hcju/isabelleps huggingface dataset:
is_pure_alt_def: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "is_pure R = (\<exists>Ri. \<forall>x y. R x y = \<up> ((y, x) \<in> Ri))"
SPEC_eq_is_RETURN_1: fixes x :: "'a" shows "SPEC ((=) x) = return x"
lift2_def: fixes h :: "'f \<Rightarrow> 'g \<Rightarrow> 'e"   and inp' :: "'d \<Rightarrow> 'f option"   and inp :: "'d \<Rightarrow> 'g option" shows "lift2 h inp' inp \<equiv> \<lambda>i. case (inp' i, inp i) of (None, b) \<Rightarrow> None | (Some v', None) \<Rightarrow> None | (Some v', Some v) \<Rightarrow> Some (h v' v)"
inv_dghm_components_1: fixes \<FF> :: "V" shows "inv_dghm \<FF>\<lparr>ObjMap\<rparr> = (\<FF>\<lparr>ObjMap\<rparr>)\<inverse>\<^sub>\<circ>"
simps_2: fixes k :: "nat" shows "H_k (Suc k) = partial_state2.ptensor_mat dims {0..<Suc k} {Suc k} (H_k k) hadamard"
avoid: shows "\<forall>p\<in>s. 0 < h p \<and> (\<forall>w\<in>cball p (h p). w \<in> s \<and> (w \<noteq> p \<longrightarrow> w \<notin> pts))"
inv_mult_on_rcosets_action: fixes H :: "'a set" assumes "subgroup H H'" shows "group_action H' (\<lambda>g. \<lambda>U\<in>rcosets\<^bsub>H'\<^esub> H. U #>\<^bsub>H'\<^esub> inv\<^bsub>H'\<^esub> g) (rcosets\<^bsub>H'\<^esub> H)"
mem_val_alt_8_mem_equal_2: fixes s1 :: "'b sparc_state"   and s2 :: "'b sparc_state"   and pa :: "word36" assumes "mem_equal s1 s2 pa" shows "mem_val_alt 8 ((pa AND 68719476732) + 2) s1 = mem_val_alt 8 ((pa AND 68719476732) + 2) s2"
greaterThanAtMost_def: fixes l :: "'b"   and u :: "'b" shows "{l<..u} \<equiv> {l<..} \<inter> {..u}"
D_sync: fixes P :: "'c process"   and A :: "'c set"   and Q :: "'c process" shows "D (P \<lbrakk>A\<rbrakk> Q) = {s. \<exists>t u r v. front_tickFree v \<and> (tickFree r \<or> v = []) \<and> s = r @ v \<and> r setinterleaves ((t, u), ev ` A \<union> {tick}) \<and> (t \<in> D P \<and> u \<in> T Q \<or> t \<in> D Q \<and> u \<in> T P)}"
measurable_lebesgue_cong: fixes S :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd"   and M :: "'d measure" assumes "\<And>x. x \<in> S \<Longrightarrow> f x = g x" shows "(f \<in> lebesgue_on S \<rightarrow>\<^sub>M M) = (g \<in> lebesgue_on S \<rightarrow>\<^sub>M M)"
intros_11: fixes s :: "'d"   and b :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and c :: "('d, 'e, 'f) com"   and s' :: "('d, 'f) xstate"   and t :: "('d, 'f) xstate" assumes "s \<in> b"   and "\<Gamma>\<turnstile> \<langle>c,Normal s\<rangle> \<Rightarrow> s'"   and "\<Gamma>\<turnstile> \<langle>While b c,s'\<rangle> \<Rightarrow> t" shows "\<Gamma>\<turnstile> \<langle>While b c,Normal s\<rangle> \<Rightarrow> t"
assms_2: shows "\<forall>\<^sub>F x in at_top. f x = 0"
blinding_of_blindable_simps2_2: fixes h :: "'c \<Rightarrow> 'd"   and bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y' :: "'d blindable\<^sub>h" shows "blinding_of_blindable h bo (Unblinded x) (Blinded y') = False"
