hcju/isabelleps huggingface dataset:
simple_path2_unsnoc: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "n \<noteq> m" obtains ns' :: "'node list" where "g \<turnstile> n-ns'\<rightarrow>last ns'" and "last ns' \<in> set (old.predecessors g m)" and "distinct ns'" and "set ns' \<subseteq> set ns" and "m \<notin> set ns'"
f'Cg: fixes s :: "'a \<times> 'b" assumes "s \<in> S" shows "isCont f' s"
UN_constant: fixes c :: "'c set"   and A :: "'d set" shows "\<mu> (\<P> (\<lambda>y. c) A) = (if A = {} then {} else c)"
intros_7: fixes iop :: "binop_i"   and c1 :: "i64"   and c2 :: "i64"   and c :: "i64" assumes "app_binop_i iop c1 c2 = Some c" shows "\<lparr>[$C ConstInt64 c1, $C ConstInt64 c2, $Binop_i T_i64 iop]\<rparr> \<leadsto> \<lparr>[$C ConstInt64 c]\<rparr>"
excess_non_negative: shows "\<forall>v\<in>V - {s, t}. (0::'a) \<le> excess f v"
absolutely_integrable_reflect_real: fixes b :: "real"   and a :: "real"   and f :: "real \<Rightarrow> 'a" shows "((\<lambda>x. f (- x)) absolutely_integrable_on {- b..- a}) = (f absolutely_integrable_on {a..b})"
wrap_def: fixes B :: "'b set" shows "wrap B = (if B = {} then {} else {B})"
lc': shows "lincomb a U = \<zero>\<^bsub>VK\<^esub>"
oalist_inv_raw_Nil: shows "tc.oalist_inv_raw []"
B_3: shows "set (rhsq r) \<subseteq> Q - (W - {q})"
dg_prod_ArrD_3: fixes f :: "V"   and I :: "V"   and \<AA> :: "V \<Rightarrow> V"   and i :: "V" assumes "f \<in>\<^sub>\<circ> dg_prod I \<AA>\<lparr>Arr\<rparr>"   and "i \<in>\<^sub>\<circ> I" shows "f\<lparr>i\<rparr> \<in>\<^sub>\<circ> \<AA> i\<lparr>Arr\<rparr>"
symmetric_cong: fixes A :: "'c set"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and r' :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<And>a b. \<lbrakk>a \<in> A; b \<in> A\<rbrakk> \<Longrightarrow> r a b = r' a b" shows "symmetric A r = symmetric A r'"
CongruenceD: fixes K :: "'c set"   and G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c"   and a :: "'c"   and b :: "'c" assumes "normal_subgroup K G composition unit"   and "(a, b) \<in> normal_subgroup.Congruence K G composition unit" shows "\<exists>k\<in>K. a = composition b k"
minus_unique: fixes y :: "'a list"   and x :: "'a list"   and y' :: "'a list" assumes "y \<oplus>\<^bsub>K [X]\<^esub> x = \<zero>\<^bsub>K [X]\<^esub>"   and "x \<oplus>\<^bsub>K [X]\<^esub> y' = \<zero>\<^bsub>K [X]\<^esub>"   and "x \<in> carrier (K [X])"   and "y \<in> carrier (K [X])"   and "y' \<in> carrier (K [X])" shows "y = y'"
empty_concatenate: fixes xs :: "brick list" shows "xs \<otimes> [] = xs"
assms_14: fixes f :: "int poly" assumes "f \<in> set res" shows "irreducible\<^sub>d f"
set_intros_13: fixes yk :: "'b"   and x112 :: "'b ltlc"   and x111 :: "'b ltlc" assumes "yk \<in> atoms_ltlc x112" shows "yk \<in> atoms_ltlc (x111 U\<^sub>c x112)"
match_result_sound_val: fixes env :: "v sem_env"   and s :: "'c state"   and v0 :: "v"   and pes :: "(pat \<times> exp0) list"   and err_v :: "v"   and e :: "exp0"   and env' :: "(char list \<times> v) list"   and ck :: "bool"   and bv :: "'c state \<times> (v, v) result" assumes "match_result env s v0 pes err_v = Rval (e, env')"   and "BigStep.evaluate ck (update_v (\<lambda>_. nsAppend (alist_to_ns env') (sem_env.v env)) env) s e bv" shows "evaluate_match ck env s v0 pes err_v bv"
