hcju/isabelleps huggingface dataset:
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "'a"   and f2 :: "bool \<Rightarrow> 'a"   and f3 :: "int \<Rightarrow> 'a"   and f4 :: "real \<Rightarrow> 'a"   and f5 :: "val \<Rightarrow> val \<Rightarrow> 'a"   and val :: "val" shows "P (case val of UnitVal \<Rightarrow> f1 | BoolVal x \<Rightarrow> f2 x | IntVal x \<Rightarrow> f3 x | RealVal x \<Rightarrow> f4 x | <|x, xa|> \<Rightarrow> f5 x xa) = ((val = UnitVal \<longrightarrow> P f1) \<and> (\<forall>x2. val = BoolVal x2 \<longrightarrow> P (f2 x2)) \<and> (\<forall>x3. val = IntVal x3 \<longrightarrow> P (f3 x3)) \<and> (\<forall>x4. val = RealVal x4 \<longrightarrow> P (f4 x4)) \<and> (\<forall>x51 x52. val = <|x51, x52|> \<longrightarrow> P (f5 x51 x52)))"
nat_of_ipv4addr_ipv4addr_of_nat_mod: fixes n :: "nat" shows "nat_of_ipv4addr (ipv4addr_of_nat n) = n mod 2 ^ 32"
sorted_filter_greater_subset_drop: fixes xs :: "'b list"   and i :: "nat" assumes "sorted xs"   and "i < length xs" shows "filter_mset ((<) (xs ! i)) (mset xs) \<subseteq># mset (drop (Suc i) xs)"
strict_monoidal_functor_axioms: shows "strict_monoidal_functor (\<cdot>) T\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<alpha>\<^sub>F\<^sub>S\<^sub>M\<^sub>C \<iota>\<^sub>F\<^sub>S\<^sub>M\<^sub>C D T\<^sub>D \<alpha>\<^sub>D \<iota>\<^sub>D F"
sylow_conjugate: fixes P :: "'a set"   and Q :: "'a set" assumes "P \<in> subgroups_of_size (p ^ a)"   and "Q \<in> subgroups_of_size (p ^ a)" obtains g :: "'a" where "g \<in> carrier G" and "Q = g <# (P #> inv g)"
negligible_def: fixes f :: "nat \<Rightarrow> real" shows "negligible f = (\<forall>c>0. f \<in> o(\<lambda>x. inverse (real x powr c)))"
tt_gr: fixes p :: "'t \<Rightarrow>\<^sub>0 'c"   and v :: "'t" assumes "\<And>u. u \<in> keys p \<Longrightarrow> v \<prec>\<^sub>t u"   and "p \<noteq> 0" shows "v \<prec>\<^sub>t tt p"
br_invar_initial: fixes \<delta> :: "('d, 'e) ta_rule set" assumes "finite \<delta>" shows "br_initial \<delta> \<in> br_invar \<delta>"
simps_2: fixes f :: "('b \<times> real) msllist \<Rightarrow> (real \<Rightarrow> real) \<Rightarrow> 'a"   and x1 :: "('b \<times> real) msllist"   and x2 :: "real \<Rightarrow> real" shows "(case MS x1 x2 of MS x xa \<Rightarrow> f x xa) = f x1 x2"
f1: shows "\<forall>x y z. x \<in> carrier rec_rng_of_frac \<longrightarrow> y \<in> carrier rec_rng_of_frac \<longrightarrow> z \<in> carrier rec_rng_of_frac \<longrightarrow> (x \<oplus>\<^bsub>rec_rng_of_frac\<^esub> y) \<otimes>\<^bsub>rec_rng_of_frac\<^esub> z = x \<otimes>\<^bsub>rec_rng_of_frac\<^esub> z \<oplus>\<^bsub>rec_rng_of_frac\<^esub> y \<otimes>\<^bsub>rec_rng_of_frac\<^esub> z"
sets_into_space: fixes x :: "'a set" assumes "x \<in> sigma_sets \<Omega> G" shows "x \<subseteq> \<Omega>"
initial_arr_unique: fixes f :: "'a"   and f' :: "'a" assumes "par f f'"   and "initial_arr f"   and "initial_arr f'" shows "f = f'"
full_def: fixes intT :: "'a \<Rightarrow> univ \<Rightarrow> bool"   and \<sigma> :: "'a" assumes "CM.Tstruct intT" shows "Tstruct.full intT \<sigma> \<equiv> \<forall>d. intT \<sigma> d"
mult_mat_vec_carrier: fixes A :: "'c mat"   and nr :: "nat"   and n :: "nat"   and v :: "'c vec" assumes "A \<in> carrier_mat nr n"   and "v \<in> carrier_vec n" shows "A *\<^sub>v v \<in> carrier_vec nr"
symmetric_subset: fixes B :: "'a set" assumes "B \<subseteq> A" shows "symmetric B (\<sim>)"