Union_iff: fixes A :: "'d"   and C :: "'d set set" shows "(A \<in> \<Union> C) = (\<exists>X\<in>C. A \<in> X)"
VSigmaE2: fixes a :: "V"   and b :: "V"   and A :: "V"   and B :: "V \<Rightarrow> V" assumes "\<langle>a, b\<rangle> \<in>\<^sub>\<circ> VSigma A B" obtains "a \<in>\<^sub>\<circ> A" and "b \<in>\<^sub>\<circ> B a"
assms_2: shows "reachable x n ts (\<lambda>_. 0) s"
wp_whileI_modset: fixes R :: "((char list \<Rightarrow> int \<Rightarrow> int) \<times> (char list \<Rightarrow> int \<Rightarrow> int)) set"   and I :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and s\<^sub>0 :: "char list \<Rightarrow> int \<Rightarrow> int"   and \<pi> :: "char list \<Rightarrow> com option"   and c :: "com"   and b :: "bexp"   and Q :: "(char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool" assumes "wf R"   and "I s\<^sub>0"   and "\<And>s. \<lbrakk>modifies (ANALYZE (lhsv \<pi> c)) s s\<^sub>0; I s; bval b s\<rbrakk> \<Longrightarrow> wp \<pi> c (\<lambda>s'. I s' \<and> (s', s) \<in> R) s"   and "\<And>s. \<lbrakk>modifies (ANALYZE (lhsv \<pi> c)) s s\<^sub>0; I s; \<not> bval b s\<rbrakk> \<Longrightarrow> Q s" shows "wp \<pi> (WHILE b DO c) Q s\<^sub>0"
natural_isomorphism_axioms: shows "natural_isomorphism CCC.comp (\<cdot>) T.ToCT T.ToTC \<alpha>'"
find_base_vectors_transfer: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "(R ===> R ===> (=)) (=) (=)" shows "((R ===> R) ===> R ===> R ===> mat_rel R ===> list_all2 (vec_rel R)) find_base_vectors_gen find_base_vectors_gen"
simps_2: fixes A :: "'a set"   and As :: "'a set list" shows "add_independentS (A # As) = (add_independentS As \<and> (\<forall>x\<in>\<Sum>B\<leftarrow>As. B. \<forall>a\<in>A. a + x = (0::'a) \<longrightarrow> a = (0::'a)))"
subst_tm_id: fixes a :: "name"   and tm :: "tm" shows "subst a (Var a) tm = tm"
alpha_d_more_eqI: fixes r :: "('c, 'd) alpha_rp_scheme"   and r' :: "('c, 'd) alpha_rp_scheme" assumes "tr r = tr r'"   and "wait r = wait r'"   and "ref r = ref r'"   and "alpha_rp.more r = alpha_rp.more r'" shows "alpha_d.more r = alpha_d.more r'"
leg0_is_map: shows "is_left_adjoint t\<^sub>0"
at2: fixes a :: "'a"   and b :: "'a"   and pi :: "('a \<times> 'a) list"   and x :: "'a" assumes "at TYPE('a)" shows "((a, b) # pi) \<bullet> x = swap (a, b) (pi \<bullet> x)"
simps_25: fixes vc :: "'a rexp"   and v :: "'a rexp"   and va :: "'a rexp" shows "local.le_rexp (Star vc) (Times v va) = True"
list_all2_reorder_right_invariance: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'a list"   and ys :: "'b list"   and ys' :: "'b list" assumes "list_all2 R xs ys"   and "mset ys' = mset ys" shows "\<exists>xs'. list_all2 R xs' ys' \<and> mset xs' = mset xs"
inversion_zero: shows "inversion 0\<^sub>h = \<infinity>\<^sub>h"
Constrains_weaken_L: fixes F :: "'b program"   and A :: "'b set"   and A' :: "'b set"   and B :: "'b set" assumes "F \<in> A Co A'"   and "B \<subseteq> A" shows "F \<in> B Co A'"