is_chain_def: fixes f :: "nat \<Rightarrow> 'a set" shows "is_chain f = (\<forall>n. f n \<subseteq> f (Suc n))"
swap_swap: fixes p :: "'k \<times> 'l" shows "prod.swap (prod.swap p) = p"
integrable_altD_1: fixes f :: "'b \<Rightarrow> 'c"   and s :: "'b set"   and a :: "'b"   and b :: "'b" assumes "f integrable_on s" shows "(\<lambda>x. if x \<in> s then f x else (0::'c)) integrable_on cbox a b"
function_ring_car_memE_1: fixes a :: "'c \<Rightarrow> 'a" assumes "a \<in> carrier F" shows "a \<in> extensional S"
set_tag_name_get_disconnected_nodes: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) element_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) document_ptr" assumes "l_set_tag_name_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_tag_name set_tag_name_locs get_disconnected_nodes get_disconnected_nodes_locs" shows "\<forall>w\<in>set_tag_name_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_disconnected_nodes_locs ptr'. r h h')"
reachable_in_verts_1: fixes u :: "'a"   and v :: "'a" assumes "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> S\<^esub> v" shows "u \<in> verts (G \<restriction> S)"
runit_simps_1: fixes f :: "'d" assumes "D.ide f" shows "D.arr \<r>\<^sub>D[f]"
StaticCastRedsFail: fixes e :: "expr"   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 a :: "nat"   and Cs :: "char list 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"   and E :: "char list \<Rightarrow> ty option"   and C :: "char list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>ref (a, Cs),s'\<rangle>"   and "C \<notin> set Cs"   and "(last Cs, C) \<notin> (subcls1 P)\<^sup>*" shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,s\<rangle> \<rightarrow>* \<langle>Throw (addr_of_sys_xcpt ClassCast, [ClassCast]),s'\<rangle>"
transversalD: fixes p :: "'a set"   and x :: "'a" assumes "transversal p"   and "x \<in> p" shows "\<forall>\<^sub>F t in at_right 0. flow0 x t \<notin> p"
update_convs_8: fixes state_var' :: "sparc_state_var \<Rightarrow> sparc_state_var"   and cpu_reg :: "CPU_register \<Rightarrow> word32"   and user_reg :: "'b word \<Rightarrow> word5 \<Rightarrow> word32"   and sys_reg :: "sys_reg \<Rightarrow> word32"   and mem :: "word8 \<Rightarrow> word36 \<Rightarrow> word8 option"   and mmu :: "MMU_state"   and cache :: "cpu_cache"   and dwrite :: "(int \<times> word32 \<times> CPU_register) list"   and state_var :: "sparc_state_var"   and traps :: "Trap set"   and undef :: "bool"   and more :: "'c" shows "state_var_update state_var' \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var, traps = traps, undef = undef, \<dots> = more\<rparr> = \<lparr>cpu_reg = cpu_reg, user_reg = user_reg, sys_reg = sys_reg, mem = mem, mmu = mmu, cache = cache, dwrite = dwrite, state_var = state_var' state_var, traps = traps, undef = undef, \<dots> = more\<rparr>"
cCons_0_Nil_eq: shows "cCons (0::'a) [] = []"
closure_of_subset_Int: fixes X :: "'c topology"   and S :: "'c set" shows "topspace X \<inter> S \<subseteq> X closure_of S"
