hcju/isabelleps huggingface dataset:
DiffE: fixes c :: "'g"   and A :: "'g set"   and B :: "'g set" assumes "c \<in> A - B" obtains "c \<in> A" and "c \<notin> B"
fmap_eqdom_induct: fixes R :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xm :: "'g \<rightharpoonup>\<^sub>f 'e"   and ym :: "'g \<rightharpoonup>\<^sub>f 'f"   and P :: "'g \<rightharpoonup>\<^sub>f 'e \<Rightarrow> 'g \<rightharpoonup>\<^sub>f 'f \<Rightarrow> bool" assumes "fmrel R xm ym"   and "fmdom xm = fmdom ym"   and "P (fmap_of_list []) (fmap_of_list [])"   and "\<And>x xm y ym i. \<lbrakk>R x y; fmrel R xm ym; fmdom xm = fmdom ym; P xm ym\<rbrakk> \<Longrightarrow> P (xm(i \<mapsto>\<^sub>f x)) (ym(i \<mapsto>\<^sub>f y))" shows "P xm ym"
mem_correct_empty_axioms_def: fixes lookup :: "'a \<Rightarrow> ('b, 'c option) state"   and P :: "'b \<Rightarrow> bool"   and empty :: "'b" shows "mem_correct_empty_axioms lookup P empty \<equiv> state_mem_defs.map_of lookup empty \<subseteq>\<^sub>m Map.empty \<and> P empty"
A: shows "((f, l), p, f', l') \<in> trcl pr_algo_lts"
adds_def: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" shows "plus.adds plus b a = (\<exists>k. a = plus b k)"
eq1: shows "bind_spmf (TRY map_spmf OK p ELSE return_spmf Fault) (\<lambda>s. obsf_oracle callee1 s x) = TRY bind_spmf p (\<lambda>s. map_spmf (map_prod OK OK) (callee1 s x)) ELSE return_spmf (Fault, Fault)"
tensor_mat_dim_row: fixes m1 :: "'a mat"   and m2 :: "'a mat" shows "dim_row (tensor_mat m1 m2) = d"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat" assumes "\<And>k m. (\<not> m \<le> k \<Longrightarrow> P (Suc k) (m - Suc k)) \<Longrightarrow> P k m" shows "P a0 a1"
Float_pos_eq_mantissa_pos: fixes m :: "int"   and e :: "int" shows "(0 < Float m e) = (0 < m)"
prj_prod_def: shows "PRJ('e \<times> 'f) = prod_map\<cdot>PRJ('e)\<cdot>PRJ('f) oo prod_prj"
generate_topology_Union: fixes I :: "'c set"   and S :: "'d set set"   and K :: "'c \<Rightarrow> 'd set" assumes "\<And>k. k \<in> I \<Longrightarrow> generate_topology S (K k)" shows "generate_topology S (\<Union> (K ` I))"
seqE: fixes g :: "('obj, 'obj) concrete_category.arr"   and f :: "('obj, 'obj) concrete_category.arr" assumes "seq g f" obtains "arr f" and "arr g" and "local.dom g = cod f"
total_onI: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" assumes "\<And>x y. \<lbrakk>x \<in> A; y \<in> A; x \<noteq> y\<rbrakk> \<Longrightarrow> (x, y) \<in> r \<or> (y, x) \<in> r" shows "total_on A r"
onb_expansion_finite: fixes T :: "'b set"   and x :: "'b" assumes "cspan T = UNIV"   and "is_ortho_set T"   and "\<And>t. t \<in> T \<Longrightarrow> norm t = 1" shows "x = (\<Sum>t\<in>T. \<langle>t, x\<rangle> *\<^sub>C t)"
limit_subset: fixes f :: "nat \<Rightarrow> 'c"   and n :: "nat" shows "limit f \<subseteq> f ` {n..}"
that_2: fixes p :: "'a" assumes "raw_has_prod f 0 p" shows "thesis"
body_def2: fixes G :: "prog"   and C :: "qtname"   and sig :: "sig" shows "body G C sig = Body (declclass (the (methd G C sig))) (stmt (mbody (mthd (the (methd G C sig)))))"
Empt: fixes z :: "NominalSequents.var"   and y :: "NominalSequents.var" shows "[z;y][] = []"
emeasure_restricted: fixes S :: "'a set"   and M :: "'a measure"   and X :: "'a set" assumes "S \<in> sets M"   and "X \<in> sets M" shows "emeasure (density M (indicator S)) X = emeasure M (S \<inter> X)"