flows_E_simp: shows "flows_fix \<T> \<union> backflows (filternew_flows_state \<T>) - (backflows (filternew_flows_state \<T>) - E) = flows_fix \<T> \<union> E"
rel_cases: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'b tag_const"   and b :: "'c tag_const" assumes "rel_tag_const R a b" obtains xa :: "'b" and ya :: "'c" where "a = \<tau>\<^sub>c\<^sub>s\<^sub>t xa" and "b = \<tau>\<^sub>c\<^sub>s\<^sub>t ya" and "R xa ya"
tfr_stp_all_split_4: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and S :: "('a, 'b) strand_step list"   and S' :: "('a, 'b) strand_step list" assumes "typed_model arity public Ana \<Gamma>"   and "list_all (typed_model.tfr\<^sub>s\<^sub>t\<^sub>p \<Gamma>) (S @ S')" shows "list_all (typed_model.tfr\<^sub>s\<^sub>t\<^sub>p \<Gamma>) S'"
FAss1Red2: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and s :: "'heap \<times> 'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and s' :: "'heap \<times> 'addr val list"   and v :: "'addr val"   and F :: "String.literal"   and D :: "String.literal" assumes "uf,P,t \<turnstile>1 \<langle>e,s\<rangle> -ta\<rightarrow> \<langle>e',s'\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>Val v\<bullet>F{D} := e,s\<rangle> -ta\<rightarrow> \<langle>Val v\<bullet>F{D} := e',s'\<rangle>"
RHS: fixes f :: "'a \<Rightarrow> complex" assumes "continuous_on S f \<and> f ` S \<subseteq> sphere 0 1" shows "\<exists>g. continuous_on S g \<and> (\<forall>x\<in>S. f x = exp (g x))"
fst_matrix_to_iarray_echelon_form_of_column_k_det: fixes k :: "nat"   and A :: "(('c, 'd) vec, 'e) vec"   and i :: "nat"   and bezout :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<times> 'c \<times> 'c \<times> 'c \<times> 'c"   and det_P :: "'c" assumes "k < ncols A"   and "i \<le> nrows A" shows "fst (echelon_form_of_column_k_det bezout (det_P, A, i) k) = fst (echelon_form_of_column_k_det_iarrays (det_P, matrix_to_iarray A, i, bezout) k)"
op_take_image_def: fixes n :: "nat"   and X :: "'c list set" shows "op_take_image n X = take n ` X"
trailIsDecreasedByDeciedUnitPropagateAndBackjump: fixes stateA :: "State"   and F0 :: "Literal list list"   and decisionVars :: "nat set"   and stateB :: "State" assumes "invariantsHoldInState stateA F0 decisionVars"   and "appliedDecide stateA stateB decisionVars \<or> appliedUnitPropagate stateA stateB \<or> appliedBackjump stateA stateB" shows "(getM stateB, getM stateA) \<in> lexLessRestricted (vars F0 \<union> decisionVars)"
locks_ticket_simps_3: fixes k :: "nat"   and k' :: "nat" shows "locks_ticket (HOLD k) k' = (k' = k)"
distinct_5: fixes x11 :: "exp"   and x12 :: "exp"   and x4 :: "Vars.var" shows "Alts x11 x12 \<noteq> Dummy x4"
div_pos_neg_trivial: fixes k :: "int"   and l :: "int" assumes "0 < k"   and "k + l \<le> 0" shows "k div l = - 1"
valid_msg_def: fixes msg :: "nat" shows "valid_msg msg \<equiv> msg < order \<G>"
extract_viewres_isNone_def: fixes r :: "('a, 'b) viewres" shows "extract_viewres_isNone r \<equiv> Rep_viewres r = None"
neutralunmodified: fixes a :: "'a" shows "a = (0::'a) \<or> \<lambda> (\<nn>, a) = \<nn>"
lcsTr5: fixes H :: "'a set"   and a :: "'a"   and b :: "'a"   and x :: "'a" assumes "G \<guillemotright> H "   and "a \<in> carrier G"   and "b \<in> carrier G"   and "\<rho> a \<cdot> b \<in> H"   and "x \<in> a \<diamondsuit> H" shows "\<rho> b \<cdot> x \<in> H"
