hcju/isabelleps huggingface dataset:
sinks_aux_single_dom: fixes I :: "('b \<times> 'b) set"   and D :: "'c \<Rightarrow> 'b"   and u :: "'b"   and xs :: "'c list" shows "sinks_aux I D {u} xs = insert u (sinks I D u xs)"
ring_pow_apow: fixes R :: "('c, 'd) Ring_scheme"   and I :: "'c set"   and n :: "nat" assumes "Ring R"   and "ideal R I" shows "I \<^bsup>\<diamondsuit>R n\<^esup> = I\<^bsup> R an n\<^esup>"
dagger_def: fixes M :: "complex Matrix.mat" shows "M\<^sup>\<dagger> \<equiv> Matrix.mat (dim_col M) (dim_row M) (\<lambda>(i, j). cnj (M $$ (j, i)))"
exec_append: fixes s :: "'a"   and rs :: "('b \<times> 'c) list"   and rs' :: "('b \<times> 'c) list" shows "(s \<star> rs) \<star> rs' = s \<star> append rs' rs"
simps_2: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'c \<Rightarrow> 'b"   and x :: "'c"   and xs :: "'c list" shows "map_index' n f (x # xs) = f n x # map_index' (Suc n) f xs"
simps_49: fixes x41 :: "('a, 'b, 'c) Stmt"   and x42 :: "('a, 'b, 'c) Stmt" shows "x41 ;; x42 \<noteq> Stop"
a1: shows "low_equal s1 s2 \<and> t1 = snd (fst (div_comp instr rs1 rd operand2 s1)) \<and> t2 = snd (fst (div_comp instr rs1 rd operand2 s2))"
sterm_continuous': fixes I :: "('a, 'b, 'c) interp"   and \<theta> :: "('a, 'c) trm"   and S :: "(real, 'c) vec set" assumes "is_interp I"   and "dfree \<theta>" shows "continuous_on S (sterm_sem I \<theta>)"
intros_1: fixes prob :: "('a, 'b, 'c, 'd) ifc_problem"   and i :: "nat"   and \<pi> :: "nat \<Rightarrow> 'a"   and k :: "nat" assumes "IFC prob"   and "IFC_def.is_cdi prob i \<pi> k" shows "(\<pi> k, \<pi> i) \<in> IFC.pdg prob"
l6_21: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and P :: "'p"   and Q :: "'p" assumes "\<not> Col A B C"   and "C \<noteq> D"   and "Col A B P"   and "Col A B Q"   and "Col C D P"   and "Col C D Q" shows "P = Q"
compl_inf: fixes x :: "'a upred"   and y :: "'a upred" shows "(\<not> (x \<and> y)) = (\<not> x \<or> \<not> y)"
S: shows "S = (\<forall>X<:S\<^sub>1. S\<^sub>2)"
in_lset_ldropD: fixes x :: "'c"   and n :: "enat"   and xs :: "'c llist" assumes "x \<in> lset (ldrop n xs)" shows "x \<in> lset xs"
wilsons_construction_index_0: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and sizes :: "int set"   and groups :: "'b set set"   and I :: "'c set"   and f :: "'b set \<Rightarrow> ('b \<times> 'c) set multiset"   and \<K>' :: "int set"   and G :: "('b \<times> 'c) set"   and X :: "'b \<times> 'c"   and Y :: "'b \<times> 'c" assumes "K_GDD point_set block_collection sizes groups"   and "\<And>B. B \<in># block_collection \<Longrightarrow> K_GDD (B \<times> I) (f B) \<K>' {{x} \<times> I |x. x \<in> B}"   and "G \<in> {GG \<times> I |GG. GG \<in> groups}"   and "X \<in> G"   and "Y \<in> G"   and "X \<noteq> Y" shows "\<Sum>\<^sub># (image_mset f block_collection) index {X, Y} = 0"
chain_split: fixes ord :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and C :: "'b set"   and x :: "'b" assumes "Complete_Partial_Order.chain ord C"   and "x \<in> C" shows "C = {y \<in> C. ord x y} \<union> {y \<in> C. ord y x}"
filter_singleton: fixes f :: "'a \<Rightarrow> bool"   and s :: "'a list"   and y :: "'a" assumes "filter f s = [y]" shows "f y \<and> y \<in> set s"
p_smaller_d: shows "enat (Suc p) < llength Ns"
