hcju/isabelleps huggingface dataset:
mult_pos_pos: fixes plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and a :: "'e"   and b :: "'e" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero a"   and "less zero b" shows "less zero (times a b)"
assms_5: fixes K1 :: "'a set"   and K2 :: "'a set" assumes "K1 \<in> s"   and "K2 \<in> s"   and "K1 \<noteq> K2" shows "interior K1 \<inter> interior K2 = {}"
prj_simps_3: shows "C.cod \<nu>\<pi>.prj\<^sub>1 = \<nu>.apex"
star_right_induct_mult_iff: fixes star :: "'c \<Rightarrow> 'c"   and one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and bot :: "'c"   and y :: "'c"   and x :: "'c" assumes "class.strong_left_kleene_algebra star one times sup less_eq less bot" shows "less_eq (times y x) y = less_eq (times y (star x)) y"
finite_chamber: fixes X :: "'b set set"   and x :: "'b set" assumes "ChamberComplex X"   and "SimplicialComplex.maxsimp X x" shows "finite x"
drop_rev: fixes n :: "nat"   and xs :: "'b list" shows "drop n (rev xs) = rev (take (length xs - n) xs)"
normal_intersection_hom_surj: shows "(#>) H ` carrier (G\<lparr>carrier := S\<rparr>) = carrier (G\<lparr>carrier := H <#> S\<rparr> Mod H)"
rank_transpose: fixes A :: "(('a, 'c) vec, 'b) vec" shows "rank (Finite_Cartesian_Product.transpose A) = rank A"
rel_cset_aux: fixes a :: "'f cset"   and b :: "'g cset"   and R :: "'f \<Rightarrow> 'g \<Rightarrow> bool" shows "((\<forall>t\<in>rcset a. \<exists>u\<in>rcset b. R t u) \<and> (\<forall>t\<in>rcset b. \<exists>u\<in>rcset a. R u t)) = ((BNF_Def.Grp {x. rcset x \<subseteq> {(a, b). R a b}} (cimage fst))\<inverse>\<inverse> OO BNF_Def.Grp {x. rcset x \<subseteq> {(a, b). R a b}} (cimage snd)) a b"
iMOD_pred: fixes x :: "nat"   and r :: "nat"   and m :: "nat" assumes "x \<in> [ r, mod m ]"   and "r < x" shows "(x - Suc 0 \<in> [ r, mod m ]) = (m = Suc 0)"
assms_3: fixes x :: "'x"   and ln :: "'c \<Rightarrow>f nat" assumes "ts t = \<lfloor>(x, ln)\<rfloor>"   and "\<not> final x" shows "thesis"
map_id: fixes t :: "'b option" shows "map_option id t = t"
swap_swaps: fixes arr' :: "'a \<Rightarrow> 'b"   and arr :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "arr' = swap arr x y" shows "arr' y = arr x \<and> arr' x = arr y"
new2_subset_frmls: fixes \<phi> :: "'d ltlr"   and \<psi> :: "'d ltlr" assumes "\<phi> \<in> new2 \<psi>" shows "\<phi> \<in> subfrmlsr \<psi>"
UP_integral: fixes p :: "nat \<Rightarrow> 'a"   and q :: "nat \<Rightarrow> 'a" assumes "p \<otimes>\<^bsub>P\<^esub> q = \<zero>\<^bsub>P\<^esub>"   and "p \<in> carrier P"   and "q \<in> carrier P" shows "p = \<zero>\<^bsub>P\<^esub> \<or> q = \<zero>\<^bsub>P\<^esub>"
closure_of_minimal: fixes S :: "'c set"   and T :: "'c set"   and X :: "'c topology" assumes "S \<subseteq> T"   and "closedin X T" shows "X closure_of S \<subseteq> T"
completely_multiplicative_function_mult: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" assumes "completely_multiplicative_function f"   and "completely_multiplicative_function g" shows "completely_multiplicative_function (\<lambda>n. f n * g n)"
