hcju/isabelleps huggingface dataset:
makes_compatible_dma_eq: fixes cms\<^sub>1 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and cms\<^sub>2 :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and mems :: "(('Var \<Rightarrow> 'Val) \<times> ('Var \<Rightarrow> 'Val)) list"   and i :: "nat"   and \<sigma> :: "'Var \<Rightarrow> 'Val option" assumes "makes_compatible (cms\<^sub>1, mem\<^sub>1) (cms\<^sub>2, mem\<^sub>2) mems"   and "i < length cms\<^sub>1"   and "dom \<sigma> = differing_vars_lists mem\<^sub>1 mem\<^sub>2 mems i" shows "dma (subst \<sigma> (fst (mems ! i))) = dma mem\<^sub>1"
hyps_3: assumes "P \<subseteq> {a. irreducible a}"   and "\<forall>c\<in>P. [a = b] (mod c)"   and "\<forall>p1 p2. p1 \<in> P \<and> p2 \<in> P \<and> p1 \<noteq> p2 \<longrightarrow> comm_monoid_mult_class.coprime p1 p2" shows "[a = b] (mod \<Prod>P)"
S1_def: fixes p :: "nat"   and a :: "int" assumes "prime p"   and "2 < p"   and "[a \<noteq> 0] (mod int p)" shows "S1 p = {0<..int p - 1}"
prems_1: shows "x \<in> lists (UNIV \<times> gens)"
take_append: fixes n :: "nat"   and xs :: "'a list"   and ys :: "'a list" shows "take n (xs @ ys) = take n xs @ take (n - |xs|) ys"
winding_number_lt_half: fixes \<gamma> :: "real \<Rightarrow> complex"   and a :: "complex"   and z :: "complex"   and b :: "real" assumes "valid_path \<gamma>"   and "a \<bullet> z \<le> b"   and "path_image \<gamma> \<subseteq> {w. b < a \<bullet> w}" shows "\<bar>Re (winding_number \<gamma> z)\<bar> < 1 / 2"
norm_triangle_ineq2: fixes a :: "'a"   and b :: "'a" shows "norm a - norm b \<le> norm (a - b)"
th: fixes a :: "real"   and x :: "real" assumes "0 \<le> a" shows "(\<bar>x\<bar> = a) = (x = a \<or> x = - a)"
monom_mult_uminus_left: fixes c :: "'c"   and t :: "'a"   and p :: "'t \<Rightarrow>\<^sub>0 'c" shows "monom_mult (- c) t p = - monom_mult c t p"
connected_space_eq_not_separated: fixes X :: "'c topology" shows "connected_space X = (\<nexists>C1 C2. C1 \<union> C2 = topspace X \<and> C1 \<noteq> {} \<and> C2 \<noteq> {} \<and> separatedin X C1 C2)"
intro: fixes U\<^sub>M :: "'d set"   and scale :: "'e \<Rightarrow> 'd \<Rightarrow> 'd" assumes "\<And>x y a. \<lbrakk>x \<in> U\<^sub>M; y \<in> U\<^sub>M\<rbrakk> \<Longrightarrow> scale a (x + y) = scale a x + scale a y"   and "\<And>x a b. x \<in> U\<^sub>M \<Longrightarrow> scale (a + b) x = scale a x + scale b x"   and "\<And>x a b. x \<in> U\<^sub>M \<Longrightarrow> scale a (scale b x) = scale (a * b) x"   and "\<And>x. x \<in> U\<^sub>M \<Longrightarrow> scale (1::'e) x = x"   and "\<And>x y. \<lbrakk>x \<in> U\<^sub>M; y \<in> U\<^sub>M\<rbrakk> \<Longrightarrow> x + y \<in> U\<^sub>M"   and "(0::'d) \<in> U\<^sub>M"   and "\<And>x a. x \<in> U\<^sub>M \<Longrightarrow> scale a x \<in> U\<^sub>M" shows "module_on U\<^sub>M scale"