compactin_fip: fixes X :: "'c topology"   and S :: "'c set" shows "compactin X S = (S \<subseteq> topspace X \<and> (\<forall>\<U>. (\<forall>C\<in>\<U>. closedin X C) \<and> (\<forall>\<F>. finite \<F> \<and> \<F> \<subseteq> \<U> \<longrightarrow> S \<inter> \<Inter> \<F> \<noteq> {}) \<longrightarrow> S \<inter> \<Inter> \<U> \<noteq> {}))"
eq_iff: fixes a :: "'x \<Rightarrow>\<^sub>0 nat"   and b :: "'x \<Rightarrow>\<^sub>0 nat" shows "(a = b) = (b \<preceq> a \<and> a \<preceq> b)"
converse_nreachable_cases: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and n :: "nat"   and v :: "'c" assumes "wf_digraph G"   and "u \<rightarrow>\<^bsup>n\<^esup>\<^bsub>G\<^esub> v" obtains "u = v" and "n = 0" and "u \<in> verts G"   | w :: "'c" and m :: "nat" where "u \<rightarrow>\<^bsub>G\<^esub> w" and "n = Suc m" and "w \<rightarrow>\<^bsup>m\<^esup>\<^bsub>G\<^esub> v"
finite_k_words: fixes k :: "nat" shows "finite {w. length w = k \<and> set w \<subseteq> L}"
simps_2: fixes f :: "'c \<Rightarrow> 'd"   and x :: "'c"   and xs :: "'c list" shows "groupF f (x # xs) = (x # filter (\<lambda>y. f x = f y) xs) # groupF f (filter (\<lambda>y. f x \<noteq> f y) xs)"
cnj_beta_l: fixes l :: "nat" shows "conjugate (beta_l l) = beta_l l"
T2: fixes w :: "'a"   and w' :: "'a"   and \<theta> :: "'a"   and u :: "'a"   and \<theta>' :: "'a"   and \<beta> :: "'a" assumes "ide w"   and "ide w'"   and "\<guillemotleft>\<theta> : t\<^sub>0 \<star> w \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<theta>' : t\<^sub>0 \<star> w' \<Rightarrow> u\<guillemotright>"   and "\<guillemotleft>\<beta> : t\<^sub>1 \<star> w \<Rightarrow> t\<^sub>1 \<star> w'\<guillemotright>"   and "(t \<star> \<theta>) \<cdot> \<a>[t, t\<^sub>0, w] \<cdot> (\<tau> \<star> w) = ((t \<star> \<theta>') \<cdot> \<a>[t, t\<^sub>0, w'] \<cdot> (\<tau> \<star> w')) \<cdot> \<beta>" shows "\<exists>!\<gamma>. \<guillemotleft>\<gamma> : w \<Rightarrow> w'\<guillemotright> \<and> \<beta> = t\<^sub>1 \<star> \<gamma> \<and> \<theta> = \<theta>' \<cdot> (t\<^sub>0 \<star> \<gamma>)"
that: fixes u :: "dB" assumes "s \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* u"   and "t \<rightarrow>\<^sub>\<beta>\<^sub>\<eta>\<^sup>* u" shows "thesis"
infer_pairE: fixes \<Gamma> :: "'a \<Rightarrow> PreSimplyTyped.type option"   and A :: "'a trm"   and B :: "'a trm"   and \<tau> :: "PreSimplyTyped.type" assumes "infer \<Gamma> (SimplyTyped.Pair A B) = Some \<tau>" shows "\<exists>T S. \<tau> = TPair T S \<and> infer \<Gamma> A = Some T \<and> infer \<Gamma> B = Some S"
\<ll>_ide_simp: fixes a :: "'d" assumes "D.ide a" shows "D.\<ll> a = D.lunit a"
cod_cod: fixes f :: "'a1" assumes "A1.arr f" shows "A1.cod (A1.cod f) = A1.cod f"
union: fixes A :: "nat \<Rightarrow> 'a set" assumes "disjoint_family A"   and "range A \<subseteq> sigma_sets \<Omega> G"   and "\<And>i. P (A i)" shows "P (\<Union> (range A))"
induct: fixes x :: "'c"   and R :: "('c, 'd) ring_scheme"   and I :: "'c set"   and J :: "'c set"   and P :: "'c \<Rightarrow> bool" assumes "x \<in> I \<cdot>\<^bsub>R\<^esub> J"   and "\<And>i j. \<lbrakk>i \<in> I; j \<in> J\<rbrakk> \<Longrightarrow> P (i \<otimes>\<^bsub>R\<^esub> j)"   and "\<And>s1 s2. \<lbrakk>s1 \<in> I \<cdot>\<^bsub>R\<^esub> J; P s1; s2 \<in> I \<cdot>\<^bsub>R\<^esub> J; P s2\<rbrakk> \<Longrightarrow> P (s1 \<oplus>\<^bsub>R\<^esub> s2)" shows "P x"
