre_digraph.apath G u p v \<and> set (pre_digraph.inner_verts G p) \<inter> V = {} \<and> p \<noteq> []"
conductor_eq_1_iff_principal: shows "(\<Phi>.conductor = 1) = (\<Phi> = principal_dchar c)"
map: fixes z :: "nat"   and L :: "real list" shows "eval (liftmap (\<lambda>x. quadratic_sub (var + x) (liftPoly 0 x a) (liftPoly 0 x b) (liftPoly 0 x c) (liftPoly 0 x d)) F (z + x1)) L = eval (liftmap (\<lambda>x. quadratic_sub (var + x1 + x) (liftPoly 0 (x + x1) a) (liftPoly 0 (x + x1) b) (liftPoly 0 (x + x1) c) (liftPoly 0 (x + x1) d)) F z) L"
finite_incidence_system_axioms: shows "finite_incidence_system \<V> (del_block bl)"
distinct_2: fixes nat' :: "nat" shows "Suc nat' \<noteq> 0"
of_nat_2p: shows "word_of_nat (2 ^ LENGTH('a)) = 0"
big_step_Pii_o_def: shows "big_step_Pii_o \<equiv> \<lambda>x. pred.Pred ((\<Rightarrow>) x)"
incl_lsbis2: fixes B1 :: "'d set"   and B2 :: "'e set"   and s1 :: "'d \<Rightarrow> ('f, 'd, 'e) F1"   and s2 :: "'e \<Rightarrow> ('g, 'd, 'e) F2"   and R1 :: "('d \<times> 'd) set"   and R2 :: "('e \<times> 'e) set" assumes "sbis B1 B2 s1 s2 R1 R2" shows "R2 \<subseteq> lsbis2 B1 B2 s1 s2"
card_insert_fset_iff: fixes x :: "'b"   and S :: "'b fset" shows "card_fset (insert_fset x S) = (if x |\<in>| S then card_fset S else Suc (card_fset S))"
rep_eq: fixes x :: "num"   and xa :: "num" shows "int_of_integer (Code_Numeral.sub x xa) = numeral x - numeral xa"
prems_3: shows "fst cf1 \<approx>s fst cf2"
matrix_matrix_mult_iarray_def: fixes A :: "'c iarray iarray"   and B :: "'c iarray iarray" shows "A **i B = tabulate2 (nrows_iarray A) (ncols_iarray B) (\<lambda>i j. \<Sum>k = 0..<ncols_iarray A. A !! i !! k * B !! k !! j)"
modular_var_2: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and composition :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and converse :: "'b \<Rightarrow> 'b"   and unit :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.relation_algebra minus uminus inf less_eq less sup bot top composition converse unit" shows "less_eq (composition x y) (composition x (inf y (composition (converse x) top)))"
len_takeWhile_set: fixes n2 :: "'a"   and xs :: "'a list"   and n1 :: "'a" assumes "length (takeWhile ((\<noteq>) n2) xs) < length (takeWhile ((\<noteq>) n1) xs)"   and "n1 \<noteq> n2"   and "n1 \<in> set xs"   and "n2 \<in> set xs" shows "set (takeWhile ((\<noteq>) n2) xs) \<subseteq> set (takeWhile ((\<noteq>) n1) xs)"
execute_parallel_operators_strips_none_if_contraposition: fixes s :: "'a \<Rightarrow> bool option"   and ops :: "'a strips_operator list"   and v :: "'a" assumes "execute_parallel_operator s ops v \<noteq> None" shows "(\<exists>op\<in>set ops. v \<in> set (add_effects_of op) \<or> v \<in> set (delete_effects_of op)) \<or> s v \<noteq> None"
wt_method_wt_lbv: fixes P :: "jvm_prog"   and Ts :: "ty list"   and p :: "jvm_prog \<Rightarrow> char list \<Rightarrow> jvm_method mdecl \<Rightarrow> bool"   and C :: "char list"   and b :: "staticb"   and T\<^sub>r :: "ty"   and mxs :: "nat"   and mxl\<^sub>0 :: "nat"   and "is" :: "bytecode"   and xt :: "ex_table"   and \<tau>s :: "(ty list \<times> ty err list) option list" assumes "start_context P Ts p C"   and "wt_method P C b Ts T\<^sub>r mxs mxl\<^sub>0 is xt \<tau>s" shows "wt_lbv P C b Ts T\<^sub>r mxs mxl\<^sub>0 xt (mk_cert P mxs T\<^sub>r xt is \<tau>s) is"