lunit_char_3: fixes f :: "'a" assumes "S.ide f" shows "\<exists>!\<mu>. \<guillemotleft>\<mu> : L f \<Rightarrow>\<^sub>S f\<guillemotright> \<and> L \<mu> = (\<iota> \<star> f) \<cdot>\<^sub>S local.inv \<a>[b, b, f]"
exexH_k_dim: fixes n :: "nat"   and f :: "nat \<Rightarrow> bool"   and dims :: "nat list"   and R :: "nat"   and K :: "nat"   and k :: "nat" assumes "grover_state_sig n f dims R K"   and "k < n" shows "grover_state_sig.exexH_k n dims K k \<in> carrier_mat (state_sig.d dims) (state_sig.d dims)"
simps_2: fixes Q :: "('a \<times> 'b, unit) HashMap.hashmap"   and W :: "('a \<times> 'b) list"   and qp :: "'a \<times> 'b"   and qs :: "('a \<times> 'b) list" shows "pa_upd_rule Q W (qp # qs) = (if \<not> hs.memb qp Q then pa_upd_rule (hs.ins qp Q) (qp # W) qs else pa_upd_rule Q W qs)"
res: shows "euclid_ext_poly_dynamic p f g = (a, b)"
orderI: fixes f :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "semilattice_order f less_eq less"   and "a = f a b" shows "less_eq a b"
simps_1: fixes n :: "nat"   and S :: "(nat \<times> nat) set" shows "is_path n S [] = False"
sorted_wrt_list_of_set_eqI: fixes B :: "'b set"   and R :: "('b \<times> 'b) set"   and A :: "'b set"   and xs :: "'b list" assumes "linorder_on B R"   and "A \<subseteq> B"   and "set xs = A"   and "distinct xs"   and "Linorder_Relations.sorted_wrt R xs" shows "sorted_wrt_list_of_set R A = xs"
S_2: shows "\<forall>t\<in>S. t \<le> s"
Par1B: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi" assumes "P \<Longrightarrow>a<\<nu>x> \<prec> P'"   and "x \<sharp> Q" shows "P \<parallel> Q \<Longrightarrow>a<\<nu>x> \<prec> P' \<parallel> Q"
J_def: shows "J = (\<lambda>x. \<Sum>n. moebius_mu (n + Suc (nat \<lfloor>x\<rfloor>)) / real ((n + Suc (nat \<lfloor>x\<rfloor>))\<^sup>2))"
ws_idecl_HasFoo: shows "ws_idecl tprg HasFoo []"
ndet_distr: fixes P :: "'b \<times> 'b \<Rightarrow> bool"   and Q :: "'b \<times> 'b \<Rightarrow> bool"   and R :: "'b \<times> 'b \<Rightarrow> bool" shows "(P \<or> Q \<or> R) = ((P \<or> Q) \<or> P \<or> R)"
chamber_distance_def: fixes C :: "'a set"   and D :: "'a set" shows "chamber_distance C D = (if C = D then 0 else Suc (length (ARG_MIN length Cs. folding.gallery (C # Cs @ [D]))))"
get_child_nodes_new_shadow_root: fixes ptr' :: "(_) object_ptr"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and r :: "(_) heap \<Rightarrow> (_) heap \<Rightarrow> bool" assumes "ptr' \<noteq> cast\<^sub>s\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>r\<^sub>o\<^sub>o\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r\<^sub>2\<^sub>o\<^sub>b\<^sub>j\<^sub>e\<^sub>c\<^sub>t\<^sub>_\<^sub>p\<^sub>t\<^sub>r new_shadow_root_ptr"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "r \<in> Shadow_DOM.get_child_nodes_locs ptr'" shows "r h h'"
fps_XD'_power: fixes a :: "'a fps"   and b :: "'a fps"   and n :: "nat" shows "fps_XD' a (b ^ n) = of_nat n * b ^ (n - 1) * fps_XD' a b"
dbm_lt_asym: fixes e :: "'b DBMEntry"   and f :: "'b DBMEntry" assumes "e \<prec> f" shows "\<not> f \<prec> e"