cong_modulus_mult_nat: fixes x :: "nat"   and y :: "nat"   and m :: "nat"   and n :: "nat" assumes "[x = y] (mod m * n)" shows "[x = y] (mod m)"
finite_measure_subalgebra_axioms: shows "finite_measure_subalgebra M F"
ceq: fixes p :: "'b" shows "OF_match_linear \<gamma> (sort_descending_key ofe_prio ft) p = OF_same_priority_match2 \<gamma> (sort_descending_key ofe_prio ft) p"
D_gchain0: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'a set"   and i :: "nat"   and j :: "nat" assumes "D_gchain G n f"   and "i \<le> n"   and "j \<le> n"   and "i < j" shows "f j \<subset> f i"
inj_Variable: fixes i :: "nat"   and j :: "nat" shows "(Variable i = Variable j) = (i = j)"
domino_finite: fixes d :: "(nat \<times> nat) set" assumes "d \<in> domino" shows "finite d"
conj_commute: fixes x :: "'a"   and y :: "'a" shows "x \<^bold>\<sqinter> y = y \<^bold>\<sqinter> x"
65231: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> (- x \<sqinter> y \<squnion> - y)) = bot"
almost_full_on_imp_homogeneous_subseq: fixes P :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and A :: "'d set"   and f :: "nat \<Rightarrow> 'd" assumes "almost_full_on P A"   and "\<forall>i. f i \<in> A" shows "\<exists>\<phi>. \<forall>i j. i < j \<longrightarrow> \<phi> i < \<phi> j \<and> P (f (\<phi> i)) (f (\<phi> j))"
sync_withI: fixes P :: "'a prog"   and E :: "('b \<times> ('c, 'b) obs_event action) llist"   and a :: "nat"   and a' :: "nat" assumes "P,E \<turnstile> a \<le>so a'"   and "P \<turnstile> (action_tid E a, action_obs E a) \<leadsto>sw (action_tid E a', action_obs E a')" shows "P,E \<turnstile> a \<le>sw a'"
distinct_lists_def: shows "Shadow_DOM.distinct_lists \<equiv> l_heap_is_wellformed\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_distinct_lists get_shadow_root"
vars1'_def: shows "a.vars1' = ind_in_set a.vars0 ` vars1"
option_while'_THE: fixes r :: "'d"   and sr' :: "'d option"   and C :: "'d \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd option" assumes "(Some r, sr') \<in> option_while' C B" shows "(THE s. (Some r, s) \<in> option_while' C B) = sr'"
take_tl: fixes n :: "nat"   and xs :: "'g list" shows "take n (tl xs) = tl (take (Suc n) xs)"
l_get_dom_component_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component_create_character_data\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.known_ptr heap_is_wellformed parent_child_rel DocumentClass.type_wf DocumentClass.known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_element_by_id get_elements_by_tag_name set_val set_val_locs get_disconnected_nodes get_disconnected_nodes_locs set_disconnected_nodes set_disconnected_nodes_locs create_character_data"
Vagree_def: fixes \<nu> :: "(real, 'b) vec \<times> (real, 'b) vec"   and \<nu>' :: "(real, 'b) vec \<times> (real, 'b) vec"   and V :: "('b + 'b) set" shows "Vagree \<nu> \<nu>' V \<equiv> (\<forall>i. Inl i \<in> V \<longrightarrow> fst \<nu> $ i = fst \<nu>' $ i) \<and> (\<forall>i. Inr i \<in> V \<longrightarrow> snd \<nu> $ i = snd \<nu>' $ i)"
assms: shows "j < card {i. i < dim_col A \<and> i \<in> J}"
dense_repr_normalize: fixes p :: "'a list list" shows "UP.dense_repr (UP.normalize p) = UP.dense_repr p"
plus_monom: fixes a :: "'a"   and p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "a \<in> carrier R"   and "a \<noteq> \<zero>"   and "p \<in> carrier P"   and "deg R p < n"   and "Q p" shows "Q (p \<oplus>\<^bsub>P\<^esub> monom P a n)"
