hcju/isabelleps huggingface dataset:
elims_2: fixes x :: "nat"   and xa :: "form"   and xb :: "float interval option list"   and xc :: "nat list" assumes "approx_form x xa xb xc" obtains prec :: "nat" and n :: "nat" and a :: "floatarith" and b :: "floatarith" and f :: "form" and bs :: "float interval option list" and ss :: "nat list" where "x = prec__" and "xa = Bound (Var n__) a__ b__ f__" and "xb = bs__" and "xc = ss__" and "case (approx prec__ a__ bs__, approx prec__ b__ bs__) of (None, b) \<Rightarrow> False | (Some ivl1, None) \<Rightarrow> False | (Some ivl1, Some ivl2) \<Rightarrow> approx_form' prec__ f__ (ss__ ! n__) n__ (sup ivl1 ivl2) bs__ ss__"   | prec :: "nat" and n :: "nat" and a :: "floatarith" and f :: "form" and bs :: "float interval option list" and ss :: "nat list" where "x = preca__" and "xa = Assign (Var n) aa__ f" and "xb = bsa__" and "xc = ssa__" and "case approx preca__ aa__ bsa__ of None \<Rightarrow> False | Some ivl \<Rightarrow> approx_form' preca__ f (ssa__ ! n) n ivl bsa__ ssa__"   | prec :: "nat" and a :: "floatarith" and b :: "floatarith" and bs :: "float interval option list" and ss :: "nat list" where "x = precb__" and "xa = Less ab__ ba__" and "xb = bsb__" and "xc = ssb__" and "case (approx precb__ ab__ bsb__, approx precb__ ba__ bsb__) of (None, b) \<Rightarrow> False | (Some ivl, None) \<Rightarrow> False | (Some ivl, Some ivl') \<Rightarrow> float_plus_up precb__ (upper ivl) (- lower ivl') < 0"   | prec :: "nat" and a :: "floatarith" and b :: "floatarith" and bs :: "float interval option list" and ss :: "nat list" where "x = precc__" and "xa = LessEqual ac__ bb__" and "xb = bsc__" and "xc = ssc__" and "case (approx precc__ ac__ bsc__, approx precc__ bb__ bsc__) of (None, b) \<Rightarrow> False | (Some ivl, None) \<Rightarrow> False | (Some ivl, Some ivl') \<Rightarrow> float_plus_up precc__ (upper ivl) (- lower ivl') \<le> 0"   | prec :: "nat" and xd :: "floatarith" and a :: "floatarith" and b :: "floatarith" and bs :: "float interval option list" and ss :: "nat list" where "x = precd__" and "xa = AtLeastAtMost xd ad__ bc__" and "xb = bsd__" and "xc = ssd__" and "case (approx precd__ xd bsd__, approx precd__ ad__ bsd__, approx precd__ bc__ bsd__) of (None, b) \<Rightarrow> False | (Some ivlx, None, ba) \<Rightarrow> False | (Some ivlx, Some ivl, None) \<Rightarrow> False | (Some ivlx, Some ivl, Some ivl') \<Rightarrow> float_plus_up precd__ (upper ivl) (- lower ivlx) \<le> 0 \<and> float_plus_up precd__ (upper ivlx) (- lower ivl') \<le> 0"   | prec :: "nat" and a :: "form" and b :: "form" and bs :: "float interval option list" and ss :: "nat list" where "x = prece__" and "xa = Conj ae__ bd__" and "xb = bse__" and "xc = sse__" and "approx_form prece__ ae__ bse__ sse__ \<and> approx_form prece__ bd__ bse__ sse__"   | prec :: "nat" and a :: "form" and b :: "form" and bs :: "float interval option list" and ss :: "nat list" where "x = prec" and "xa = Disj a b" and "xb = bs" and "xc = ss" and "approx_form prec a bs ss \<or> approx_form prec b bs ss"
assms_3: shows "F exposed_face_of {x + y |x y. x \<in> S \<and> y \<in> T}"
downset_inj: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.order less_eq less" shows "inj (ord.downset less_eq)"
