hcju/isabelleps huggingface dataset:
bounded_linear_right: fixes a :: "'a" shows "bounded_linear ((**) a)"
nonempty_factor_def: fixes u :: "'b list"   and v :: "'b list" shows "u \<le>nf v \<equiv> u \<noteq> \<epsilon> \<and> (\<exists>p s. p \<cdot> u \<cdot> s = v)"
conjr1_iff: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x \<rhd> y \<le> z) = (y \<le> - (x \<cdot> - z))"
diagonalize_tensor: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set"   and g :: "'b local.term set" assumes "free_monoidal_category C"   and "partial_magma.arr (free_monoidal_category.comp C) f"   and "partial_magma.arr (free_monoidal_category.comp C) g" shows "free_monoidal_category.D C (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C f g) = free_monoidal_category.D C (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C (free_monoidal_category.D C f) (free_monoidal_category.D C g))"
gbinomial_factors: fixes a :: "'a"   and k :: "nat" shows "a + (1::'a) gchoose Suc k = (a + (1::'a)) / of_nat (Suc k) * (a gchoose k)"
K_distrib_K_imp: fixes i :: "'b"   and G :: "'b fm list"   and q :: "'b fm" assumes "\<turnstile> K i (imply G q)" shows "\<turnstile> imply (map (K i) G) (K i q)"
c_pair_at_0: shows "c_pair 0 0 = 0"
and_pre_def: fixes P1 :: "'b \<Rightarrow> bool"   and P2 :: "'b \<Rightarrow> bool" shows "and_pre P1 P2 \<equiv> \<lambda>x. P1 x \<and> P2 x"
inputInduct: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and F :: "'a \<Rightarrow> pi \<Rightarrow> name \<Rightarrow> name \<Rightarrow> pi \<Rightarrow> bool"   and C :: "'a" assumes "P \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l P'"   and "x \<sharp> P"   and "\<And>a x P C. F C (a<x>.P) a x P"   and "\<And>P a x P' b C. \<lbrakk>P \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l P'; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C ([b\<frown>b]P) a x P'"   and "\<And>P a x P' b c C. \<lbrakk>P \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l P'; \<And>C. F C P a x P'; b \<noteq> c\<rbrakk> \<Longrightarrow> F C ([b\<noteq>c]P) a x P'"   and "\<And>P Q a x P' C. \<lbrakk>P \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l P'; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a x P'"   and "\<And>P Q a x Q' C. \<lbrakk>Q \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l Q'; \<And>C. F C Q a x Q'\<rbrakk> \<Longrightarrow> F C (P \<oplus> Q) a x Q'"   and "\<And>P P' Q a x C. \<lbrakk>P \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l P'; x \<sharp> P; x \<sharp> Q; x \<noteq> a; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a x (P' \<parallel> Q)"   and "\<And>P Q Q' a x C. \<lbrakk>Q \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l Q'; x \<sharp> P; x \<sharp> Q; x \<noteq> a; \<And>C. F C Q a x Q'\<rbrakk> \<Longrightarrow> F C (P \<parallel> Q) a x (P \<parallel> Q')"   and "\<And>P P' a x y C. \<lbrakk>P \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l P'; y \<noteq> a; y \<noteq> x; y \<sharp> C; \<And>C. F C P a x P'\<rbrakk> \<Longrightarrow> F C (<\<nu>y>P) a x (<\<nu>y>P')"   and "\<And>P a x P' C. \<lbrakk>P \<parallel> !P \<longmapsto>\<^sub>l a<x> \<prec>\<^sub>l P'; \<And>C. F C (P \<parallel> !P) a x P'\<rbrakk> \<Longrightarrow> F C (!P) a x P'" shows "F C P a x P'"