indmhom: fixes N :: "('g, 'b, 'h) Module_scheme"   and f :: "'a \<Rightarrow> 'g" assumes "R module N"   and "f \<in> mHom R M N" shows "f\<^sup>\<flat>\<^bsub>R M, N\<^esub> \<in> mHom R (M /\<^sub>m (ker\<^bsub>M,N\<^esub> f)) N"
inline1_in_sub_gpvs_callee: fixes out :: "'call'"   and callee' :: "('ret \<times> 's, 'call', 'ret') rpv"   and rpv' :: "('b, 'call, 'ret) rpv"   and gpv :: "('b, 'call, 'ret) gpv"   and s :: "'s" assumes "Inr (out, callee', rpv') \<in> set_spmf (inline1 callee gpv s)"   and "\<I> \<turnstile>g gpv \<surd>"   and "I s" shows "\<exists>call\<in>outs_\<I> \<I>. \<exists>s. I s \<and> (\<forall>x\<in>responses_\<I> \<I>' out. callee' x \<in> sub_gpvs \<I>' (callee s call))"
DirProds_comm_group_iff: fixes G :: "'e \<Rightarrow> ('f, 'g) monoid_scheme"   and I :: "'e set" shows "comm_group (DirProds G I) = (\<forall>i\<in>I. comm_group (G i))"
inject: fixes x1 :: "'h"   and x2 :: "'i"   and y1 :: "'h"   and y2 :: "'i" shows "((x1, x2) = (y1, y2)) = (x1 = y1 \<and> x2 = y2)"
finite_reduced_GB_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (component_of_term ` Keys F)"   and "F \<subseteq> dgrad_p_set d m" shows "finite (reduced_GB F)"
Carmichael_number_imp_lower_bound_on_strong_fermat_witness_2: fixes n :: "nat" assumes "Carmichael_number n" shows "real (card {a. 2 \<le> a \<and> a < n \<and> strong_fermat_liar a n}) < real (n - 2) / 2"
tllist_all2_TNil: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x1 :: "'c"   and y1 :: "'d" shows "tllist_all2 R1 R2 (TNil x1) (TNil y1) = R2 x1 y1"
wf_type_Type_imp_mgd: fixes \<Sigma> :: "(String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)"   and n :: "String.literal"   and Ts :: "typ list" assumes "wf_sig \<Sigma>"   and "wf_type \<Sigma> (Ty n Ts)" shows "tcsigs (osig \<Sigma>) n \<noteq> None"
indRelLTPO_impl_TRel_is_strong_reduction_bisimulation: fixes TRel :: "('procT \<times> 'procT) set" assumes "(\<forall>P Q P'. P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q \<and> P \<longmapsto>STCal Source Target P' \<longrightarrow> (\<exists>Q'. Q \<longmapsto>STCal Source Target Q' \<and> P' \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q')) \<and> (\<forall>P Q Q'. P \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q \<and> Q \<longmapsto>STCal Source Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>STCal Source Target P' \<and> P' \<lesssim>\<lbrakk>\<cdot>\<rbrakk>LT<TRel> Q'))" shows "strong_reduction_simulation (TRel\<^sup>+) Target \<and> (\<forall>P Q Q'. (P, Q) \<in> TRel\<^sup>+ \<and> Q \<longmapsto>Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>Target P' \<and> (P', Q') \<in> TRel\<^sup>+))"
admissible: fixes f :: "'a \<Rightarrow> ('d, 'e) state" shows "adm_wf {(nxt x, x) |x. cnt x} (\<lambda>rec x. if cnt x then f x \<bind> (\<lambda>_. rec (nxt x)) else State_Monad.return ())"
b: shows "(p, q, u, v, d) = bezout (A $ from_nat i $ from_nat i) (A $ from_nat k $ from_nat k)"
abs_fiX_bound: fixes f :: "nat \<Rightarrow> real"   and p :: "nat"   and l :: "nat" shows "\<bar>fiX f p l - f p\<bar> \<le> \<Delta>"
simps_5: fixes test :: "nat \<Rightarrow> 'b \<Rightarrow> bool"   and r :: "'b regex" shows "match test (Star r) = (match test r)\<^sup>*\<^sup>*"