crename_interesting2: fixes a :: "coname"   and c :: "coname"   and d :: "coname"   and b :: "coname"   and M :: "trm" assumes "a \<noteq> c"   and "a \<noteq> d"   and "a \<noteq> b"   and "c \<noteq> d"   and "b \<noteq> c" shows "M[a\<turnstile>c>b][c\<turnstile>c>d] = M[c\<turnstile>c>d][a\<turnstile>c>b]"
assms_1: shows "u \<in> borel_measurable lebesgue"
simps_3: fixes v :: "char"   and va :: "char list" shows "lleq_charlist (v # va) [] = False"
comp_char: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" shows "VVV.comp g f = (if VVV.arr f \<and> VVV.arr g \<and> VxVxV.seq g f then VxVxV.comp g f else VxVxV.null)"
IH: fixes t :: "nat \<Rightarrow> nat" assumes "t \<in> set ts" shows "0 \<le> f (t x)"
contractible_imp_connected_space: fixes X :: "'a topology" assumes "contractible_space X" shows "connected_space X"
word_bw_lcs_1: fixes y :: "'a word"   and x :: "'a word"   and z :: "'a word" shows "y && x && z = x && y && z"
not_ordLess_ordIso: fixes r :: "('f \<times> 'f) set"   and r' :: "('g \<times> 'g) set" assumes "r <o r'" shows "(r, r') \<notin> ordIso"
equality: fixes r :: "('a, 'b, 'c) graph_scheme"   and r' :: "('a, 'b, 'c) graph_scheme" assumes "nodes r = nodes r'"   and "edges r = edges r'"   and "graph.more r = graph.more r'" shows "r = r'"
assert_all_state_sat_atoms_equiv_bounds: fixes t :: "(nat \<times> linear_poly) list"   and as :: "('i \<times> 'a atom) list"   and s' :: "('i, 'a) state" assumes "\<triangle> t"   and "assert_all_state t as = s'"   and "\<not> \<U> s'" shows "(snd ` set as) \<doteq> \<B> s'"
filterlim_at_top_add_at_top: fixes f :: "'a \<Rightarrow> real"   and F :: "'a filter"   and g :: "'a \<Rightarrow> real" assumes "filterlim f at_top F"   and "filterlim g at_top F" shows "LIM x F. f x + g x :> at_top"
sums: fixes p :: "nat" assumes "prime p" shows "(\<lambda>n. (fds_nth f p / nat_power p s) ^ Suc n) sums ((1::'a) / ((1::'a) - fds_nth f p / nat_power p s) - (1::'a))"
inline2_fixp_induct: fixes P :: "(('b, 'c2, 'r2) gpv \<Rightarrow> 's2 \<Rightarrow> 's1 \<Rightarrow> ('b \<times> 's2 \<times> 's1 + 'c \<times> ('r1 \<times> 's1, 'c, 'r) rpv \<times> ('r2 \<times> 's2, 'c1, 'r1) rpv \<times> ('b, 'c2, 'r2) rpv) spmf) \<Rightarrow> bool" assumes "spmf.admissible (\<lambda>inline2. P (\<lambda>gpv s2 s1. inline2 ((gpv, s2), s1)))"   and "P (\<lambda>_ _ _. return_pmf None)"   and "\<And>inline2'. P inline2' \<Longrightarrow> P (\<lambda>gpv s2 s1. the_gpv gpv \<bind> (\<lambda>generat. case generat of Pure x \<Rightarrow> return_spmf (Inl (x, s2, s1)) | IO out rpv \<Rightarrow> inline1 callee1 (callee2 s2 out) s1 \<bind> (\<lambda>lr. case lr of Inl ((r2, s2), c) \<Rightarrow> inline2' (rpv r2) s2 c | Inr (x, rpv'', rpv') \<Rightarrow> return_spmf (Inr (x, rpv'', rpv', rpv)))))" shows "P local.inline2"