IH: fixes ws' :: "val list" assumes "map Val vs' = map Val ws' @ throw ex # es" shows "False"
not_is_all_imp_strip_all_body_unchanged: fixes t :: "Core.term" assumes "\<not> is_all t" shows "strip_all_body t = t"
ty: shows "\<Gamma> \<turnstile> LAM [x].t : T"
sqrt_def: fixes n :: "nat" shows "local.sqrt n = Max {m. m\<^sup>2 \<le> n}"
antisym_conv2: fixes x :: "'a Word.word"   and y :: "'a Word.word" assumes "x \<le>s y" shows "(\<not> x <s y) = (x = y)"
ltake_trans_fusion: fixes g :: "'d \<Rightarrow> ('c, 'd) step"   and n :: "enat"   and s :: "'d" shows "lunstream (ltake_trans g) (n, s) = ltake n (lunstream g s)"
less_trans: assumes "frac x < (1::'a)"   and "(1::'a) < x" shows "frac x < x"
Cons_eq_append_conv: fixes x :: "'b"   and xs :: "'b list"   and ys :: "'b list"   and zs :: "'b list" shows "(x # xs = ys \<cdot> zs) = (ys = \<epsilon> \<and> x # xs = zs \<or> (\<exists>ys'. x # ys' = ys \<and> xs = ys' \<cdot> zs))"
correct: fixes \<G> :: "'d cyclic_group"   and x0 :: "'d"   and x1 :: "'d"   and \<sigma> :: "bool" assumes "ot \<G>"   and "x0 \<in> carrier \<G>"   and "x1 \<in> carrier \<G>" shows "funct_OT_12 (x0, x1) \<sigma> = ot_base.protocol_ot \<G> (x0, x1) \<sigma>"
ts_inv_okE: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and I :: "'f \<Rightarrow> 'i option" assumes "ts_inv_ok ts I" obtains "\<forall>t. (ts t = None) = (I t = None)"
None_lfinite: fixes t :: "'b llist"   and i :: "nat" assumes "t !! i = None" shows "t \<in> UNIV\<^sup>\<star>"
pOp_assocTr42: fixes A :: "('e, 'f) aGroup_scheme"   and a :: "'e"   and b :: "'e"   and c :: "'e"   and d :: "'e" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A"   and "c \<in> carrier A"   and "d \<in> carrier A" shows "a \<plusminus>\<^bsub>A\<^esub> b \<plusminus>\<^bsub>A\<^esub> c \<plusminus>\<^bsub>A\<^esub> d = a \<plusminus>\<^bsub>A\<^esub> (b \<plusminus>\<^bsub>A\<^esub> c) \<plusminus>\<^bsub>A\<^esub> d"
atoms_BigAnd: fixes Fs :: "'a formula list" shows "atoms \<^bold>\<And>Fs = \<Union> (atoms ` set Fs)"
red2_simps_4: fixes u :: "'a local.term"   and v :: "'a local.term" assumes "Ide u"   and "Nml u"   and "Ide v"   and "Nml v"   and "Src u = Trg v" shows "Cod (u \<^bold>\<Down> v) = \<^bold>\<lfloor>u \<^bold>\<star> v\<^bold>\<rfloor>"
phi0: assumes "wtl ins c 0 s\<^sub>0 \<noteq> \<top>"   and "0 < length ins" shows "s\<^sub>0 \<sqsubseteq>\<^bsub>r\<^esub> \<tau>s ! 0"
tmod2: fixes mn :: "char list"   and s1 :: "'b state"   and ds :: "dec list"   and ck :: "bool"   and env :: "v sem_env"   and s2 :: "'b state"   and err :: "v error_result"   and specs :: "spec list option" assumes "[mn] \<notin> defined_mods s1 \<and> no_dup_types ds \<and> evaluate_decs ck [mn] env s1 ds (s2, Rerr err)" shows "evaluate_top ck env s1 (Tmod mn specs ds) (update_defined_mods (\<lambda>_. {[mn]} \<union> defined_mods s2) s2, Rerr err)"
