hcju/isabelleps huggingface dataset:
rep_eq: fixes x :: "'c"   and xa :: "('c, 'd) mapping" shows "mapping.rep (Mapping.delete x xa) = (mapping.rep xa)(x := None)"
phiDefNodes_\<alpha>n: fixes n :: "'node"   and g :: "'g"   and v :: "'var" assumes "n \<in> phiDefNodes g v" shows "n \<in> set (\<alpha>n g)"
decides_def: fixes \<F> :: "nat set set"   and S :: "nat set"   and M :: "nat set" shows "decides \<F> S M \<equiv> rejects \<F> S M \<or> strongly_accepts \<F> S M"
Abs_Store_induct: fixes P :: "Store \<Rightarrow> bool"   and x :: "Store" assumes "\<And>y. y \<in> Store \<Longrightarrow> P (Abs_Store y)" shows "P x"
C: shows "Partial_Cost_Model.config' (rTS h0) ([x, y], x # h) v = ([x, y], rev v @ x # h)"
ps_2: shows "pders (pderiv p) = pderiv p # ps"
timpl_closure'_inv: fixes s :: "(('e, 'f, 'g) prot_fun, 'h) Term.term"   and t :: "(('e, 'f, 'g) prot_fun, 'h) Term.term"   and TI :: "('g set \<times> 'g set) set" assumes "(s, t) \<in> timpl_closure' TI" shows "(\<exists>x. s = Var x \<and> t = Var x) \<or> (\<exists>f g S T. s = Fun f S \<and> t = Fun g T \<and> length S = length T)"
BSIA2: shows "BSIA \<rho>2 \<V>2 Tr\<^bsub>ES2\<^esub>"
flat_ordI: fixes x :: "'e"   and a :: "'e"   and y :: "'e" assumes "x \<noteq> a \<Longrightarrow> x = y" shows "flat_ord a x y"
cap_simps_3: fixes yo :: "'v vertex" shows "cap (SOURCE, yo) = (if yo = \<langle>b\<rangle> then weight \<Gamma> b - d_IN (h 0) b else 0)"
annir: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and zero :: "'b"   and x :: "'b" assumes "class.ab_near_semiring_one_zero plus times one zero" shows "times x zero = zero"
image_group_one: fixes f :: "'d \<Rightarrow> 'c"   and G :: "('d, 'e) monoid_scheme" shows "\<one>\<^bsub>image_group f G\<^esub> = f \<one>\<^bsub>G\<^esub>"
SPEC_rule: fixes \<Phi> :: "'b \<Rightarrow> bool"   and \<Phi>' :: "'b \<Rightarrow> bool" assumes "\<And>x. \<Phi> x \<Longrightarrow> \<Phi>' x" shows "SPEC \<Phi> \<le> SPEC \<Phi>'"
get_disconnected_nodes_def: shows "get_disconnected_nodes \<equiv> l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_disconnected_nodes"
singleton: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e" assumes "semilattice_set f" shows "semilattice_set.F f {x} = x"
X_def: shows "X = {x \<in> space M. real_cond_exp M F f x \<le> c}"
cgenideal_ideal: fixes a :: "('c multiset \<Rightarrow> 'a) list" assumes "a \<in> carrier (poly_ring L)" shows "ideal (PIdl\<^bsub>poly_ring L\<^esub> a) (poly_ring L)"
integrable_continuous_interval: fixes a :: "'c"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "continuous_on {a..b} f" shows "f integrable_on {a..b}"
that_2: shows "num_params (tm_poly t2) \<le> X"
L3_Boiler: fixes x :: "nat \<Rightarrow> bit list"   and s :: "nat \<Rightarrow> nat list"   and y :: "nat \<Rightarrow> nat list" assumes "SteamBoiler x s y"   and "ts x" shows "200 \<le> hd (s 0)"
\<psi>_char_1: shows "\<guillemotleft>\<psi> : \<I>\<^sub>D \<rightarrow>\<^sub>D F \<I>\<^sub>C\<guillemotright>"
ev_p: shows "\<forall>\<^sup>\<infinity>n. 0 < p n \<and> p n < 1"