mat_mult_left_mono: fixes C :: "'b mat"   and n :: "nat"   and A :: "'b mat"   and B :: "'b mat" assumes "C \<ge>\<^sub>m 0\<^sub>m n n"   and "A \<ge>\<^sub>m B"   and "A \<in> carrier_mat n n"   and "B \<in> carrier_mat n n"   and "C \<in> carrier_mat n n" shows "A * C \<ge>\<^sub>m B * C"
simps: fixes y :: "nat"   and x :: "nat" shows "c_assoc_have_key y x = (if y = 0 then 1 else if c_fst (c_hd y) = x then 0 else c_assoc_have_key (c_tl y) x)"
1: shows "regular v \<and> regular (v * v\<^sup>T) \<and> regular ((v \<squnion> minarc (v * - v\<^sup>T \<sqinter> g)\<^sup>T * top) * (v \<squnion> minarc (v * - v\<^sup>T \<sqinter> g)\<^sup>T * top)\<^sup>T) \<and> regular (top * minarc (v * - v\<^sup>T \<sqinter> g))"
is_node_s1_s2: shows "prod_is_node A B (0, 0)"
inject: fixes x1 :: "'l"   and x2 :: "'m"   and y1 :: "'l"   and y2 :: "'m" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
induct: fixes x :: "'a"   and R :: "('a, 'b) ring_scheme"   and H :: "'a set"   and P :: "'a \<Rightarrow> bool" assumes "x \<in> generate_field R H"   and "P \<one>\<^bsub>R\<^esub>"   and "\<And>h. h \<in> H \<Longrightarrow> P h"   and "\<And>h. \<lbrakk>h \<in> generate_field R H; P h\<rbrakk> \<Longrightarrow> P (\<ominus>\<^bsub>R\<^esub> h)"   and "\<And>h. \<lbrakk>h \<in> generate_field R H; P h; h \<noteq> \<zero>\<^bsub>R\<^esub>\<rbrakk> \<Longrightarrow> P (inv\<^bsub>R\<^esub> h)"   and "\<And>h1 h2. \<lbrakk>h1 \<in> generate_field R H; P h1; h2 \<in> generate_field R H; P h2\<rbrakk> \<Longrightarrow> P (h1 \<oplus>\<^bsub>R\<^esub> h2)"   and "\<And>h1 h2. \<lbrakk>h1 \<in> generate_field R H; P h1; h2 \<in> generate_field R H; P h2\<rbrakk> \<Longrightarrow> P (h1 \<otimes>\<^bsub>R\<^esub> h2)" shows "P x"
the_Sup_def: fixes A :: "'b set" shows "the_Sup A = (THE a. a \<in> A)"
map_le_upd: fixes f :: "'c \<Rightarrow> 'd option"   and g :: "'c \<Rightarrow> 'd option"   and a :: "'c"   and b :: "'d option" assumes "f \<subseteq>\<^sub>m g" shows "f(a := b) \<subseteq>\<^sub>m g(a := b)"
rA: shows "r `\<^sub>\<circ> A = r `\<^sub>\<circ> (\<D>\<^sub>\<circ> r \<inter>\<^sub>\<circ> A)"
card_of_Sigma_mono1: fixes I :: "'b set"   and A :: "'b \<Rightarrow> 'c set"   and B :: "'b \<Rightarrow> 'd set" assumes "\<forall>i\<in>I. |A i| \<le>o |B i|" shows "|Sigma I A| \<le>o |Sigma I B|"
Fresh_eqvt: fixes h :: "'a \<Rightarrow> 'b"   and p :: "perm" assumes "\<exists>a. atom a \<sharp> (h, h a)" shows "p \<bullet> Fresh h = Fresh (p \<bullet> h)"
Tree_def: fixes G :: "('e, 'f) Graph_scheme" shows "Tree G \<equiv> Graph G \<and> Tree_axioms G"
finite_measure_mono: fixes A :: "real set"   and B :: "real set" assumes "A \<subseteq> B"   and "B \<in> sets M" shows "Sigma_Algebra.measure M A \<le> Sigma_Algebra.measure M B"
res_one_eq: shows "\<one>\<^bsub>residue_ring (int p)\<^esub> = 1"
cases: fixes x :: "'a \<times> 'b \<times> 'c" obtains t :: "'a" and a1 :: "'b" and a2 :: "'c" where "x = (t, a1, a2)"
z_3: shows "- z \<notin> \<real>\<^sub>\<le>\<^sub>0"
Gcd_eq_Gcd_factorial: fixes A :: "'a set" shows "Gcd A = Gcd_factorial A"
chain_imp_rtrancl: fixes S :: "nat \<Rightarrow> 'b"   and r :: "('b \<times> 'b) set"   and i :: "nat"   and j :: "nat" assumes "\<forall>i. (S i, S (Suc i)) \<in> r"   and "i \<le> j" shows "(S i, S j) \<in> r\<^sup>*"