prod_list_neutral: fixes xs :: "'b list" assumes "\<And>x. x \<in> set xs \<Longrightarrow> x = (1::'b)" shows "prod_list xs = (1::'b)"
prems_2: fixes k :: "'t"   and v :: "'t" assumes "k \<in> set ks"   and "v \<in> set vs" shows "v \<prec>\<^sub>t k"
new_shadow_root_get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and new_shadow_root_ptr :: "(_) shadow_root_ptr"   and ptr :: "(_) element_ptr"   and getter :: "('z, 'x, 'y, 'aa, 'ad option) RElement_scheme \<Rightarrow> 'aj" assumes "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>h h'"   and "h \<turnstile> new\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t_M \<rightarrow>\<^sub>r new_shadow_root_ptr" shows "preserved (get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
assms_1: shows "(\<gamma> \<longrightarrow>\<^sup>c \<gamma>') t n"
rel_cases: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'c llist"   and b :: "'d llist" assumes "llist_all2 R a b" obtains "a = LNil" and "b = LNil"   | x1 :: "'c" and x2 :: "'c llist" and y1 :: "'d" and y2 :: "'d llist" where "a = LCons x1 x2" and "b = LCons y1 y2" and "R x1 y1" and "llist_all2 R x2 y2"
that: fixes T :: "'t set" assumes "finite T"   and "T \<subseteq> S"   and "\<And>s. s \<in> S \<Longrightarrow> \<exists>t\<in>T. t adds\<^sub>t s" shows "thesis"
asym: fixes a :: "'a"   and b :: "'a" assumes "cmp.lt a b"   and "cmp.lt b a" shows "False"
bar3: fixes x :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent"   and xs :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and xa :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators" assumes "x \<in> dom (Cp (list2FWpolicy (xs @ [xa])))" shows "x \<in> dom (Cp (list2FWpolicy xs)) \<or> x \<in> dom (Cp xa)"
distinct_1: fixes x1 :: "char list"   and x2 :: "char list" shows "Repeats x1 \<noteq> Shares x2"
sharing_consistent_takeWhile: fixes \<S> :: "nat \<Rightarrow> bool option"   and \<O> :: "nat set"   and sb :: "'b memref list"   and P :: "'b memref \<Rightarrow> bool" assumes "sharing_consistent \<S> \<O> sb" shows "sharing_consistent \<S> \<O> (takeWhile P sb)"
acomplete_if_braun: fixes t :: "'b tree" assumes "braun t" shows "acomplete t"
FAss_\<tau>red1t_xt2: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and h :: "'heap"   and e :: "'addr expr1"   and xs :: "'addr val list"   and e' :: "'addr expr1"   and xs' :: "'addr val list"   and v :: "'addr val"   and F :: "String.literal"   and D :: "String.literal" assumes "\<tau>red1gt uf P t h (e, xs) (e', xs')" shows "\<tau>red1gt uf P t h (Val v\<bullet>F{D} := e, xs) (Val v\<bullet>F{D} := e', xs')"
simps_2: fixes r' :: "nat"   and u :: "vname aexp"   and t :: "(nat \<times> vname aexp) list"   and r :: "nat" shows "no_illegal_updates_code ((r', u) # t) r = (r \<noteq> r' \<and> no_illegal_updates_code t r)"
low_rank_det_zero: fixes A :: "'a mat" assumes "A \<in> carrier_mat n n"   and "det A \<noteq> (0::'a)" shows "rank A = n"
assms_8: shows "qf2 x2 y2 = 0"
vrestr_union: fixes f :: "'a \<Rightarrow> 'a"   and A :: "'a set"   and B :: "'a set" shows "f --` (A \<union> B) = f --` A \<union> f --` B"
iT_Mult_def: fixes I :: "nat set"   and k :: "nat" shows "I \<otimes> k \<equiv> (\<lambda>n. n * k) ` I"
expand: fixes hmultiset :: "hmultiset"   and hmultiset' :: "hmultiset" assumes "hmsetmset hmultiset = hmsetmset hmultiset'" shows "hmultiset = hmultiset'"