comm_diagrams: fixes X :: "'d set"   and is_open :: "'d set \<Rightarrow> bool"   and \<FF> :: "'d set \<Rightarrow> 'e set"   and \<rho> :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'e \<Rightarrow> 'e"   and b :: "'e"   and add_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and mult_str :: "'d set \<Rightarrow> 'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero_str :: "'d set \<Rightarrow> 'e"   and one_str :: "'d set \<Rightarrow> 'e"   and \<FF>' :: "'d set \<Rightarrow> 'f set"   and \<rho>' :: "'d set \<Rightarrow> 'd set \<Rightarrow> 'f \<Rightarrow> 'f"   and b' :: "'f"   and add_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and mult_str' :: "'d set \<Rightarrow> 'f \<Rightarrow> 'f \<Rightarrow> 'f"   and zero_str' :: "'d set \<Rightarrow> 'f"   and one_str' :: "'d set \<Rightarrow> 'f"   and fam_morphisms :: "'d set \<Rightarrow> 'e \<Rightarrow> 'f"   and U :: "'d set"   and V :: "'d set"   and x :: "'e" assumes "morphism_presheaves_of_rings X is_open \<FF> \<rho> b add_str mult_str zero_str one_str \<FF>' \<rho>' b' add_str' mult_str' zero_str' one_str' fam_morphisms"   and "is_open U"   and "is_open V"   and "V \<subseteq> U"   and "x \<in> \<FF> U" shows "(\<rho>' U V \<circ> fam_morphisms U) x = (fam_morphisms V \<circ> \<rho> U V) x"
UINF_atLeast_first: fixes P :: "nat \<Rightarrow> ('a, 'b) uexpr"   and n :: "nat" shows "P n \<sqinter> (\<Sqinter> i \<in> {Suc n..} \<bullet> P i) = (\<Sqinter> i \<in> {n..} \<bullet> P i)"
set_eqI: fixes A :: "'a set"   and R :: "('a, 'b) eq_object_scheme"   and B :: "'a set" assumes "\<And>a. a \<in> A \<Longrightarrow> a .\<in>\<^bsub>R\<^esub> B"   and "\<And>b. b \<in> B \<Longrightarrow> b .\<in>\<^bsub>R\<^esub> A" shows "A {.=}\<^bsub>R\<^esub> B"
lookup_except: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and S :: "'a set" shows "lookup (except p S) = (\<lambda>t. if t \<in> S then 0::'b else lookup p t)"
eventually_inf_principal: fixes P :: "'c \<Rightarrow> bool"   and F :: "'c filter"   and s :: "'c set" shows "eventually P (inf F (principal s)) = (\<forall>\<^sub>F x in F. x \<in> s \<longrightarrow> P x)"
Red1CASSucceed: fixes h :: "'heap"   and a :: "'addr"   and D :: "String.literal"   and F :: "String.literal"   and v :: "'addr val"   and v' :: "'addr val"   and h' :: "'heap"   and uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and l :: "'addr val list" assumes "heap_read h a (CField D F) v"   and "heap_write h a (CField D F) v' h'" shows "uf,P,t \<turnstile>1 \<langle>addr a\<bullet>compareAndSwap(D\<bullet>F, Val v, Val v'),(h, l)\<rangle> -\<lbrace>ReadMem a (CField D F) v, WriteMem a (CField D F) v'\<rbrace>\<rightarrow> \<langle>true,(h', l)\<rangle>"
Koszul_syz_nonzero_lt_1: fixes a :: "'t \<Rightarrow>\<^sub>0 'b"   and b :: "'t \<Rightarrow>\<^sub>0 'b" assumes "rep_list a \<noteq> 0"   and "rep_list b \<noteq> 0"   and "component_of_term (lt a) < component_of_term (lt b)" shows "rep_list a \<odot> b - rep_list b \<odot> a \<noteq> 0"
m1a_inv2i_serv_def: shows "m1a_inv2i_serv \<equiv> {s. \<forall>A B Ra Kab nla. A \<notin> bad \<longrightarrow> runs s Ra = Some (Init, [A, B], aKey Kab # nla) \<longrightarrow> (\<exists>Rs. Kab = sesK (Rs $ sk) \<and> runs s Rs = Some (Serv, [A, B], aNon (Ra $ na) # take is_len nla))}"
rball: shows "\<forall>x\<in>proots r. \<not> P x"
AE_mp: fixes P :: "'d \<Rightarrow> bool"   and M :: "'d measure"   and Q :: "'d \<Rightarrow> bool" assumes "almost_everywhere M P"   and "AE x in M. P x \<longrightarrow> Q x" shows "almost_everywhere M Q"