swap_image_eq: fixes a :: "'c"   and A :: "'c set"   and b :: "'c"   and f :: "'c \<Rightarrow> 'd" assumes "a \<in> A"   and "b \<in> A" shows "Fun.swap a b f ` A = f ` A"
inf_principal: fixes A :: "'b set"   and B :: "'b set" shows "inf (principal A) (principal B) = principal (A \<inter> B)"
suntil_inv: fixes f :: "'a stream \<Rightarrow> 'b stream"   and P :: "'b stream \<Rightarrow> bool"   and Q :: "'b stream \<Rightarrow> bool"   and s :: "'a stream" assumes "\<And>s. f (stl s) = stl (f s)" shows "(P suntil Q) (f s) = ((\<lambda>x. P (f x)) suntil (\<lambda>x. Q (f x))) s"
p_coprime_right_int: fixes a :: "int" shows "coprime a (int p) = (\<not> int p dvd a)"
int_of_uint16_code: fixes x :: "uint16" shows "int_of_uint16 x = int_of_integer (integer_of_uint16 x)"
UL_eq_D: shows "\<Union> (list.set L) = D"
intros_18: fixes x :: "name"   and P :: "pi" shows "<\<nu>x>\<tau>.P \<equiv>\<^sub>e \<tau>.(<\<nu>x>P)"
r_step_Cn_def: shows "r_step_Cn \<equiv> Cn 1 r_ifeq [r_e2lenls, Cn 1 r_length [Cn 1 r_pdec222 [r_e2i]], Cn 1 r_ifz [r_e2rv, Cn 1 r_prod_encode [Cn 1 r_cons [r_triple_encode (Cn 1 r_pdec122 [r_e2i]) r_e2ls Z, r_e2stack], Z], Cn 1 r_prod_encode [r_e2tail, r_e2rv]], Cn 1 r_ifz [r_e2rv, Cn 1 r_ifless [r_e2lenls, Cn 1 r_length [Cn 1 r_pdec222 [r_e2i]], Cn 1 r_prod_encode [Cn 1 r_cons [r_triple_encode (Cn 1 r_nth [Cn 1 r_pdec222 [r_e2i], r_e2lenls]) r_e2xs Z, r_e2stack], Z], Cn 1 r_prod_encode [r_e2tail, r_e2rv]], Cn 1 r_prod_encode [Cn 1 r_cons [r_triple_encode r_e2i r_e2xs (Cn 1 r_snoc [r_e2ls, Cn 1 r_dec [r_e2rv]]), r_e2tail], Z]]]"
dot_lmul_matrix: fixes x :: "(real, 'c) vec"   and A :: "((real, 'b) vec, 'c) vec"   and y :: "(real, 'b) vec" shows "(x v* A) \<bullet> y = x \<bullet> (A *v y)"
Eps_psimp: fixes P :: "'b \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "P x"   and "\<not> P y"   and "Eps P = y" shows "Eps P = x"
set_relD1: fixes s :: "'a set"   and s' :: "'b set"   and R :: "('a \<times> 'b) set"   and x :: "'a" assumes "(s, s') \<in> \<langle>R\<rangle>set_rel"   and "x \<in> s" shows "\<exists>x'\<in>s'. (x, x') \<in> R"
prefixCases: fixes \<Psi> :: "'b"   and \<alpha> :: "'a prefix"   and P :: "('a, 'b, 'c) psi"   and \<beta> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and Prop :: "'a prefix \<Rightarrow> 'a action \<Rightarrow> ('a, 'b, 'c) psi \<Rightarrow> bool" assumes "\<Psi> \<rhd> \<alpha>\<cdot>P \<longmapsto> \<beta> \<prec> P'"   and "\<And>M xvec N K Tvec. \<lbrakk>\<Psi> \<turnstile> M \<leftrightarrow> K; set xvec \<subseteq> supp N; length xvec = length Tvec; distinct xvec\<rbrakk> \<Longrightarrow> Prop (pInput M xvec N) (K\<lparr>N[xvec::=Tvec]\<rparr>) (P[xvec::=Tvec])"   and "\<And>M N K. \<Psi> \<turnstile> M \<leftrightarrow> K \<Longrightarrow> Prop (pOutput M N) (K\<langle>N\<rangle>) P"   and "\<Psi> \<rhd> P \<sim> P' \<Longrightarrow> Prop pTau (\<tau>) P'" shows "Prop \<alpha> \<beta> P'"
in_nlists_UNIV: fixes xs :: "'a list"   and n :: "nat" shows "(xs \<in> nlists UNIV n) = (length xs = n)"
CD_on_def2: fixes ds :: "'d set"   and A :: "'x set" shows "CD_on ds A = (\<Union>d\<in>ds. Cd d (A \<inter> Field (Pd d)))"