mult_col_div_row_carrier_3: fixes A :: "'a mat"   and n :: "nat"   and a :: "'a"   and k :: "nat" assumes "A \<in> carrier_mat n n" shows "mult_col_div_row a k A \<in> carrier_mat n n"
v1: shows "\<forall>x\<in>set (digitToNlist sf1 @ ts @ digitToNlist pr2). is_leveln_node n x \<and> is_measured_node x"
bacc_step_def: shows "bacc_step \<equiv> \<lambda>\<delta> Q. {x. bacc_stepp (\<lambda>x. x \<in> \<delta>) Q x}"
WT_calleeD: fixes \<I> :: "('b, 'd) \<I>"   and callee :: "'b \<Rightarrow> ('d \<times> 'e) spmf"   and ret :: "'d"   and s :: "'e"   and out :: "'b" assumes "\<I> \<turnstile>c callee \<surd>"   and "(ret, s) \<in> set_spmf (callee out)"   and "out \<in> outs_\<I> \<I>" shows "ret \<in> responses_\<I> \<I> out"
simps_2: fixes a :: "'c"   and t1 :: "'c bt"   and t2 :: "'c bt" shows "depth (Br a t1 t2) = Suc (max (depth t1) (depth t2))"
hyps: shows "diag_block_mat As ^\<^sub>m n = diag_block_mat (map (\<lambda>A. A ^\<^sub>m n) As)"
cone_mono: fixes Vs :: "'a vec set"   and Ws :: "'a vec set" assumes "Vs \<subseteq> Ws" shows "cone Vs \<subseteq> cone Ws"
norm_triangle_ineq4: fixes a :: "'a"   and b :: "'a" shows "norm (a - b) \<le> norm a + norm b"
reflexive_sup_closed: fixes x :: "'a"   and y :: "'a" assumes "reflexive x" shows "reflexive (x \<squnion> y)"
p: fixes p :: "pat" assumes "p \<in> set ps" shows "is_cupcake_pat p"
gfp_upperbound: fixes X :: "'b \<times> 'c"   and f :: "'b \<times> 'c \<Rightarrow> 'b \<times> 'c" assumes "less_eq_prod X (f X)" shows "less_eq_prod X (prod.gfp f)"
Int_Diff_Un: fixes A :: "'b set"   and B :: "'b set" shows "A \<inter> B \<union> (A - B) = A"
one_div_2_pow_eq: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and division_segment :: "'a \<Rightarrow> 'a"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "divide one (power.power one times (numeral.numeral one plus (num.Bit0 num.One)) n) = zero_neq_one.of_bool one zero (n = 0)"
clop_ext: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and cl_op :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.cl_op less_eq cl_op" shows "less_eq x (cl_op x)"
ax3: shows "\<exists>A B C D. distinct4 A B C D \<and> (\<forall>l. (incid A l \<and> incid B l \<longrightarrow> \<not> incid C l \<and> \<not> incid D l) \<and> (incid A l \<and> incid C l \<longrightarrow> \<not> incid B l \<and> \<not> incid D l) \<and> (incid A l \<and> incid D l \<longrightarrow> \<not> incid B l \<and> \<not> incid C l) \<and> (incid B l \<and> incid C l \<longrightarrow> \<not> incid A l \<and> \<not> incid D l) \<and> (incid B l \<and> incid D l \<longrightarrow> \<not> incid A l \<and> \<not> incid C l) \<and> (incid C l \<and> incid D l \<longrightarrow> \<not> incid A l \<and> \<not> incid B l))"
assms: shows "Lyndon_rec (a # w) \<^bold>|w\<^bold>|"
edge'_simps_4: fixes y :: "'v vertex ??.MFMC_Flow_Attainability.flow_attainability.node"   and x :: "'v vertex" shows "??.local.\<Psi>.edge' y (??.MFMC_Flow_Attainability.flow_attainability.node.Inner x) = (y = ??.MFMC_Flow_Attainability.flow_attainability.node.SOURCE \<and> x = source \<Psi> \<or> (\<exists>xa. y = ??.MFMC_Flow_Attainability.flow_attainability.node.Inner xa \<and> edge \<Psi> xa x))"