divide_nat_diff_div_nat_less_one: fixes x :: "nat"   and b :: "nat" shows "real x / real b - real (x div b) < 1"
sum_fun_insert: fixes A :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "finite A" shows "\<Sum> (f ` insert x A) = f x + \<Sum> (f ` A)"
null_set: shows "\<Union> (range N) \<union> N' \<in> null_sets M"
aneg_le: fixes x :: "ant"   and y :: "ant" shows "(\<not> x \<le> y) = (y < x)"
eq_set_perm_remdups: fixes xs :: "'b list"   and ys :: "'b list" assumes "set xs = set ys" shows "remdups xs <~~> remdups ys"
expand: fixes sum :: "'a + 'b"   and sum' :: "'a + 'b" assumes "isl sum = isl sum'"   and "\<lbrakk>isl sum; isl sum'\<rbrakk> \<Longrightarrow> projl sum = projl sum'"   and "\<lbrakk>\<not> isl sum; \<not> isl sum'\<rbrakk> \<Longrightarrow> projr sum = projr sum'" shows "sum = sum'"
\<tau>: shows "m\<tau>move (ls, (ts, m), ws, is) (t', \<lbrace>\<rbrace>) (ls', (ts', m'), ws', is')"
Ints_def: 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" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.Ints one plus zero minus = range (ring_1.of_int one plus zero minus)"
sup_eq_bot_iff: fixes x :: "'a"   and y :: "'a" shows "(x + y = (0::'a)) = (x = (0::'a) \<and> y = (0::'a))"
assms_3: fixes i :: "nat" assumes "i < hd (dims A)" shows "subtensor A i = subtensor B i"
assms_3: shows "\<forall>k<length \<pi>. are_all_operators_non_interfering (\<pi> ! k)"
one_plus_numeral_commute: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and x :: "num" assumes "class.numeral plus" shows "plus one (numeral.numeral one plus x) = plus (numeral.numeral one plus x) one"
sgnx_values: fixes f :: "real \<Rightarrow> real"   and F :: "real filter" assumes "f sgnx_able F"   and "F \<noteq> bot" shows "sgnx f F = - 1 \<or> sgnx f F = 0 \<or> sgnx f F = 1"
wf_inv_image: fixes r :: "('c \<times> 'c) set"   and f :: "'d \<Rightarrow> 'c" assumes "wf r" shows "wf (inv_image r f)"
monom_mult_zero_left: fixes pair_of_term :: "'c \<Rightarrow> 'd \<times> 'e"   and term_of_pair :: "'d \<times> 'e \<Rightarrow> 'c"   and t :: "'d"   and p :: "'c \<Rightarrow>\<^sub>0 'f" assumes "term_powerprod pair_of_term term_of_pair" shows "term_powerprod.monom_mult pair_of_term term_of_pair (0::'f) t p = 0"
cd_nnn: fixes n :: "nat" assumes "0 < n" shows "cd n n n"
ebind_rule_iff: fixes m :: "('c + 'e) nres"   and f :: "'e \<Rightarrow> ('c + 'd) nres"   and \<Phi> :: "'c \<Rightarrow> bool"   and \<Psi> :: "'d \<Rightarrow> bool" shows "(ebind m f \<le> ESPEC \<Phi> \<Psi>) = (m \<le> ESPEC \<Phi> (\<lambda>x. f x \<le> ESPEC \<Phi> \<Psi>))"
rel_mono_strong: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and x :: "('d, 'f, 'h) gpv"   and y :: "('e, 'g, 'h) gpv"   and R1a :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2a :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "rel_gpv R1 R2 x y"   and "\<And>z1 y1. \<lbrakk>z1 \<in> results'_gpv x; y1 \<in> results'_gpv y; R1 z1 y1\<rbrakk> \<Longrightarrow> R1a z1 y1"   and "\<And>z2 y2. \<lbrakk>z2 \<in> outs'_gpv x; y2 \<in> outs'_gpv y; R2 z2 y2\<rbrakk> \<Longrightarrow> R2a z2 y2" shows "rel_gpv R1a R2a x y"
simps_3: shows "real_of_ereal (- \<infinity>) = 0"
assms_4: shows "SubRel.subpath g' v1 es v2 subs"