le_iff_diff_le_0: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a - b \<le> (0::'a))"
simps_9: fixes prec :: "nat"   and ord :: "nat"   and I :: "float interval list"   and a :: "float interval list"   and f :: "floatarith"   and env :: "taylor_model list" shows "approx_tm prec ord I a (Arctan f) env = compute_tm_by_comp prec ord I a (Arctan (Var 0)) (approx_tm prec ord I a f env) (\<lambda>x. True)"
orthogonal_matrix_orthonormal_rows: fixes A :: "((real, 'a) vec, 'a) vec" shows "orthogonal_matrix A = ((\<forall>i. norm (row i A) = 1) \<and> (\<forall>i j. i \<noteq> j \<longrightarrow> orthogonal (row i A) (row j A)))"
wp_SetDC_Bind: fixes a :: "'b \<Rightarrow> bool \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and S :: "'a \<Rightarrow> 'b set" shows "SetDC a S = Bind S (\<lambda>S. SetDC a (\<lambda>_. S))"
listrel_iff_zip: fixes xs :: "'g list"   and ys :: "'h list"   and r :: "('g \<times> 'h) set" shows "((xs, ys) \<in> listrel r) = (length xs = length ys \<and> (\<forall>(x, y)\<in>set (zip xs ys). (x, y) \<in> r))"
assms: shows "disjoint (\<A> \<union> \<B>)"
get_owner_document_def: fixes ptr :: "(_) object_ptr" shows "get_owner_document ptr = invoke a_get_owner_document_tups ptr ()"
fls_nonzeroI: fixes f :: "'b fls"   and n :: "int" assumes "fls_nth f n \<noteq> (0::'b)" shows "f \<noteq> 0"
isCont_sum: fixes A :: "'a set"   and a :: "'b"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c" assumes "\<forall>i\<in>A. isCont (f i) a" shows "isCont (\<lambda>x. \<Sum>i\<in>A. f i x) a"
abs: fixes Fr :: "'a filter"   and fr :: "'a \<Rightarrow> real" shows "\<Theta>[Fr](\<lambda>x. \<bar>fr x\<bar>) = \<Theta>[Fr](fr)"
coprime_divisors: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and a :: "'b"   and c :: "'b"   and b :: "'b"   and d :: "'b" assumes "class.algebraic_semidom divide plus minus zero times one"   and "dvd.dvd times a c"   and "dvd.dvd times b d"   and "algebraic_semidom.coprime times one c d" shows "algebraic_semidom.coprime times one a b"
f: fixes x :: "'a" assumes "x \<in> space M" shows "sets (f x) = sets N"
stoch_procs_def: fixes M :: "'e measure"   and N :: "'f measure" shows "stoch_procs M N = {X. \<forall>t. X t \<in> M \<rightarrow>\<^sub>M N}"
simps_6: fixes a :: "('a, 'b) rbt"   and k :: "'a"   and x :: "'b"   and b :: "('a, 'b) rbt"   and va :: "('a, 'b) rbt"   and vb :: "'a"   and vc :: "'b"   and vd :: "('a, 'b) rbt" shows "combine (Branch R a k x b) (Branch B va vb vc vd) = Branch R a k x (combine b (Branch B va vb vc vd))"
tminus_nneg: fixes a :: "real"   and b :: "real" shows "0 \<le> a \<ominus> b"
commute: fixes a :: "'c word"   and b :: "'c word" shows "signed.min a b = signed.min b a"
rev_is_rev_conv: fixes xs :: "'b list"   and ys :: "'b list" shows "(rev xs = rev ys) = (xs = ys)"
swap0in2: fixes l :: "'c list"   and x :: "'c"   and y :: "'c" assumes "set l = {x, y}"   and "x \<noteq> y"   and "length l = 2"   and "dist_perm l l" shows "x < y in swap 0 l = (\<not> x < y in l)"
mset_subset_eq_insertD: fixes x :: "'c"   and A :: "'c multiset"   and B :: "'c multiset" assumes "add_mset x A \<subseteq># B" shows "x \<in># B \<and> A \<subset># B"
mset_remove1: fixes a :: "'a"   and xs :: "'a list" shows "mset (remove1 a xs) = mset xs - {#a#}"
prob_space: fixes i :: "'i" shows "Y.prob i (space (distr M borel (Y i))) = 1"
