hcju/isabelleps huggingface dataset:
lemma_1_i: fixes s :: "('a, 'b) fmap"   and PROB :: "(('a, 'b) fmap \<times> ('a, 'b) fmap) set"   and a :: "('a, 'b) fmap \<times> ('a, 'b) fmap" assumes "s \<in> valid_states PROB"   and "a \<in> PROB" shows "state_succ s a \<in> valid_states PROB"
even_plus_one_iff: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.semiring_parity plus minus zero times one divide modulo" shows "dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) (plus a one) = (\<not> dvd.dvd times (numeral.numeral one plus (num.Bit0 num.One)) a)"
hmset_pos_neg_dual: fixes a :: "zhmultiset"   and b :: "zhmultiset" shows "hmset_pos a + hmset_pos b + (hmset_neg a - hmset_pos b) + (hmset_neg b - hmset_pos a) = hmset_neg a + hmset_neg b + (hmset_pos a - hmset_neg b) + (hmset_pos b - hmset_neg a)"
SumRes: fixes x :: "name"   and P :: "pi"   and Q :: "pi" shows "<\<nu>x>(P \<oplus> Q) \<equiv>\<^sub>e <\<nu>x>P \<oplus> <\<nu>x>Q"
h2: shows "h \<turnstile> set_shadow_root ptr None \<rightarrow>\<^sub>h h2"
icard_infinite: fixes A :: "'a set" assumes "infinite A" shows "icard A = \<infinity>"
ccNeighbors_ccProd: fixes x :: "Vars.var"   and S :: "Vars.var set"   and S' :: "Vars.var set" shows "ccNeighbors x (S G\<times> S') = (if x \<in> S then S' else {}) \<union> (if x \<in> S' then S else {})"
infinite_chain_arbitrary_esize: fixes C :: "'a set"   and n :: "nat" assumes "Complete_Partial_Order.chain (\<le>) C"   and "infinite C" obtains x :: "'a" where "x \<in> C" and "enat n \<le> esize x"
Lbot: shows "([],  \<LM> ff  \<RM> \<Rightarrow>* Em) \<in> Ax"
locate_unique: fixes al :: "'a list"   and f :: "'a \<Rightarrow> nat"   and n :: "nat"   and j :: "nat"   and i :: "nat" assumes "\<And>a. a \<in> set al \<Longrightarrow> 0 < f a"   and "n < length al"   and "j < f (al ! n)"   and "i = lsum f (take n al) + j" shows "n = locate1 f al i \<and> j = locate2 f al i"
Empty: fixes sp :: "'b set"   and A :: "'b set set" shows "{} \<in> sigma_sets sp A"
card_of_def: fixes A :: "'a set" shows "|A| = (SOME r. card_order_on A r)"
simps_2: fixes t :: "nat"   and volatile :: "bool"   and a :: "nat" shows "valid_sops_expr t (Mem volatile a) = True"
red_rtrancl_ord: fixes F :: "('a \<Rightarrow>\<^sub>0 'c) set"   and p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" assumes "(punit.red F)\<^sup>*\<^sup>* p q" shows "punit.ord_p q p"
aff_dim_simplex: fixes n :: "int"   and S :: "'b set" assumes "n simplex S" shows "aff_dim S = n"
IH_2: fixes l :: "float poly"   and r :: "float poly"   and ord :: "nat" assumes "(l, r) = split_by_degree ord p" shows "(maxdegree l \<le> ord \<and> Ipoly x (map_poly real_of_float p) = Ipoly x (map_poly real_of_float l) + Ipoly x (map_poly real_of_float r)) \<and> num_params l \<le> num_params p \<and> num_params r \<le> num_params p"
is_category_with_pullbacks: shows "category_with_pullbacks (\<odot>)"
inext_nth_eq_Min_conv: fixes I :: "nat set"   and n :: "nat" assumes "I \<noteq> {}" shows "(I \<rightarrow> n = iMin I) = (n = 0 \<or> (\<exists>a. I = {a}))"
face_of_disjoint_rel_interior: fixes T :: "'b set"   and S :: "'b set" assumes "T face_of S"   and "T \<noteq> S" shows "T \<inter> rel_interior S = {}"