quasi_sturm_seq_axioms: shows "quasi_sturm_seq (p # ps)"
lipschitz_on_transform: fixes C :: "real"   and U :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e" assumes "C-lipschitz_on U f"   and "\<And>x. x \<in> U \<Longrightarrow> g x = f x" shows "C-lipschitz_on U g"
eqvtI: fixes P :: "pi"   and Rs :: "residual"   and perm :: "(name \<times> name) list" assumes "P \<Longrightarrow>\<^sub>l\<^sup>^Rs" shows "perm \<bullet> P \<Longrightarrow>\<^sub>l\<^sup>^perm \<bullet> Rs"
AC: shows "A \<inter> C \<in> null_sets lborel"
induct: fixes P :: "'a fm \<Rightarrow> bool"   and a0 :: "'a fm" assumes "\<And>\<phi>\<^sub>1 \<phi>\<^sub>2. \<lbrakk>P \<phi>\<^sub>1; P \<phi>\<^sub>2\<rbrakk> \<Longrightarrow> P (And \<phi>\<^sub>1 \<phi>\<^sub>2)"   and "\<And>\<phi>\<^sub>1 \<phi>\<^sub>2. \<lbrakk>P \<phi>\<^sub>1; P \<phi>\<^sub>2\<rbrakk> \<Longrightarrow> P (Or \<phi>\<^sub>1 \<phi>\<^sub>2)"   and "P (Neg TrueF)"   and "P (Neg FalseF)"   and "\<And>\<phi>. P \<phi> \<Longrightarrow> P (Neg (Neg \<phi>))"   and "\<And>\<phi>\<^sub>1 \<phi>\<^sub>2. \<lbrakk>P (Neg \<phi>\<^sub>1); P (Neg \<phi>\<^sub>2)\<rbrakk> \<Longrightarrow> P (Neg (And \<phi>\<^sub>1 \<phi>\<^sub>2))"   and "\<And>\<phi>\<^sub>1 \<phi>\<^sub>2. \<lbrakk>P (Neg \<phi>\<^sub>1); P (Neg \<phi>\<^sub>2)\<rbrakk> \<Longrightarrow> P (Neg (Or \<phi>\<^sub>1 \<phi>\<^sub>2))"   and "\<And>a. P (Neg (Atom a))"   and "P TrueF"   and "P FalseF"   and "\<And>v. P (Atom v)"   and "\<And>va. P (Neg (ExQ va))"   and "\<And>v. P (ExQ v)" shows "P a0"
LLL_inv_imp_w: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" assumes "LLL_invariant upw i fs" shows "LLL_invariant_weak fs"
set_eq_rule: fixes a :: "'a set"   and b :: "'a set" assumes "\<And>x. x \<in> a \<Longrightarrow> x \<in> b"   and "\<And>x. x \<in> b \<Longrightarrow> x \<in> a" shows "a = b"
last_fst: fixes P :: "'b \<Rightarrow> bool"   and a :: "'b"   and x :: "'b list" assumes "P ((a # x) ! length x)"   and "\<not> P a" shows "P (x ! (length x - Suc 0))"
gh: shows "src\<^sub>B g = trg\<^sub>B h"
left_neutral: fixes a :: "'a" shows "\<^bold>1 \<^bold>* a = a"
bij_betwE: fixes f :: "'f \<Rightarrow> 'g"   and A :: "'f set"   and B :: "'g set" assumes "bij_betw f A B" shows "\<forall>a\<in>A. f a \<in> B"
1: shows "equivalence (forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e))"
min_states_def: shows "min_states \<equiv> card (UNIV // eq_app_right language)"
ik\<^sub>s\<^sub>t_update\<^sub>s\<^sub>t_subset_2: fixes x :: "('fun, 'var) strand_step"   and S :: "('fun, 'var) strand_step list"   and \<S> :: "('fun, 'var) strand_step list set" assumes "x # S \<in> \<S>" shows "\<Union> (assignment_rhs\<^sub>s\<^sub>t ` update\<^sub>s\<^sub>t \<S> (x # S)) \<subseteq> \<Union> (assignment_rhs\<^sub>s\<^sub>t ` \<S>)"
ffmember_filter: fixes x :: "'e"   and P :: "'e \<Rightarrow> bool"   and A :: "'e fset" shows "(x |\<in>| ffilter P A) = (x |\<in>| A \<and> P x)"
op_map_run_def: shows "op_map_run \<equiv> (\<circ>)"
image_iff: fixes z :: "'x"   and f :: "'y \<Rightarrow> 'x"   and A :: "'y set" shows "(z \<in> f ` A) = (\<exists>x\<in>A. z = f x)"
mono_Inf: fixes f :: "'a \<Rightarrow> 'c"   and A :: "'a set" assumes "mono f" shows "f (\<Sqinter> A) \<le> \<Sqinter> (f ` A)"
