hcju/isabelleps huggingface dataset:
wf_redE: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action"   and x'' :: "'x"   and m'' :: "'m" assumes "s \<in> wf_state"   and "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "t \<turnstile> \<langle>x, shr s\<rangle> -ta\<rightarrow> \<langle>x'', m''\<rangle>"   and "\<not> waiting (wset s t)" obtains ta' :: "('l, 't, 'x, 'm, 'w, 'o) thread_action" and x' :: "'x" and m' :: "'m" where "t \<turnstile> \<langle>x, shr s\<rangle> -ta'__\<rightarrow> \<langle>x'__, m'__\<rangle>" and "actions_ok' s t ta'__" and "actions_subset ta'__ ta"   | ta' :: "('l, 't, 'x, 'm, 'w, 'o) thread_action" and x' :: "'x" and m' :: "'m" where "t \<turnstile> \<langle>x, shr s\<rangle> -ta'\<rightarrow> \<langle>x', m'\<rangle>" and "actions_ok s t ta'"
in_the_default_empty_conv: fixes M_empty :: "'a"   and M_update :: "'b \<Rightarrow> ('b \<times> nat) list \<Rightarrow> 'a \<Rightarrow> 'a"   and M_delete :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and M_lookup :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> nat) list option"   and M_invar :: "'a \<Rightarrow> bool"   and x :: "'c"   and m :: "'c list option" assumes "wgraph_by_map M_empty M_update M_delete M_lookup M_invar" shows "(x \<in> set (the_default [] m)) = (\<exists>xs. m = Some xs \<and> x \<in> set xs)"
rvars_eq_subst_var_eq_supset: fixes eq :: "nat \<times> linear_poly"   and x\<^sub>j :: "nat"   and lp :: "linear_poly" shows "rvars_eq eq - {x\<^sub>j} - vars lp \<subseteq> rvars_eq (subst_var_eq x\<^sub>j lp eq)"
HF_quot_coding_tm: fixes t :: "tm"   and e :: "(name, hf) finfun" assumes "coding_tm t" shows "Goedel_I.HF \<lbrakk>t\<rbrakk>e = \<guillemotleft>t\<guillemotright>"
left_lesseq: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (a l\<rightarrow> b = (1::'a))"
completenessForUNSAT: fixes F0 :: "Literal list list"   and decisionVars :: "nat set"   and state0 :: "State"   and state :: "State" assumes "vars F0 \<subseteq> decisionVars"   and "\<not> satisfiable F0"   and "isInitialState state0 F0"   and "(state0, state) \<in> transitionRelation decisionVars"   and "isFinalState state decisionVars" shows "formulaFalse (getF state) (elements (getM state)) \<and> decisions (getM state) = []"
T_mtf_def: fixes n :: "nat" shows "T_mtf n = sum t_mtf {..<n}"
not_zero: shows "1 mod int CARD('a) \<noteq> 0"
expands_to_imp_eventually_pos: fixes basis :: "(real \<Rightarrow> real) list"   and f :: "real \<Rightarrow> real"   and F :: "'a" assumes "basis_wf basis"   and "(f expands_to F) basis"   and "trimmed_pos F" shows "\<forall>\<^sub>F x in at_top. 0 < f x"
bounded_pos: fixes S :: "'c set" shows "bounded S = (\<exists>b>0. \<forall>x\<in>S. norm x \<le> b)"
IdD: fixes a :: "'h"   and b :: "'h" assumes "(a, b) \<in> Id" shows "a = b"
cSup_atLeastAtMost: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "less_eq y x" shows "Sup (ord.atLeastAtMost less_eq y x) = x"
llcp_lprefix1: fixes xs :: "'c llist"   and ys :: "'c llist" assumes "xs \<sqsubseteq> ys" shows "llcp xs ys = llength xs"
prv_exiI: fixes x :: "'var"   and \<phi> :: "'fmla"   and t :: "'trm" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "t \<in> trm"   and "bprv (subst \<phi> t x)" shows "bprv (exi x \<phi>)"