trans_opar_comp: fixes s :: "((nat \<Rightarrow> 'd) \<times> 'e, 'g seq_action) automaton"   and i :: "nat"   and t :: "('f, 'g seq_action) automaton" shows "automaton.trans (s \<langle>\<langle>\<^bsub>i\<^esub> t) = oparp_sos i (automaton.trans s) (automaton.trans t)"
poly_add_length_eq: fixes K :: "'a list set"   and p1 :: "'a list list"   and p2 :: "'a list list" assumes "subring K (K [X])"   and "polynomial\<^bsub>K [X]\<^esub> K p1"   and "polynomial\<^bsub>K [X]\<^esub> K p2"   and "length p1 \<noteq> length p2" shows "length (UP.poly_add p1 p2) = max (length p1) (length p2)"
not_le_imp_less: fixes y :: "'a word"   and x :: "'a word" assumes "\<not> y \<le>s x" shows "x <s y"
FOREACHc_rule': fixes S :: "'a set"   and I :: "'a set \<Rightarrow> 'b \<Rightarrow> bool"   and \<sigma>0 :: "'b"   and c :: "'b \<Rightarrow> bool"   and f :: "'a \<Rightarrow> 'b \<Rightarrow> 'b nres"   and P :: "'b \<Rightarrow> bool" assumes "finite S"   and "I S \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>x \<in> it; it \<subseteq> S; I it \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> SPEC (I (it - {x}))"   and "\<And>\<sigma>. \<lbrakk>I {} \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>"   and "\<And>it \<sigma>. \<lbrakk>it \<subseteq> S; I it \<sigma>; \<not> c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>" shows "FOREACH\<^sub>C S c f \<sigma>0 \<le> SPEC P"
sup_bot_right: fixes x :: "'e multiset" shows "x \<union># {#} = x"
simps_2: fixes satisfies :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and transition :: "'a \<Rightarrow> ('c, 'a) residual \<Rightarrow> bool"   and effect_apply :: "'d \<Rightarrow> 'a \<Rightarrow> 'a"   and L :: "'c \<times> 'd fs_set \<times> 'd \<Rightarrow> 'd fs_set"   and F :: "'d fs_set"   and P :: "'a"   and \<alpha>P' :: "(('c, 'd) L_action, ('a, 'd) L_state) residual" assumes "effect_nominal_ts satisfies transition effect_apply L" shows "effect_nominal_ts.L_transition transition effect_apply L (EF (F, P)) \<alpha>P' = (\<exists>f. f \<in>\<^sub>f\<^sub>s F \<and> \<alpha>P' = \<langle>Eff f,AC (f, F, effect_apply f P)\<rangle>)"
trans_refl_imp_rtrancl_id: fixes r :: "('a \<times> 'a) set" assumes "trans r"   and "refl r" shows "r\<^sup>* = r"
pred_spmf_spmf_of_pmf: fixes P :: "'c \<Rightarrow> bool"   and p :: "'c pmf" shows "pred_spmf P (spmf_of_pmf p) = pred_pmf P p"
tm_norm_poly_range: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and t :: "taylor_model" assumes "x \<in>\<^sub>i range_tm e t" shows "x \<in>\<^sub>i range_tm e (tm_norm_poly t)"
mem_WWa_Chain: fixes W :: "'a set"   and g :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "W \<in> WWa D g a" shows "Chain D W"
arr_char: fixes f :: "('o, 'a) arr" shows "arr f = (f \<noteq> Null \<and> Dom f \<in> Obj \<and> Cod f \<in> Obj \<and> Map f \<in> Hom (Dom f) (Cod f))"
lenG: shows "length (map gs1.gso [0..<m]) = m"
4: shows "degree (p1 - p2) \<le> length zs_ws - 1"
wf: shows "web_flow (bipartite_web_of \<Gamma>) f"
comm_semiring_0_ow_def: fixes U :: "'b set"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" shows "comm_semiring_0_ow U plus zero times \<equiv> comm_semiring_ow U plus times \<and> comm_monoid_add_ow U plus zero \<and> mult_zero_ow U times zero"
always_imp_refl: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and \<sigma> :: "nat \<Rightarrow> 'b" shows "(\<box>(\<lambda>s. P s \<longrightarrow> P s)) \<sigma>"