spectral_radius_mem_max_1: fixes A :: "complex mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "0 < n" shows "spectral_radius A \<in> cmod ` spectrum A"
cotr_init: fixes c :: "('p, 's, 'm) configuration" shows "cotrace c LNil"
poly_deg_map_indets_le: fixes f :: "'d \<Rightarrow> 'b"   and p :: "('d \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" shows "poly_deg (map_indets f p) \<le> poly_deg p"
legs_form_cospan_2: shows "C.cospan \<nu>.cod.leg0 \<pi>.cod.leg1"
pdom_graph_pfun: fixes R :: "('a \<times> 'b) set" shows "pdom (graph_pfun R) = Domain R"
comp_cancel: fixes z :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "z \<cdot> u \<bowtie> z \<cdot> v" shows "u \<bowtie> v"
Compl: fixes S :: "'a set set" assumes "S \<in> fba" shows "- S \<in> fba"
borel_measurable_nn_cond_exp: fixes M :: "'b measure"   and F :: "'b measure"   and f :: "'b \<Rightarrow> ennreal" shows "nn_cond_exp M F f \<in> borel_measurable F"
nnegD2: fixes P :: "'b \<Rightarrow> 'c" assumes "nneg P" shows "(\<lambda>s. 0::'c) \<le> P"
abs_split_full: fixes xs :: "('c \<times> 'd) list"   and p :: "'d" assumes "\<forall>(uu_, s)\<in>set xs. s < p" shows "abs_split xs p = (xs, [])"
rCase: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<phi> :: "'c"   and Cs :: "('c \<times> ('a, 'b, 'c) psi) list"   and C :: "'d" assumes "\<Psi> \<rhd> P \<longmapsto> \<tau> \<prec> P'"   and "\<And>C. Prop C \<Psi> P P'"   and "(\<phi>, P) mem Cs"   and "\<Psi> \<turnstile> \<phi>"   and "guarded P" shows "Prop C \<Psi> (Cases Cs) P'"
quot_to_fract_quot_of_fract: fixes x :: "'a fract" shows "quot_to_fract (quot_of_fract x) = x"
properfactorE2: fixes G :: "('c, 'd) monoid_scheme"   and a :: "'c"   and b :: "'c" assumes "properfactor G a b" obtains "a divides\<^bsub>G\<^esub> b" and "\<not> a \<sim>\<^bsub>G\<^esub> b"
column_Gram_Schmidt_upt_k_preserves: fixes i :: "'a"   and k :: "nat"   and A :: "(('b, 'a) vec, 'c) vec" assumes "mod_type_class.to_nat i < Suc k"   and "Suc k < CARD('a)" shows "column i (Gram_Schmidt_upt_k A (Suc k)) = column i (Gram_Schmidt_upt_k A k)"
Mxs_1: shows "loc_imps_fw c loc1 loc2 M xs"
form_of_cnf_is_nnf: fixes R :: "'a literal list list" shows "is_nnf (form_of_cnf R)"
UN_finite_bound: fixes A :: "'b set"   and f :: "nat \<Rightarrow> 'b set" assumes "finite A"   and "A \<subseteq> \<Union> (range f)" shows "\<exists>m. A \<subseteq> \<Union> (f ` {..m})"
ahm_ops_def: shows "ahm_ops \<equiv> ahm_basic.dflt_ops"
assms: fixes Q :: "ccs" shows "(\<lparr>\<nu>x\<rparr>\<lparr>\<nu>y\<rparr>Q, \<lparr>\<nu>y\<rparr>\<lparr>\<nu>x\<rparr>Q) \<in> Rel"
Deriv_pderiv: fixes c :: "'b"   and r :: "'b rexp" shows "Deriv c (lang r) = \<Union> (lang ` pderiv c r)"
UN_I: fixes a :: "'a"   and A :: "'a set"   and b :: "'b"   and B :: "'a \<Rightarrow> 'b set" assumes "a \<in> A"   and "b \<in> B a" shows "b \<in> \<Union> (B ` A)"