add_zerol: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and one :: "'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and x :: "'c" assumes "CKA.class.dioid plus less_eq less one zero times" shows "plus zero x = x"
semiring_normalization_rules_27: fixes times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and x :: "'a"   and q :: "nat" assumes "class.comm_semiring_1 times one plus zero" shows "times x (power.power one times x q) = power.power one times x (Suc q)"
fam_morphisms_are_maps: fixes U :: "'a set" assumes "is_open U" shows "Set_Theory.map (fam_morphisms U) (\<FF> U) (\<FF>' U)"
mHom_func: fixes f :: "'g \<Rightarrow> 'h"   and M :: "('g, 'a, 'i) Module_scheme"   and N :: "('h, 'a, 'j) Module_scheme" assumes "f \<in> mHom R M N" shows "f \<in> carrier M \<rightarrow> carrier N"
ctermsl_ex_stermsl: fixes q :: "('a, 'b, 'c, 'd) seqp"   and p :: "('a, 'b, 'c, 'd) seqp" assumes "q \<in> ctermsl p" shows "\<exists>ps\<in>stermsl p. q \<in> ctermsl ps"
exhaustiveUnitPropagatePreservesCurrentLevel: fixes state :: "State" assumes "exhaustiveUnitPropagate_dom state"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)" shows "let state' = exhaustiveUnitPropagate state in currentLevel (getM state') = currentLevel (getM state)"
pop_hd: fixes a :: "'a"   and u :: "'a list" shows "h (a # u) = h [a] \<cdot> h u"
T_def: shows "T = {x. infdist x S \<le> r}"
intros: fixes rest :: "('a, 'd, 'b, 'e, 'c, 'f, 'a) rest_scheme"   and \<I>_adv :: "('b, 'c) \<I>"   and I :: "'a \<Rightarrow> bool"   and \<I>_usr :: "('e, 'f) \<I>" assumes "\<And>s x y es s'. \<lbrakk>((y, es), s') \<in> set_spmf (rfunc_adv rest s x); x \<in> outs_\<I> \<I>_adv; I s\<rbrakk> \<Longrightarrow> y \<in> responses_\<I> \<I>_adv x \<and> I s'"   and "\<And>s x y es s'. \<lbrakk>((y, es), s') \<in> set_spmf (rfunc_usr rest s x); x \<in> outs_\<I> \<I>_usr; I s\<rbrakk> \<Longrightarrow> y \<in> responses_\<I> \<I>_usr x \<and> I s'"   and "I (rinit rest)" shows "WT_rest \<I>_adv \<I>_usr I rest"
blocking_on_imp_not_stable: fixes ds :: "'b set"   and X :: "'a set"   and h :: "'c"   and X'' :: "'a set" assumes "Pdds'.blocking_on ds X h X''" shows "\<not> Pdds'.stable_on ds X"
res_mult_eq: fixes x :: "int"   and y :: "int" shows "x \<otimes>\<^bsub>R\<^esub> y = x * y mod int p"
order_gt_0_iff_finite: shows "(0 < order (DirProds G I)) = finite (carrier (DirProds G I))"
less_le: fixes x :: "'a"   and y :: "'a" shows "cmp.lt x y = (cmp.le x y \<and> x \<noteq> y)"
cil_cons1: fixes w :: "'a list"   and wa :: "'a list"   and \<alpha> :: "'a \<Rightarrow> 'b set \<times> 'b set"   and wb :: "'a list"   and e :: "'a" assumes "w \<in> wa \<otimes>\<^bsub>\<alpha>\<^esub> wb"   and "fst (\<alpha> e) \<inter> mon_pl (map \<alpha> wb) = {}" shows "e # w \<in> e # wa \<otimes>\<^bsub>\<alpha>\<^esub> wb"
F2map_comp_id: fixes g1 :: "'g \<Rightarrow> 'd"   and g2 :: "'h \<Rightarrow> 'e"   and g3 :: "'i \<Rightarrow> 'f"   and f2 :: "'j \<Rightarrow> 'h"   and f3 :: "'k \<Rightarrow> 'i"   and x :: "('g, 'j, 'k) F2" shows "F2map g1 g2 g3 (F2map id f2 f3 x) = F2map g1 (g2 \<circ> f2) (g3 \<circ> f3) x"
id_is_hom: shows "ring_hom_cring R R (\<lambda>x. x)"