circ_one: shows "(1::'a)\<^sup>\<circ> = (1::'a)\<^sup>\<circ>\<^sup>\<circ>"
min_product_comp: fixes a :: "'b fset"   and A :: "'b fset set"   and b :: "'b fset"   and B :: "'b fset set" assumes "a \<in> A"   and "b \<in> B" shows "\<exists>c. c |\<subseteq>| a |\<union>| b \<and> c \<in> A \<otimes>\<^sub>m B"
restriction_un: fixes v :: "view"   and u :: "view"   and w :: "view"   and f :: "cars \<Rightarrow> nat_int"   and c :: "cars" shows "v=u--w \<longrightarrow> restrict v f c = restrict u f c \<squnion> restrict w f c"
start_conc: fixes L :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and R :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)" shows "start (RegExp2NA.conc L R) = True # start L"
h': shows "h3 \<turnstile> set_shadow_root element_ptr (Some new_shadow_root_ptr) \<rightarrow>\<^sub>h h'"
Im_sin: fixes z :: "complex" shows "Im (sin z) = cos (Re z) * (exp (Im z) - exp (- Im z)) / 2"
second_vertices_new_path: fixes xs :: "'a list" assumes "hd (tl xs) \<notin> second_vertices" shows "xs \<notin> paths"
PSimQ: shows "P \<leadsto>\<guillemotleft>Rel\<guillemotright> Q"
subdegree_fps_const: fixes c :: "'a" shows "subdegree (fps_const c) = 0"
primitive_def: fixes f :: "'a poly" shows "primitive f = (\<forall>x. (\<forall>y\<in>set (coeffs f). x dvd y) \<longrightarrow> x dvd (1::'a))"
weak_partial_order_bottom_axioms: shows "weak_partial_order_bottom (fpl \<P> \<H>)"
exhaust: fixes y :: "'e prot_atom" obtains x1 :: "'e" where "y = prot_atom.Atom x1"   | "y = Value"   | "y = SetType"   | "y = AttackType"   | "y = Bottom"   | "y = OccursSecType"
union_inter: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and z :: "'b"   and A :: "'c set"   and B :: "'c set"   and g :: "'c \<Rightarrow> 'b" assumes "comm_monoid_set f z"   and "finite A"   and "finite B" shows "f (comm_monoid_set.F f z g (A \<union> B)) (comm_monoid_set.F f z g (A \<inter> B)) = f (comm_monoid_set.F f z g A) (comm_monoid_set.F f z g B)"
ltlc_to_ltln'_correct_2: fixes w :: "nat \<Rightarrow> 'a set"   and \<phi> :: "'a ltlc" shows "w \<Turnstile>\<^sub>n ltlc_to_ltln' False \<phi> = w \<Turnstile>\<^sub>c \<phi>"
inj_map_def: fixes \<pi> :: "'d \<Rightarrow> 'e option" shows "inj_map \<pi> = (\<forall>x y. \<pi> x = \<pi> y \<and> \<pi> x \<noteq> None \<longrightarrow> x = y)"
to_fract_eq_iff: fixes x :: "'a"   and y :: "'a" shows "(to_fract x = to_fract y) = (x = y)"
coeff_0_power: fixes p :: "'a poly"   and n :: "nat" shows "coeff (p ^ n) 0 = coeff p 0 ^ n"
C_eq_RS3_aux: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "not_MT Cp p" shows "Cp (list2FWpolicy p) x = Cp (list2FWpolicy (rm_MT_rules Cp p)) x"
prefixToLevel_def: fixes l :: "nat"   and t :: "('b \<times> bool) list" shows "prefixToLevel l t \<equiv> prefixToLevel_aux t l 0"
mod_mult_cong: 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 a :: "'a"   and c :: "'a"   and a' :: "'a"   and b :: "'a"   and b' :: "'a" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size"   and "modulo a c = modulo a' c"   and "modulo b c = modulo b' c" shows "modulo (times a b) c = modulo (times a' b') c"
kSup_distl: fixes f :: "'a \<Rightarrow> 'c set"   and G :: "('c \<Rightarrow> 'b set) set" shows "f \<circ>\<^sub>K \<Squnion> G = \<Squnion> \<P> ((\<circ>\<^sub>K) f) G"