borel_measurable_const: fixes c :: "'c"   and M :: "'b measure" shows "(\<lambda>x. c) \<in> borel_measurable M"
iT_inext_2: fixes t :: "nat"   and n :: "nat" assumes "t < n" shows "inext t [\<dots>n] = Suc t"
subst_unrest: fixes x :: "'b \<Longrightarrow> 'c"   and P :: "('d, 'c) uexpr"   and \<sigma> :: "'e \<Rightarrow> 'c"   and v :: "('b, 'e) uexpr" assumes "x \<sharp> P" shows "\<sigma>(x \<mapsto>\<^sub>s v) \<dagger> P = \<sigma> \<dagger> P"
tendsto_unique: fixes "open" :: "'c set \<Rightarrow> bool"   and F :: "'d filter"   and f :: "'d \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.t2_space open"   and "F \<noteq> bot"   and "filterlim f (topological_space.nhds open a) F"   and "filterlim f (topological_space.nhds open b) F" shows "a = b"
distinct_7: fixes x5 :: "'b" shows "Unit \<noteq> Addr x5"
exhaust: fixes y :: "subject_Rep" obtains x1 :: "name" where "y = InputS_Rep x1"   | x2 :: "name" where "y = BoundOutputS_Rep x2"
assms_3: shows "p \<leadsto>\<^bsub>\<Gamma>\<^esub>\<^sup>* q"
typeof_Integer: 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and v :: "val" assumes "P \<turnstile> typeof\<^bsub>h\<^esub> v = \<lfloor>Integer\<rfloor>" shows "\<exists>i. v = Intg i"
c_tl_less: fixes u :: "nat" assumes "0 < u" shows "c_tl u < u"
left_uvar_indep_pre_uvar: fixes x :: "'d \<Longrightarrow> 'f"   and y :: "'h \<Longrightarrow> 'e" shows "0-x \<bowtie> y\<^sub><"
iso_char: fixes f :: "'c local.term set \<times> 'c local.term set" shows "\<F>C.CC.iso f = (\<F>C.iso (fst f) \<and> \<F>C.iso (snd f))"
\<psi>_minus_\<theta>_bigo: shows "(\<lambda>x. \<psi> x - \<theta> x) \<in> O(\<lambda>x. ln x * sqrt x)"
trms\<^sub>s\<^sub>s\<^sub>t_append: fixes A :: "('e, 'f) stateful_strand_step list"   and B :: "('e, 'f) stateful_strand_step list" shows "trms\<^sub>s\<^sub>s\<^sub>t (A @ B) = trms\<^sub>s\<^sub>s\<^sub>t A \<union> trms\<^sub>s\<^sub>s\<^sub>t B"
size_fun_lt: fixes s :: "('a, 'b) tm" assumes "is_App s" shows "size (fun s) < size s"
span_span: fixes scale :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and A :: "'e set" assumes "module scale" shows "module.span scale (module.span scale A) = module.span scale A"
cases: fixes a :: "ed_ou_witness set" assumes "open a" obtains "a = {}"   | "a = {ed_ou_witness.a}"   | "a = {ed_ou_witness.b}"   | "a = {e}"   | "a = {ed_ou_witness.a, ed_ou_witness.c}"   | "a = {ed_ou_witness.b, d}"   | "a = {ed_ou_witness.a, ed_ou_witness.c, e}"   | "a = {ed_ou_witness.a, ed_ou_witness.b}"   | "a = {ed_ou_witness.a, e}"   | "a = {ed_ou_witness.b, e}"   | "a = {ed_ou_witness.a, ed_ou_witness.b, ed_ou_witness.c}"   | "a = {ed_ou_witness.a, ed_ou_witness.b, d}"   | "a = {ed_ou_witness.a, ed_ou_witness.b, e}"   | "a = {ed_ou_witness.b, d, e}"   | "a = {ed_ou_witness.a, ed_ou_witness.b, ed_ou_witness.c, d}"   | "a = {ed_ou_witness.a, ed_ou_witness.b, ed_ou_witness.c, e}"   | "a = {ed_ou_witness.a, ed_ou_witness.b, d, e}"   | "a = {ed_ou_witness.a, ed_ou_witness.b, ed_ou_witness.c, d, e}"
homeomorphism_ident: fixes T :: "'c set" shows "homeomorphism T T (\<lambda>a. a) (\<lambda>a. a)"