CConcEnd2: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and a :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and xs :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" assumes "x \<in> dom (Cp a)" shows "Cp (list2FWpolicy (xs @ [a])) x = Cp a x"
bind_assoc: fixes x :: "'m"   and f :: "'a \<Rightarrow> 'm"   and g :: "'a \<Rightarrow> 'm" shows "bind (bind x f) g = bind x (\<lambda>y. bind (f y) g)"
UP_zero_closed: fixes R :: "('c, 'd) ring_scheme" assumes "UP_ring R" shows "\<zero>\<^bsub>UP R\<^esub> \<in> carrier (UP R)"
conversep_mono: fixes r :: "'b \<Rightarrow> 'a \<Rightarrow> bool"   and s :: "'b \<Rightarrow> 'a \<Rightarrow> bool" shows "(r\<inverse>\<inverse> \<le> s\<inverse>\<inverse>) = (r \<le> s)"
sup_right_isotone: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "class.semilattice_sup sup less_eq less"   and "less_eq x y" shows "less_eq (sup z x) (sup z y)"
totalizer_init: fixes f :: "nat \<Rightarrow> nat option"   and d :: "nat"   and i :: "nat"   and n :: "nat" assumes "f \<in> \<R>" shows "totalizer d i (f \<triangleright> n) = (if the (delay i (f \<triangleright> n)) = 0 then Some d else \<phi> i (f \<triangleright> (the (delay i (f \<triangleright> n)) - 1)))"
rel_adv_eq_dis_log_adv: fixes \<A> :: "'a \<Rightarrow> nat spmf" shows "Schnorr_\<Sigma>_commit.rel_advantage \<A> = discrete_log.advantage \<A>"
language_state_for_input_take: fixes io :: "('e \<times> 'f) list"   and M :: "('e, 'f, 'g) FSM"   and q :: "'g"   and xs :: "'e list"   and n :: "nat" assumes "io \<in> language_state_for_input M q xs" shows "take n io \<in> language_state_for_input M q (take n xs)"
bit_xor_iff: fixes a :: "'a"   and b :: "'a"   and n :: "nat" shows "bit (a XOR b) n = (bit a n \<noteq> bit b n)"
infinite_Card_order_limit: fixes r :: "('a \<times> 'a) set"   and a :: "'a" assumes "Card_order r"   and "infinite (Field r)"   and "a \<in> Field r" shows "\<exists>b\<in>Field r. a \<noteq> b \<and> (a, b) \<in> r"
separation_closedin_Un_gen: fixes X :: "'b topology"   and S :: "'b set"   and T :: "'b set" shows "separatedin X S T = (S \<subseteq> topspace X \<and> T \<subseteq> topspace X \<and> disjnt S T \<and> closedin (subtopology X (S \<union> T)) S \<and> closedin (subtopology X (S \<union> T)) T)"
reduced_calculus_axioms_def: fixes Inf :: "'a inference set"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" shows "reduced_calculus_axioms Inf Red_I Red_F \<equiv> \<forall>N. inference_system.Inf_between Inf UNIV (Red_F N) \<subseteq> Red_I N"
tab_simps_5: shows "cod \<tau> = t \<star> t\<^sub>0"
Tw_mem: fixes E :: "('e, 'f) Order_scheme"   and x :: "'a" assumes "Worder E"   and "x \<in> carrier D"   and "\<forall>a\<in>carrier D. \<exists>b\<in>carrier E. ord_equiv (Iod D (segment D a)) (Iod E (segment E b))" shows "(Tw\<^bsub>D,E\<^esub>) x \<in> carrier E"
from_nat_to_nat_greatest: fixes k :: "nat"   and A :: "(('b, 'c) vec, 'a) vec" shows "mod_type_class.from_nat (Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A))) = (GREATEST n. \<not> is_zero_row_upt_k n k A) + (1::'a)"
OrdP_IN_SUCC_D_lemma: fixes x :: "tm"   and y :: "tm" shows "{OrdP x, SUCC y IN SUCC x} \<turnstile> y IN x"
finite_Psegments_included: fixes P :: "real \<Rightarrow> bool"   and a :: "real"   and d :: "real"   and b :: "real"   and c :: "real" assumes "finite_Psegments P a d"   and "a \<le> b"   and "c \<le> d" shows "finite_Psegments P b c"