dg_in_ArrE: fixes \<alpha> :: "V"   and \<CC> :: "V"   and f :: "V" assumes "digraph \<alpha> \<CC>"   and "f \<in>\<^sub>\<circ> \<CC>\<lparr>Arr\<rparr>" obtains a :: "V" and b :: "V" where "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" and "a \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>" and "b \<in>\<^sub>\<circ> \<CC>\<lparr>Obj\<rparr>"
intro: fixes get_host :: "(_) shadow_root_ptr \<Rightarrow> ((_) heap, exception, (_) element_ptr) prog"   and get_host_locs :: "((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "\<And>element_ptr h. pure (get_host element_ptr) h"   and "\<And>node_ptr h h'. reads get_host_locs (get_host node_ptr) h h'" shows "l_get_host get_host get_host_locs"
gmft_correct: fixes t :: "('c, 'd) FingerTreeStruc" assumes "is_measured_ftree t" shows "gmft t = sum_list (map snd (toList t))"
eq: shows "H <#>\<^bsub>B\<^esub> K = carrier B"
p'_def: shows "p' = (\<Sum>i<n. monom (1::'a) (n - Suc i) * c i)"
G_Inf_reductive: fixes \<iota> :: "'a literal multiset inference"   and M :: "'a literal multiset set" assumes "\<iota> \<in> G_Inf M" shows "concl_of \<iota> < main_prem_of \<iota>"
unset_bit_less_eq: fixes n :: "nat"   and k :: "int" shows "unset_bit n k \<le> k"
find_nu_def: shows "find_nu \<equiv> (\<lambda>s. s\<lparr>nu := 1, tortoise := f x0, hare := f (f x0)\<rparr>) ;; while (\<lambda>s. hare s \<noteq> tortoise s) (\<lambda>s. s\<lparr>nu := nu s + 1, tortoise := f (tortoise s), hare := f (f (hare s))\<rparr>)"
formulaFalseIffContainsFalseClause: fixes formula :: "Literal list list"   and valuation :: "Literal list" shows "formulaFalse formula valuation = (\<exists>clause. clause el formula \<and> clauseFalse clause valuation)"
IH_1: shows "\<not> t1 \<prec> t1"
dfs_while_algo: fixes R :: "('d \<times> 'd) set"   and \<Sigma>i :: "'d set" assumes "finite (R\<^sup>* `` \<Sigma>i)" shows "while_algo (dfs_algo \<Sigma>i R)"
cases: fixes a :: "'d set" assumes "finite a" obtains "aa__ = {}"   | A :: "'d set" and a :: "'d" where "aa__ = insert a A" and "finite A"
set_2: fixes x21 :: "'c"   and x22 :: "'c list" shows "set (x21 # x22) = Set.insert x21 (set x22)"
simps_3: fixes x21 :: "'d"   and x22 :: "('d, 'e) Term.term list"   and x23 :: "('d, 'e) ctxt"   and x24 :: "('d, 'e) Term.term list" shows "More x21 x22 x23 x24 \<noteq> \<box>"
mult_cancel_left1: fixes one :: "'d"   and times :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and plus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and zero :: "'d"   and minus :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and uminus :: "'d \<Rightarrow> 'd"   and c :: "'d"   and b :: "'d" assumes "class.ring_1_no_zero_divisors one times plus zero minus uminus" shows "(c = times c b) = (c = zero \<or> b = one)"
subpath_def: fixes g :: "('d, 'e) rgraph_scheme"   and v1 :: "'d"   and es :: "'d edge list"   and v2 :: "'d" shows "Graph.subpath g v1 es v2 \<equiv> Graph.ces v1 es v2 \<and> v1 \<in> Graph.vertices g \<and> set es \<subseteq> edges g"
cod_char: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b local.term set" assumes "free_monoidal_category C" shows "partial_magma.cod (free_monoidal_category.comp C) f = (if partial_magma.arr (free_monoidal_category.comp C) f then free_monoidal_category.mkarr C (monoidal_language.Cod C (free_monoidal_category.rep C f)) else partial_magma.null (free_monoidal_category.comp C))"