substC_def: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'e trm"   and c :: "('e, 'f) lit list" shows "substC \<pi> c = map (substL \<pi>) c"
atlas_def: shows "atlas = {c. domain c \<subseteq> carrier \<and> (\<forall>c'\<in>charts. k-smooth_compat c c')}"
brn_Par_simp: fixes cl :: "('test, 'atom, 'choice) cmd list" shows "brn (Par cl) = brnL cl (length cl)"
assms_2: shows "op \<in> set (\<Pi>\<^sub>\<O>)"
simps_2: fixes d :: "int"   and i :: "int"   and ks :: "int list"   and xs :: "int list" shows "I\<^sub>Z (Dvd d i ks) xs = (d dvd i + \<langle>ks,xs\<rangle>)"
inv1_rbtreeify_g: fixes n :: "nat"   and kvs :: "('a \<times> 'b) list" assumes "n \<le> Suc (length kvs)" shows "inv1 (fst (rbtreeify_g n kvs))"
map_family_triangle: fixes I :: "'c set"   and A :: "'c \<Rightarrow> ('d, 'e) aGroup_scheme"   and S :: "('f, 'g) aGroup_scheme"   and g :: "'c \<Rightarrow> 'f \<Rightarrow> 'd" assumes "\<forall>j\<in>I. aGroup (A j)"   and "aGroup S"   and "\<forall>j\<in>I. g j \<in> aHom S (A j)" shows "\<exists>!f. f \<in> aHom S (a\<Pi>\<^bsub>I\<^esub> A) \<and> (\<forall>j\<in>I. compos S (\<pi>\<^bsub>I,A,j\<^esub>) f = g j)"
emeasure_pair_measure_alt: fixes X :: "('c \<times> 'b) set"   and N :: "'c measure" assumes "X \<in> sets (N \<Otimes>\<^sub>M count_space B)" shows "emeasure (N \<Otimes>\<^sub>M count_space B) X = \<integral>\<^sup>+ x. emeasure (count_space B) (Pair x -` X) \<partial>N"
product_swap: fixes A :: "'c set"   and B :: "'b set" shows "prod.swap ` (A \<times> B) = B \<times> A"
not_zero_satisfies: fixes c :: "'a ns_constraint"   and v :: "nat \<Rightarrow> 'a" assumes "poly c = 0"   and "\<not> zero_satisfies c" shows "\<not> v \<Turnstile>\<^sub>n\<^sub>s c"
int_nat_eq: fixes z :: "int" shows "int (nat z) = (if 0 \<le> z then z else 0)"
rep_eq: fixes x :: "'a" shows "blinfun_apply (blinfun_mult_left x) = (\<lambda>y. y * x)"
phi'_eq: fixes d :: "nat" assumes "d dvd n" shows "phi' d = card {m \<in> {1..n}. n div gcd m n = d}"
reachable_def: fixes succs :: "'a \<Rightarrow> 'a list"   and is_node :: "'a \<Rightarrow> bool"   and invariant :: "'b \<Rightarrow> bool"   and ins :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and memb :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and empt :: "'b"   and xs :: "'a set" assumes "DFS succs is_node invariant ins memb empt" shows "DFS.reachable succs xs = {(x, y). y \<in> set (succs x)}\<^sup>* `` xs"
start_steps_plusD: fixes A :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and r :: "bool list"   and w :: "'b list" assumes "(start A, r) \<in> steps (RegExp2NA.plus A) w" shows "\<exists>us v. w = concat us @ v \<and> Ball (set us) (accepts A) \<and> (start A, r) \<in> steps A v"
list_comb_free: fixes f :: "'c"   and xs :: "'c list" assumes "is_free (list_comb f xs)" shows "is_free f"
intro: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "(c, s) \<rightarrow>c (c', s') \<and> mayDiverge c' s'" shows "mayDiverge c s"
ifields_def: fixes P :: "'a prog"   and C :: "char list" shows "ifields P C \<equiv> filter (\<lambda>((F, D), b, T). b = NonStatic) (fields P C)"
publicKey_image_eq: fixes b :: "bool"   and A :: "agent"   and c :: "bool"   and AS :: "agent set" shows "(publicKey b A \<in> publicKey c ` AS) = (b = c \<and> A \<in> AS)"