IH_1: fixes w :: "nat \<Rightarrow> 'a set" assumes "\<forall>\<psi>\<in>X. w \<Turnstile>\<^sub>n G\<^sub>n (F\<^sub>n \<psi>)"   and "w \<Turnstile>\<^sub>n \<phi>1[X]\<^sub>\<nu>" shows "w \<Turnstile>\<^sub>n \<phi>1"
c_def: shows "c = (\<lambda>n. if even n then 2 * (1 - 1 / real (Suc n)) else 0)"
two_div_two: shows "2 div Suc (Suc 0) = 1"
simps_3: fixes x1 :: "'b" shows "extT.Top \<noteq> extT.Val x1"
externals_of_par: fixes A1 :: "('e set \<times> 'e set \<times> 'e set) \<times> 'f set \<times> ('f \<times> 'e \<times> 'f) set"   and A2 :: "('e set \<times> 'e set \<times> 'e set) \<times> 'g set \<times> ('g \<times> 'e \<times> 'g) set" shows "externals (asig_of (A1 || A2)) = externals (asig_of A1) \<union> externals (asig_of A2)"
null_eq: shows "null_space A = null_space (Finite_Cartesian_Product.transpose A ** A)"
Compl_Collect: fixes b :: "'b \<Rightarrow> bool" shows "- Collect b = \<lbrace>\<not> \<acute>b\<rbrace>"
ntrp_mon_s: fixes s :: "'n list"   and c :: "'n list multiset"   and e :: "('p, 'ba) label list el_step"   and s' :: "'n list"   and c' :: "'n list multiset" assumes "((s, c), e, s', c') \<in> ntrp fg" shows "mon_c fg (add_mset s' c') = mon_c fg (add_mset s c) \<union> fst (\<alpha>nl fg e)"
in_rel_Collect_case_prod_eq: fixes X :: "'a \<Rightarrow> 'b \<Rightarrow> bool" shows "in_rel {(x, y). X x y} = X"
simps: fixes a1 :: "('d \<times> ('c, real) cconstraint \<times> 'e \<times> 'c list \<times> 'd) set \<times> ('d \<Rightarrow> ('c, real) cconstraint)"   and a2 :: "'d"   and a3 :: "('c \<Rightarrow> real) set"   and a4 :: "'d"   and a5 :: "('c \<Rightarrow> real) set" shows "a1 \<turnstile> \<langle>a2, a3\<rangle> \<leadsto>\<^sub>\<beta>* \<langle>a4, a5\<rangle> = ((\<exists>A l Z. a1 = A \<and> a2 = l \<and> a3 = Z \<and> a4 = l \<and> a5 = Z) \<or> (\<exists>A l Z l' Z' l'' Z''. a1 = A \<and> a2 = l \<and> a3 = Z \<and> a4 = l'' \<and> a5 = Z'' \<and> A \<turnstile> \<langle>l, Z\<rangle> \<leadsto>\<^sub>\<beta>* \<langle>l', Z'\<rangle> \<and> A \<turnstile> \<langle>l', Z'\<rangle> \<leadsto>\<^sub>\<beta> \<langle>l'', Z''\<rangle>))"
frangeE: fixes b :: "V"   and r :: "V" assumes "b \<in>\<^sub>\<circ> \<R>\<^sub>\<bullet> r" obtains a :: "V" where "[a, b]\<^sub>\<circ> \<in>\<^sub>\<circ> r"
a_1: shows "f' (real_of_float l) \<le> real_of_float a"
discI_1: fixes llist :: "'b llist" assumes "llist = LNil" shows "lnull llist"
induct: fixes x :: "'a"   and S :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> \<langle>S\<rangle>"   and "P (0::'a)"   and "\<And>s. s \<in> S \<Longrightarrow> P s"   and "\<And>w w'. \<lbrakk>w \<in> \<langle>S\<rangle>; P w; w' \<in> \<langle>S\<rangle>; P w'\<rbrakk> \<Longrightarrow> P (w - w')" shows "P x"
flip: shows "bounded_cbilinear (\<lambda>x y. y *\<^sub>V x)"
invalid_maybe_and: fixes x :: "trilean" shows "invalid \<and>? x = invalid"