a: shows "length (int_to_bv i) - 1 \<le> k - 1"
pointwise_Append: fixes T :: "(symbol \<times> character list) set"   and k :: "nat" shows "pointwise (Append T k)"
inject_1: fixes x1 :: "lname"   and y1 :: "lname" shows "(LVar x1 = LVar y1) = (x1 = y1)"
ll_def: shows "local.ll = (let n = sqrt_int_ceiling (\<parallel>f\<parallel>\<^sup>2 ^ (2 * j') * 2 ^ (5 * j' * j')); ll' = find_exponent p n in if ll' < l then ll' else l)"
that: fixes x :: "nat" assumes "0 < x"   and "x \<le> (p - 1) div 2"   and "[x\<^sup>2 + 1 = 0] (mod p)" shows "thesis"
connected_by_dist: fixes v :: "nat"   and v' :: "nat" shows "connected v v' = (\<exists>d. dist v d v')"
CompDiag_Cod_Diag: fixes t :: "'c local.term" assumes "Diag t" shows "Cod t \<^bold>\<lfloor>\<^bold>\<cdot>\<^bold>\<rfloor> t = t"
induct: fixes x1 :: "'node SDG_node"   and x2 :: "'node SDG_node list"   and x3 :: "'node SDG_node"   and P :: "'node SDG_node \<Rightarrow> 'node SDG_node list \<Rightarrow> 'node SDG_node \<Rightarrow> bool" assumes "x1 irs-x2\<rightarrow>\<^sub>d* x3"   and "\<And>n. valid_SDG_node n \<Longrightarrow> P n [] n"   and "\<And>n'' ns n' n. \<lbrakk>n'' irs-ns\<rightarrow>\<^sub>d* n'; P n'' ns n'; n s\<longrightarrow>\<^bsub>cd\<^esub> n''\<rbrakk> \<Longrightarrow> P n (n # ns) n'"   and "\<And>n'' ns n' n V. \<lbrakk>n'' irs-ns\<rightarrow>\<^sub>d* n'; P n'' ns n'; n s-V\<rightarrow>\<^sub>d\<^sub>d n''; n \<noteq> n''\<rbrakk> \<Longrightarrow> P n (n # ns) n'"   and "\<And>n'' ns n' n p. \<lbrakk>n'' irs-ns\<rightarrow>\<^sub>d* n'; P n'' ns n'; n s-p\<rightarrow>\<^bsub>sum\<^esub> n''\<rbrakk> \<Longrightarrow> P n (n # ns) n'"   and "\<And>n'' ns n' n p. \<lbrakk>n'' irs-ns\<rightarrow>\<^sub>d* n'; P n'' ns n'; n s-p\<rightarrow>\<^bsub>ret\<^esub> n''\<rbrakk> \<Longrightarrow> P n (n # ns) n'"   and "\<And>n'' ns n' n p V. \<lbrakk>n'' irs-ns\<rightarrow>\<^sub>d* n'; P n'' ns n'; n s-p:V\<rightarrow>\<^bsub>out\<^esub> n''\<rbrakk> \<Longrightarrow> P n (n # ns) n'" shows "P x1 x2 x3"
Atm_transC_invert: fixes tval :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and aval :: "'c \<Rightarrow> 'b \<Rightarrow> 'b"   and atm :: "'c"   and s :: "'b"   and cf :: "('a, 'c) com \<times> 'b"   and phi :: "bool" assumes "PL.transC tval aval (Atm atm, s) cf" shows "phi"
simps_7: fixes uw :: "'d" shows "inoclosed uw \<tau>\<^sub>n = True"
largest: shows "- largest TYPE(('e, 'f) IEEE.float) \<le> valof a"
basic_prop_2: fixes v :: "i" shows "[Contingent q\<^sub>0 in v]"
blinding_of_fun_hash: fixes bo :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and rh :: "'c \<Rightarrow> 'd" assumes "bo \<le> vimage2p rh rh (=)" shows "rel_fun (=) bo \<le> vimage2p ((\<circ>) rh) ((\<circ>) rh) (=)"
sup_inf_absorb: fixes inf :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and sup :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'e"   and y :: "'e" assumes "class.lattice inf less_eq less sup" shows "sup x (inf x y) = x"
orth_at__ncop1: fixes U :: "'p"   and X :: "'p"   and A :: "'p"   and B :: "'p"   and C :: "'p"   and V :: "'p" assumes "U \<noteq> X"   and "X OrthAt A B C U V" shows "\<not> Coplanar A B C U"
lesser_than_two_values: fixes a :: "'b"   and b :: "'b"   and z :: "'b flat_complete_lattice" assumes "a \<noteq> b"   and "z \<le> flat_complete_lattice.Value a"   and "z \<le> flat_complete_lattice.Value b" shows "z = flat_complete_lattice.Bot"