WhileAnnoFix: fixes P :: "'d set"   and I :: "'e \<Rightarrow> 'd set"   and b :: "'d set"   and Q :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'g) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'g set"   and c :: "'e \<Rightarrow> ('d, 'f, 'g) com"   and A :: "'d set"   and V :: "'e \<Rightarrow> ('d \<times> 'd) set" assumes "P \<subseteq> {s. \<exists>Z. s \<in> I Z \<and> I Z \<inter> - b \<subseteq> Q}"   and "\<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>(I Z \<inter> b) c Z (I Z),A"   and "\<forall>Z. c Z = c undefined" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P whileAnnoFix b I V c Q,A"
odisconnect_completeTE: fixes \<sigma> :: "nat \<Rightarrow> 'd"   and s :: "'e net_state"   and i :: "nat"   and i' :: "nat"   and \<sigma>' :: "nat \<Rightarrow> 'd"   and s' :: "'e net_state"   and S :: "(((nat \<Rightarrow> 'd) \<times> 'e net_state) \<times> 'f node_action \<times> (nat \<Rightarrow> 'd) \<times> 'e net_state) set" assumes "((\<sigma>, s), disconnect(i, i'), \<sigma>', s') \<in> ocnet_sos S" obtains "((\<sigma>, s), disconnect(i, i'), \<sigma>', s') \<in> S" and "\<forall>j. j \<notin> net_ips s \<longrightarrow> \<sigma>' j = \<sigma> j"
norm_cblinfun_of_matrix: fixes a :: "'b \<Rightarrow> 'a \<Rightarrow> complex" shows "norm (cblinfun_of_matrix a) \<le> (\<Sum>i\<in>CBasis. \<Sum>j\<in>CBasis. cmod (a i j))"
less_subset_eq_Union_mset: fixes i :: "nat"   and CAi :: "'b multiset list" assumes "i < length CAi" shows "CAi ! i \<subseteq># \<Sum>\<^sub># (mset CAi)"
graph_rel_def: fixes Rv :: "('c \<times> 'e) set"   and Rw :: "('d \<times> 'f) set" shows "\<langle>Rv, Rw\<rangle>graph_rel \<equiv> {(g, g'). (nodes g, nodes g') \<in> \<langle>Rv\<rangle>set_rel \<and> (edges g, edges g') \<in> \<langle>Rv \<times>\<^sub>r Rw \<times>\<^sub>r Rv\<rangle>set_rel}"
inline1_fixp_induct: fixes P :: "(('d, 'e, 'f) gpv \<Rightarrow> 'g \<Rightarrow> ('d \<times> 'g + 'h \<times> ('f \<times> 'g, 'h, 'i) rpv \<times> ('d, 'e, 'f) rpv) spmf) \<Rightarrow> bool"   and callee :: "'g \<Rightarrow> 'e \<Rightarrow> ('f \<times> 'g, 'h, 'i) gpv" assumes "spmf.admissible (\<lambda>inline1'. P (\<lambda>gpv s. inline1' (gpv, s)))"   and "P (\<lambda>_ _. return_pmf None)"   and "\<And>inline1'. P inline1' \<Longrightarrow> P (\<lambda>gpv s. the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). inline1' (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv)))))" shows "P (inline1 callee)"
set_mset_minus_replicate_mset_2: fixes n :: "nat"   and A :: "'a multiset"   and a :: "'a" assumes "n < count A a" shows "set_mset (A - replicate_mset n a) = set_mset A"
assms_7: shows "A\<^sub>P\<^sub>Q \<sharp>* M"
Sample_sending_data: fixes stop :: "nat \<Rightarrow> stopType list"   and lose :: "nat \<Rightarrow> bool list"   and req :: "nat \<Rightarrow> reqType list"   and a1 :: "nat \<Rightarrow> aType list"   and d :: "nat"   and t :: "nat"   and ack :: "nat \<Rightarrow> GatewayStatus list"   and dt :: "nat \<Rightarrow> ECall_Info list"   and i1 :: "nat \<Rightarrow> ECall_Info list"   and vc :: "nat \<Rightarrow> vcType list"   and x :: "nat" assumes "msg (Suc 0) stop"   and "ts lose"   and "msg (Suc 0) req"   and "msg (Suc 0) a1"   and "\<forall>j\<le>2 * d. lose (t + j) = [False]"   and "ack t = [sending_data]"   and "Sample req dt a1 stop lose ack i1 vc"   and "x \<le> d + d"   and "\<forall>t4\<le>t + d + d. a1 t4 = []" shows "ack (t + x) = [sending_data]"