assms: fixes x :: "Value"   and y :: "C \<rightarrow> CValue" assumes "x \<triangleleft>\<triangleright>\<^bsub>n\<^esub> y\<cdot>C\<^sup>\<infinity>" shows "f\<cdot>x \<triangleleft>\<triangleright>\<^bsub>n\<^esub> g\<cdot>y\<cdot>C\<^sup>\<infinity>"
lConsts_def: fixes w :: "'b \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and pr_strict :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and pr_weak :: "'b \<times> nat \<Rightarrow> 'b \<times> nat \<Rightarrow> bool"   and least :: "'b \<Rightarrow> bool"   and scf :: "'b \<times> nat \<Rightarrow> nat \<Rightarrow> nat" assumes "admissible_kbo w w0 pr_strict pr_weak least scf" shows "admissible_kbo.lConsts least = {(f, 0) |f. least f}"
graph_map_union: fixes g :: "nat \<Rightarrow> ('g \<times> 'f) set"   and X :: "('e, 'g) labeled_graph"   and Y :: "('e, 'f) labeled_graph" assumes "\<And>i. graph_union (map_graph (g i) X) Y = Y"   and "\<And>i j. i \<le> j \<Longrightarrow> g i \<subseteq> g j" shows "graph_union (map_graph (\<Union> (range g)) X) Y = Y"
sp: shows "nmu (sports m) \<longrightarrow> nmu (sports m1) \<or> nmu (sports m2)"
set_sel_1: fixes a :: "'b list" assumes "a \<noteq> []" shows "hd a \<in> set a"
compile_correct_unhalt: fixes ly :: "nat list"   and ap :: "abc_inst list"   and tp :: "(action \<times> nat) list"   and lm :: "nat list"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list"   and off :: "nat"   and n :: "nat" assumes "ly = layout_of ap"   and "tp = tm_of ap"   and "crsp ly (0, lm) (1, l, r) ires"   and "off = length tp div 2"   and "\<forall>stp. case abc_steps_l (0, lm) ap stp of (as, am) \<Rightarrow> as < length ap" shows "\<forall>stp. \<not> is_final (steps0 (1, l, r) (tp @ shift (mopup n) off) stp)"
galois_property: fixes x :: "'a"   and y :: "'b" assumes "x \<in> carrier \<X>"   and "y \<in> carrier \<Y>" shows "(\<pi>\<^sup>* x \<sqsubseteq>\<^bsub>\<Y>\<^esub> y) = (x \<sqsubseteq>\<^bsub>\<X>\<^esub> \<pi>\<^sub>* y)"
precompact_open_coverE: obtains U :: "nat \<Rightarrow> 'a set" where "\<Union> (range U) = carrier" and "\<And>i. open (U i)" and "\<And>i. compact (closure (U i))" and "\<And>i. closure (U i) \<subseteq> carrier"
represents_degree: fixes p :: "int poly"   and x :: "'a" assumes "p represents x" shows "degree p \<noteq> 0"
exhaust: fixes y :: "'b formula" obtains x1 :: "'b" where "(y::'b formula) = Atom (x1::'b)"   | "(y::'b formula) = \<bottom>"   | x3 :: "'b formula" where "(y::'b formula) = \<^bold>\<not> (x3::'b formula)"   | x41 :: "'b formula" and x42 :: "'b formula" where "(y::'b formula) = (x41::'b formula) \<^bold>\<and> (x42::'b formula)"   | x51 :: "'b formula" and x52 :: "'b formula" where "(y::'b formula) = (x51::'b formula) \<^bold>\<or> (x52::'b formula)"   | x61 :: "'b formula" and x62 :: "'b formula" where "(y::'b formula) = (x61::'b formula) \<^bold>\<rightarrow> (x62::'b formula)"
P_mod_0: fixes P :: "'a set"   and R' :: "('a set, 'e) Ring_scheme"   and Y :: "'a set"   and g :: "'a" assumes "Idomain S"   and "ideal S P"   and "PolynRg R' (S /\<^sub>r P) Y"   and "g \<in> carrier R" shows "(erH R S X R' (S /\<^sub>r P) Y (pj S P) g = \<zero>\<^bsub>R'\<^esub>) = P_mod R S X P g"