hyps_3: assumes "\<forall>a\<in>A'. finite (B a)" shows "finite {f. \<forall>a. if a \<in> A' then f a \<in> B a else f a = f0 a}"
wp_seq: fixes x :: "'a"   and y :: "'a"   and q :: "'a" shows "wp (x \<cdot> y) q = wp x (wp y q)"
ivl_disj_un_two_touch_1: fixes l :: "'b"   and m :: "'b"   and u :: "'b" assumes "l < m"   and "m < u" shows "{l<..m} \<union> {m..<u} = {l<..<u}"
intros: fixes ts :: "'h \<Rightarrow> ('f \<times> 'g \<Rightarrow>f nat) option"   and t :: "'h" assumes "ts t = None" shows "free_thread_id ts t"
compact_eq_totally_bounded: fixes s :: "'a set" shows "compact s = (complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>x\<in>k. ball x e)))"
invar_rderiv_and_add_def: fixes \<Sigma> :: "nat \<Rightarrow> 'c set"   and wf_atom :: "nat \<Rightarrow> 'd \<Rightarrow> bool"   and project :: "'c \<Rightarrow> 'c"   and lookup :: "'d \<Rightarrow> 'c \<Rightarrow> bool"   and embed :: "'c \<Rightarrow> 'c list"   and singleton :: "'c \<Rightarrow> 'd"   and as :: "'c"   and r :: "'d rexp"   and brs :: "bool \<times> 'd rexp list" assumes "embed2 \<Sigma> wf_atom project lookup embed singleton" shows "embed2.invar_rderiv_and_add lookup embed as r brs \<equiv> (if fst brs then True else \<guillemotleft>embed.rderiv lookup embed as (hd (snd brs))\<guillemotright> \<in> set (snd brs)) \<and> snd brs \<noteq> [] \<and> distinct (snd brs) \<and> (\<forall>i<length (snd brs). snd brs ! i = \<guillemotleft>embed.rderivs lookup embed (replicate (length (snd brs) - 1 - i) as) r\<guillemotright>)"
eq_len_iff: fixes x :: "'b list"   and y :: "'b list"   and u :: "'b list"   and v :: "'b list" assumes "x \<cdot> y = u \<cdot> v" shows "(\<^bold>|x\<^bold>| \<le> \<^bold>|u\<^bold>|) = (\<^bold>|v\<^bold>| \<le> \<^bold>|y\<^bold>|)"
case_distrib: fixes h :: "'i \<Rightarrow> 'h"   and f1 :: "'i"   and f2 :: "'j \<Rightarrow> 'i"   and option :: "'j option" shows "h (case option of None \<Rightarrow> f1 | Some x \<Rightarrow> f2 x) = (case option of None \<Rightarrow> h f1 | Some x \<Rightarrow> h (f2 x))"
induct: fixes P :: "('f, 'h, 'fun, 'op, 'opinl) Inca.instr \<Rightarrow> 'g option list \<Rightarrow> bool"   and a0 :: "('f, 'h, 'fun, 'op, 'opinl) Inca.instr"   and a1 :: "'g option list" assumes "\<And>x \<Sigma>. P (Inca.instr.IPush x) \<Sigma>"   and "\<And>uu_ \<Sigma>. P Inca.instr.IPop (uu_ # \<Sigma>)"   and "\<And>x \<Sigma>. P (Inca.instr.ILoad x) (None # \<Sigma>)"   and "\<And>x \<Sigma>. P (Inca.instr.IStore x) (None # None # \<Sigma>)"   and "\<And>op \<Sigma>. P (Inca.instr.IOp op) \<Sigma>"   and "\<And>opinl \<Sigma>. P (Inca.instr.IOpInl opinl) \<Sigma>"   and "\<And>f \<Sigma>. P (Inca.instr.ICall f) \<Sigma>"   and "P Inca.instr.IPop []"   and "\<And>v. P (Inca.instr.ILoad v) []"   and "\<And>v vc vb. P (Inca.instr.ILoad v) (Some vc # vb)"   and "\<And>v. P (Inca.instr.IStore v) []"   and "\<And>v vc vb. P (Inca.instr.IStore v) (Some vc # vb)"   and "\<And>v va. P (Inca.instr.IStore v) [va]"   and "\<And>v va ve vd. P (Inca.instr.IStore v) (va # Some ve # vd)"   and "\<And>v uw_. P (Inca.instr.ICJump v) uw_" shows "P a0 a1"
in_mult: shows "(mset xs, mset ys) \<in> mult {(x, y). x \<in> A \<and> y \<in> A \<and> gt y x}"