\<delta>'_def: shows "(\<delta>', p\<^sub>1) = find_closest_code p\<^sub>0 \<delta> (p\<^sub>2 # ps)"
comp_def: fixes g :: "'c"   and f :: "'c" shows "Fop.Bop.comp g f \<equiv> f \<cdot>\<^sup>o\<^sup>p g"
option_of: shows "option_of = inv_image is_option_of Rep_game"
EndPhase1_HInv3_p: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and q :: "Proc"   and d :: "Disk" assumes "EndPhase1 s s' p"   and "HInv3_L s' p q d" shows "HInv3_R s' p q d"
assms: shows "ground\<^sub>t\<^sub>s (snd l)"
uminus_finfun_apply: fixes g :: "'a \<Rightarrow>f 'b" shows "($) (- g) = - ($) g"
square_free_int_imp_resultant_non_zero_mod_ring: fixes f :: "int poly" assumes "square_free f"   and "separable_bound f < int CARD('a)" shows "resultant (of_int_poly f) (pderiv (of_int_poly f)) \<noteq> 0 \<and> of_int_poly f \<noteq> 0"
hcomp_simps\<^sub>W\<^sub>C_2: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<nu> :: "'b"   and \<mu> :: "'b" assumes "weak_composition V H"   and "H \<nu> \<mu> \<noteq> partial_magma.null V" shows "partial_magma.dom V (H \<nu> \<mu>) = H (partial_magma.dom V \<nu>) (partial_magma.dom V \<mu>)"
simps_4: fixes uw :: "pdf_operator"   and e :: "expr" shows "randomfree (uw $$ e) = randomfree e"
prems_1: fixes args' :: "'a list" assumes "t = name $$ args'"   and "list_all2 P args' (xs @ [x])" shows "thesis"
closed_segment_translation: fixes c :: "'b"   and a :: "'b"   and b :: "'b" shows "closed_segment (c + a) (c + b) = (+) c ` closed_segment a b"
unit_char_2: fixes a :: "'a" assumes "C.obj a" shows "D.iso (unit a)"
ipset_from_netmask_minusone: fixes ip :: "'a word" shows "ipset_from_netmask ip (- 1) = {ip}"
lemma_2_5_12_a: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<le> b" shows "c l\<rightarrow> a \<le> c l\<rightarrow> b"
lQuots_def: fixes xs :: "'b list"   and A :: "'b list set" shows "lQuots xs A = {ys. xs @ ys \<in> A}"
a: shows "liminf (\<lambda>x. ereal (real (card (A \<inter> {..<x})) / real x)) \<le> 1"
poly_add_degree_eq: fixes K :: "'a list set"   and p1 :: "'a list list"   and p2 :: "'a list list" assumes "subring K (poly_ring R)"   and "polynomial\<^bsub>poly_ring R\<^esub> K p1"   and "polynomial\<^bsub>poly_ring R\<^esub> K p2"   and "degree p1 \<noteq> degree p2" shows "degree (UP.poly_add p1 p2) = max (degree p1) (degree p2)"
map_mat_def: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b mat" shows "map_mat f A \<equiv> mat (dim_row A) (dim_col A) (\<lambda>ij. f (A $$ ij))"
vars_pivot_eq: fixes pivot_eq :: "nat \<times> linear_poly \<Rightarrow> nat \<Rightarrow> nat \<times> linear_poly"   and x\<^sub>j :: "nat"   and eq :: "nat \<times> linear_poly" assumes "PivotEq TYPE('a) pivot_eq"   and "x\<^sub>j \<in> rvars_eq eq"   and "lhs eq \<notin> rvars_eq eq" shows "let eq' = pivot_eq eq x\<^sub>j in lhs eq' = x\<^sub>j \<and> rvars_eq eq' = {lhs eq} \<union> (rvars_eq eq - {x\<^sub>j})"
prime_nat_numeral_eq: fixes m :: "num" shows "prime (numeral m) = (1 < numeral m \<and> (\<forall>n\<in>set [2..<numeral m]. \<not> n dvd numeral m))"