identify_blocks_iff: fixes k :: "nat"   and A :: "'c mat" assumes "k < n" shows "set (identify_blocks A k) = {(i, j) |i j. i \<le> j \<and> j < k \<and> (\<forall>l\<ge>i. l < j \<longrightarrow> A $$ (l, Suc l) = (1::'c)) \<and> (Suc j \<noteq> k \<longrightarrow> A $$ (j, Suc j) \<noteq> (1::'c)) \<and> (0 < i \<longrightarrow> A $$ (i - 1, i) \<noteq> (1::'c))}"
hmset_neg_plus: fixes A :: "zhmultiset"   and B :: "zhmultiset" shows "hmset_neg (A + B) = hmset_neg A - hmset_pos B + (hmset_neg B - hmset_pos A)"
ind_in_set_less: fixes A :: "nat set"   and x :: "nat" assumes "finite A"   and "x \<in> A" shows "ind_in_set A x < card A"
prv_softSubst_imp_subst: fixes x :: "'var"   and t :: "'trm"   and \<phi> :: "'fmla" assumes "x \<in> var"   and "t \<in> trm"   and "\<phi> \<in> fmla"   and "x \<notin> FvarsT t" shows "bprv (imp (softSubst \<phi> t x) (subst \<phi> t x))"
gso_carrier: fixes i :: "nat" assumes "i < length (map of_int_hom.vec_hom fs)" shows "gs1.gso i \<in> Rn"
is_unit_gcd_iff: fixes one :: "'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and gcd :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and lcm :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.semiring_gcd one zero times gcd lcm divide plus minus unit_factor normalize" shows "dvd.dvd times (gcd a b) one = (gcd a b = one)"
Res: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi"   and x :: "name" assumes "(\<Psi>, P, Q) \<in> {(\<Psi>, \<lparr>\<nu>*xvec\<rparr>P \<parallel> R, \<lparr>\<nu>*xvec\<rparr>Q \<parallel> R) |xvec \<Psi> P Q R. xvec \<sharp>* \<Psi> \<and> (\<forall>A\<^sub>R \<Psi>\<^sub>R. extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle> \<and> A\<^sub>R \<sharp>* \<Psi> \<and> A\<^sub>R \<sharp>* P \<and> A\<^sub>R \<sharp>* Q \<longrightarrow> \<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<approx> Q)} \<union> weakBisim"   and "x \<sharp> \<Psi>" shows "(\<Psi>, \<lparr>\<nu>x\<rparr>P, \<lparr>\<nu>x\<rparr>Q) \<in> {(\<Psi>, \<lparr>\<nu>*xvec\<rparr>P \<parallel> R, \<lparr>\<nu>*xvec\<rparr>Q \<parallel> R) |xvec \<Psi> P Q R. xvec \<sharp>* \<Psi> \<and> (\<forall>A\<^sub>R \<Psi>\<^sub>R. extractFrame R = \<langle>A\<^sub>R, \<Psi>\<^sub>R\<rangle> \<and> A\<^sub>R \<sharp>* \<Psi> \<and> A\<^sub>R \<sharp>* P \<and> A\<^sub>R \<sharp>* Q \<longrightarrow> \<Psi> \<otimes> \<Psi>\<^sub>R \<rhd> P \<approx> Q)} \<union> weakBisim"
assms_2: shows "\<lbrace>c\<rbrace> Q \<lbrace>d\<^sub>2\<rbrace>\<^sub>u"
lin_def: shows "lin = (\<Sum>g\<in>T. f g *\<^sub>C g)"
length_append: fixes xs :: "'b list"   and ys :: "'b list" shows "length (xs @ ys) = length xs + length ys"
integrable_eq: fixes f :: "'b \<Rightarrow> 'c"   and s :: "'b set"   and g :: "'b \<Rightarrow> 'c" assumes "f integrable_on s"   and "\<And>x. x \<in> s \<Longrightarrow> f x = g x" shows "g integrable_on s"
conn: shows "\<forall>x y. (hl x \<le> y) = (x \<le> h y)"
tickFree_rev: fixes t :: "'c event list" shows "tickFree (rev t) = tickFree t"
horner_sum_bit_eq_take_bit: fixes a :: "'a"   and n :: "nat" shows "horner_sum of_bool (2::'a) (map (bit a) [0..<n]) = take_bit n a"
Abs_float_inverse: fixes y :: "1 word \<times> 'a word \<times> 'b word" assumes "y \<in> UNIV" shows "Rep_float (Abs_float y) = y"
arr_def: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d" assumes "partial_magma C" shows "partial_magma.arr C f \<equiv> partial_magma.domains C f \<noteq> {} \<or> partial_magma.codomains C f \<noteq> {}"