cblinfun_compose_zero_2: fixes x :: "'c \<Rightarrow>\<^sub>L 'b" shows "x o\<^sub>L 0 = 0"
comp_cod_arr: fixes f :: "('a, 'a set) Maps.arr"   and b :: "('a, 'a set) Maps.arr" assumes "Maps.arr f"   and "Maps.cod f = b" shows "b \<odot> f = f"
sem\<^sub>e\<^sub>s\<^sub>t_c_decomps\<^sub>e\<^sub>s\<^sub>t_append: fixes \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and A :: "('fun, 'var) extstrand_step list"   and D :: "('fun, 'var) extstrand_step list"   and \<A> :: "('fun, 'var) extstrand_step list" assumes "sem\<^sub>e\<^sub>s\<^sub>t_c {} \<I> A"   and "D \<in> decomps\<^sub>e\<^sub>s\<^sub>t (ik\<^sub>e\<^sub>s\<^sub>t A) (assignment_rhs\<^sub>e\<^sub>s\<^sub>t \<A>) \<I>" shows "sem\<^sub>e\<^sub>s\<^sub>t_c {} \<I> (A @ D)"
Cidm: fixes C :: "('g, 'h, 'i) Category_scheme"   and X :: "'g" assumes "Category C"   and "X \<in> obj\<^bsub>C\<^esub>" shows "id\<^bsub>C\<^esub> X maps\<^bsub>C\<^esub> X to X"
subject_reduction2: fixes P :: "(char list list \<times> char list exp) prog"   and e :: "char list exp"   and h :: "heap"   and l :: "sfields"   and sh :: "sheap"   and b :: "bool"   and e' :: "char list exp"   and h' :: "heap"   and l' :: "sfields"   and sh' :: "sheap"   and b' :: "bool"   and E :: "char list \<Rightarrow> ty option"   and T :: "ty" assumes "wf_J_prog P"   and "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow> \<langle>e',(h', l', sh'),b'\<rangle>"   and "P,E \<turnstile> (h, l, sh) \<surd>"   and "iconf sh e"   and "P,E,h,sh \<turnstile> e : T" shows "\<exists>T'. P,E,h',sh' \<turnstile> e' : T' \<and> P \<turnstile> T' \<le> T"
that: fixes g :: "nat \<Rightarrow> 'a" assumes "bij_betw g UNIV A" shows "thesis"
safe_formula_mmonitorable_exec: fixes \<phi> :: "'b MFOTL.formula" assumes "safe_formula \<phi>"   and "MFOTL.future_reach \<phi> \<noteq> \<infinity>" shows "mmonitorable_exec \<phi>"
leg1_simps_3: shows "src (tab\<^sub>1 (Maps.REP S\<^sub>1)) = src (tab\<^sub>0 (Maps.REP S\<^sub>1))"
sat_fv_cong: fixes v :: "'a list"   and v' :: "'a list"   and \<sigma> :: "'b trace"   and i :: "nat" assumes "\<And>x. x \<in> fv \<Longrightarrow> v ! x = v' ! x" shows "sat \<sigma> v i = sat \<sigma> v' i"
map_poly_Re_of_real: fixes p :: "real poly" shows "map_poly Re (map_poly complex_of_real p) = p"
cond_def: shows "cond = (\<lambda>W. Suc (card W) = n \<and> lin_indpt W \<and> W \<subseteq> X)"
1: shows "\<ominus> (deriv f a \<otimes> b) = deriv f a \<otimes> \<ominus> b"
distinct_1: fixes x1 :: "fresh_t"   and x2 :: "nat" shows "nonce_fresh x1 \<noteq> nonce_atk x2"
sel_1: fixes x1 :: "'c"   and x2 :: "'d" shows "fst (x1, x2) = x1"
alphaAbs_ex_distinct_qFresh_def: fixes xs :: "'f"   and x :: "'g"   and X :: "('h, 'i, 'f, 'g, 'j) qTerm"   and xs' :: "'f"   and x' :: "'g"   and X' :: "('h, 'i, 'f, 'g, 'j) qTerm" shows "alphaAbs_ex_distinct_qFresh xs x X xs' x' X' \<equiv> xs = xs' \<and> (\<exists>y. y \<notin> {x, x'} \<and> qFresh xs y X \<and> qFresh xs y X' \<and> X #[[y \<and> x]]_xs #= X' #[[y \<and> x']]_xs)"
has_bochner_integral_eq: fixes M :: "'b measure"   and f :: "'b \<Rightarrow> 'c"   and x :: "'c"   and y :: "'c" assumes "has_bochner_integral M f x"   and "has_bochner_integral M f y" shows "x = y"