start_nth0_unstable_auxi: shows "\<not> [0] \<sqsubseteq>\<^bsub>r\<^esub> rev [0..<n]"
fold_A_im_min_gallery': fixes Cs :: "'a set list" assumes "SimplicialComplex.min_maxsimpchain (fold_A \<turnstile> A) (C # Cs)" shows "SimplicialComplex.min_maxsimpchain A (C # Cs)"
index_Basis_list_axis1: fixes i :: "'e" shows "index Basis_list (axis i 1) = index enum_class.enum i"
simps_2: fixes x21 :: "'b Tree"   and x22 :: "'b"   and x23 :: "'b Tree" shows "Tip \<noteq> T x21 x22 x23"
antipar_1: shows "trg d\<^sub>0 = src e\<^sub>0"
list_rel_id_simp: shows "\<langle>Id\<rangle>list_rel = Id"
symmetric_subset: fixes A :: "'c set"   and equiv :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'c set" assumes "symmetric A equiv"   and "B \<subseteq> A" shows "symmetric B equiv"
D_set_def: fixes x :: "'a set"   and y :: "'a set" shows "D x y \<equiv> x \<inter> y = {}"
assms: shows "rel_spmf (R \<upharpoonleft> P \<otimes> Q) x y"
bind_spmf_parametric: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'f \<Rightarrow> bool" shows "rel_fun (rel_spmf A) (rel_fun (rel_fun A (rel_spmf B)) (rel_spmf B)) (\<bind>) (\<bind>)"
bij_lens_equiv_id: fixes X :: "'a \<Longrightarrow> 'b" shows "bij_lens X = X \<approx>\<^sub>L \<Sigma>"
t: shows "t = (p, is, \<theta>, sb, \<D>, \<O>, \<R>)"
neq_def: fixes c :: "int"   and t :: "Ferrack.num" shows "neq c t = (if c = 0 then NEq t else if 0 < c then NEq (CN 0 c t) else NEq (CN 0 (- c) (Neg t)))"
lookup_pair_eq_0: fixes xs :: "('c \<times> 'd) list"   and k :: "'c" assumes "tc.oalist_inv_raw xs" shows "(lookup_pair_tc xs k = (0::'d)) = (k \<notin> fst ` set xs)"
that: fixes e :: "real" assumes "0 < e"   and "ball x e \<inter> {y. (y - a) \<bullet> rot (a - b) = 0} \<subseteq> {a<--<b}" shows "thesis"
results_gpv_def: fixes \<Gamma> :: "('e, 'f) \<I>"   and gpv :: "('d, 'e, 'f) gpv" shows "results_gpv \<Gamma> gpv \<equiv> {x. resultsp_gpv \<Gamma> x gpv}"
Extend_in_C: fixes C :: "'b fm set set"   and S :: "'b fm set"   and f :: "nat \<Rightarrow> 'b fm" assumes "consistency C"   and "finite_char C"   and "S \<in> C" shows "Extend S C f \<in> C"
valid_refs_imp_reachable_snapshot_inv: fixes s :: "'e process_name \<Rightarrow> ('f, 'e, 'g, 'h) Model.local_state"   and m :: "'e" assumes "valid_refs s"   and "obj_mark ` ran (sys_heap s) \<subseteq> {sys_fM s}"   and "\<forall>p. sys_mem_store_buffers p s = []"   and "\<forall>m. ghost_honorary_root (s (mutator m)) = {}" shows "mut_m.reachable_snapshot_inv m s"
tab_simps_2: shows "src Tfg.\<rho>\<sigma>.tab = src (tab\<^sub>0 g \<star> f\<^sub>0g\<^sub>1.p\<^sub>0)"
simps_1: fixes x :: "('b, 'c) Combinators list" shows "separate (DenyAll # x) = DenyAll # separate x"
Union_Strong_dlHPP_Bisim: fixes d :: "'d"   and PP :: "nat set"   and R :: "('com list \<times> 'com list) set"   and R' :: "('com list \<times> 'com list) set" assumes "SdlHPPB d PP R"   and "SdlHPPB d PP R'"   and "Arefl R \<inter> Arefl R' \<subseteq> {[]}" shows "SdlHPPB d PP (R \<union> R')"