distinct_11: fixes x3 :: "nat"   and x4 :: "nat" shows "HOLD x3 \<noteq> REL x4"
retract_of_subset: fixes T :: "'a set"   and S :: "'a set"   and s' :: "'a set" assumes "T retract_of S"   and "T \<subseteq> s'"   and "s' \<subseteq> S" shows "T retract_of s'"
Inf_exp_greatest: fixes S :: "('a \<Rightarrow> real) set"   and Q :: "'a \<Rightarrow> real" assumes "S \<noteq> {}"   and "\<forall>P\<in>S. Q \<tturnstile> P" shows "Q \<tturnstile> Inf_exp S"
assms_1: shows "finite {a. g a \<noteq> \<^bold>1}"
Rel_eq_refl: fixes x :: "'a" shows "Transfer.Rel (=) x x"
Abs_map_def: fixes g :: "('c, unit) Old_Datatype.node set \<Rightarrow> 'b"   and M :: "('c, unit) Old_Datatype.node set" shows "Abs_map g M = List_rec M [] (\<lambda>N L. (#) (g N))"
countable_Diff: fixes A :: "'b set"   and B :: "'b set" assumes "countable A" shows "countable (A - B)"
generateI: fixes G :: "('f, 'g) monoid_scheme"   and E :: "'f set"   and H :: "'f set" assumes "Group.group G"   and "subgroup E G"   and "H \<subseteq> E"   and "\<And>K. \<lbrakk>subgroup K G; H \<subseteq> K\<rbrakk> \<Longrightarrow> E \<subseteq> K" shows "E = generate G H"
assms: shows "\<S>\<bullet>None \<tturnstile>_ i vs;es : ts"
isom4b: fixes G :: "('e, 'f) Group_scheme"   and N :: "'e set"   and H :: "'e set" assumes "Group G"   and "G \<triangleright> N"   and "G \<guillemotright> H " shows "(\<natural>\<^bsub>G\<^esub>N \<diamondop>\<^bsub>G\<^esub> H / N) \<cong> (\<natural>\<^bsub>G\<^esub>H / (H \<inter> N))"
Q: shows "UNpart cl P = (\<Union>n<length cl. local.shift cl n ` P n)"
conjugate_exponent_ennreal_1_2_PInf_3: shows "conjugate_exponent \<top> = 1"
vwalkI: fixes p :: "'a list"   and G :: "('a, 'b) pre_digraph" assumes "set p \<subseteq> verts G"   and "set (vwalk_arcs p) \<subseteq> arcs_ends G"   and "p \<noteq> []" shows "vwalk p G"
card_spEdges_le: shows "card spEdges \<le> card uE"
relpowp_imp_rtranclp: fixes n :: "nat"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "(P ^^ n) x y" shows "P\<^sup>*\<^sup>* x y"
log_def: fixes a :: "real"   and x :: "real" shows "log a x = real_ln x / real_ln a"
snd_lens_def: shows "snd\<^sub>L = \<lparr>lens_get = snd, lens_put = \<lambda>(\<sigma>, \<rho>). Pair \<sigma>\<rparr>"
inv_inv_tR: shows "invertible (matrix_inv (Finite_Cartesian_Product.transpose (snd (QR_decomposition A))))"
disj_CSP2: fixes P :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool"   and Q :: "('c, 'd) alpha_rp_scheme \<times> ('c, 'd) alpha_rp_scheme \<Rightarrow> bool" assumes "P is CSP2 healthy"   and "Q is CSP2 healthy" shows "P \<or> Q is CSP2 healthy"