to_abs_list_remove_all: fixes n :: "nat \<times> bool"   and t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and D :: "((('a, 'b, 'c) prot_fun, 'd) Term.term \<times> (('f, 'g, 'e) prot_fun, 'h) Term.term) list"   and s :: "(('f, 'g, 'e) prot_fun, 'h) Term.term" assumes "Fun (Val n) [] \<noteq> t" shows "\<alpha>\<^sub>0 D n = \<alpha>\<^sub>0 (removeAll (t, s) D) n"
assms: fixes b :: "'t \<Rightarrow>\<^sub>0 'b" assumes "b \<in> B"   and "b \<noteq> 0" shows "lc b = (1::'b)"
fmdrop_idle': fixes x :: "'e"   and B :: "'e \<rightharpoonup>\<^sub>f 'f" assumes "x \<notin> fmdom' B" shows "fmdrop x B = B"
ts_okI: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and P :: "'f \<Rightarrow> 'g \<Rightarrow> 'i \<Rightarrow> bool"   and m :: "'i" assumes "\<And>t x ln. ts t = \<lfloor>(x, ln)\<rfloor> \<Longrightarrow> P t x m" shows "ts_ok P ts m"
lset_into_lsetp: fixes x :: "'c"   and xs :: "'c llist" assumes "x \<in> lset xs" shows "Coinductive_List.lsetp xs x"
dist_0_norm: fixes x :: "'a" shows "dist (0::'a) x = norm x"
right_idem: fixes a :: "'a multiset"   and b :: "'a multiset" shows "a \<union># b \<union># b = a \<union># b"
r_consist_phi_init: fixes f :: "nat \<Rightarrow> nat option"   and i :: "nat"   and n :: "nat" assumes "f \<in> \<R>"   and "\<phi> i \<in> \<R>" shows "eval r_consist_phi [i, f \<triangleright> n] \<down>= (if \<forall>k\<le>n. \<phi> i k = f k then 0 else 1)"
hiding_advantage_ind_cpa_def: fixes \<A> :: "(nat \<times> nat \<Rightarrow> ((nat \<times> nat) \<times> 'b) spmf) \<times> (nat \<Rightarrow> 'b \<Rightarrow> bool spmf)" shows "rivest_commit.hiding_advantage_ind_cpa \<A> \<equiv> \<bar>spmf (rivest_commit.hiding_game_ind_cpa \<A>) True - 1 / 2\<bar>"
vdiff_vinsert_vsingleton: fixes a :: "V"   and A :: "V" shows "vinsert a A -\<^sub>\<circ> set {a} = A -\<^sub>\<circ> set {a}"
quotient_of_submodules_def: fixes R :: "('e, 'f) Ring_scheme"   and M :: "('g, 'e, 'h) Module_scheme"   and N :: "'g set"   and P :: "'g set" shows "N \<^bsub>R\<ddagger>M\<^esub> P = {x |x. x \<in> carrier R \<and> linear_span R M (R \<diamondsuit>\<^sub>p x) P \<subseteq> N}"
conj_mono_left: fixes c\<^sub>0 :: "'a::refinement_lattice"   and c\<^sub>1 :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" assumes "(c\<^sub>0::'a::refinement_lattice) \<sqsubseteq> (c\<^sub>1::'a::refinement_lattice)" shows "(c\<^sub>0::'a::refinement_lattice) \<iinter> (d::'a::refinement_lattice) \<sqsubseteq> (c\<^sub>1::'a::refinement_lattice) \<iinter> d"
split_beta': fixes f :: "'f \<Rightarrow> 'g \<Rightarrow> 'h" shows "(\<lambda>(x, y). f x y) = (\<lambda>x. f (fst x) (snd x))"
initiali_toy: fixes i :: "nat" shows "initiali i (init (optoy i)) (init (ptoy i))"
UnOp: fixes G :: "prog"   and s0 :: "st"   and e :: "expr"   and v :: "val"   and s1 :: "state"   and unop :: "unop" assumes "G\<turnstile>Norm s0 \<midarrow>e-\<succ>v\<rightarrow> s1" shows "G\<turnstile>Norm s0 \<midarrow>UnOp unop e-\<succ>eval_unop unop v\<rightarrow> s1"