rComm1: fixes P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and xvec :: "name list"   and P' :: "('a, 'b, 'c) psi"   and Q' :: "('a, 'b, 'c) psi"   and \<Psi>\<^sub>Q :: "'b"   and M :: "'a"   and N :: "'a"   and A\<^sub>P :: "name list"   and \<Psi>\<^sub>P :: "'b"   and K :: "'a"   and A\<^sub>Q :: "name list" assumes "cP = P \<parallel> Q"   and "cRs = \<tau> \<prec> \<lparr>\<nu>*xvec\<rparr>P' \<parallel> Q'"   and "\<Psi> \<otimes> \<Psi>\<^sub>Q \<rhd> P \<longmapsto> M\<lparr>N\<rparr> \<prec> P'"   and "extractFrame P = \<langle>A\<^sub>P, \<Psi>\<^sub>P\<rangle>"   and "distinct A\<^sub>P"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<rhd> Q \<longmapsto> K\<lparr>\<nu>*xvec\<rparr>\<langle>N\<rangle> \<prec> Q'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "distinct A\<^sub>Q"   and "\<Psi> \<otimes> \<Psi>\<^sub>P \<otimes> \<Psi>\<^sub>Q \<turnstile> M \<leftrightarrow> K"   and "A\<^sub>P \<sharp>* \<Psi>"   and "A\<^sub>P \<sharp>* \<Psi>\<^sub>Q"   and "A\<^sub>P \<sharp>* P"   and "A\<^sub>P \<sharp>* M"   and "A\<^sub>P \<sharp>* N"   and "A\<^sub>P \<sharp>* P'"   and "A\<^sub>P \<sharp>* Q"   and "A\<^sub>P \<sharp>* Q'"   and "A\<^sub>P \<sharp>* A\<^sub>Q"   and "A\<^sub>P \<sharp>* xvec"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* \<Psi>\<^sub>P"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* K"   and "A\<^sub>Q \<sharp>* N"   and "A\<^sub>Q \<sharp>* P'"   and "A\<^sub>Q \<sharp>* Q"   and "A\<^sub>Q \<sharp>* Q'"   and "A\<^sub>Q \<sharp>* xvec"   and "xvec \<sharp>* \<Psi>"   and "xvec \<sharp>* \<Psi>\<^sub>P"   and "xvec \<sharp>* \<Psi>\<^sub>Q"   and "xvec \<sharp>* P"   and "xvec \<sharp>* M"   and "xvec \<sharp>* Q"   and "xvec \<sharp>* K"   and "A\<^sub>P \<sharp>* C"   and "A\<^sub>Q \<sharp>* C"   and "xvec \<sharp>* C"   and "distinct xvec" shows "Prop"
h_def: shows "h = (\<lambda>x. - g z0 + Ln (f z0) + g x)"
expand_init_def: shows "expand_init \<equiv> 0::'a"
c: assumes "n \<noteq> 0" shows "eval c [f \<triangleright> n] = inconsist (amalg01 f) (f \<triangleright> n)"
markedElementIsMemberOfItsMarkedElementsTo: fixes M :: "('b \<times> bool) list"   and e :: "'b \<times> bool" assumes "uniq (elements M)"   and "marked e"   and "e \<in> set M" shows "element e \<in> set (markedElementsTo (element e) M)"
ins_eq: fixes x :: "'a"   and y :: "'a"   and S :: "'b" assumes "is_node x"   and "is_node y"   and "invariant S"   and "\<not> memb y S" shows "memb x (ins y S) = (x = y \<or> memb x S)"
Cons_eq_appendI: fixes x :: "'f"   and xs1 :: "'f list"   and ys :: "'f list"   and xs :: "'f list"   and zs :: "'f list" assumes "x # xs1 = ys"   and "xs = xs1 @ zs" shows "x # xs = ys @ zs"
pres_type_def: fixes step :: "nat \<Rightarrow> 'c \<Rightarrow> (nat \<times> 'c) list"   and n :: "nat"   and A :: "'c set" shows "pres_type step n A = (\<forall>\<tau>\<in>A. \<forall>p<n. \<forall>(q, \<tau>')\<in>set (step p \<tau>). \<tau>' \<in> A)"
dim_vec_conjugate: fixes v :: "'b vec" shows "dim_vec (conjugate v) = dim_vec v"
oiw: shows "openin (top_of_set (f ` S)) W"
mem_diff: fixes c' :: "'Com"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val"   and lc :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and mem'' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>c', mds', mem'\<rangle> \<in> loc_reach lc"   and "\<forall>x. var_asm_not_written mds' x \<longrightarrow> mem' x = mem'' x \<and> dma mem' x = dma mem'' x" shows "\<langle>c', mds', mem''\<rangle> \<in> loc_reach lc"