trms\<^sub>s\<^sub>t_update\<^sub>s\<^sub>t_eq_eq: fixes a :: "poscheckvariant"   and t :: "('fun, 'var) Term.term"   and t' :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set"   and \<S>' :: "('fun, 'var) strand_step list set"   and \<A>' :: "('fun, 'var) extstrand_step list"   and \<A> :: "('fun, 'var) extstrand_step list" assumes "\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>"   and "\<S>' = update\<^sub>s\<^sub>t \<S> (\<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S)"   and "\<A>' = \<A> @ [Step \<langle>a: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t]" shows "\<Union> (trms\<^sub>s\<^sub>t ` \<S>) \<union> \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set \<A>) = \<Union> (trms\<^sub>s\<^sub>t ` \<S>') \<union> \<Union> (trms\<^sub>e\<^sub>s\<^sub>t\<^sub>p ` set \<A>')"
pochhammer_rec_if: fixes a :: "'a"   and n :: "nat" shows "pochhammer a n = (if n = 0 then 1::'a else a * pochhammer (a + (1::'a)) (n - 1))"
permute_eq_iff: fixes permute :: "perm \<Rightarrow> 'c \<Rightarrow> 'c"   and p :: "perm"   and x :: "'c"   and y :: "'c" assumes "class.pt permute" shows "(permute p x = permute p y) = (x = y)"
all_rot: fixes m :: "nat" shows "rotate m w \<in> {rotate n w |n. 0 \<le> n \<and> n < \<^bold>|w\<^bold>|}"
inverse_transformation_axioms: shows "inverse_transformation (\<cdot>\<^sup>o\<^sup>p) (\<cdot>\<^sub>S) (\<Phi>'.Y a') F \<Phi>'"
g'_def: shows "g' = dirichlet_prod (\<lambda>n. moebius_mu n * f n) (\<lambda>n. if n = 0 then 0::'a else (1::'a))"
gcd_primitive_part: fixes f :: "'b poly"   and g :: "'b poly" shows "gcd (primitive_part f) (primitive_part g) = normalize (primitive_part (gcd f g))"
j: shows "(j < length A2 * length B2) = (j < length (A1 \<circ> A2 \<otimes> B1 \<circ> B2))"
ereal_right_mult_cong: fixes c :: "ereal"   and d :: "ereal"   and a :: "ereal"   and b :: "ereal" assumes "c = d"   and "d \<noteq> 0 \<Longrightarrow> a = b" shows "c * a = d * b"
feq: shows "\<forall>x\<in>S \<inter> K. f x = h x"
rename_inv_rename: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program" assumes "bij h" shows "rename (inv h) (rename h F) = F"
th2: shows "Inum (real_of_int x # bs) (CN 0 (fst (zsplit0 t)) (snd (zsplit0 t))) = Inum (real_of_int x # bs) t \<and> numbound0 (snd (zsplit0 t))"
additive_congruence: shows "normal_subgroup.Congruence I R (+) \<zero> = contained.Ring_Congruence"
opp_chambers_surj: shows "f \<turnstile> (folding.\<C> - f \<turnstile> folding.\<C>) = f \<turnstile> folding.\<C>"
disjoint_right_eq: fixes A :: "'a set"   and B1 :: "'a set"   and B2 :: "'a set" assumes "A \<union> B1 = A \<union> B2"   and "A \<inter> B1 = {}"   and "A \<inter> B2 = {}" shows "B1 = B2"
pfn_2: fixes x :: "'a set" assumes "x \<in> I" shows "d fine pfn x"
inf_sup: fixes a :: "'a"   and b :: "'a" shows "a - a \<sqinter> b + b = a \<squnion> b"
card_of_Times_ordLess_infinite: fixes C :: "'d set"   and A :: "'e set"   and B :: "'f set" assumes "infinite C"   and "|A| <o |C|"   and "|B| <o |C|" shows "|A \<times> B| <o |C|"
fps_const_mult_apply_left: fixes c :: "'a"   and a :: "'a fps"   and b :: "'a fps" shows "fps_const c * (a oo b) = fps_const c * a oo b"