domain_compose: fixes a :: "wall" assumes "domain_wall a = 0 \<and> is_tangle_diagram a" shows "a ~ basic [] \<circ> a"
min_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" shows "ord.min less_eq a b = (if less_eq a b then a else b)"
permutes_in_image: fixes p :: "'d \<Rightarrow> 'd"   and S :: "'d set"   and x :: "'d" assumes "p permutes S" shows "(p x \<in> S) = (x \<in> S)"
a_2_var: fixes x :: "'a"   and y :: "'a" shows "(ad x \<cdot> d y = (zero_class.zero::'a)) = (ad x \<le> ad y)"
finite_range_updI: fixes f :: "'d \<Rightarrow> 'c option"   and a :: "'d"   and b :: "'c" assumes "finite (range f)" shows "finite (range (f(a \<mapsto> b)))"
case_log: fixes m :: "'a match_expr" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>Rule m Log # rs, s\<rangle> \<Rightarrow> t = \<Gamma>,\<gamma>,p\<turnstile> \<langle>rs, s\<rangle> \<Rightarrow> t"
range_right: shows "ix p d < (ix (child b dir d) d + 1) * 2 ^ (lv p d - lv (child b dir d) d)"
unit_factor_gcd: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "unit_factor (gcd a b) = (if a = zero \<and> b = zero then zero else one)"
select_convs_2: fixes nothing :: "unit"   and more :: "'a" shows "RObject.more \<lparr>RObject.nothing = nothing, \<dots> = more\<rparr> = more"
assms_1: shows "\<forall>y\<in>set ys. \<forall>x\<in>set xs. gt y x \<longrightarrow> gt' y x"
map_simp_ide: fixes a :: "'c" assumes "\<tau>.A.ide a" shows "\<tau>'\<tau>.map a = B (\<tau>' a) (\<tau> a)"
r2f2r_inv1: shows "\<R> \<circ> \<F> = id"
rk_\<alpha>\<beta>\<gamma>_special_case_2: fixes A :: "Points"   and B :: "Points"   and C :: "Points"   and A' :: "Points"   and B' :: "Points"   and C' :: "Points"   and P :: "Points"   and \<alpha> :: "Points"   and \<beta> :: "Points"   and \<gamma> :: "Points" assumes "desargues_config_2D A B C A' B' C' P \<alpha> \<beta> \<gamma>"   and "rk {B', P} = 1" shows "rk {\<alpha>, \<beta>, \<gamma>} \<le> 2"
select_convs_2: fixes red :: "('d \<times> nat) rgraph"   and black :: "('d, 'e, 'f) lts"   and subs :: "(('d \<times> nat) \<times> 'd \<times> nat) set"   and init_conf :: "('e, 'f) conf"   and confs :: "'d \<times> nat \<Rightarrow> ('e, 'f) conf"   and marked :: "'d \<times> nat \<Rightarrow> bool"   and strengthenings :: "'d \<times> nat \<Rightarrow> ('e \<Rightarrow> 'f) \<Rightarrow> bool"   and more :: "'g" shows "black \<lparr>red = red, black = black, subs = subs, init_conf = init_conf, confs = confs, marked = marked, strengthenings = strengthenings, \<dots> = more\<rparr> = black"
preserves_dom: fixes f :: "'d" assumes "D.arr f" shows "D.dom (GF.map f) = GF.map (D.dom f)"
set_insert: fixes x :: "'c"   and xs :: "'c list" shows "set (List.insert x xs) = insert x (set xs)"
left_adjoint_functor_respects_naturally_isomorphic: fixes D :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and F' :: "'b \<Rightarrow> 'c" assumes "left_adjoint_functor D C F"   and "naturally_isomorphic D C F F'" shows "left_adjoint_functor D C F'"