subgroup_of_subgroup: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme"   and U :: "'c set" assumes "subgroup H G"   and "Group.group G"   and "subgroup U (G\<lparr>carrier := H\<rparr>)" shows "subgroup U G"
map_scale_eq_monom_mult: fixes c :: "'c"   and p :: "'b \<Rightarrow>\<^sub>0 'c" shows "c \<cdot> p = MPoly_Type_Class.punit.monom_mult c (0::'b) p"
ltrm_of_sum_diff_deg: fixes q :: "nat \<Rightarrow> 'a"   and a :: "'a"   and n :: "nat"   and p :: "nat \<Rightarrow> 'a" assumes "q \<in> carrier P"   and "a \<in> carrier R"   and "a \<noteq> \<zero>"   and "deg R q < n"   and "p = q \<oplus>\<^bsub>P\<^esub> monom P a n" shows "monom P (p (deg R p)) (deg R p) = monom P a n"
facE': fixes u :: "'b list"   and v :: "'b list" assumes "u \<le>f v" obtains p :: "'b list" and s :: "'b list" where "p \<cdot> u \<cdot> s = v"
gext_objD: fixes s :: "st"   and s' :: "st"   and r :: "oref"   and obj :: "obj" assumes "s\<le>|s'"   and "globs s r = Some obj" shows "\<exists>obj'. globs s' r = Some obj' \<and> tag obj' = tag obj"
poincare_between_x_axis_uvw: fixes x :: "real"   and y :: "real"   and z :: "real" assumes "- 1 < x"   and "x < 1"   and "- 1 < y"   and "y < 1"   and "y \<noteq> x"   and "- 1 < z"   and "z < 1"   and "z \<noteq> x" shows "poincare_between (of_complex (cor x)) (of_complex (cor y)) (of_complex (cor z)) = (y < x \<and> z < x \<and> z \<le> y \<or> x < y \<and> x < z \<and> y \<le> z)"
mono_step'_top: fixes C :: "'av st option acom"   and C' :: "'av st option acom" assumes "C \<le> C'" shows "step' \<top> C \<le> step' \<top> C'"
subgroup_mult_id: fixes H :: "'a set" assumes "subgroup H (add_monoid G)" shows "H <#>\<^bsub>add_monoid G\<^esub> H = H"
aff_dim_eq_dim: fixes a :: "'b"   and S :: "'b set" assumes "a \<in> affine hull S" shows "aff_dim S = int (dim ((+) (- a) ` S))"
median_5_def: fixes a :: "'b"   and b :: "'b"   and c :: "'b"   and d :: "'b"   and e :: "'b" shows "median_5 a b c d e = (let (x1, x2) = if a \<le> b then (a, b) else (b, a); (x4, x5) = if d \<le> e then (d, e) else (e, d) in if x1 \<le> x4 then median_5_aux x1 x2 c x4 x5 else median_5_aux x4 x5 c x1 x2)"
log_pow_cancel: fixes a :: "real"   and b :: "nat" assumes "0 < a"   and "a \<noteq> 1" shows "Transcendental.log a (a ^ b) = real b"
prime_ideal_def: fixes R :: "('d, 'e) Ring_scheme"   and p :: "'d set" shows "prime_ideal R p = (ideal R p \<and> 1\<^sub>r\<^bsub>R\<^esub> \<notin> p \<and> (\<forall>x\<in>carrier R. \<forall>y\<in>carrier R. x \<cdot>\<^sub>r\<^bsub>R\<^esub> y \<in> p \<longrightarrow> x \<in> p \<or> y \<in> p))"
uniform_limitI: fixes S :: "'d set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and l :: "'d \<Rightarrow> 'e"   and F :: "'c filter" assumes "\<And>e. 0 < e \<Longrightarrow> \<forall>\<^sub>F n in F. \<forall>x\<in>S. dist (f n x) (l x) < e" shows "uniform_limit S f l F"
linear_surj_imp_inj: fixes f :: "('d, 'e) vec \<Rightarrow> ('d, 'e) vec" assumes "Vector_Spaces.linear (*s) (*s) f"   and "surj f" shows "inj f"