scalars_zero_if_independent: fixes A :: "('a, 'b) vec set"   and f :: "('a, 'b) vec \<Rightarrow> 'a" assumes "finite A"   and "vec.independent A"   and "(\<Sum>x\<in>A. f x *s x) = 0" shows "\<forall>x\<in>A. f x = (0::'a)"
PO_a0i_inv1_iagree_init: shows "init a0i \<subseteq> a0i_inv1_iagree"
shift2_invD_4: fixes d :: "nat"   and qs :: "((('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) \<times> 'x set) list"   and d0 :: "nat" assumes "shift2_inv d qs"   and "d0 < d" shows "card {q \<in> set qs. poly_deg (fst q) = d0 \<and> m < card (snd q)} \<le> 1"
simps_2: fixes x :: "Vars.var"   and S :: "stack_elem list" shows "Astack (Arg x # S) = inc\<cdot>(Astack S)"
val_Zp_ultrametric_eq'': fixes x :: "nat \<Rightarrow> int"   and y :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "y \<in> carrier Zp"   and "val_Zp y < val_Zp x" shows "val_Zp (y \<ominus> x) = val_Zp y"
supp_rel_def: fixes R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'d" shows "supp_rel R x = {a. infinite {b. \<not> R ((a \<rightleftharpoons> b) \<bullet> x) x}}"
divide_eq_1_iff: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inverse :: "'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c" assumes "class.field divide inverse times one plus zero minus uminus" shows "(divide a b = one) = (b \<noteq> zero \<and> a = b)"
elements_mat_transpose_mat: fixes A :: "'b mat" shows "elements_mat A\<^sup>T = elements_mat A"
writerT_cases: fixes y :: "'e\<cdot>('g, 'f) writerT" obtains k :: "'e\<cdot>'f writer\<cdot>'g" where "y = WriterT\<cdot>k"
isometry_on_subset: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and Y :: "'c set" assumes "isometry_on X f"   and "Y \<subseteq> X" shows "isometry_on Y f"
gState_progress_relI: fixes prog :: "program"   and g :: "'a gState_scheme"   and g' :: "'a gState_scheme" assumes "gState_inv prog g"   and "gState_inv prog g'"   and "length (gState.channels g) \<le> length (gState.channels g')"   and "dom (lm.\<alpha> (gState.vars g)) = dom (lm.\<alpha> (gState.vars g'))" shows "(g, g') \<in> gState_progress_rel prog"
simps_2: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ"   and a :: "('fsym, 'psym) atm" shows "U.satL \<xi> (Neg a) = (\<not> U.satA \<xi> a)"
sum_divide_distrib: fixes f :: "'c \<Rightarrow> 'b"   and A :: "'c set"   and r :: "'b" shows "sum f A / r = (\<Sum>n\<in>A. f n / r)"
completeness': fixes T :: "tree"   and Cs :: "fterm literal set set" assumes "closed_tree T Cs"   and "\<forall>C\<in>Cs. finite C" shows "\<exists>Cs'. resolution_deriv Cs Cs' \<and> {} \<in> Cs'"
h1: shows "insertion f p = insertion f (\<Sum>i\<le>MPoly_Type.degree p v. isolate_variable_sparse p v i * Var v ^ i)"
simple_def: fixes p :: "('b \<times> 'b) list" shows "simple p \<equiv> distinct (map uedge p)"
simps_7: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and x1 :: "'a"   and x2 :: "'a stream"   and y1 :: "'b"   and y2 :: "'b stream" shows "stream_all2 R (x1 ## x2) (y1 ## y2) = (R x1 y1 \<and> stream_all2 R x2 y2)"
surjective: fixes r :: "('a, 'b, 'c, 'd) interp_scheme" shows "r = \<lparr>Functions = Functions r, Predicates = Predicates r, Contexts = Contexts r, Programs = Programs r, ODEs = ODEs r, ODEBV = ODEBV r, \<dots> = interp.more r\<rparr>"
nonzero_memI: fixes a :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "a \<noteq> \<zero>" shows "a \<in> nonzero Zp"