cases: fixes a :: "real mat convnet" assumes "shared_weight_net a" obtains M :: "nat" where "a = Input M"   | m :: "real mat convnet" and A :: "real mat" where "a = Conv A m" and "shared_weight_net m"   | m1 :: "real mat convnet" and m2 :: "real mat convnet" where "a = Pool m1 m2" and "shared_weight_net m1" and "shared_weight_net m2" and "count_weights True (remove_weights m1) = count_weights True (remove_weights m2)" and "\<And>x. x < count_weights True (remove_weights m1) \<Longrightarrow> extract_weights True m1 x = extract_weights True m2 x"
assms_2: shows "1 < \<^bold>|us\<^bold>|"
Fvars_eql: fixes t1 :: "'trm"   and t2 :: "'trm" assumes "t1 \<in> trm"   and "t2 \<in> trm" shows "Fvars (eql t1 t2) = FvarsT t1 \<union> FvarsT t2"
Ici_subset_Ioi_iff: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "class.linorder less_eq less" shows "(ord.atLeast less_eq a \<subseteq> ord.greaterThan less b) = less b a"
rawpsubstT_tms: fixes r1 :: "'trm"   and r2 :: "'trm"   and txs :: "('trm \<times> 'var) list" assumes "r1 \<in> trm"   and "r2 \<in> trm"   and "snd ` set txs \<subseteq> var"   and "fst ` set txs \<subseteq> trm" shows "rawpsubstT (tms r1 r2) txs = tms (rawpsubstT r1 txs) (rawpsubstT r2 txs)"
weakPar1: fixes \<Psi> :: "'b"   and \<Psi>\<^sub>Q :: "'b"   and R :: "('a, 'b, 'c) psi"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and A\<^sub>Q :: "name list" assumes "\<Psi> \<otimes> \<Psi>\<^sub>Q : R \<rhd> P \<Longrightarrow>\<alpha> \<prec> P'"   and "extractFrame Q = \<langle>A\<^sub>Q, \<Psi>\<^sub>Q\<rangle>"   and "bn \<alpha> \<sharp>* Q"   and "A\<^sub>Q \<sharp>* \<Psi>"   and "A\<^sub>Q \<sharp>* P"   and "A\<^sub>Q \<sharp>* \<alpha>"   and "A\<^sub>Q \<sharp>* R" shows "\<Psi> : R \<parallel> Q \<rhd> P \<parallel> Q \<Longrightarrow>\<alpha> \<prec> P' \<parallel> Q"
wadjust_loop_on_move_no_Oc_4: fixes m :: "nat"   and rs :: "nat"   and c :: "cell list"   and list :: "cell list" shows "wadjust_on_left_moving_B m rs (c, Oc # list) = False"
assms_2: shows "sub \<in> \<lbrakk> time-relation \<lfloor>c\<^sub>1, c\<^sub>2\<rfloor> \<in> R \<rbrakk>\<^sub>T\<^sub>E\<^sub>S\<^sub>L"
simps: fixes a1 :: "string_ternop"   and a2 :: "'a OCL_Types.type"   and a3 :: "'b OCL_Types.type"   and a4 :: "'c OCL_Types.type"   and a5 :: "'d OCL_Types.type" shows "string_ternop_type a1 a2 a3 a4 a5 = (\<exists>\<sigma> \<rho>. a1 = SubstringOp \<and> a2 = String[1] \<and> a3 = \<sigma> \<and> a4 = \<rho> \<and> a5 = String[1] \<and> \<sigma> = UnlimitedNatural[1]\<midarrow>Integer[1] \<and> \<rho> = UnlimitedNatural[1]\<midarrow>Integer[1])"
infinite_finite_induct: fixes P :: "'e set \<Rightarrow> bool"   and A :: "'e set" assumes "\<And>A. infinite A \<Longrightarrow> P A"   and "P {}"   and "\<And>x F. \<lbrakk>finite F; x \<notin> F; P F\<rbrakk> \<Longrightarrow> P (insert x F)" shows "P A"
fls_residue_add: fixes f :: "'a fls"   and g :: "'a fls" shows "fls_residue (f + g) = fls_residue f + fls_residue g"