k_not_0: fixes k :: "nat"   and as :: "real list"   and bs :: "real list" assumes "akra_bazzi_params k as bs" shows "k \<noteq> 0"
iT_Minus_empty_iff: fixes k :: "nat"   and I :: "nat set" shows "(k \<ominus> I = {}) = (I = {} \<or> k < iMin I)"
vector2_append1_non_zero: fixes v :: "(real, 2) vec" shows "vector2_append1 v \<noteq> 0"
eq_inc_simp: fixes n :: "Arity"   and v1 :: "Value"   and v2 :: "Value" shows "eq (inc\<cdot>n) v1 v2 = (\<forall>v. eq n (v1 \<down>Fn v) (v2 \<down>Fn v))"
zcount_inI: fixes M :: "'b zmultiset"   and x :: "'b" assumes "zcount M x = 0 \<Longrightarrow> False" shows "x \<in>#\<^sub>z M"
MinkowskiUnreachable_axioms: shows "MinkowskiUnreachable \<E> \<P> betw"
size_multiset_overloaded_def: shows "size = size_multiset (\<lambda>_. 0)"
q_min: fixes a :: "real"   and v :: "real"   and t :: "real"   and s0 :: "real" assumes "braking_movement a v"   and "0 \<le> t" shows "s0 \<le> movement.q v s0 t"
prj_tuple_1: fixes f :: "'a"   and g :: "'a"   and h :: "'a"   and k :: "'a" assumes "C.commutative_square f g h k" shows "\<p>\<^sub>0[f, g] \<cdot> \<langle>h \<lbrakk>f, g\<rbrakk> k\<rangle> = k"
node_deliver_messages_distinct: fixes xs :: "'msg event list"   and i :: "nat" assumes "xs prefix of i" shows "distinct (node_deliver_messages xs)"
norm_def: fixes V :: "'b set"   and norm :: "'b \<Rightarrow> real" shows "Normed_Space.norm V norm \<equiv> seminorm V norm \<and> norm_axioms V norm"
spec_ops_distinct: fixes ops :: "('a \<times> 'b) list"   and deps :: "'b \<Rightarrow> 'a set" assumes "spec_ops ops deps" shows "distinct ops"
card_2_iff: fixes S :: "'b set" shows "(card S = 2) = (\<exists>x y. S = {x, y} \<and> x \<noteq> y)"
AC_idem: fixes a :: "bool \<Rightarrow> ('c \<Rightarrow> real) \<Rightarrow> 'c \<Rightarrow> real" shows "a \<Squnion> a = a"
prems: shows "wf\<^sub>s\<^sub>t V (x # S)"
FOREACH_inv_def: fixes xs :: "'c list"   and \<Phi> :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and s :: "'c list \<times> 'd" shows "FOREACH_inv xs \<Phi> s \<equiv> case s of (it, \<sigma>) \<Rightarrow> \<exists>xs'. xs = xs' @ it \<and> \<Phi> (set it) \<sigma>"
wfG_empty: shows "wf_graph \<lparr>nodes = nodes G, edges = {}\<rparr>"
MClkbusy: fixes rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and p :: "PrIds"   and send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState" shows "\<turnstile> $Calling rcv p \<longrightarrow> \<not> MClkNext send rcv cst p"
iprefix_eq_iprefix_take_ex: fixes xs :: "'d list"   and f :: "nat \<Rightarrow> 'd" shows "(xs \<sqsubseteq> f) = (\<exists>n. f \<Down> n = xs)"
exp_rewrite: fixes w1 :: "nat"   and w2 :: "nat" shows "\<^bold>g [^] w1 \<otimes> g' [^] w2 = \<^bold>g [^] (w1 + x * w2)"
disc_127: fixes x81 :: "formula"   and x82 :: "formula" shows "is_And (And x81 x82)"
p_def: fixes t :: "real" shows "delayed_safe_distance.other.p t = other.s \<delta> + other.s' \<delta> * t + 1 / 2 * a\<^sub>o * t\<^sup>2"
gle: shows "\<guillemotleft> G \<guillemotright> && (wp a Q \<circ> \<phi>) \<tturnstile> wp b (Q \<circ> \<phi>)"
dimacs_model_def: fixes ls :: "int list"   and cs :: "int list list" shows "dimacs_model ls cs \<equiv> (\<forall>c\<in>set cs. \<exists>l\<in>set ls. l \<in> set c) \<and> distinct (map dimacs_lit_to_var ls)"
simps_17: fixes uy :: "'a"   and v :: "'a rexp"   and va :: "'a rexp" shows "local.le_rexp (Atom uy) (Plus v va) = True"