healthy_fixes_bot: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" assumes "healthy t" shows "t (\<lambda>s. 0) = (\<lambda>s. 0)"
completeness_game_def: fixes h :: "'a"   and w :: "nat \<times> nat"   and e :: "nat" shows "local.\<Sigma>_protocols_base.completeness_game h w e = init h w \<bind> (\<lambda>(r, a). response r w e \<bind> (\<lambda>z. return_spmf (check h a e z)))"
WTrtCASNT: fixes P :: "'addr J_prog"   and h :: "'heap"   and E :: "String.literal \<Rightarrow> ty option"   and e1 :: "'addr expr"   and e2 :: "'addr expr"   and T2 :: "ty"   and e3 :: "'addr expr"   and T3 :: "ty"   and D :: "String.literal"   and F :: "String.literal" assumes "P,E,h \<turnstile> e1 : NT"   and "P,E,h \<turnstile> e2 : T2"   and "P,E,h \<turnstile> e3 : T3" shows "P,E,h \<turnstile> e1\<bullet>compareAndSwap(D\<bullet>F, e2, e3) : Boolean"
unsat: shows "\<forall>G. \<not> eval\<^sub>c\<^sub>s HFun G Cs"
sel_3: fixes x1 :: "le_rel"   and x2 :: "linear_poly"   and x3 :: "'a" shows "lec_const (Le_Constraint x1 x2 x3) = x3"
Pyz_nn: fixes y :: "'c"   and z :: "'d" assumes "y \<in> space T"   and "z \<in> space P" shows "0 \<le> Pyz (y, z)"
connT: fixes y :: "'a" assumes "y \<in> T" shows "connected (S \<inter> f -` {y})"
get_return_edges_valid: fixes a :: "'edge"   and a' :: "'edge" assumes "valid_edge a"   and "a' \<in> get_return_edges a" shows "valid_edge a'"
simps_16: fixes x2 :: "'a" shows "set_Literal (Neg x2) = {x2}"
rep_eq: fixes x :: "'b \<Rightarrow>\<^sub>0 'a" shows "Poly_Mapping.range x = range (lookup x) - {0::'a}"
one_closed: fixes H :: "'c set"   and G :: "('c, 'd) monoid_scheme" assumes "submonoid H G" shows "\<one>\<^bsub>G\<^esub> \<in> H"
apaths_finite: fixes u :: "'a"   and v :: "'a" shows "finite {p. apath u p v}"
mult_zero_ow_axioms_def: fixes U :: "'b set"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b" shows "mult_zero_ow_axioms U times zero \<equiv> (\<forall>a. a \<in> U \<longrightarrow> times zero a = zero) \<and> (\<forall>a. a \<in> U \<longrightarrow> times a zero = zero)"
monom_pow: fixes v :: "nat"   and n0 :: "nat"   and a :: "'b"   and n :: "nat" shows "MPoly_Type.monom (Poly_Mapping.single v n0) a ^ n = MPoly_Type.monom (Poly_Mapping.single v (n0 * n)) (a ^ n)"
iMODb_Min: fixes r :: "nat"   and m :: "nat"   and c :: "nat" shows "iMin [ r, mod m, c ] = r"
map\<^sub>0_simps_2: fixes a :: "'c" assumes "C.obj a" shows "src\<^sub>D (\<sigma>'.map\<^sub>0 a) = map\<^sub>0 a"
real_sqrt_mult: fixes x :: "real"   and y :: "real" shows "sqrt (x * y) = sqrt x * sqrt y"
proj_id: fixes wtFsym :: "'a \<Rightarrow> bool"   and wtPsym :: "'b \<Rightarrow> bool"   and arOf :: "'a \<Rightarrow> 'c list"   and resOf :: "'a \<Rightarrow> 'c"   and parOf :: "'b \<Rightarrow> 'c list"   and \<Phi> :: "('a, 'b) lit list set"   and infTp :: "'c \<Rightarrow> bool"   and pol :: "'c \<Rightarrow> 'b \<Rightarrow> epol"   and grdOf :: "('a, 'b) lit list \<Rightarrow> ('a, 'b) lit \<Rightarrow> Preliminaries.var \<Rightarrow> ('a, 'b) lit"   and intT :: "'c \<Rightarrow> univ \<Rightarrow> bool"   and intF :: "'a \<Rightarrow> univ list \<Rightarrow> univ"   and intP :: "'b \<Rightarrow> univ list \<Rightarrow> bool"   and \<sigma> :: "'c"   and a :: "univ" assumes "ModelIkPolMcalc2C wtFsym wtPsym arOf resOf parOf \<Phi> infTp pol grdOf intT intF intP"   and "intT \<sigma> a" shows "ModelIkPolMcalc2C.proj intT \<sigma> a = a"