vsubstAbs_substAbs_compose2: fixes A :: "('d, 'e, 'varSort, 'var, 'f) qAbs set"   and Y :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and z1 :: "'var" assumes "goodAbs A"   and "good Y"   and "ys \<noteq> zs \<or> y \<notin> {z, z1}" shows "A $[Y / y]_ys $[z1 // z]_zs = A $[z1 // z]_zs $[Y #[z1 // z]_zs / y]_ys"
simps_5: fixes f1 :: "'b \<Rightarrow> 'a"   and f2 :: "'c \<Rightarrow> 'a"   and x1 :: "'b" shows "(case Inl x1 of Inl x \<Rightarrow> f1 x | Inr x \<Rightarrow> f2 x) = f1 x1"
inorder_delete: fixes t :: "('b \<times> bal) tree"   and x :: "'b" assumes "avl t"   and "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (delete x t) = del_list x (Tree2.inorder t)"
system_needed_asses_include_action_needed_asses_i: fixes A :: "'a set"   and B :: "'a set"   and f :: "('a, 'b) fmap" assumes "A \<subseteq> B" shows "fmrestrict_set A (fmrestrict_set B f) = fmrestrict_set A f"
comp_fun_idem: fixes x :: "'a" shows "f x \<circ> f x = f x"
lower_or_eq_monotonic: fixes t1 :: "'a trm"   and s1 :: "'a trm"   and t2 :: "'a trm"   and s2 :: "'a trm" assumes "lower_or_eq t1 s1"   and "lower_or_eq t2 s2" shows "lower_or_eq (t1 \<cdot> t2) (s1 \<cdot> s2)"
Collect_empty_eq: fixes P :: "'f \<Rightarrow> bool" shows "(Collect P = {}) = (\<forall>x. \<not> P x)"
nat_inc_prod': fixes m :: "nat"   and n :: "nat" shows "[(m * n)] \<cdot> \<one> = [m] \<cdot> \<one> \<otimes> [n] \<cdot> \<one>"
option_bindE: fixes x :: "'d option"   and f :: "'d \<Rightarrow> 'c option"   and a :: "'c" assumes "x \<bind> f = Some a" obtains x' :: "'d" where "x = Some x'" and "f x' = Some a"
DERIV_sin: fixes x :: "'b" shows "(sin has_field_derivative cos x) (at x)"
ID: shows "case_sum (\<lambda>e. undefined) (\<lambda>a. a) \<circ> Inr = id"
wf_mdecl_def: fixes wf_md :: "'b prog \<Rightarrow> char list \<Rightarrow> 'b mdecl \<Rightarrow> bool"   and P :: "'b prog"   and C :: "char list" shows "wf_mdecl wf_md P C \<equiv> \<lambda>(M, b, Ts, T, m). (\<forall>T\<in>set Ts. is_type P T) \<and> is_type P T \<and> wf_md P C (M, b, Ts, T, m)"
inv_10: shows "gram_schmidt_fs.reduced n (map of_int_hom.vec_hom fs) \<alpha> i"
inj_semigroup_mult_hom_axioms: shows "inj_semigroup_mult_hom hom"
conjugate_square_positive: fixes conjugate :: "'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b" assumes "class.conjugatable_ordered_ring conjugate plus zero minus uminus times less_eq less" shows "less_eq zero (times a (conjugate a))"
sets_2: shows "B \<subseteq> Pi\<^sub>E J S"
fixing_ide_gives_functor_1: fixes A1 :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and A2 :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'a \<times> 'c \<Rightarrow> 'd"   and a1 :: "'a" assumes "binary_functor A1 A2 B F"   and "partial_magma.ide A1 a1" shows "functor A2 B (\<lambda>f2. F (a1, f2))"
cSUP_upper2: fixes Inf :: "'d set \<Rightarrow> 'd"   and Sup :: "'d set \<Rightarrow> 'd"   and inf :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sup :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'e \<Rightarrow> 'd"   and A :: "'e set"   and x :: "'e"   and u :: "'d" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "preorder.bdd_above less_eq (f ` A)"   and "x \<in> A"   and "less_eq u (f x)" shows "less_eq u (Sup (f ` A))"
times_num_def: fixes m :: "num"   and n :: "num" shows "m * n = num_of_nat (nat_of_num m * nat_of_num n)"