subgraph_subset_2: fixes G\<^sub>1 :: "('a, 'b) labeled_graph"   and G\<^sub>2 :: "('a, 'b) labeled_graph" assumes "subgraph G\<^sub>1 G\<^sub>2" shows "edges (restrict G\<^sub>1) \<subseteq> edges G\<^sub>2"
cinfinite_csum: fixes r1 :: "('k \<times> 'k) set"   and r2 :: "('l \<times> 'l) set" assumes "cinfinite r1 \<or> cinfinite r2" shows "cinfinite (r1 +c r2)"
prems_1: shows "\<forall>u s. insert\<langle>u,s\<rangle> \<in> set (B @ [b]) \<longrightarrow> Fun (Val n) [] \<noteq> u \<cdot> \<I>"
to_nat_from_nat: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and j :: "'b"   and k :: "nat" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "mod_type.to_nat Rep j = k" shows "mod_type.from_nat Abs k = j"
AddSA_CompFun_the3: fixes S' :: "'d"   and SA :: "('d, 'e, 'f) seqauto"   and S :: "'d"   and A :: "('d, 'e, 'f) hierauto" assumes "S' \<notin> States SA"   and "S \<noteq> S'"   and "States SA \<inter> HAStates A = {}"   and "S \<in> HAStates A" shows "the (CompFun (A [++] (S, SA)) S') = the (CompFun A S')"
Comp1: fixes c1 :: "OBJ"   and c2 :: "OBJ" assumes "secure c1"   and "secure c2" shows "secure (Comp c1 c2)"
mult_add_distrib_mat_vec: fixes A :: "'b mat"   and nr :: "nat"   and nc :: "nat"   and v\<^sub>1 :: "'b Matrix.vec"   and v\<^sub>2 :: "'b Matrix.vec" assumes "A \<in> carrier_mat nr nc"   and "v\<^sub>1 \<in> carrier_vec nc"   and "v\<^sub>2 \<in> carrier_vec nc" shows "A *\<^sub>v (v\<^sub>1 + v\<^sub>2) = A *\<^sub>v v\<^sub>1 + A *\<^sub>v v\<^sub>2"
drop_f_join_if: fixes n :: "nat"   and xs :: "'b list"   and I :: "nat set" shows "xs \<Join>\<^sub> I \<up> n = (if n < card (I \<down>< length xs) then xs \<Join>\<^sub> (I \<down>\<ge> (I \<rightarrow> n)) else [])"
delete_map: fixes f :: "'d \<times> 'e \<Rightarrow> 'd \<times> 'f"   and k :: "'d"   and ps :: "('d \<times> 'e) list" assumes "\<And>kv. fst (f kv) = fst kv" shows "delete k (map f ps) = map f (delete k ps)"
holomorphic_iff_power_series: fixes f :: "complex \<Rightarrow> complex"   and z :: "complex"   and r :: "real" shows "(f holomorphic_on ball z r) = (\<forall>w\<in>ball z r. (\<lambda>n. (deriv ^^ n) f z / fact n * (w - z) ^ n) sums f w)"
tiny_dg_Obj_in_Vset: shows "\<CC>\<lparr>Obj\<rparr> \<in>\<^sub>\<circ> Vset \<alpha>"
ide_mkIde: fixes A :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr set" assumes "A \<subseteq> Univ"   and "|A| <o natLeq" shows "ide (mkIde A)"
is_irred_plus: fixes F :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set"   and p :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b" assumes "\<not> punit.is_red F p"   and "\<not> punit.is_red F q" shows "\<not> punit.is_red F (p + q)"
ord_eq_0: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.group G"   and "x \<in> carrier G" shows "(group.ord G x = 0) = (\<forall>n. n \<noteq> 0 \<longrightarrow> x [^]\<^bsub>G\<^esub> n \<noteq> \<one>\<^bsub>G\<^esub>)"
decs_add_to_counter: fixes ck :: "bool"   and mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and d :: "dec list"   and res :: "'b state \<times> (v sem_env, v) result" assumes "evaluate_decs ck mn env s d res" shows "\<forall>s' r extra. res = (s', r) \<and> ck = True \<and> r \<noteq> Rerr (Rabort Rtimeout_error) \<longrightarrow> evaluate_decs True mn env (update_clock (\<lambda>_. clock s + extra) s) d (update_clock (\<lambda>_. clock s' + extra) s', r)"