Red_I_of_Inf_to_N: fixes \<iota> :: "'a literal multiset inference"   and N :: "'a literal multiset set" assumes "\<iota> \<in> F_Inf"   and "concl_of \<iota> \<in> N" shows "\<iota> \<in> F.Red_I_\<G> N"
lm132: fixes Y :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and X :: "'a set" assumes "Y \<subseteq> f -` {0::'b}"   and "finite X" shows "sum f X = sum f (X - Y)"
both_mono1': fixes t :: "'a ttree"   and t' :: "'a ttree"   and t'' :: "'a ttree" assumes "t \<sqsubseteq> t'" shows "t \<otimes>\<otimes> t'' \<sqsubseteq> t' \<otimes>\<otimes> t''"
qSwap_3commute: fixes ys :: "'d"   and y :: "'e"   and X :: "('f, 'g, 'd, 'e, 'h) qTerm"   and y0 :: "'e"   and zs :: "'d"   and z1 :: "'e"   and z2 :: "'e"   and x :: "'e" assumes "qAFresh ys y X"   and "qAFresh ys y0 X"   and "ys \<noteq> zs \<or> y0 \<notin> {z1, z2}" shows "X #[[z1 \<and> z2]]_zs #[[y0 \<and> x @ys[z1 \<and> z2]_zs]]_ys = X #[[y \<and> x]]_ys #[[y0 \<and> y]]_ys #[[z1 \<and> z2]]_zs"
from_mregex_to_mregex: fixes m :: "modality"   and g :: "safety"   and r :: "Formula.formula Regex.regex" assumes "safe_regex m g r" shows "(case to_mregex r of (x, xa) \<Rightarrow> from_mregex x xa) = r"
rel_ANNOT_eq: fixes t :: "'a"   and R :: "('c \<times> 'a) set" shows "t \<equiv> t ::: R"
insert_DiffM: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<in># M" shows "add_mset x (remove1_mset x M) = M"
permutes_domain_partition_eq: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and B :: "'d set"   and p\<^sub>A :: "'c \<Rightarrow> 'c"   and b :: "'d" assumes "f \<in> A \<rightarrow> B"   and "p\<^sub>A permutes A"   and "b \<in> B" shows "p\<^sub>A ` {x \<in> A. f x = b} = {x \<in> A. f (inv p\<^sub>A x) = b}"
R2_R3_commute: shows "R2 \<circ> R3 = R3 \<circ> R2"
f3: shows "\<forall>m p. lookup (mapping_of m) p \<noteq> (0::'a) \<or> p \<notin> monomials m"
WT_converter_parallel_converter2: fixes \<I>1 :: "('h, 'i) \<I>"   and \<I>2 :: "('j, 'k) \<I>"   and conv1 :: "('h, 'i, 'j, 'k) converter"   and \<I>1' :: "('l, 'm) \<I>"   and \<I>2' :: "('n, 'o) \<I>"   and conv2 :: "('l, 'm, 'n, 'o) converter" assumes "\<I>1, \<I>2 \<turnstile>\<^sub>C conv1 \<surd>"   and "\<I>1', \<I>2' \<turnstile>\<^sub>C conv2 \<surd>" shows "\<I>1 \<oplus>\<^sub>\<I> \<I>1', \<I>2 \<oplus>\<^sub>\<I> \<I>2' \<turnstile>\<^sub>C conv1 |\<^sub>= conv2 \<surd>"
c_remove_df: shows "c_remove = (\<lambda>x u. set_to_nat (nat_to_set u - {x}))"
le: shows "(pp \<le> x + x') = (int_of_uint64 pp \<le> int_of_uint64 x + int_of_uint64 x')"
disc_nodes_h3: shows "h3 \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes_h3"
loop_stepE: fixes P\<^sub>1 :: "'a set set"   and P\<^sub>2 :: "'a set set"   and B\<^sub>1 :: "'a set"   and B\<^sub>2 :: "'a set"   and S :: "'a set"   and L :: "'a set"   and u :: "'a" assumes "inv\<^sub>3 P\<^sub>1 P\<^sub>2 B\<^sub>1 B\<^sub>2 S L"   and "B\<^sub>1 \<noteq> {}"   and "u \<in> S"   and "real c < W B\<^sub>1 + w u"   and "real c < W B\<^sub>2 + w u" shows "inv\<^sub>3 (P\<^sub>1 \<union> wrap B\<^sub>1) (P\<^sub>2 \<union> wrap B\<^sub>2) {} {u} (S - {u}) L"