coprime_iff_gcd_eq_1: fixes a :: "'b"   and b :: "'b" shows "algebraic_semidom_class.coprime a b = (gcd a b = (1::'b))"
least_prefixpoint_unique: fixes f :: "'a \<Rightarrow> 'a" assumes "has_least_prefixpoint f" shows "\<exists>!x. is_least_prefixpoint f x"
Greatest_equality: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and P :: "'b \<Rightarrow> bool"   and x :: "'b" assumes "class.order less_eq less"   and "P x"   and "\<And>y. P y \<Longrightarrow> less_eq y x" shows "order.Greatest less_eq P = x"
simps_18: fixes n :: "float"   and M :: "(floatarith, nat) mapping"   and slp :: "floatarith list" shows "slp_of_fa (Num n) M slp = slp_of_fa_cnst (Num n) (Num n) M slp"
intro: fixes hom :: "'c \<Rightarrow> 'd" assumes "Ring_Hom.ring_hom hom"   and "zero_hom_0 hom" shows "inj_ring_hom hom"
to_fun_Fun_one: shows "to_fun \<one>\<^bsub>P\<^esub> = \<one>\<^bsub>function_ring (carrier R) R\<^esub>"
i_set0_i_set_mem_conv: fixes I :: "nat set" shows "(I \<in> i_set0) = (I \<in> i_set \<or> I = {})"
map_ms_aux_def: fixes f :: "'b \<Rightarrow> 'a"   and xs :: "('b \<times> real) msllist" shows "map_ms_aux f xs = mslmap (\<lambda>(c, e). (f c, e)) xs"
hyp_port_h_for_def: fixes t :: "('form, 'rule, 'subst, 'var) itnode rose_tree"   and "is" :: "nat list"   and f :: "'form" shows "hyp_port_h_for t is f = snd (snd (hyp_port_for' t is f))"
intro: fixes \<S> :: "nat \<Rightarrow> bool option"   and ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" assumes "valid_ownership \<S> ts"   and "outstanding_non_volatile_writes_unshared \<S> ts"   and "sharing_consis \<S> ts"   and "no_outstanding_write_to_read_only_memory \<S> ts" shows "valid_ownership_and_sharing \<S> ts"
use_fps: fixes x :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "x \<in> fps L f" shows "f x .=\<^bsub>L\<^esub> x"
step: shows "\<Gamma>\<turnstile> (cs, asscss, s) \<rightarrow> (cs', css', t)"
gOr_commute: fixes a :: "'a gexp"   and b :: "'a gexp"   and s :: "'a \<Rightarrow> value option" shows "gval (gOr a b) s = gval (gOr b a) s"
BasicNegTT: fixes G :: "('a, 'b) form list" shows "\<stileturn> Neg \<top> # G"
congruent_triangleI_aas: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" assumes "angle a1 b1 c1 = angle a2 b2 c2"   and "angle b1 c1 a1 = angle b2 c2 a2"   and "dist a1 b1 = dist a2 b2"   and "\<not> collinear {a1, b1, c1}" shows "congruent_triangle a1 b1 c1 a2 b2 c2"
simps: fixes rha :: "'e \<Rightarrow> 'f"   and ma :: "'e \<Rightarrow> 'e \<Rightarrow> 'e option"   and rhb :: "'g \<Rightarrow> 'h"   and mb :: "'g \<Rightarrow> 'g \<Rightarrow> 'g option"   and x :: "(('e, 'f) T\<^sub>m, 'f T\<^sub>h, 'g, 'h) F\<^sub>m"   and y' :: "('e, 'f, 'g, 'h) G\<^sub>m" shows "merge_G rha ma rhb mb (G\<^sub>m x) y' = (case y' of G\<^sub>m y \<Rightarrow> map_option G\<^sub>m (merge_F (root_hash_T rha) (merge_T rha ma) rhb mb x y))"
im_diff: fixes G :: "'a set"   and T :: "'a \<Rightarrow> 'b"   and g :: "'a"   and g' :: "'a" assumes "GroupHom G T"   and "g \<in> G"   and "g' \<in> G" shows "T (g - g') = T g - T g'"
point_mult_add: fixes a :: "'b"   and b :: "'b"   and p :: "'b point"   and m :: "nat"   and n :: "nat" assumes "on_curve a b p"   and "nonsingular a b" shows "point_mult a (m + n) p = ell_field_class.add a (point_mult a m p) (point_mult a n p)"
map_simp_ide: fixes a :: "'j" assumes "J.ide a" shows "Da'oDa.map a = at a' D a \<cdot>\<^sub>B at a D a"