psubst_imp: fixes \<phi>1 :: "'fmla"   and \<phi>2 :: "'fmla"   and txs :: "('trm \<times> 'var) list" assumes "\<phi>1 \<in> fmla"   and "\<phi>2 \<in> fmla"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm"   and "distinct (map snd txs)" shows "psubst (imp \<phi>1 \<phi>2) txs = imp (psubst \<phi>1 txs) (psubst \<phi>2 txs)"
uniform_limit_on_Un: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and I :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and F :: "'b filter"   and J :: "'c set" assumes "uniform_limit I f g F"   and "uniform_limit J f g F" shows "uniform_limit (I \<union> J) f g F"
letter_neq: fixes a :: "letter"   and b :: "letter"   and c :: "letter" assumes "a \<noteq> b"   and "c \<noteq> a" shows "c = b"
cotangent_space_def: fixes p :: "'a" shows "src.cotangent_space p = dual_space (src.tangent_space p)"
upclose_def: fixes f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> real" shows "upclose f g = (\<lambda>t. max (f t) (g t))"
power_sum: fixes c :: "'c"   and f :: "'d \<Rightarrow> nat"   and A :: "'d set" shows "c ^ sum f A = (\<Prod>a\<in>A. c ^ f a)"
notStuck_WhileTrueD1: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and b :: "'e set"   and c :: "('e, 'd, 'f) com"   and s :: "'e" assumes "\<Gamma>\<turnstile> \<langle>While b c,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"   and "s \<in> b" shows "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck}"
circ_sup_2: fixes x :: "'a"   and y :: "'a" shows "(x \<squnion> y)\<^sup>\<star> \<le> (x\<^sup>\<star> * y\<^sup>\<star>)\<^sup>\<star>"
kronecker_factorization_rat_2: fixes p :: "rat poly" assumes "kronecker_factorization_rat p = None"   and "1 \<le> degree p" shows "irreducible\<^sub>d p"
div_int_then_rqp_2: fixes p :: "int poly"   and q :: "int poly"   and r :: "int poly" assumes "div_int_poly p q = Some r" shows "q \<noteq> 0"
hyps: fixes tab :: "'a \<Rightarrow> 'b option"   and ys :: "'b list"   and z :: "'b"   and tab' :: "'a \<Rightarrow> 'b option" assumes "(tab(xs [\<mapsto>] ys)) vn = Some z"   and "tab vn \<noteq> Some z" shows "(tab'(xs [\<mapsto>] ys)) vn = Some z"
contour_integral_subpath_refl: fixes u :: "real"   and g :: "real \<Rightarrow> complex"   and f :: "complex \<Rightarrow> complex" shows "contour_integral (subpath u u g) f = 0"
fs: shows "conf_fs P h sh \<Phi> C M (length Ts) T frs"
basisI: fixes X :: "(nat \<times> int \<times> nat) set" assumes "forest \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = X\<rparr> \<and> subgraph \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = X\<rparr> \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr>"   and "\<And>x. x \<in> E - X \<Longrightarrow> \<not> (forest \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = insert x X\<rparr> \<and> subgraph \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = insert x X\<rparr> \<lparr>nodes = fst ` E \<union> (snd \<circ> snd) ` E, edges = E\<rparr>)" shows "s.SpanningForest X"