fund_sol_code: fixes D :: "nat" assumes "\<not> is_square D" shows "pell.fund_sol D = projr (while isl (find_fund_sol_step D) (Inl (Suc 0, Suc D)))"
domain_eq: shows "Domainp (pcr_key_order (=)) = comparator"
analz_image_priEK: fixes evs :: "event list"   and i :: "nat"   and KK :: "nat set" assumes "evs \<in> set_mr" shows "(Key (priEK (CA i)) \<in> analz (Key ` KK \<union> knows Spy evs)) = (priEK (CA i) \<in> KK \<or> CA i \<in> bad)"
locks_activate_cond_actions2: fixes s1 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x1 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm1) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and s2 :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x2 \<times> 'l \<Rightarrow>f nat) option) \<times> 'm2) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and cts :: "'t conditional_action list" shows "locks (activate_cond_actions2 s1 s2 cts) = locks s2"
I1: shows "((\<lambda>x. c / max b \<bar>x - a\<bar> ^ n) has_integral F1 x1 - F1 l) {l..x1}"
dfree_Var_simps: fixes x :: "'b" shows "dfree (Var x) = True"
dl_iso_inv_dense: fixes xf :: "('c, 'd) lifted_distrib_lattice_top" shows "dense (Abs_lifted_pair (\<lambda>f. (top, up_filter (Rep_lifted_distrib_lattice_top xf f))))"
Suc_1: fixes A :: "'a mat" assumes "A \<in> carrier_mat n m"   and "J < m" shows "swap_col_to_front A J = mat n m (\<lambda>a. case a of (i, j) \<Rightarrow> if j = 0 then A $$ (i, J) else if j \<le> J then A $$ (i, j - 1) else A $$ (i, j))"
set_mult_card_eq_impl_empty_inter': fixes H :: "'a set"   and J :: "'a set"   and a :: "'a"   and b :: "'a" assumes "finite H"   and "finite J"   and "H \<subseteq> carrier G"   and "J \<subseteq> carrier G"   and "card (H <#> J) = card H * card J"   and "a \<in> H"   and "b \<in> H"   and "a \<noteq> b" shows "(a <# J) \<inter> (b <# J) = {}"
less_term_def: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" shows "(s < t) = strict (\<le>) s t"
1: fixes x :: "'a"   and y :: "'a" assumes "x \<in> span A"   and "y \<in> B" shows "x \<bullet> y = 0"
simps_1: fixes insts :: "((variable \<times> String.literal set) \<times> typ) list"   and a :: "String.literal"   and Ts :: "typ list" shows "subst_typ insts (Ty a Ts) = Ty a (map (subst_typ insts) Ts)"
eqvt: fixes x1 :: "lam"   and x2 :: "lam"   and pi :: "(name \<times> name) list" assumes " x1 \<longrightarrow>\<^sub>1 x2" shows " pi \<bullet> x1 \<longrightarrow>\<^sub>1 pi \<bullet> x2"
mult_ac_2: fixes a :: "'a"   and b :: "'a" shows "a * b = b * a"
simps_2: fixes bs :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "insertion bs (poly.Bound n) = bs n"
sv_round_def: fixes r :: "nat"   and S :: "process set"   and v :: "val"   and r_decisions :: "process \<Rightarrow> val option" shows "sv_round r S v r_decisions = {(s, s'). r = v_state.next_round s \<and> (S \<noteq> {} \<longrightarrow> safe s r v) \<and> d_guard r_decisions (const_map v S) \<and> s' = s\<lparr>v_state.next_round := Suc r, votes := (votes s)(r := const_map v S), v_state.decisions := v_state.decisions s ++ r_decisions\<rparr>}"