wf_osig_imp_wf_subclass_loc: fixes oss :: "(String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)" assumes "wf_osig oss" shows "wf_subclass_loc (subclass oss)"
VPow_in_Vset_revD: fixes A :: "V"   and i :: "V" assumes "VPow A \<in>\<^sub>\<circ> Vset i" shows "A \<in>\<^sub>\<circ> Vset i"
strict_trans: fixes a :: "'c list"   and b :: "'c list"   and c :: "'c list" assumes "strict_prefix a b"   and "strict_prefix b c" shows "strict_prefix a c"
isEmpty_def: fixes q :: "('a, 'b) SkewBinomialHeap" shows "SkewBinomialHeap_loc.isEmpty q \<equiv> Rep_SkewBinomialHeap q = BsSkewBinomialHeapStruc.bs_empty"
alpha_tNot: fixes t1 :: "('e, 'f, 'g) Tree"   and t2 :: "('e, 'f, 'g) Tree" shows "(tNot t1 =\<^sub>\<alpha> tNot t2) = (t1 =\<^sub>\<alpha> t2)"
False: shows "\<not> continuous_map (nsphere p) (nsphere p) (\<lambda>x. a)"
set_iterator_image_filter_def: fixes g :: "'c \<Rightarrow> 'b option"   and it :: "('a \<Rightarrow> bool) \<Rightarrow> ('c \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a" shows "set_iterator_image_filter g it \<equiv> \<lambda>c f. it c (\<lambda>x \<sigma>. case g x of None \<Rightarrow> \<sigma> | Some x' \<Rightarrow> f x' \<sigma>)"
inf2: fixes B :: "'b"   and k :: "'b"   and b :: "'b"   and a :: "'b" assumes "B \<equiv> inf k b" shows "inf a B \<equiv> inf k (inf a b)"
subsumes_accessible_globally_def: fixes subst :: "('a, 'b, 'c) expr \<Rightarrow> 'c \<Rightarrow> ('a, 'b, 'c) expr \<Rightarrow> ('a, 'b, 'c) expr"   and s :: "'a \<Rightarrow> (('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('b \<Rightarrow> ('a, 'b, 'c) val option) \<times> ('a, 'b, 'c) expr) option" assumes "substitution subst" shows "substitution.subsumes_accessible_globally s = (\<forall>r\<^sub>1 r\<^sub>2. r\<^sub>1 \<in> dom s \<longrightarrow> r\<^sub>2 \<in> dom s \<longrightarrow> substitution.subsumes_accessible r\<^sub>1 r\<^sub>2 s)"
ab: shows "a - b = zerosign (if is_zero a \<and> is_zero b \<and> sign a \<noteq> sign b then sign a else 0) (IEEE.round To_nearest (valof a - valof b))"
some1_equality: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b" assumes "\<exists>!x. P x"   and "P a" shows "(SOME x. P x) = a"
subgraph_del_arc: fixes a :: "'b" shows "subgraph (H.del_arc a) (del_arc u)"
reflexive: fixes P :: "pi" shows "P \<sim>\<^sub>l P"
integrable_add: fixes M :: "'d measure"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "integrable M f"   and "integrable M g" shows "integrable M (\<lambda>x. f x + g x)"
real_alg_2_list_convert: fixes xs :: "real_alg_2_list" shows "real_alg_2_list_convert xs = real_alg_2_list_convert_id xs"
not_neg_one_le_neg_numeral_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sgn :: "'c \<Rightarrow> 'c"   and abs :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and m :: "num" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one" shows "(\<not> less_eq (uminus one) (uminus (numeral.numeral one plus m))) = (m \<noteq> num.One)"
Super: fixes B :: "lname set"   and Env :: "env" assumes "This \<in> B" shows "Env\<turnstile> B \<guillemotright>\<langle>Super\<rangle>\<guillemotright> \<lparr>nrm = B, brk = \<lambda>l. UNIV\<rparr>"
prems: fixes k :: "nat" assumes "k < Suc n" shows "f k = f' k"
trelPS: shows "rel_weakly_preserves_barb_set TRel TWB {success}"