strategyproofness_1: shows "(pmf (sds R29) a + pmf (sds R29) b < pmf (sds R39) d + pmf (sds R39) c \<or> pmf (sds R29) a + (pmf (sds R29) b + pmf (sds R29) d) < pmf (sds R39) d + (pmf (sds R39) c + pmf (sds R39) a)) \<or> pmf (sds R29) a + pmf (sds R29) b = pmf (sds R39) d + pmf (sds R39) c \<and> pmf (sds R29) a + (pmf (sds R29) b + pmf (sds R29) d) = pmf (sds R39) d + (pmf (sds R39) c + pmf (sds R39) a)"
distinct_25: fixes x31 :: "'b ptrm"   and x32 :: "'b ptrm"   and x51 :: "'b ptrm"   and x52 :: "'b ptrm" shows "PApp x31 x32 \<noteq> PPair x51 x52"
Carmichael_number_def: fixes n :: "nat" shows "Carmichael_number n = (1 < n \<and> \<not> prime n \<and> (\<forall>a. coprime a n \<longrightarrow> [a ^ (n - 1) = 1] (mod n)))"
Nsub0: fixes x :: "int \<times> int"   and y :: "int \<times> int" assumes "isnormNum x"   and "isnormNum y" shows "(x -\<^sub>N y = 0\<^sub>N) = (x = y)"
sees_field_decl_above: fixes P :: "'b prog"   and C :: "char list"   and F :: "char list"   and b :: "staticb"   and T :: "ty"   and D :: "char list" assumes "P \<turnstile> C sees F,b:T in D" shows "P \<turnstile> C \<preceq>\<^sup>* D"
BinOpThrow1: fixes P :: "(char list list \<times> char list exp) prog"   and e\<^sub>1 :: "char list exp"   and s\<^sub>0 :: "heap \<times> sfields \<times> sheap"   and e :: "char list exp"   and s\<^sub>1 :: "heap \<times> sfields \<times> sheap"   and bop :: "bop"   and e\<^sub>2 :: "char list exp" assumes "P \<turnstile> \<langle>e\<^sub>1,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>1\<rangle>" shows "P \<turnstile> \<langle>e\<^sub>1 \<guillemotleft>bop\<guillemotright> e\<^sub>2,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e,s\<^sub>1\<rangle>"
fds_const_of_int: fixes c :: "int" shows "fds_const (of_int c) = of_int c"
groupWIs_same_fw_not2: fixes A :: "'b wordinterval list"   and c :: "parts_connection"   and rs :: "'b simple_rule list"   and B :: "'b wordinterval list" assumes "A \<in> set (groupWIs c rs)"   and "B \<in> set (groupWIs c rs)"   and "A \<noteq> B" shows "\<forall>ip1\<in>wordinterval_list_to_set A. \<forall>ip2\<in>wordinterval_list_to_set B. \<not> same_fw_behaviour_one ip1 ip2 c rs"
assms_2: shows "outs_\<I> \<I> \<subseteq> A"
prod_algebraI: fixes J :: "'d set"   and I :: "'d set"   and E :: "'d \<Rightarrow> 'e set"   and M :: "'d \<Rightarrow> 'e measure" assumes "finite J"   and "J \<noteq> {} \<or> I = {}"   and "J \<subseteq> I"   and "\<And>i. i \<in> J \<Longrightarrow> E i \<in> sets (M i)" shows "prod_emb I M J (Pi\<^sub>E J E) \<in> prod_algebra I M"
dense_pp: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" shows "matrix_p_algebra.dense x = (\<ominus> \<ominus> x = mtop)"
maxl_fun_def: fixes D :: "('c, 'd) Order_scheme" shows "maxl_fun D = (\<lambda>x\<in>carrier D. if \<exists>y. y \<in> upper_bounds D {x} \<and> y \<noteq> x then SOME z. z \<in> upper_bounds D {x} \<and> z \<noteq> x else x)"
h_def: fixes z :: "complex" shows "h z = (if z = 0 then deriv f 0 else f z / z)"
no_bounded_connected_component_imp_winding_number_zero: fixes g :: "real \<Rightarrow> complex"   and S :: "complex set"   and z :: "complex" assumes "path g"   and "path_image g \<subseteq> S"   and "pathfinish g = pathstart g"   and "z \<notin> S"   and "\<And>z. bounded (connected_component_set (- S) z) \<Longrightarrow> z \<in> S" shows "winding_number g z = 0"
fps_inverse_one': assumes "inverse (1::'b) = (1::'b)" shows "inverse 1 = 1"