simps_1: fixes x :: "'b" shows "PermutationLemmas.count x [] = 0"
spr_ii_simRels_def: shows "spr_ii_simRels \<equiv> \<lambda>a. {(s, s') |s s'. envObs a (snd s) = envObs a (snd s') \<and> fst s = fst s'}"
nsum_split1: fixes A :: "('d, 'e) aGroup_scheme"   and m :: "nat"   and f :: "nat \<Rightarrow> 'd"   and n :: "nat" assumes "aGroup A"   and "\<forall>j\<le>m. f j \<in> carrier A"   and "n < m" shows "\<Sigma>\<^sub>e A f m = \<Sigma>\<^sub>e A f n \<plusminus>\<^bsub>A\<^esub> \<Sigma>\<^sub>f A f Suc n m"
foreachc_mono: fixes f\<^sub>1 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool"   and f\<^sub>2 :: "'c \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> bool"   and c :: "'d \<Rightarrow> bool"   and S :: "'c set"   and a\<^sub>0 :: "'d"   and a\<^sub>1 :: "'d" assumes "\<And>x a\<^sub>0 a\<^sub>1. f\<^sub>1 x a\<^sub>0 a\<^sub>1 \<Longrightarrow> f\<^sub>2 x a\<^sub>0 a\<^sub>1"   and "foreach\<^sub>C c f\<^sub>1 S a\<^sub>0 a\<^sub>1" shows "foreach\<^sub>C c f\<^sub>2 S a\<^sub>0 a\<^sub>1"
lea__nlta: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and E :: "'p"   and F :: "'p" assumes "A B C LeA D E F" shows "\<not> D E F LtA A B C"
unsigned_of_int: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and k :: "int" assumes "class.semiring_1 one times plus zero" shows "semiring_1.unsigned one plus zero (word_of_int k) = semiring_1.of_nat one plus zero (nat (take_bit LENGTH('d) k))"
\<Omega>: shows "(\<lambda>a. emeasure (K a) \<Omega>) \<in> borel_measurable M"
boundOutputChainEq'': fixes xvec :: "name list"   and M :: "'d"   and P :: "('d, 'e, 'f) psi"   and yvec :: "name list"   and N :: "'d"   and Q :: "('d, 'e, 'f) psi" assumes "\<lparr>\<nu>*xvec\<rparr>M \<prec>' P = \<lparr>\<nu>*yvec\<rparr>N \<prec>' Q"   and "xvec \<sharp>* yvec"   and "distinct xvec"   and "distinct yvec" obtains p :: "(name \<times> name) list" where "set p \<subseteq> set xvec \<times> set (p \<bullet> xvec)" and "distinctPerm p" and "yvec = p \<bullet> xvec" and "N = p \<bullet> M" and "Q = p \<bullet> P" and "xvec \<sharp>* N" and "xvec \<sharp>* Q" and "(p \<bullet> xvec) \<sharp>* M" and "(p \<bullet> xvec) \<sharp>* P"
vimage_set_def: fixes Aa :: "V"   and f :: "V \<Rightarrow> V"   and A :: "V" shows "VLambda Aa f `\<^sub>\<circ> A = set {b. \<exists>a\<in>elts (A \<inter>\<^sub>\<circ> \<D>\<^sub>\<circ> (VLambda Aa f)). VLambda Aa f\<lparr>a\<rparr> = b}"
decseq_SucD: fixes A :: "nat \<Rightarrow> 'c"   and i :: "nat" assumes "decseq A" shows "A (Suc i) \<le> A i"
assms_1: shows "a \<noteq> 0\<^sub>h"
run_subseteq_reach_2: fixes w :: "nat \<Rightarrow> 'd"   and \<Sigma> :: "'d set"   and \<delta> :: "'e \<Rightarrow> 'd \<Rightarrow> 'e"   and q\<^sub>0 :: "'e" assumes "range w \<subseteq> \<Sigma>" shows "range (run\<^sub>t \<delta> q\<^sub>0 w) \<subseteq> reach\<^sub>t \<Sigma> \<delta> q\<^sub>0"
psubstEnv_idEnv_id: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option" assumes "goodEnv rho" shows "rho &[idEnv] = rho"
bounded: fixes x :: "(real, 'sz) vec" shows "bounded_linear (\<lambda>a. dterm_sem (extendf I x) \<theta>\<^sub>1 \<nu> * extendf_deriv I i \<theta>\<^sub>2 \<nu> x a + extendf_deriv I i \<theta>\<^sub>1 \<nu> x a * dterm_sem (extendf I x) \<theta>\<^sub>2 \<nu>)"