bounded_scaleR_image: fixes f :: "'b \<Rightarrow> real"   and s :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "bounded (f ` s)"   and "bounded (g ` s)" shows "bounded ((\<lambda>x. f x *\<^sub>R g x) ` s)"
prob_space_distr: fixes f :: "'b \<Rightarrow> 'd"   and M' :: "'d measure" assumes "X.random_variable M' f" shows "prob_space (distr (density S Px) M' f)"
mann8: fixes x :: "'a" shows "- (- 3 \<times> x \<squnion> x) \<squnion> 2 \<times> x = - (- (- (- 3 \<times> x \<squnion> x) \<squnion> - 3 \<times> x \<squnion> 2 \<times> x) \<squnion> - 3 \<times> x)"
prv_all_cong: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<in> var"   and "prv (imp \<phi> \<chi>)" shows "prv (imp (all x \<phi>) (all x \<chi>))"
chine_simps_3: shows "C.cod \<mu>.chine = \<mu>.cod.apex"
anonymous_profile_def: shows "anonymous_profile (R(i := Ri')) = image_mset (weak_ranking \<circ> R(i := Ri')) (mset_set agents)"
dvd_int_poly: fixes q :: "int poly"   and p :: "int poly" shows "dvd_int_poly q p = (q dvd p)"
injD: fixes f :: "'d \<Rightarrow> 'e"   and x :: "'d"   and y :: "'d" assumes "inj f"   and "f x = f y" shows "x = y"
assms_2: shows "i < Suc (Suc (length x + length y))"
unsat_def: fixes \<A> :: "('a \<times> ('fun, 'var) strand_step) list" shows "unsat \<A> \<equiv> \<forall>\<I>. subst_domain \<I> = \<V> \<and> ground (subst_range \<I>) \<longrightarrow> \<not> \<I> \<Turnstile> \<langle>unlabel \<A>\<rangle>"
ring_tOp_commute: fixes R :: "('e, 'f) Ring_scheme"   and x :: "'e"   and y :: "'e" assumes "Ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<cdot>\<^sub>r\<^bsub>R\<^esub> y = y \<cdot>\<^sub>r\<^bsub>R\<^esub> x"
chine_assoc_props_2: shows "trg_trg_f.Prj\<^sub>1 \<cdot> trg_trg_f.chine_assoc = trg_trg_f.Prj\<^sub>1\<^sub>1"
relcompp_assoc: fixes r :: "'d \<Rightarrow> 'g \<Rightarrow> bool"   and s :: "'g \<Rightarrow> 'f \<Rightarrow> bool"   and t :: "'f \<Rightarrow> 'e \<Rightarrow> bool" shows "(r OO s) OO t = r OO s OO t"
norm_CBasis: fixes scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and scaleC :: "complex \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and cinner :: "'c \<Rightarrow> 'c \<Rightarrow> complex"   and CBasis :: "'c set"   and u :: "'c" assumes "class.ceuclidean_space scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner CBasis"   and "u \<in> CBasis" shows "norm u = 1"
implements_def: fixes pi :: "'e \<Rightarrow> 'f"   and Sa :: "('g, 'f) spec"   and Sc :: "('h, 'e) spec" shows "implements pi Sa Sc \<equiv> map pi ` obeh Sc \<subseteq> obeh Sa"
vimage_Inl_image_Inr: fixes A :: "'j set" shows "Inl -` Inr ` A = {}"
order_refl: fixes x :: "'t" shows "le_of_nat_term_order cmp_term x x"
assms: fixes v :: "'a llist" assumes "v \<le> w"   and "lfinite v" shows "lproject A v \<le> x"
supp_append: fixes xs :: "'b list"   and ys :: "'b list" shows "supp (xs @ ys) = supp xs \<union> supp ys"
RedCallNull: fixes E :: "char list \<Rightarrow> ty option"   and Copt :: "char list option"   and M :: "char list"   and vs :: "val list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and 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" shows "P,E \<turnstile> \<langle>Call null Copt M (map Val vs),s\<rangle> \<rightarrow> \<langle>Throw (addr_of_sys_xcpt NullPointer, [NullPointer]),s\<rangle>"