A2: shows "\<forall>x. \<bar>g x\<bar> \<le> d * \<bar>f x\<bar>"
step_6: shows "P,E \<turnstile> s \<surd>"
Re_Arcsin: fixes z :: "complex" shows "Re (Arcsin z) = Im (Ln (\<i> * z + csqrt (1 - z\<^sup>2)))"
inf_top_right: fixes x :: "'c" shows "brouwer.comp_inf.vector x"
HSem_subset_below: fixes ESem :: "'d \<Rightarrow> ('e \<Rightarrow> 'f) \<rightarrow> 'f"   and \<Gamma> :: "('e \<times> 'd) list"   and \<Delta> :: "('e \<times> 'd) list"   and \<rho> :: "'e \<Rightarrow> 'f" assumes "has_ignore_fresh_ESem ESem"   and "atom ` domA \<Gamma> \<sharp>* \<Delta>" shows "has_ESem.HSem ESem \<Delta>\<cdot>(\<rho> f|` (- domA \<Gamma>)) \<sqsubseteq> has_ESem.HSem ESem (\<Delta> @ \<Gamma>)\<cdot>\<rho> f|` (- domA \<Gamma>)"
alpha_d_more_eqE: fixes r :: "('c, 'd) alpha_rp_scheme"   and r' :: "('c, 'd) alpha_rp_scheme" assumes "alpha_d.more r = alpha_d.more r'" obtains "tr r = tr r'" and "wait r = wait r'" and "ref r = ref r'" and "alpha_rp.more r = alpha_rp.more r'"
atLeast0_lessThan_Suc_shift: fixes g :: "nat \<Rightarrow> 'a"   and n :: "nat" shows "prod g {0..<Suc n} = g 0 * prod (g \<circ> Suc) {0..<n}"
ASSERT_rule: fixes \<Phi> :: "bool"   and \<Psi> :: "unit \<Rightarrow> bool" assumes "\<Phi>"   and "\<Phi> \<Longrightarrow> \<Psi> ()" shows "ASSERT \<Phi> \<le> SPEC \<Psi>"
arr_lunit: fixes a :: "'c local.term set" assumes "ide a" shows "arr \<l>[a]"
inject: fixes x :: "(nat, 'a \<times> QDelta) mapping \<times> (nat, 'a \<times> QDelta) mapping"   and ya :: "(nat, 'a \<times> QDelta) mapping \<times> (nat, 'a \<times> QDelta) mapping" shows "(Simplex_Checkpoint x = Simplex_Checkpoint ya) = (x = ya)"
le_mtx_norm: fixes m :: "real"   and A :: "'a sq_mtx" assumes "m \<in> {\<parallel>A *\<^sub>V x\<parallel> |x. \<parallel>x\<parallel> = 1}" shows "m \<le> \<parallel>A\<parallel>"
OP_def: fixes x :: "'a" shows "OP x \<equiv> x"
wffp_hdI: fixes sl :: "('St, 'U, 'C) state list"   and s :: "('St, 'U, 'C) state" assumes "wffp sl"   and "hd sl \<in> \<delta> s" shows "wffp (s # sl)"
mtriples_def: fixes P :: "'c \<Rightarrow> 'd \<Rightarrow> 'b assn"   and tf :: "'c \<Rightarrow> 'd \<Rightarrow> expr"   and Q :: "'c \<Rightarrow> 'd \<Rightarrow> 'b assn"   and ms :: "('c \<times> 'd) set" shows "{{P} tf-\<succ> {Q} | ms} = (\<lambda>(C, sig). {Normal (P C sig)} tf C sig-\<succ> {Q C sig}) ` ms"
reversepath_simps_3: fixes g :: "real \<Rightarrow> 'a" shows "pathstart (reversepath g) = pathfinish g"
assms_1: shows "homeomorphism (cbox a b) (cbox u v) f1 g1"
n_6: shows "length (renamings_apart (DA0 # CAs0)) = Suc n"
map_poly_hom_monom: fixes a :: "'a"   and i :: "nat" shows "map_poly hom (monom a i) = monom (hom a) i"
mono_read_ref: fixes P :: "'b \<Rightarrow> 'c process"   and Q :: "'b \<Rightarrow> 'c process"   and c :: "'b \<Rightarrow> 'c" assumes "\<And>x. P x \<sqsubseteq> Q x" shows "read c UNIV P \<sqsubseteq> read c UNIV Q"