finite_psupset_wf: fixes S :: "'b set" assumes "finite S" shows "wf (finite_psupset S)"
pac_ideal_alt_def: fixes A :: "int mpoly set" shows "pac_ideal A = More_Modules.ideal (A \<union> More_Modules.ideal polynomial_bool)"
weak_respection_of_barbs_and_closures_4: fixes Rel :: "('b \<times> 'b) set"   and CWB :: "('b, 'c) calculusWithBarbs" assumes "rel_weakly_respects_barbs Rel CWB" shows "rel_weakly_respects_barbs (symcl (Rel\<^sup>=)) CWB"
that: fixes q1 :: "enat"   and q2 :: "enat" assumes "q1 + q2 \<le> q"   and "\<And>pk. interaction_bounded_by consider (\<A>1 pk) q1"   and "\<And>cipher \<sigma>. interaction_bounded_by consider (\<A>2 cipher \<sigma>) q2" shows "thesis"
borel_measurable_enn2real: fixes f :: "'b \<Rightarrow> ennreal"   and M :: "'b measure" assumes "f \<in> borel_measurable M" shows "(\<lambda>x. enn2real (f x)) \<in> borel_measurable M"
intro: fixes bc :: "nat \<Rightarrow> nat"   and mining :: "nat \<Rightarrow> bool" assumes "\<And>n. bc (Suc n) \<le> Suc (bc n)"   and "\<And>n. bc n < bc (Suc n) \<Longrightarrow> mining (Suc n)" shows "dishonest_growth bc mining"
while_eq_nfoldli: fixes c :: "'a \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'a \<Rightarrow> 'a nres"   and l :: "'b list"   and \<sigma> :: "'a" shows "WHILE\<^sub>T (FOREACH_cond c) (\<lambda>x. ASSERT (FOREACH_cond c x) \<bind> (\<lambda>_. FOREACH_body f x)) (l, \<sigma>) \<bind> (\<lambda>(uu_, \<sigma>). RETURN \<sigma>) = nfoldli l c f \<sigma>"
enab_alpha1: shows "\<turnstile> $pc1 = #a \<longrightarrow> #0 < $sem \<longrightarrow> Enabled alpha1"
row_mat_of_row: fixes y :: "'b Matrix.vec" shows "Matrix.row (mat_of_row y) 0 = y"
bit_1_iff: fixes n :: "nat" shows "bit (1::'a) n = ((1::'a) \<noteq> (0::'a) \<and> n = 0)"
ctr: fixes \<omega> :: "(real \<times> 'd) stream"   and j :: "real"   and \<omega>' :: "(real \<times> 'd) stream" shows "merge_at \<omega> j \<omega>' = (case \<omega> of a ## \<omega> \<Rightarrow> if fst a \<le> j then (fst a, snd a) else shd \<omega>') ## (case \<omega> of a ## \<omega> \<Rightarrow> if fst a \<le> j then merge_at \<omega> j \<omega>' else stl \<omega>')"
intro: fixes Smith_1x2 :: "'b mat \<Rightarrow> 'b mat \<times> 'b mat"   and Smith_2x2 :: "'b mat \<Rightarrow> 'b mat \<times> 'b mat \<times> 'b mat"   and div_op :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<forall>A\<in>carrier_mat 1 2. is_SNF A (1\<^sub>m 1, Smith_1x2 A)"   and "\<forall>A\<in>carrier_mat 2 2. is_SNF A (Smith_2x2 A)"   and "is_div_op div_op" shows "Smith_Impl Smith_1x2 Smith_2x2 div_op"
assms: shows "fold (\<lambda>(x, y) option. Option.bind option (f x y)) (xs @ ys) start = Some res"
iT_Plus_Div_distrib_mod_less: fixes I :: "nat set"   and m :: "nat"   and n :: "nat" assumes "\<forall>x\<in>I. x mod m + n mod m < m" shows "I \<oplus> n \<oslash> m = I \<oslash> m \<oplus> n div m"
disjoint_up_mod1: fixes M :: "'b"   and N :: "'b" shows "upRules \<inter> p_e modRules2 M N = {}"
mat_of_row_index: fixes x :: "nat"   and y :: "'a Matrix.vec" assumes "x < dim_vec y" shows "mat_of_row y $$ (0, x) = y $v x"
set_3: fixes x3 :: "'b" shows "atoms_ltlc prop\<^sub>c(x3) = {x3}"