H_face_cycle_succ_G_not_in: fixes a :: "'b" assumes "a \<in> arcs G" shows "H.face_cycle_succ a \<notin> {rev_H uw, rev_H vw}"
induct: fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and x1 :: "'b"   and x2 :: "'b"   and P :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "r\<^sup>+\<^sup>+ x1 x2"   and "\<And>a b. r a b \<Longrightarrow> P a b"   and "\<And>a b c. \<lbrakk>r\<^sup>+\<^sup>+ a b; P a b; r b c\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
restrict_apply': fixes x :: "'a"   and A :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "x \<in> A" shows "restrict f A x = f x"
new_addrs_def: fixes h :: "heap"   and as :: "nat set"   and h' :: "heap" shows "new_addrs h as h' = as \<union> {a. lim h \<le> a \<and> a < lim h'}"
elims: fixes x :: "'a \<Rightarrow> 'b"   and xa :: "'a list"   and y :: "'a" assumes "argmin x xa = y" obtains f :: "'a \<Rightarrow> 'b" and xb :: "'a" and xs :: "'a list" where "x = f" and "xa = xb # xs" and "y = (if xs = [] then xb else let m = argmin f xs in if f xb < f m then xb else m)"   | a :: "'a \<Rightarrow> 'b" where "x = a" and "xa = []" and "y = undefined"
x1Cond: shows "p x1 = Max (p ` fst ` f) \<and> x1 \<in> fst ` f"
square_free_reflect_poly: fixes f :: "'a poly" assumes "square_free f"   and "coeff f 0 \<noteq> (0::'a)" shows "square_free (reflect_poly f)"
splitFace_def: fixes g :: "graph"   and ram\<^sub>1 :: "nat"   and ram\<^sub>2 :: "nat"   and oldF :: "face"   and newVs :: "nat list" shows "splitFace g ram\<^sub>1 ram\<^sub>2 oldF newVs \<equiv> let fs = faces g; n = countVertices g; Fs = faceListAt g; h = heights g; vs\<^sub>1 = between (vertices oldF) ram\<^sub>1 ram\<^sub>2; vs\<^sub>2 = between (vertices oldF) ram\<^sub>2 ram\<^sub>1; (f\<^sub>1, f\<^sub>2) = split_face oldF ram\<^sub>1 ram\<^sub>2 newVs; Fs = replacefacesAt vs\<^sub>1 oldF [f\<^sub>1] Fs; Fs = replacefacesAt vs\<^sub>2 oldF [f\<^sub>2] Fs; Fs = replacefacesAt [ram\<^sub>1] oldF [f\<^sub>2, f\<^sub>1] Fs; Fs = replacefacesAt [ram\<^sub>2] oldF [f\<^sub>1, f\<^sub>2] Fs; Fs = Fs @ replicate |newVs| [f\<^sub>1, f\<^sub>2] in (f\<^sub>1, f\<^sub>2, Graph (replace oldF [f\<^sub>2] fs @ [f\<^sub>1]) (n + |newVs|) Fs (h @ heightsNewVertices (h ! ram\<^sub>1) (h ! ram\<^sub>2) |newVs|))"
Limit_ordertype_imp_Field_Restr: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "Limit (ordertype A r)"   and "wf r"   and "total_on A r"   and "small A" shows "Field (Restr r A) = A"
Z'_def: shows "Z' \<equiv> abs_var (BPS (bsingleton (True, undefined)))"
extensibleI: fixes R2 :: "('e, 'f) labeled_graph"   and G :: "('e, 'g) labeled_graph"   and g :: "('f \<times> 'g) set"   and R1 :: "('e, 'f) labeled_graph"   and f :: "('f \<times> 'g) set" assumes "graph_homomorphism R2 G g"   and "agree_on R1 f g" shows "extensible (R1, R2) G f"
preserves_hom: fixes f :: "'a \<times> 'a \<times> 'a"   and a :: "'a \<times> 'a \<times> 'a"   and b :: "'a \<times> 'a \<times> 'a" assumes "S.VxVxV.in_hom f a b" shows "S.in_hom (S.\<alpha> f) (M.T.ToTC a) (M.T.ToCT b)"