monad_state_writerT': assumes "monad_state Monad_Overloading.return Monad_Overloading.bind get put" shows "monad_state Monad_Overloading.return Monad_Overloading.bind get put"
inf_left_commute: fixes inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.semilattice_inf inf less_eq less" shows "inf x (inf y z) = inf y (inf x z)"
lsc_hull_def: fixes f :: "'b \<Rightarrow> ereal" shows "lsc_hull f = (SOME g. Epigraph UNIV g = closure (Epigraph UNIV f))"
vars_list_def: shows "vars_list \<equiv> map_fun Rep_linear_poly id fun_vars_list"
term_subst_eq: fixes t :: "('d, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('d, 'e) Term.term"   and \<tau> :: "'c \<Rightarrow> ('d, 'e) Term.term" assumes "\<And>x. x \<in> vars_term t \<Longrightarrow> \<sigma> x = \<tau> x" shows "t \<cdot> \<sigma> = t \<cdot> \<tau>"
eq_onp_to_eq: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "eq_onp P x y" shows "x = y"
hfcomp: fixes f :: "'e \<Rightarrow> 'f Heap"   and g :: "'g \<Rightarrow> 'h nres"   and P :: "'g \<Rightarrow> bool"   and RR' :: "('g \<Rightarrow> 'e \<Rightarrow> assn) \<times> ('g \<Rightarrow> 'e \<Rightarrow> assn)"   and S :: "'h \<Rightarrow> 'f \<Rightarrow> assn"   and h :: "'i \<Rightarrow> 'j nres"   and Q :: "'i \<Rightarrow> bool"   and T :: "('g \<times> 'i) set"   and U :: "('h \<times> 'j) set" assumes "(f, g) \<in> [P]\<^sub>a RR' \<rightarrow> S"   and "(g, h) \<in> [Q]\<^sub>f T \<rightarrow> \<langle>U\<rangle>nres_rel" shows "(f, h) \<in> [\<lambda>a. Q a \<and> (\<forall>a'. (a', a) \<in> T \<longrightarrow> P a')]\<^sub>a hrp_comp RR' T \<rightarrow> hr_comp S U"
prod_ennreal: fixes A :: "'c set"   and f :: "'c \<Rightarrow> real" assumes "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i" shows "(\<Prod>i\<in>A. ennreal (f i)) = ennreal (prod f A)"
intro: fixes G :: "('c, 'd) ring_scheme"   and H :: "('e, 'f) ring_scheme"   and h :: "'c \<Rightarrow> 'e" assumes "abelian_group G"   and "abelian_group H"   and "abelian_group_hom_axioms G H h" shows "abelian_group_hom G H h"
CD_on_XD_smallest: fixes ds :: "'b set"   and X :: "'a set" assumes "stable_on ds X" shows "CD_on ds (XD_smallest ds X) = X"
ModuleEnd_def: fixes smult :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and M :: "'b set"   and T :: "'b \<Rightarrow> 'b" shows "ModuleEnd smult M T \<equiv> ModuleHom smult M smult T \<and> ModuleEnd_axioms M T"
simplefact: fixes i :: "nat" assumes "i + 1 < length \<HH>" shows "simple_group (G\<lparr>carrier := \<HH> ! (i + 1)\<rparr> Mod \<HH> ! i)"
distinct_1: fixes x1 :: "nat"   and x2 :: "(val \<times> val) fset" shows "VNat x1 \<noteq> val.Fun x2"
set_intros_1: fixes x21 :: "'c"   and x22 :: "'c list" shows "x21 \<in> set (x21 # x22)"
dot_square_norm: fixes x :: "'b" shows "x \<bullet> x = (\<parallel>x\<parallel>)\<^sup>2"