select_convs_3: fixes orderA :: "('d, 'f) gorder_scheme"   and orderB :: "('e, 'g) gorder_scheme"   and lower :: "'d \<Rightarrow> 'e"   and upper :: "'e \<Rightarrow> 'd"   and more :: "'h" shows "\<pi>\<^sup>*\<^bsub>\<lparr>orderA = orderA, orderB = orderB, lower = lower, upper = upper, \<dots> = more\<rparr>\<^esub> = lower"
117: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<squnion> (x \<sqinter> - y \<squnion> z) = x \<squnion> z"
distinct_159: shows "GreaterOrEqual \<noteq> ShiftRightZeros"
intros: fixes eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and min_lvar_not_in_bounds :: "('a, 'b) state \<Rightarrow> nat option"   and min_rvar_incdec_eq :: "('a, 'b) Direction \<Rightarrow> ('a, 'b) state \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a list + nat"   and pivot_and_update :: "nat \<Rightarrow> nat \<Rightarrow> 'b \<Rightarrow> ('a, 'b) state \<Rightarrow> ('a, 'b) state"   and s :: "('a, 'b) state" assumes "PivotUpdateMinVars eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update"   and "\<And>x\<^sub>i. \<lbrakk>\<not> \<U> s; Some x\<^sub>i = min_lvar_not_in_bounds s; \<langle>\<V> s\<rangle> x\<^sub>i <\<^sub>l\<^sub>b \<B>\<^sub>l s x\<^sub>i\<rbrakk> \<Longrightarrow> PivotUpdateMinVars.check_dom eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update (PivotUpdateMinVars.check' eq_idx_for_lvar min_rvar_incdec_eq pivot_and_update Positive x\<^sub>i s)"   and "\<And>x\<^sub>i. \<lbrakk>\<not> \<U> s; Some x\<^sub>i = min_lvar_not_in_bounds s; \<not> \<langle>\<V> s\<rangle> x\<^sub>i <\<^sub>l\<^sub>b \<B>\<^sub>l s x\<^sub>i\<rbrakk> \<Longrightarrow> PivotUpdateMinVars.check_dom eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update (PivotUpdateMinVars.check' eq_idx_for_lvar min_rvar_incdec_eq pivot_and_update Negative x\<^sub>i s)" shows "PivotUpdateMinVars.check_dom eq_idx_for_lvar min_lvar_not_in_bounds min_rvar_incdec_eq pivot_and_update s"
mkState_def: fixes s0 :: "(Var \<times> Val) list \<times> (Addr \<times> Class \<times> (Field \<times> Val) list) list" shows "mkState s0 = ([], fst s0, snd s0)"
check_def: fixes h :: "'a"   and a :: "'a"   and e :: "nat"   and z :: "nat" shows "check h a e z = (a \<otimes> h [^] e = \<^bold>g [^] z \<and> a \<in> carrier \<G>)"
mv_box_correct: fixes m :: "nat"   and n :: "nat"   and lm :: "nat list" assumes "max m n < length lm"   and "m \<noteq> n" shows "{\<lambda>nl. nl = lm} mv_box m n {\<lambda>nl. nl = lm[n := lm ! m + lm ! n, m := 0]}"
assms: fixes x :: "'a" assumes "x \<in> S" shows "norm x \<le> B"
inject_2: fixes x2 :: "'c"   and y2 :: "'c" shows "(Inr x2 = Inr y2) = (x2 = y2)"
return_bind: fixes x :: "'d"   and g :: "'d \<Rightarrow> ('a, 'b, 'c) prog" shows "return x \<bind> g = g x"
one_mat_def: fixes n :: "nat" shows "1\<^sub>m n \<equiv> mat n n (\<lambda>(i, j). if i = j then 1::'a else (0::'a))"
infinite_growing: fixes less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and X :: "'e set" assumes "class.linorder less_eq less"   and "X \<noteq> {}"   and "\<And>x. x \<in> X \<Longrightarrow> \<exists>y\<in>X. less x y" shows "infinite X"