simps: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and lm :: "nat list"   and n :: "nat"   and ires :: "cell list" shows "mopup_bef_erase_a (s, l, r) lm n ires = (\<exists>ln m rn. l = Bk \<up> ln @ Bk # Bk # ires \<and> r = Oc \<up> m @ Bk # <drop ((s + 1) div 2) lm> @ Bk \<up> rn)"
polytope_eq_bounded_polyhedron: fixes S :: "'b set" shows "polytope S = (polyhedron S \<and> bounded S)"
stateless_callee_def: fixes callee :: "'k \<Rightarrow> ('l, 'n, 'o) gpv"   and s :: "'m" shows "stateless_callee callee s = map_gpv (\<lambda>b. (b, s)) id \<circ> callee"
of_nat_less_two_power: fixes n :: "nat" shows "of_nat n < (2::'b) ^ n"
falsifies\<^sub>g_empty: fixes C :: "fterm literal set" assumes "unification"   and "falsifies\<^sub>g [] C" shows "C = {}"
axioms_2: fixes Bot :: "'a set"   and Inf :: "'a inference set"   and entails :: "'a set \<Rightarrow> 'a set \<Rightarrow> bool"   and Red_I :: "'a set \<Rightarrow> 'a inference set"   and Red_F :: "'a set \<Rightarrow> 'a set" assumes "reducedly_statically_complete_reduced_calculus Bot Inf entails Red_I Red_F" shows "reducedly_statically_complete_reduced_calculus_axioms Bot Inf entails Red_I Red_F"
dgrad_sig_set_closed_sig_red_single: fixes d :: "'a \<Rightarrow> nat"   and p :: "'t \<Rightarrow>\<^sub>0 'b"   and f :: "'t \<Rightarrow>\<^sub>0 'b"   and sing_red :: "'t \<Rightarrow> 't \<Rightarrow> bool"   and top_tail :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and q :: "'t \<Rightarrow>\<^sub>0 'b"   and t :: "'a" assumes "dickson_grading d"   and "p \<in> dgrad_sig_set' (length fs) d"   and "f \<in> dgrad_sig_set' (length fs) d"   and "sig_red_single sing_red top_tail p q f t" shows "q \<in> dgrad_sig_set' (length fs) d"
irrefl: fixes a :: "'c multiset" shows "\<not> a \<subset># a"
run_catch_env: fixes catch :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and m1 :: "('e, 'd) envT"   and m2 :: "('e, 'd) envT"   and r :: "'e" shows "run_env (catch_env catch m1 m2) r = catch (run_env m1 r) (run_env m2 r)"
clamp_continuous_at: fixes a :: "'b"   and b :: "'b"   and f :: "'b \<Rightarrow> 'c"   and x :: "'b" assumes "continuous_on (cbox a b) f" shows "isCont (\<lambda>x. f (clamp a b x)) x"
simps_1: fixes uu :: "'c"   and uv :: "'c"   and F :: "'b" shows "Map (MkArr uu uv F) = F"
SNa_preserved_renaming2: fixes N :: "trm"   and x :: "name"   and y :: "name" assumes "SNa N" shows "SNa (N[x\<turnstile>n>y])"
sum_list_filter_coll_ex_scale: fixes z :: "real \<times> real"   and zs :: "(real \<times> real) list" assumes "z \<noteq> 0" shows "\<exists>r. sum_list (filter (coll 0 z) zs) = r *\<^sub>R z"
minim_least: fixes r :: "('c \<times> 'c) set"   and B :: "'c set"   and b :: "'c" assumes "wo_rel r"   and "B \<subseteq> Field r"   and "b \<in> B" shows "(minim r B, b) \<in> r"
pr_id1_1': shows "(\<lambda>x. x) \<in> PrimRec1'"
exhaust_sel: fixes float10 :: "float10" obtains "float10 = (mantissa10 float10) \<e> (exponent10 float10)"
ordIso_symmetric: fixes r :: "('f \<times> 'f) set"   and r' :: "('g \<times> 'g) set" assumes "r =o r'" shows "r' =o r"
good_def: fixes t :: "('sequent \<times> 'rule) tree" shows "good t \<equiv> \<forall>steps. ipath t steps \<longrightarrow> ev (\<lambda>steps'. \<exists>bs Ms. follow bs Ms steps' \<and> infDecr bs) steps"