new_document_def: shows "new_document = Heap_Error_Monad.bind get_heap (\<lambda>h. Heap_Error_Monad.bind (return (new\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t h)) (\<lambda>(new_ptr, h'). Heap_Error_Monad.bind (return_heap h') (\<lambda>_. return new_ptr)))"
fisher_yates_alt_aux_altdef: fixes i :: "nat"   and xs :: "'a list" assumes "i < length xs" shows "fisher_yates_alt_aux i xs = map_pmf rev (fisher_yates_aux (length xs - i - 1) (rev xs))"
set_iteratei_def: fixes \<alpha> :: "'c \<Rightarrow> 'd set"   and invar :: "'c \<Rightarrow> bool"   and iteratei :: "'c \<Rightarrow> ('e \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'e \<Rightarrow> 'e) \<Rightarrow> 'e \<Rightarrow> 'e" shows "set_iteratei \<alpha> invar iteratei \<equiv> finite_set \<alpha> invar \<and> set_iteratei_axioms \<alpha> invar iteratei"
\<I>: shows "subst_domain \<I> = \<V> \<and> ground (subst_range \<I>)"
vmatch_env: fixes pat :: "pat"   and v :: "value"   and env :: "(name, value) fmap" assumes "vmatch pat v = Some env"   and "not_shadows_vconsts v" shows "not_shadows_vconsts_env env"
assms: shows "t \<in>\<^sub>A frontier M1 - frontier M0"
represents_def: fixes B :: "'b set"   and r :: "'b \<Rightarrow>\<^sub>0 'a"   and x :: "'b" shows "represents B r x = (keys r \<subseteq> B \<and> rep r = x)"
sign_extended_weaken: fixes n :: "nat"   and w :: "'b word"   and m :: "nat" assumes "sign_extended n w"   and "n \<le> m" shows "sign_extended m w"
valid_store_sops: fixes i :: "nat"   and p :: "'a"   and "is" :: "instr list"   and \<theta> :: "nat \<Rightarrow> nat option"   and sb :: "'a memref list"   and \<D> :: "bool"   and \<O> :: "nat set"   and \<R> :: "nat \<Rightarrow> bool option" assumes "i < length ts\<^sub>s\<^sub>b"   and "ts\<^sub>s\<^sub>b ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)" shows "\<forall>sop\<in>store_sops is. valid_sop sop"
perm_sing_eq: fixes ys :: "'a list"   and y :: "'a" shows "(ys <~~> [y]) = (ys = [y])"
vconst_onI: fixes a :: "V"   and A :: "V"   and c :: "V" assumes "a \<in>\<^sub>\<circ> A" shows "\<langle>a, c\<rangle> \<in>\<^sub>\<circ> vconst_on A c"
prime: fixes p :: "int" assumes "poly_mod_prime p" shows "prime p"
SINV_inv_def: fixes par :: "Var list"   and T :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<times> Val \<Rightarrow> bool"   and I :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> Val list \<times> (Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> (Addr \<times> Class \<times> (Field \<times> Val) list) list \<Rightarrow> bool" shows "SINV_inv par T I = (\<lambda>s0 s h'. \<forall>ops1 ops2 S R h k w. (ops1, par, R, ops2) \<in> Frame \<longrightarrow> T (R, k) (h, w) \<longrightarrow> s = (w # ops2, S, h) \<longrightarrow> I s0 (ops1, S, k) h')"
indexed_fmember_fnth: fixes x :: "'b"   and i :: "nat"   and s :: "'b fset" shows "(x |\<in>|\<^bsub>i\<^esub> s) = (s |!| i = x \<and> i < size s)"
P1_total_imp_R1: fixes f :: "nat \<Rightarrow> nat option" assumes "f \<in> \<P>"   and "total1 f" shows "f \<in> \<R>"
iT_Plus_neg_Plus_le_Min_eq: fixes a :: "nat"   and b :: "nat"   and I :: "nat set" assumes "a \<le> b"   and "a \<le> iMin I" shows "I \<oplus>- a \<oplus> b = I \<oplus> b - a"
bintr_cat_same: fixes n :: "nat"   and b :: "int"   and a :: "int" shows "take_bit n (concat_bit n b a) = take_bit n b"