exists_at_height: fixes t :: "'a tree" assumes "consistent t" shows "\<exists>a\<in>alphabet t. depth t a = height t"
antimono_funpow: fixes Q :: "'c \<Rightarrow> 'c" assumes "mono Q" shows "decseq (\<lambda>i. (Q ^^ i) top)"
supteq_not_supt: fixes s :: "('a, 'b) Term.term"   and t :: "('a, 'b) Term.term" assumes "s \<unrhd> t"   and "(s, t) \<notin> {\<rhd>}" shows "s = t"
Inf_lower2: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and u :: "'c"   and A :: "'c set"   and v :: "'c" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "u \<in> A"   and "less_eq u v" shows "less_eq (Inf A) v"
neg: fixes V :: "'b set"   and f :: "'b \<Rightarrow> real"   and x :: "'b" assumes "linearform V f"   and "vectorspace V"   and "x \<in> V" shows "f (- x) = - f x"
representation_extend: fixes basis :: "'b set"   and v :: "'b"   and basis' :: "'b set" assumes "independent basis"   and "v \<in> span basis'"   and "basis' \<subseteq> basis" shows "representation basis v = representation basis' v"
assms: shows "(f, g) \<in> Id \<rightarrow> \<langle>Id\<rangle>nres_rel"
\<phi>: fixes y :: "'b" assumes "y \<in> \<Union> (B ` X)" shows "\<phi> y \<in> X \<and> y \<in> B (\<phi> y)"
inext_nth_atMost: fixes a :: "nat"   and n :: "nat" assumes "a \<le> n" shows "{..n} \<rightarrow> a = a"
sums_split_initial_segment: fixes f :: "nat \<Rightarrow> 'a"   and s :: "'a"   and n :: "nat" assumes "f sums s" shows "(\<lambda>i. f (i + n)) sums (s - sum f {..<n})"
union_commute: fixes M :: "'b multiset"   and N :: "'b multiset" shows "M + N = N + M"
seq_comp_failures_def: shows "seq_comp_failures \<equiv> \<lambda>P Q. {(xa, x). seq_comp_failuresp P Q xa x}"
LT: fixes x :: "'b"   and y :: "'b" shows "(cmp x y = LT) = (x < y)"
min_key_val_raw_in: fixes xs :: "('b \<times> 'c) list \<times> 'b nat_term_order"   and ko :: "'b nat_term_order" assumes "fst xs \<noteq> []" shows "min_key_val_ko_ntm ko xs \<in> set (fst xs)"
guessed_frame_Resp_xgny: fixes R :: "fid_t" assumes "role (guessed_runs R) = Resp" shows "guessed_frame R xgny = Some (Exp Gen (NonceF (R $ ny)))"
triangle_in_hom_1: shows "\<guillemotleft>(\<epsilon> \<star>\<^sub>C f) \<cdot>\<^sub>C \<a>\<^sub>C\<^sup>-\<^sup>1[f, g, f] \<cdot>\<^sub>C (f \<star>\<^sub>C \<eta>) : f \<star>\<^sub>C src\<^sub>C f \<Rightarrow>\<^sub>C trg\<^sub>C f \<star>\<^sub>C f\<guillemotright>"
assms_6: shows "\<forall>\<tau>\<in>{0..t}. \<tau> \<le> - (ln (Tmin / T) / a)"
resAct: fixes x :: "name"   and \<alpha> :: "act"   and P :: "ccs" assumes "x \<sharp> \<alpha>" shows "\<lparr>\<nu>x\<rparr>\<alpha>.P \<sim> \<alpha>.(\<lparr>\<nu>x\<rparr>P)"
gcs_plus_right: fixes s :: "'a"   and u :: "'a"   and t :: "'a" shows "gcs (s + u) (t + u) = gcs s t + u"
Complement_def: fixes A :: "'b set" shows "Complement A = - A"
measure_pmf_eq_1_iff: fixes \<mu> :: "'a pmf"   and x :: "'a" shows "(emeasure (measure_pmf \<mu>) {x} = 1) = (\<mu> = return_pmf x)"