Collect_case_prod_in_rel_leI: fixes X :: "('d \<times> 'e) set"   and Y :: "('d \<times> 'e) set" assumes "X \<subseteq> Y" shows "X \<subseteq> {(x, y). in_rel Y x y}"
Ixy: shows "integrable (S \<Otimes>\<^sub>M T) (\<lambda>x. Pxy x * log b (Pxy x))"
right_idem: fixes a :: "'b"   and b :: "'b" shows "sup (sup a b) b = sup a b"
boundary_chain_def: fixes s :: "('a \<times> 'b) set" shows "boundary_chain s \<equiv> \<forall>(k, \<gamma>)\<in>s. k = (1::'a) \<or> k = - (1::'a)"
invar_empty: fixes empty :: "'b"   and enq :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and first :: "'b \<Rightarrow> 'c"   and deq :: "'b \<Rightarrow> 'b"   and is_empty :: "'b \<Rightarrow> bool"   and list :: "'b \<Rightarrow> 'c list"   and invar :: "'b \<Rightarrow> bool" assumes "Queue empty enq first deq is_empty list invar" shows "invar empty"
factorization_is_sound: fixes trm_ord :: "('b trm \<times> 'b trm) set"   and sel :: "'b literal set \<Rightarrow> 'b literal set"   and pos_ord :: "'b eclause \<Rightarrow> 'b trm \<Rightarrow> (indices list \<times> indices list) set"   and vars :: "'b set"   and filter_trms :: "'b literal set \<Rightarrow> 'b trm set \<Rightarrow> 'b trm set"   and C :: "'b eclause"   and D :: "'b eclause"   and \<sigma> :: "('b \<times> 'b trm) list"   and k :: "inferences"   and C' :: "'b literal set" assumes "basic_superposition trm_ord sel pos_ord vars filter_trms"   and "finite (cl_ecl C)"   and "basic_superposition.factorization trm_ord sel filter_trms C D \<sigma> k C'" shows "clause_entails_clause (cl_ecl C) (cl_ecl D)"
IH: shows "map (\<lambda>t. vars_term_ms (t \<cdot> \<sigma>)) ts = map (\<lambda>t. \<Sum>x\<in>#vars_term_ms t. vars_term_ms (\<sigma> x)) ts"
set_borel_measurable_def: fixes M :: "'b measure"   and A :: "'b set"   and f :: "'b \<Rightarrow> 'c" shows "set_borel_measurable M A f \<equiv> (\<lambda>x. indicat_real A x *\<^sub>R f x) \<in> borel_measurable M"
secret_object3_def: shows "\<beta> = \<iota> \<squnion> \<iota>"
prod_assn_def: fixes P1 :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and P2 :: "'c \<Rightarrow> 'd \<Rightarrow> assn"   and a :: "'a \<times> 'c"   and c :: "'b \<times> 'd" shows "(P1 \<times>\<^sub>a P2) a c \<equiv> case (a, c) of ((a1, a2), c1, c2) \<Rightarrow> P1 a1 c1 * P2 a2 c2"
less_le_not_le: fixes y :: "'b"   and x :: "'b" shows "(y < x) = (y \<le> x \<and> \<not> x \<le> y)"
wellformed_transaction_def: fixes T :: "('e, 'f, 'g, 'h) prot_transaction" shows "wellformed_transaction T \<equiv> list_all stateful_strand_step.is_Receive (unlabel (transaction_receive T)) \<and> list_all is_Assignment (unlabel (transaction_selects T)) \<and> list_all is_Check (unlabel (transaction_checks T)) \<and> list_all is_Update (unlabel (transaction_updates T)) \<and> list_all stateful_strand_step.is_Send (unlabel (transaction_send T)) \<and> set (transaction_fresh T) \<subseteq> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_updates T) \<union> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_send T) \<and> set (transaction_fresh T) \<inter> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_receive T) = {} \<and> set (transaction_fresh T) \<inter> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_selects T) = {} \<and> fv_transaction T \<inter> bvars_transaction T = {} \<and> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_checks T) \<subseteq> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_receive T) \<union> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_selects T) \<and> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_updates T) \<union> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_send T) - set (transaction_fresh T) \<subseteq> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_receive T) \<union> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_selects T) \<and> (\<forall>x\<in>set (unlabel (transaction_selects T)). stateful_strand_step.is_Equality x \<longrightarrow> fv (the_rhs x) \<subseteq> fv\<^sub>l\<^sub>s\<^sub>s\<^sub>t (transaction_receive T))"
tss_list'_def_1: shows "linearize tss = tss_list @ tss_list'"
intro: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" assumes "ast_problem.well_formed problem" shows "wf_ast_problem problem"
minus_one_power_iff: fixes n :: "nat" shows "(- (1::'a)) ^ n = (if even n then 1::'a else - (1::'a))"
g_group: fixes xs :: "'a list" shows "g (local.group xs) = g xs"
closed_empty: fixes "open" :: "'c set \<Rightarrow> bool" assumes "class.topological_space open" shows "topological_space.closed open {}"
interval_size_3: fixes x :: "real_alg_3"   and i :: "nat" shows "upper_3 x i - lower_3 x i = (upper_3 x 0 - lower_3 x 0) / 2 ^ i"
singleton_iff: fixes b :: "'a"   and a :: "'a" shows "(b \<in> {a}) = (b = a)"
sum_transp: fixes R1 :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'f \<Rightarrow> bool" assumes "transp R1"   and "transp R2" shows "transp (rel_sum R1 R2)"
map_Val_throw_eq: fixes vs :: "val list"   and ex :: "expr"   and es :: "expr list"   and ws :: "val list"   and ex' :: "expr"   and es' :: "expr list" assumes "map Val vs @ throw ex # es = map Val ws @ throw ex' # es'" shows "vs = ws \<and> ex = ex' \<and> es = es'"
gbinomial_partial_sum_poly_xpos: fixes m :: "nat"   and a :: "'a"   and x :: "'a"   and y :: "'a" shows "(\<Sum>k\<le>m. (of_nat m + a gchoose k) * x ^ k * y ^ (m - k)) = (\<Sum>k\<le>m. (of_nat k + a - (1::'a) gchoose k) * x ^ k * (x + y) ^ (m - k))"
simps_22: fixes x5 :: "('b, 'c) fm"   and x2 :: "'c" shows "(\<^bold>\<diamond> x5) \<noteq> Nom x2"
finite_gpvI: fixes gpv :: "('f, 'd, 'e) gpv"   and \<I> :: "('d, 'e) \<I>" assumes "\<And>out c input. \<lbrakk>IO out c \<in> set_spmf (the_gpv gpv); input \<in> responses_\<I> \<I> out\<rbrakk> \<Longrightarrow> finite_gpv \<I> (c input)" shows "finite_gpv \<I> gpv"
triv_phis'_def: fixes g :: "'g"   and "next" :: "'node \<times> 'val"   and triv_phis :: "('node \<times> 'val) set"   and nodes_of_phis :: "('val, ('node \<times> 'val) set) mapping" shows "triv_phis' g next triv_phis nodes_of_phis = Set.remove next triv_phis \<union> Set.filter (\<lambda>n. uninst_code.ssa.trivial_code (snd n) (the (Mapping.lookup (phis g) n))) (case Mapping.lookup nodes_of_phis (snd next) of None \<Rightarrow> {} | Some x \<Rightarrow> Set.remove next x)"
simps_3: fixes a :: "nat"   and b :: "nat"   and At :: "atom" shows "swap_fm a b (fm.Atom At) = fm.Atom (swap_atom a b At)"
pmpt_def: fixes M :: "'c measure"   and T :: "'c \<Rightarrow> 'c" shows "pmpt M T \<equiv> fmpt M T \<and> prob_space M"
prems_2: shows "dom F3 = dom (F(x \<mapsto> y))"