gallery_def: fixes X :: "'b set set"   and xs :: "'b set list" assumes "ChamberComplex X" shows "SimplicialComplex.maxsimpchain X xs \<equiv> (\<forall>x\<in>set xs. SimplicialComplex.maxsimp X x) \<and> adjacentchain xs"
action_proj_idempot': fixes a :: "('d, 'e) fmap \<times> ('d, 'f) fmap"   and vs :: "'d set" assumes "action_dom (fst a) (snd a) \<subseteq> vs" shows "action_proj a vs = a"
set_eqI: fixes A :: "'b set"   and B :: "'b set" assumes "\<And>x. (x \<in> A) = (x \<in> B)" shows "A = B"
q_def: shows "q \<equiv> Poly (map (\<lambda>k. (\<Sum>i\<le>min N k. c (N - i) * f (k - i)) - g k) [0..<N + m])"
cm_unchanged_worklist: fixes c0 :: "('loc, 't) configuration"   and c1 :: "('loc, 't) configuration"   and loc :: "'loc"   and t :: "'t"   and n :: "int"   and loc' :: "'loc" assumes "next_change_multiplicity' c0 c1 loc t n"   and "loc' \<noteq> loc" shows "c_work c1 loc' = c_work c0 loc'"
coprime_mi_mj: shows "\<forall>i\<in>{i. i < n}. \<forall>j\<in>{i. i < n}. i \<noteq> j \<longrightarrow> comm_monoid_mult_class.coprime (m i) (m j)"
left_uniqueI: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool" assumes "\<And>x y z. \<lbrakk>A x z; A y z\<rbrakk> \<Longrightarrow> x = y" shows "left_unique A"
read_only_unacquired_share': fixes \<O> :: "nat set"   and S :: "nat \<Rightarrow> bool option"   and sb :: "'b memref list"   and a :: "nat" assumes "\<O> \<inter> read_only S = {}"   and "weak_sharing_consistent \<O> sb"   and "a \<in> read_only S"   and "a \<notin> all_shared sb"   and "a \<notin> acquired True sb \<O>" shows "a \<in> read_only (share sb S)"
fact: shows "\<forall>i<m. \<forall>j<n. \<exists>no. \<forall>n\<ge>no. P n i j"
metric_CauchyI: fixes dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and X :: "nat \<Rightarrow> 'b" assumes "class.metric_space dist uniformity open"   and "\<And>e. 0 < e \<Longrightarrow> \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. dist (X m) (X n) < e" shows "uniform_space.Cauchy uniformity X"
action_order_is_new_actionD: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and a :: "nat"   and a' :: "nat" assumes "E \<turnstile> a \<le>a a'"   and "is_new_action (action_obs E a')" shows "is_new_action (action_obs E a)"
sinr_below: fixes x :: "'d"   and y :: "'d" shows "(sinr\<cdot>x \<sqsubseteq> sinr\<cdot>y) = (x \<sqsubseteq> y)"
fiX_ubound: fixes f :: "nat \<Rightarrow> real"   and p :: "nat"   and l :: "nat" shows "fiX f p l \<le> f p + \<Delta>"
Pure: fixes generat :: "('a, 'b, ('a, 'b, 'c) rpv) generat"   and gpv :: "('a, 'b, 'c) gpv" assumes "generat \<in> set_spmf (the_gpv gpv)"   and "x \<in> generat_pures generat" shows "P gpv"
conjunctProtoD2: fixes b :: "8 word"   and a :: "protocol"   and x :: "protocol" assumes "simple_proto_conjunct (Proto b) a = Some x" shows "x = Proto b \<and> (a = ProtoAny \<or> a = Proto b)"
prems_1: shows "\<R> = {region X I r |I r. valid_region X k I r}"
TERMINAL_not_LINEAR: fixes a :: "'h"   and b :: "'g"   and r :: "('f + 'g + 'h, 'i) Old_Datatype.node set" shows "TERMINAL a \<noteq> LINEAR b r"
stable_wti: fixes pc :: "nat" assumes "stable r step \<tau>s pc"   and "pc < length \<tau>s" shows "wti c pc (\<tau>s ! pc) \<noteq> \<top>"
dim_eq_card: fixes B :: "'b set"   and V :: "'b set" assumes "vs1.span B = vs1.span V"   and "vs1.independent B" shows "vs1.dim V = card B"