outs'_GPV: fixes r :: "('e, 'd, ('e, 'd, 'f) rpv) generat spmf" shows "outs'_gpv (GPV r) = (set_spmf r \<bind> generat_outs) \<union> (set_spmf r \<bind> generat_conts \<bind> outs'_rpv)"
Acc_to_mojmir_accept: fixes \<psi> :: "'a ltl"   and \<pi> :: "'a ltl \<Rightarrow> nat option"   and \<phi> :: "'a ltl" assumes "G \<psi> \<in> dom \<pi>"   and "dom \<pi> \<subseteq> \<^bold>G \<phi>"   and "accepting_pair\<^sub>R (\<A>.\<delta>\<^sub>\<A> \<Sigma>) (\<A>.\<iota>\<^sub>\<A> \<phi>) (\<A>.Acc\<^sub>\<A> \<Sigma> \<pi> (G \<psi>)) w"   and "the (\<pi> (G \<psi>)) < semi_mojmir_def.max_rank \<Sigma> \<up>af\<^sub>G (Abs \<psi>)" shows "mojmir_def.accept \<up>af\<^sub>G (Abs \<psi>) w {q. dom \<pi> \<up>\<Turnstile>\<^sub>P q}"
cop_F_rejected_inv: fixes ds :: "'b set" shows "valid ds (\<lambda>ds fp. cop_F_range_inv ds fp \<and> cop_F_closed_inv ds fp \<and> inj_on Xd (CH fp)) cop_F_rejected_inv"
Polys_closed_monom_mult: fixes t :: "'c \<Rightarrow>\<^sub>0 nat"   and X :: "'c set"   and p :: "('c \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'd"   and c :: "'d" assumes "t \<in> .[X]"   and "p \<in> P[X]" shows "MPoly_Type_Class.punit.monom_mult c t p \<in> P[X]"
norm_in_cf: fixes x :: "'b itrm" shows "norm x \<in> CF"
inflsts_def: fixes A :: "'b set" shows "A\<^sup>\<omega> \<equiv> A\<^sup>\<infinity> - UNIV\<^sup>\<star>"
assms_6: fixes s :: "(nat list \<times> nat list) pmf"   and h :: "nat list" assumes "a \<noteq> b"   and "{a, b} = {x, y}"   and "TS_inv s a [x, y]"   and "set qs \<subseteq> {a, b}"   and "qs \<in> pattern" shows "TS_inv (Partial_Cost_Model.config'_rand (Partial_Cost_Model.embed (rTS h)) s qs) (last qs) [x, y] \<and> T\<^sub>p_on_rand' (Partial_Cost_Model.embed (rTS h)) s qs = ts"
independent_card_le_dim: fixes B :: "'a set"   and V :: "'a set" assumes "B \<subseteq> V"   and "independent B" shows "card B \<le> dim V"
p_iff_p: fixes P :: "'a upred" shows "(P \<Leftrightarrow> P) = true"
P_deadend_v0_LCons: assumes "deadend v" shows "P = LCons v LNil"
diff_minus: fixes a :: "'b"   and b :: "'b" shows "a - b = a + - b"
SD_inefficientI': fixes q :: "'alt pmf"   and p :: "'alt pmf" assumes "q \<in> lotteries"   and "\<And>i. i \<in> agents \<Longrightarrow> SD (R i) p q"   and "\<exists>i\<in>agents. p \<prec>[SD (R i)] q" shows "\<not> SD_efficient R p"
oSuc_eq_oSuc: fixes x :: "ordinal"   and y :: "ordinal" shows "(oSuc x = oSuc y) = (x = y)"
closure_ofI2: fixes S :: "('c, 'd) eq_object_scheme"   and a :: "'c"   and a' :: "'c"   and A :: "'c set" assumes "a .=\<^bsub>S\<^esub> a'"   and "a' \<in> A"   and "a \<in> carrier S" shows "a \<in> closure_of\<^bsub>S\<^esub> A"
insert_DiffM2: fixes x :: "'c"   and M :: "'c multiset" assumes "x \<in># M" shows "remove1_mset x M + {#x#} = M"
Lambert_W'_geI: fixes x :: "real"   and y :: "real" assumes "x \<le> y * exp y"   and "y \<le> - 1"   and "- exp (- 1) \<le> x" shows "y \<le> Lambert_W' x"