ope: fixes T :: "'a set" assumes "closedin (top_of_set S) T" shows "closedin (top_of_set (f ` S)) (f ` T)"
intpow_def: fixes x :: "'b"   and n :: "int" shows "x ^\<^sub>Z n = (if n < 0 then inverse (x ^ nat (- n)) else x ^ nat n)"
pair_sigma_finite_def: fixes M1 :: "'a measure"   and M2 :: "'b measure" shows "pair_sigma_finite M1 M2 \<equiv> sigma_finite_measure M1 \<and> sigma_finite_measure M2"
set_attribute_ok: fixes h :: "(_) heap"   and element_ptr :: "(_) element_ptr"   and k :: "char list"   and v :: "char list option" assumes "DocumentClass.type_wf h"   and "element_ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok set_attribute element_ptr k v"
symmetricI_app_iso: fixes hom :: "('a, 'b, 'e, 'f) digraph_isomorphism" assumes "digraph_isomorphism hom"   and "symmetric G" shows "symmetric (app_iso hom G)"
locally_similar_sym: fixes \<sigma> :: "nat \<Rightarrow> 'f \<Rightarrow> 'g"   and \<tau> :: "nat \<Rightarrow> 'f \<Rightarrow> 'g" assumes "locally_similar \<sigma> \<tau>" shows "locally_similar \<tau> \<sigma>"
accessible_on_imp_inductive_on: fixes A :: "'b set"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<forall>x\<in>A. accessible_on P A x" shows "inductive_on P A"
nonneg_Reals_minus_one_I: shows "- (1::'b) \<notin> \<real>\<^sub>\<ge>\<^sub>0"
simps_3: fixes v :: "'b" shows "le_rexp (Atom v) Zero = False"
shift_ms_aux_def: fixes e :: "real"   and F :: "('a \<times> real) msllist" shows "shift_ms_aux e F = scale_shift_ms_aux (const_expansion 1, e) F"
vinsertE1: fixes x :: "V"   and y :: "V"   and A :: "V" assumes "x \<in>\<^sub>\<circ> vinsert y A" obtains "x = y"   | "x \<in>\<^sub>\<circ> A"
alw_sconst: fixes h :: "'a"   and t :: "'a stream" shows "alw (\<lambda>xs. shd xs = h) t = (t = sconst h)"
impl_of_inverse: fixes x :: "('a, 'b) alist" shows "Alist (impl_of x) = x"
mult_left_isotone: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "x * z \<le> y * z"
ordered_semiring_record_simps_1: fixes carrier :: "'c set"   and more :: "'d" shows "carrier \<lparr>carrier = carrier, \<dots> = more\<rparr> = carrier"
of_nat_prod_mset: fixes A :: "nat multiset" shows "\<Prod>\<^sub># (image_mset of_nat A) = of_nat (\<Prod>\<^sub># A)"
enat_le_plus_same_2: fixes x :: "enat"   and y :: "enat" shows "x \<le> y + x"
entails_trans: fixes P :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real"   and R :: "'c \<Rightarrow> real" assumes "P \<tturnstile> Q"   and "Q \<tturnstile> R" shows "P \<tturnstile> R"
solution_of_nat_of_nat': fixes D :: "nat"   and z :: "nat \<times> nat" assumes "pell D" shows "pell.solution D (case z of (a, b) \<Rightarrow> (of_nat a, of_nat b)) = pell.solution D z"
lit_fun_simps_3: fixes g :: "'c \<Rightarrow> 'd \<Rightarrow> 'a"   and x :: "'c"   and y :: "'d" shows "\<guillemotleft>g x y\<guillemotright> = bop g \<guillemotleft>x\<guillemotright> \<guillemotleft>y\<guillemotright>"
Int_Un_eq_1: fixes S :: "'f set"   and T :: "'f set" shows "S \<inter> T \<union> S = S"
a1: fixes i :: "nat" shows "f (Suc i) \<sqsupset>p f i"
adjacent_atomics_nonempty_reusables: fixes w :: "nat \<times> nat"   and t :: "'a tree"   and x :: "'a tree"   and xs :: "'a tree list" assumes "0 < fst w"   and "l t \<le> fst w"   and "r t \<le> snd w"   and "local.valid t"   and "reusables t w = x # xs" shows "adjacent (Suc (r x), snd w) (local.atomics (max (fst w) (Suc (r t))) (snd w))"