triangle_right': shows "\<r>[g] \<cdot> (g \<star> \<epsilon>') \<cdot> \<a>[g, f, g] \<cdot> (\<eta> \<star> g) \<cdot> \<l>\<^sup>-\<^sup>1[g] = g"
offs_enum_next_cons: fixes index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and xs :: "'c list"   and mi :: "'d"   and ma :: "'d"   and i :: "nat"   and x :: "'c"   and n :: "nat"   and k :: "nat" assumes "index_less index key"   and "\<And>x. x \<in> set xs \<Longrightarrow> key x \<in> {mi..ma}" shows "(if i < index key x n mi ma then (\<le>) else (<)) (offs_next (offs (enum xs index key n mi ma) k) (length xs + k) xs index key mi ma i) (offs_next (offs ((enum xs index key n mi ma)[index key x n mi ma := Suc (enum xs index key n mi ma ! index key x n mi ma)]) k) (Suc (length xs + k)) (x # xs) index key mi ma i)"
simps: fixes C_info :: "(name, dt_def) fmap"   and a :: "sterm" shows "constructors.is_value C_info a = ((\<exists>cs. a = Sabs cs) \<or> (\<exists>vs name. a = name $$ vs \<and> list_all (constructors.is_value C_info) vs \<and> name |\<in>| constructors.C C_info))"
gamma_plus': fixes n1 :: "int"   and a1 :: "'av"   and n2 :: "int"   and a2 :: "'av" assumes "n1 \<in> \<gamma> a1"   and "n2 \<in> \<gamma> a2" shows "n1 + n2 \<in> \<gamma> (plus' a1 a2)"
is_type_class: fixes P :: "'b prog"   and C :: "String.literal" shows "is_type P (Class C) = is_class P C"
LETI: fixes A :: "type_scheme list"   and e1 :: "expr"   and t1 :: "typ"   and e2 :: "expr"   and t :: "typ" assumes "A |- e1 :: t1"   and "gen A t1 # A |- e2 :: t" shows "A |- LET e1 e2 :: t"
huntington: fixes x :: "'a"   and y :: "'a" shows "x = - (- x \<squnion> y) \<squnion> - (- x \<squnion> - y)"
abs_eq: fixes xa :: "'f \<Rightarrow> 'g"   and x :: "'e \<Longrightarrow> 'g" shows "\<langle>xa\<rangle>\<^sub>s x = mk\<^sub>e (\<lambda>b. get\<^bsub>x\<^esub> (xa b))"
assms: fixes \<Psi>' :: "'b" shows "\<exists>Q' Q''. \<Psi> \<rhd> Q \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q' \<and> insertAssertion (extractFrame P) \<Psi> \<hookrightarrow>\<^sub>F insertAssertion (extractFrame Q') \<Psi> \<and> \<Psi> \<otimes> \<Psi>' \<rhd> Q' \<Longrightarrow>\<^sup>^\<^sub>\<tau> Q'' \<and> (\<Psi> \<otimes> \<Psi>', P, Q'') \<in> Rel"
action_closed: fixes m :: "'c"   and g :: "'a" assumes "m \<in> M"   and "g \<in> carrier G" shows "\<phi> g m \<in> M"
push_bit_of_0: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and n :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit n zero = zero"
trans': fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a \<in> S"   and "b \<in> S"   and "c \<in> S"   and "a \<noteq> b"   and "b \<noteq> c"   and "a \<noteq> c"   and "le a b"   and "le b c" shows "le a c"
scc_disj: fixes c :: "('a, 'b) pre_digraph"   and d :: "('a, 'b) pre_digraph" assumes "c \<in> sccs"   and "d \<in> sccs"   and "c \<noteq> d" shows "verts c \<inter> verts d = {}"
finite_lists_length_eq': fixes xs :: "'a list"   and P :: "'a \<Rightarrow> 'b \<Rightarrow> bool" assumes "\<And>x. x \<in> set xs \<Longrightarrow> finite {y. P x y}" shows "finite {ys. length xs = length ys \<and> (\<forall>y\<in>set ys. \<exists>x\<in>set xs. P x y)}"
strict_order_conv_closed: fixes x :: "'a" assumes "strict_order x" shows "strict_order (x\<^sup>T)"