prems_3: shows "A4 \<in> carrier_mat (Suc m) (Suc m)"
xt1: shows "(x, return_time {x \<in> S. s x = 0} x) \<in> Sigma X existence_ivl0"
mset_single_iff: fixes xs :: "'k list"   and x :: "'k" shows "(mset xs = {#x#}) = (xs = [x])"
insert_remove: fixes A :: "'a word set"   and x :: "'a word" assumes "finite A" shows "signed.Max (insert x A) = (if A - {x} = {} then x else signed.max x (signed.Max (A - {x})))"
pdevs_val_pdevs_of_list_append: fixes f :: "nat \<Rightarrow> real"   and I :: "real set"   and g :: "nat \<Rightarrow> real"   and xs :: "'a list"   and ys :: "'a list" assumes "f \<in> UNIV \<rightarrow> I"   and "g \<in> UNIV \<rightarrow> I" obtains e :: "nat \<Rightarrow> real" where "pdevs_val f (pdevs_of_list xs) + pdevs_val g (pdevs_of_list ys) = pdevs_val e (pdevs_of_list (xs @ ys))" and "e \<in> UNIV \<rightarrow> I"
same_length_prefixes_distinct: fixes pfx1 :: "'b prefix_match"   and pfx2 :: "'b prefix_match"   and w :: "'b word" assumes "valid_prefix pfx1"   and "valid_prefix pfx2"   and "pfx1 \<noteq> pfx2"   and "pfxm_length pfx1 = pfxm_length pfx2"   and "prefix_match_semantics pfx1 w"   and "prefix_match_semantics pfx2 w" shows "False"
IH: shows "(x [^]\<^bsub>DirProds G I\<^esub> k) i = x i [^]\<^bsub>G i\<^esub> k"
IH_1: fixes X :: "'c set" assumes "reduced t1 X"   and "t1 \<noteq> Trueif"   and "t1 \<noteq> Falseif" shows "\<exists>s1 s2. val_ifex t1 s1 \<noteq> val_ifex t1 s2"
assms_2: fixes xs :: "'a stream" assumes "\<phi> xs" shows "\<psi> xs"
sig_red_red_rtrancl: fixes sing_reg :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and F :: "('t \<Rightarrow>\<^sub>0 'b) set"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and q :: "'t \<Rightarrow>\<^sub>0 'b" assumes "(sig_red sing_reg top_tail F)\<^sup>*\<^sup>* p q" shows "(punit.red (rep_list ` F))\<^sup>*\<^sup>* (rep_list p) (rep_list q)"
square_free_monic_poly_def: fixes p :: "'b poly" shows "square_free_monic_poly p = p div gcd p (pderiv p)"
lookup_monom_of_set: fixes X :: "nat set"   and i :: "nat" shows "lookup (monom_of_set X) i = (if finite X \<and> i \<in> X then 1 else 0)"
pathstart_compose: fixes f :: "'b \<Rightarrow> 'a"   and p :: "real \<Rightarrow> 'b" shows "pathstart (f \<circ> p) = f (pathstart p)"
simps_2: fixes a :: "'a"   and n :: "nat" shows "a\<^bsup>Suc n\<^esup> = a \<cdot> a\<^bsup>n\<^esup>"
power2_eq_1_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(power.power one times a 2 = one) = (a = one \<or> a = uminus one)"
simps_4: fixes t :: "('c, 'd) Term.term"   and t' :: "('c, 'd) Term.term" shows "trms\<^sub>s\<^sub>s\<^sub>t\<^sub>p (insert\<langle>t,t'\<rangle>) = {t, t'}"
simps: fixes a1 :: "('procS, 'procT) Proc"   and a2 :: "('procS, 'procT) Proc" shows "a1 \<sim>\<lbrakk>\<cdot>\<rbrakk> a2 = ((\<exists>S. S \<in>S a1 \<and> \<lbrakk>S\<rbrakk> \<in>T a2) \<or> (\<exists>S. \<lbrakk>S\<rbrakk> \<in>T a1 \<and> S \<in>S a2) \<or> (\<exists>T. T \<in>T a1 \<and> T \<in>T a2) \<or> (\<exists>P Q R. a1 = P \<and> a2 = R \<and> P \<sim>\<lbrakk>\<cdot>\<rbrakk> Q \<and> Q \<sim>\<lbrakk>\<cdot>\<rbrakk> R))"
invol_dual: shows "dual_prod \<circ> dual_prod = id"
fsmult_def: fixes a :: "'f"   and v :: "'v" shows "a \<sharp>\<cdot> v \<equiv> a \<delta>\<delta> (0::'g) \<cdot> v"