divides_mult_lI: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a divides b"   and "a \<in> carrier G"   and "c \<in> carrier G" shows "c \<otimes> a divides c \<otimes> b"
union_iff: fixes a :: "'c"   and A :: "'c multiset"   and B :: "'c multiset" shows "(a \<in># A + B) = (a \<in># A \<or> a \<in># B)"
nprv_clear4_3: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and \<phi>4 :: "'fmla"   and \<psi> :: "'fmla"   and \<phi>3 :: "'fmla" assumes "nprv {\<phi>1, \<phi>2, \<phi>4} \<psi>"   and "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "\<phi>3 \<in> fmla"   and "\<phi>4 \<in> fmla"   and "\<psi> \<in> fmla" shows "nprv {\<phi>1, \<phi>2, \<phi>3, \<phi>4} \<psi>"
fact3: assumes "\<exists>v1. fetch_instruction s'' = Inr v1 \<and> (\<nexists>v2. decode_instruction v1 = Inr v2)" shows "False"
rename_transient: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program"   and A :: "'c set" assumes "bij h" shows "(rename h F \<in> transient (h ` A)) = (F \<in> transient A)"
bind_is_OK_E: fixes h :: "'a"   and f :: "('a, 'b, 'd) prog"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog" assumes "h \<turnstile> ok (f \<bind> g)" obtains x :: "'d" and h' :: "'a" where "h \<turnstile> f \<rightarrow>\<^sub>r x" and "h \<turnstile> f \<rightarrow>\<^sub>h h'" and "h' \<turnstile> ok g x"
set_intros_2: fixes y :: "'a"   and x22 :: "'a list"   and x21 :: "'a" assumes "y \<in> set x22" shows "y \<in> set (x21 # x22)"
assms_2: shows "\<GG> : op_dg \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
distinct_21: fixes x2 :: "'b"   and x7 :: "'b ptrm" shows "PVar x2 \<noteq> PSnd x7"
new_AddrD: fixes s :: "state"   and v :: "val" assumes "new_Addr s = v" shows "(\<exists>a. v = Addr a \<and> heap s a = None) \<or> v = Null"
min_elems_def: fixes X :: "'b set" shows "min_elems X = {s \<in> X. \<forall>t. t \<in> X \<longrightarrow> \<not> t < s}"
Reals_of_real: fixes r :: "real" shows "of_real r \<in> \<real>"
infinite_super: fixes S :: "'b set"   and T :: "'b set" assumes "order.greater_eq T S"   and "infinite S" shows "infinite T"
ac_eq_iff: fixes A :: "'a antichain"   and B :: "'a antichain" shows "(A = B) = (\<forall>x. (x \<in>\<^sub>A A) = (x \<in>\<^sub>A B))"
tl_replicate: fixes n :: "nat"   and x :: "'g" shows "tl (replicate n x) = replicate (n - 1) x"
lookup_raw_eq_value: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and v :: "'c"   and k :: "'a" assumes "oalist_inv xs"   and "v \<noteq> (0::'c)" shows "(lookup_raw xs k = v) = ((k, v) \<in> set (fst xs))"
le_infI1: fixes a :: "'a multiset"   and x :: "'a multiset"   and b :: "'a multiset" assumes "a \<subseteq># x" shows "a \<inter># b \<subseteq># x"
Reds_dom_lcl: 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" assumes "wwf_J_prog P"   and "P \<turnstile> \<langle>e,(h, l, sh),b\<rangle> \<rightarrow>* \<langle>e',(h', l', sh'),b'\<rangle>" shows "dom l' \<subseteq> dom l \<union> fv e"