tree_path_iff_parenthesis: fixes v :: "'v"   and w :: "'v" assumes "v \<in> dom (finished s)"   and "w \<in> dom (finished s)" shows "((v, w) \<in> (tree_edges s)\<^sup>+) = (\<delta> s v < \<delta> s w \<and> \<phi> s w < \<phi> s v)"
span_is_submodule: fixes S :: "rat vec set" assumes "S \<subseteq> Rn" shows "LinearCombinations.submodule class_ring (gs.span S) gs.V"
get_root_node_si_root_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr"   and root :: "(_) object_ptr" assumes "heap_is_wellformed h"   and "type_wf h"   and "known_ptrs h"   and "h \<turnstile> get_root_node_si ptr \<rightarrow>\<^sub>r root" shows "root |\<in>| object_ptr_kinds h"
cases: fixes a1 :: "('p, 's, 'm) configuration"   and a2 :: "('p, 's, 'm) event list"   and a3 :: "('p, 's, 'm) configuration" assumes "trace a1 a2 a3" obtains c :: "('p, 's, 'm) configuration" where "a1 = c__" and "a2 = []" and "a3 = c__"   | c :: "('p, 's, 'm) configuration" and ev :: "('p, 's, 'm) event" and c' :: "('p, 's, 'm) configuration" and t :: "('p, 's, 'm) event list" and c'' :: "('p, 's, 'm) configuration" where "a1 = c" and "a2 = ev # t" and "a3 = c''" and "c \<turnstile> ev \<mapsto> c'" and "trace c' t c''"
seqE: fixes g :: "'a \<times> 'a \<times> 'a"   and f :: "'a \<times> 'a \<times> 'a" assumes "VVV.seq g f" obtains "VVV.arr f" and "VVV.arr g" and "VVV.dom g = VVV.cod f"
sum_case_non: fixes a :: "V"   and f :: "V \<Rightarrow> 'a"   and g :: "V \<Rightarrow> 'a" assumes "\<not> is_sum a" shows "sum_case f g a = undefined"
lookup_HSem_other: fixes y :: "Vars.var"   and h :: "(Vars.var \<times> exp) list"   and \<rho> :: "Vars.var \<Rightarrow> 'Value" assumes "y \<notin> domA h" shows "(\<lbrace>h\<rbrace>\<rho>) y = \<rho> y"
subst_Bool_the_BoolI: fixes P :: "'b assn"   and b :: "val"   and s :: "state"   and Z :: "'b"   and Y :: "vals" assumes "P (In1 b) s Z" shows "(P\<leftarrow>=the_Bool b) Y s Z"
mod_mult_mult2: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and euclidean_size :: "'b \<Rightarrow> nat"   and a :: "'b"   and c :: "'b"   and b :: "'b" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "modulo (times a c) (times b c) = times (modulo a b) c"
S_def: shows "S \<equiv> subresultant_mat j p q"
that: fixes R :: "('a, 'b, 'c) psi" assumes "P = \<lparr>\<nu>z\<rparr>R"   and "\<lparr>\<nu>*xvec\<rparr>M \<prec>' R = \<lparr>\<nu>*yvec\<rparr>N \<prec>' Q" shows "thesis"
simps_1: fixes a :: "('a, 'b) atm" shows "local.wtL (Pos a) = local.wtA a"
simps_2: fixes k :: "nat"   and u :: "nat" shows "\<T> k (Suc u) = Tokens k (\<T> k u) (\<J> k u)"
equivalence_respects_iso: fixes f :: "'a"   and g :: "'a"   and \<eta> :: "'a"   and \<epsilon> :: "'a"   and \<phi> :: "'a"   and f' :: "'a"   and \<psi> :: "'a"   and g' :: "'a" assumes "internal_equivalence f g \<eta> \<epsilon>"   and "\<guillemotleft>\<phi> : f \<Rightarrow> f'\<guillemotright>"   and "local.iso \<phi>"   and "\<guillemotleft>\<psi> : g \<Rightarrow> g'\<guillemotright>"   and "local.iso \<psi>" shows "internal_equivalence f' g' ((g' \<star> \<phi>) \<cdot> (\<psi> \<star> f) \<cdot> \<eta>) (\<epsilon> \<cdot> (local.inv \<phi> \<star> g) \<cdot> (f' \<star> local.inv \<psi>))"