mat_of_cols_Cons_index_Suc: fixes i :: "nat"   and n :: "nat"   and w :: "'b vec"   and ws :: "'b vec list"   and j :: "nat" assumes "i < n" shows "mat_of_cols n (w # ws) $$ (i, Suc j) = mat_of_cols n ws $$ (i, j)"
K: shows "K \<in> M \<rightarrow>\<^sub>M prob_algebra M"
inv_equality: fixes y :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set"   and x :: "(((nat \<Rightarrow> real) \<Rightarrow> 'a) \<Rightarrow>\<^sub>0 int) set" assumes "y \<otimes>\<^bsub>homology_group p X\<^esub> x = \<one>\<^bsub>homology_group p X\<^esub>"   and "x \<in> carrier (homology_group p X)"   and "y \<in> carrier (homology_group p X)" shows "inv\<^bsub>homology_group p X\<^esub> x = y"
zero_im_sheaf_def: shows "zero_im_sheaf \<equiv> \<lambda>V. zero_sheaf_spec (identity Spec \<^sup>\<inverse> Spec V)"
insertAt_def: fixes n :: "nat"   and x :: "'b"   and ls :: "'b list" shows "insertAt n x ls = take n ls @ x # drop n ls"
add_minus_2: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and s :: "'c"   and t :: "'c" assumes "class.comm_powerprod plus minus zero" shows "minus (plus s t) s = t"
update_matchF_step_def: fixes I :: "\<I>"   and mr :: "mregex"   and mrs :: "mregex list"   and nt :: "nat" shows "update_matchF_step I mr mrs nt = (\<lambda>(t, rels', rel) (aux', X). let Y = mrtabulate mrs (l\<delta> id X rels') in ((t, rels', if mem (nt - t) I then rel \<union> lookup Y mr else rel) # aux', Y))"
fv_list\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_append: fixes F :: "(('d, 'c) Term.term \<times> ('e, 'c) Term.term) list"   and G :: "(('d, 'c) Term.term \<times> ('e, 'c) Term.term) list" shows "fv_list\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F @ G) = fv_list\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F @ fv_list\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s G"
soundnessForUNSAT: fixes state0 :: "State"   and F0 :: "Literal list list"   and state :: "State"   and decisionVars :: "nat set" assumes "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation F0 decisionVars"   and "formulaFalse F0 (elements (getM state))"   and "decisions (getM state) = []" shows "\<not> satisfiable F0"
pfun_upd_ext: fixes x :: "'a"   and f :: "('a, 'b) pfun" assumes "x \<in> pdom f" shows "f(x \<mapsto> f(x)\<^sub>p)\<^sub>p = f"
union_assoc: fixes M :: "'f multiset"   and N :: "'f multiset"   and K :: "'f multiset" shows "M + N + K = M + (N + K)"
any_sdominate_0: fixes G :: "nat graph_rec"   and n :: "nat" assumes "cfg_doms G"   and "n \<in> set (g_V G)" shows "cfg_doms.non_strict_dominate G n 0"
f1'_le_f2': shows "f1' \<le> f2'"
carrier_lucas_lehmer_ring: shows "carrier lucas_lehmer_ring = UNIV"
oconf_new: fixes 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 h :: "nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option"   and obj :: "char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set"   and a :: "nat"   and x :: "char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set" assumes "P,h \<turnstile> obj \<surd>"   and "h a = None" shows "P,h(a \<mapsto> x) \<turnstile> obj \<surd>"
min: shows "\<forall>g. (m, g) \<in> gseq \<longrightarrow> good (list_emb P) g"
not_mem_eq_mem_not: fixes a :: "'c"   and A :: "'c set" shows "(a \<notin> A) = (a \<in> - A)"