that: fixes e' :: "Syntax.term"   and eV' :: "Syntax.term" assumes "{$$} \<turnstile> e', eP', eV' : \<tau>"   and "\<lless>[], e\<ggreater> I\<rightarrow>i \<lless>[], e'\<ggreater>"   and "\<lless>\<pi>, eV\<ggreater> V\<rightarrow>i \<lless>[], eV'\<ggreater>" shows "thesis"
monom_mult_is_smult: fixes R :: "('a, 'b) ring_scheme"   and a :: "'a"   and p :: "nat \<Rightarrow> 'a" assumes "UP_ring R"   and "a \<in> carrier R"   and "p \<in> carrier (UP R)" shows "monom (UP R) a 0 \<otimes>\<^bsub>UP R\<^esub> p = a \<odot>\<^bsub>UP R\<^esub> p"
enm_def: fixes \<eta> :: "nat" shows "MAC.enm \<eta> \<equiv> CNV (\<lambda>bs m. if bs then Done ((), True) else Pause (Inl m) Done \<bind> (\<lambda>r. lift_spmf (mac \<eta> (projl r) m) \<bind> (\<lambda>a. Pause (Inr (a, m)) Done \<bind> (\<lambda>_. Done ((), True))))) False"
ty_of_def: fixes x :: "Atoms.var" shows "ty_of x = inv sort_of_ty (sort_of (atom x))"
Cnext_solve_strictly_precedes: fixes K\<^sub>2 :: "clock"   and n :: "nat"   and K\<^sub>1 :: "clock"   and \<Gamma> :: "'c constr list"   and \<Psi> :: "'c TESL_atomic list"   and \<Phi> :: "'c TESL_atomic list" shows "{\<lceil>#\<^sup>\<le> K\<^sub>2 n, #\<^sup>< K\<^sub>1 n\<rceil> \<in> \<lambda>(x, y). x \<le> y # \<Gamma>, n \<turnstile> \<Psi> \<triangleright> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Phi>} \<subseteq> \<C>\<^sub>n\<^sub>e\<^sub>x\<^sub>t \<Gamma>, n \<turnstile> (K\<^sub>1 strictly precedes K\<^sub>2) # \<Psi> \<triangleright> \<Phi>"
Inequality: fixes X :: "'var list"   and F :: "(('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set"   and \<A> :: "('fun, 'var) strand_step list" assumes "\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S \<in> \<S>" shows "(\<S>, \<A>) \<Rightarrow>\<^sup>\<bullet> (??.local.update\<^sub>s\<^sub>t \<S> (\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t # S), \<A> @ [\<forall>X\<langle>\<or>\<noteq>: F\<rangle>\<^sub>s\<^sub>t])"
unit_disc_compl_def: shows "unit_disc_compl = disc_compl ounit_circle"
Image_mono: fixes r' :: "('b \<times> 'c) set"   and r :: "('b \<times> 'c) set"   and A' :: "'b set"   and A :: "'b set" assumes "r' \<subseteq> r"   and "A' \<subseteq> A" shows "r' `` A' \<subseteq> r `` A"
fsubsetD: fixes A :: "'b fset"   and B :: "'b fset"   and c :: "'b" assumes "A |\<subseteq>| B"   and "c |\<in>| A" shows "c |\<in>| B"
hyps_3: assumes "\<And>x. x \<in> A \<Longrightarrow> f x = \<one>" shows "finprod G f A = \<one>"
last_smods_nzero: fixes p :: "real poly"   and q :: "real poly" assumes "p \<noteq> 0" shows "last (smods p q) \<noteq> 0"
signed_0: 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.signed one plus zero minus 0 = zero"
vsv_vimage_vinsert: fixes r :: "V"   and a :: "V"   and A :: "V" assumes "vsv r"   and "a \<in>\<^sub>\<circ> \<D>\<^sub>\<circ> r" shows "r `\<^sub>\<circ> vinsert a A = vinsert (r\<lparr>a\<rparr>) (r `\<^sub>\<circ> A)"