u_5: fixes z :: "'a" assumes "z \<in> cball x e" shows "u z \<in> existence_ivl0 z"
\<beta>'_def: shows "\<beta>' = (\<lambda>Y. \<Sum>f\<in>tuples Y. \<Prod>g\<in>X \<rightarrow>\<^sub>E Roots. g (f g))"
C1: fixes R :: "ccs"   and S :: "ccs" assumes "(R, S) \<in> Rel" shows "R \<leadsto>[Rel] S"
j: shows "j < dim_col (submatrix A I UNIV * submatrix B UNIV J)"
pred_sexp_CONS_I2: fixes M :: "('b, unit) Old_Datatype.node set"   and N :: "('b, unit) Old_Datatype.node set" assumes "M \<in> sexp"   and "N \<in> sexp" shows "(N, CONS M N) \<in> pred_sexp\<^sup>+"
hyps_1: shows "\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
rel_mono_strong: fixes R1 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2 :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'c \<times> 'e"   and y :: "'d \<times> 'f"   and R1a :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and R2a :: "'e \<Rightarrow> 'f \<Rightarrow> bool" assumes "rel_prod R1 R2 x y"   and "\<And>z1 y1. \<lbrakk>z1 \<in> Basic_BNFs.fsts x; y1 \<in> Basic_BNFs.fsts y; R1 z1 y1\<rbrakk> \<Longrightarrow> R1a z1 y1"   and "\<And>z2 y2. \<lbrakk>z2 \<in> Basic_BNFs.snds x; y2 \<in> Basic_BNFs.snds y; R2 z2 y2\<rbrakk> \<Longrightarrow> R2a z2 y2" shows "rel_prod R1a R2a x y"
has_val_one_neq_zero: fixes v :: "'a \<Rightarrow> ant" assumes "valuation K v" shows "1\<^sub>r \<noteq> \<zero>"
gtrp_env_5: shows "((s_, add_mset (u # s @ r) c_), e_, s'_, add_mset (u # s @ r) c'_) \<in> ntrs fg"
simps: fixes stop :: "'a"   and eval :: "((('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) \<times> ('a \<times> (Mode \<Rightarrow> 'b set)) \<times> ('b \<Rightarrow> 'c)) set"   and some_val :: "'c"   and some_val' :: "'c"   and dma :: "'b \<Rightarrow> Sec"   and cms\<^sub>1 :: "('a \<times> (Mode \<Rightarrow> 'b set)) list"   and mem\<^sub>1 :: "'b \<Rightarrow> 'c"   and cms\<^sub>2 :: "('a \<times> (Mode \<Rightarrow> 'b set)) list"   and mem\<^sub>2 :: "'b \<Rightarrow> 'c"   and mems :: "(('b \<Rightarrow> 'c) \<times> ('b \<Rightarrow> 'c)) list" assumes "sifum_security stop eval some_val some_val'" shows "sifum_security.makes_compatible dma eval (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems = (length cms\<^sub>1 = length cms\<^sub>2 \<and> length cms\<^sub>1 = length mems \<and> (\<forall>i<length cms\<^sub>1. (\<forall>\<sigma>. dom \<sigma> = sifum_security.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i \<longrightarrow> ((cms\<^sub>1 ! i, subst \<sigma> (fst (mems ! i))), cms\<^sub>2 ! i, subst \<sigma> (snd (mems ! i))) \<in> sifum_security.mm_equiv dma eval) \<and> (\<forall>x. mem\<^sub>1 x = mem\<^sub>2 x \<or> dma x = High \<longrightarrow> x \<notin> sifum_security.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i)) \<and> (length cms\<^sub>1 = 0 \<and> sifum_security.low_eq dma mem\<^sub>1 mem\<^sub>2 \<or> (\<forall>x. \<exists>i<length cms\<^sub>1. x \<notin> sifum_security.differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i)))"
extensional_merge: fixes I :: "'d set"   and J :: "'d set"   and x :: "'d \<Rightarrow> 'e"   and y :: "'d \<Rightarrow> 'e" shows "merge I J (x, y) \<in> extensional (I \<union> J)"
snapshot_stable: fixes c :: "('p, 's, 'm) configuration"   and t :: "('p, 's, 'm) event list"   and c' :: "('p, 's, 'm) configuration"   and p :: "'p" assumes "trace c t c'"   and "ps c p \<noteq> None" shows "ps c' p \<noteq> None"