simplify_preserves_semantic: fixes T :: "'at Literal set set"   and S :: "'at Literal set set"   and I :: "'at set" assumes "T = simplify S"   and "all_fulfill finite S" shows "I \<Turnstile> S = I \<Turnstile> T"
assms_2: fixes x :: "'a"   and x' :: "'a"   and y :: "'b"   and y' :: "'b" assumes "\<lfloor>x\<rfloor> = \<lfloor>x'\<rfloor>"   and "\<lfloor>y\<rfloor> = \<lfloor>y'\<rfloor>" shows "g x y = g x' y'"
map_fun2_id: fixes f :: "'a \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'b"   and x :: "'c \<Rightarrow> 'd" shows "map_fun f g x = g \<circ> map_fun f id x"
list_slice_length: fixes xs :: "'b list"   and k :: "nat" shows "length (list_slice xs k) = length xs div k"
wt_sym_ge: fixes f :: "'s" shows "of_nat (\<epsilon> - the_enat (of_nat \<delta> * arity_sym f)) \<le> wt_sym f"
gen_akiyama_tanigawa_fps: fixes f :: "nat \<Rightarrow> real" shows "Abs_fps (\<lambda>n. gen_akiyama_tanigawa f n 0 / fact n) = fps_exp 1 * (Abs_fps f oo 1 - fps_exp 1)"
restriction_card_mon1: fixes v :: "view"   and u :: "view"   and w :: "view"   and f :: "cars \<Rightarrow> nat_int"   and c :: "cars" shows "v=u--w \<longrightarrow> |restrict u f c| \<le> |restrict v f c|"
add_inv_exists_mat: fixes A :: "'a mat"   and nr :: "nat"   and nc :: "nat" assumes "A \<in> carrier_mat nr nc" shows "\<exists>B\<in>carrier_mat nr nc. B + A = 0\<^sub>m nr nc \<and> A + B = 0\<^sub>m nr nc"
simps_7: fixes i :: "int"   and a :: "Ferrack.num" shows "numbound0 (Mul i a) = numbound0 a"
two_pred_def: shows "two_pred = (\<Lambda> x. if x \<sqsubseteq> once then none else x)"
norm_assertion_simps_13: fixes a :: "assn"   and b :: "assn"   and c :: "assn" shows "a * (b * c) = a * b * c"
viewR_correct_empty: shows "FingerTreeStruc.viewR Empty = None"
epi_implies_arr: fixes f :: "'a \<times> 'a" assumes "CC.epi f" shows "CC.arr f"
assms_2: shows "\<forall>a\<in>set as. depends\<^sub>d\<^sub>l\<^sub>o a"
division_wellfounded: shows "wf {(x, y). x \<in> carrier mk_monoid \<and> y \<in> carrier mk_monoid \<and> comm_semiring_1_class.properfactor x y}"
enat_defs_6: fixes m :: "enat"   and n :: "enat" shows "(m < n) = (case m of enat m1 \<Rightarrow> case n of enat n1 \<Rightarrow> m1 < n1 | \<infinity> \<Rightarrow> True | \<infinity> \<Rightarrow> False)"
stalk_is_ring: fixes V :: "'a set" assumes "is_open\<^sub>X V"   and "x \<in> V" shows "ring stx.carrier_stalk stx.add_stalk stx.mult_stalk (stx.zero_stalk V) (stx.one_stalk V)"
get_min_in: fixes h :: "'b heap" assumes "h \<noteq> Empty" shows "get_min h \<in># mset_heap h"
r2f_f2r_galois_var: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e set"   and R :: "'c \<Rightarrow> ('d \<times> 'e) set" shows "(\<R> \<circ> f = R) = (\<F> \<circ> R = f)"
pos_poly_abs_poly: fixes p :: "int poly" shows "(0 < lead_coeff (abs_int_poly p)) = (p \<noteq> 0)"
prv_imp_mp: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "bprv (imp \<phi> \<chi>)"   and "bprv \<phi>" shows "bprv \<chi>"
abs_eq: shows "1 = Abs_float (0, 2 ^ (LENGTH('a) - 1) - 1, 0)"