modify_outside_partial_heap: fixes heap :: "int \<Rightarrow> 'a"   and heap' :: "int \<Rightarrow> 'a"   and m :: "int"   and r :: "int"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and l :: "int" assumes "heap = heap' on {m..<r}"   and "is_partial_heap cmp heap l m r" shows "is_partial_heap cmp heap' l m r"
Crypt: fixes X :: "msg"   and H :: "msg set"   and K :: "nat" assumes "X \<in> synth H"   and "Key K \<in> H" shows "Crypt K X \<in> synth H"
pivotal_xdictates: fixes i :: "indi"   and b :: "alt" assumes "pivotal i b" shows "i dictates_except b"
cgenideal_self: fixes i :: "'a" assumes "i \<in> carrier R" shows "i \<in> PIdl i"
simulation2_\<tau>rtrancl3p: fixes s2 :: "'s2"   and tls2 :: "'tl2 list"   and s2' :: "'s2"   and s1 :: "'s1" assumes "s2 -\<tau>2-tls2\<rightarrow>* s2'"   and "s1 \<approx> s2" shows "\<exists>tls1 s1'. s1 -\<tau>1-tls1\<rightarrow>* s1' \<and> s1' \<approx> s2' \<and> tls1 [\<sim>] tls2"
iSince_True_if: fixes I :: "nat set"   and P :: "nat \<Rightarrow> bool" shows "(P t'. t' \<S> t I. True) = (if finite I then I \<noteq> {} else \<diamond> t1 I. \<box> t2 I \<down>> t1. P t2)"
is_loc_confluent_def: shows "is_loc_confluent \<equiv> \<forall>a b1 b2. a \<rightarrow> b1 \<and> a \<rightarrow> b2 \<longrightarrow> b1 \<down>\<^sup>* b2"
cfun_belowI: fixes f :: "'d \<rightarrow> 'e"   and g :: "'d \<rightarrow> 'e" assumes "\<And>x. f\<cdot>x \<sqsubseteq> g\<cdot>x" shows "f \<sqsubseteq> g"
cases: fixes x :: "('a \<times> 'a rose_tree list \<times> 'a rose_tree list) list \<times> 'a rose_tree" obtains p :: "('a \<times> 'a rose_tree list \<times> 'a rose_tree list) list" and a :: "'a" and ts :: "'a rose_tree list" where "x = (p, Tree (a, ts))"
emptyI: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c"   and z :: "'c" shows "fold_graph f z {} z"
sum_image: fixes X :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "finite X" shows "sum f X = \<Sum> (f ` X)"
Cons: fixes A :: "'a \<Rightarrow> bool"   and a :: "'a"   and l :: "'a list" assumes "A a"   and "listsp A l" shows "listsp A (a # l)"
bounded_linear_right: fixes a :: "'d \<Rightarrow>\<^sub>L 'e" shows "bounded_linear (blinfun_apply a)"
symp_eqvt: fixes R :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and p :: "perm" assumes "symp R" shows "symp (p \<bullet> R)"
b: shows "emeasure M (induced_map A --` WB) = emeasure M WB"
underS_Field: fixes i :: "'a"   and R :: "('a \<times> 'a) set"   and j :: "'a" assumes "i \<in> underS R j" shows "i \<in> Field R"
bcontfun\<^sub>N_space: shows "space\<^sub>N bcontfun\<^sub>N = bcontfun"
scale: fixes r :: "real"   and b :: "'a" shows "f (r *\<^sub>R b) = r *\<^sub>R f b"
d_Suc: fixes i :: "nat" shows "gs.d (Suc i) = gs.\<mu>' i i"
compat_add_triple: fixes a :: "'b"   and b :: "'b"   and p :: "'b point" assumes "nonsingular a b"   and "on_curve a b p"   and "p \<noteq> opp p"   and "ell_field_class.add a p p \<noteq> opp p" shows "ell_field_class.add a (ell_field_class.add a p p) (opp p) = p"
simps_4: fixes x21 :: "pat"   and x22 :: "pat"   and x11 :: "name"   and x12 :: "ty" shows "\<langle>\<langle>x21, x22\<rangle>\<rangle> \<noteq> PVar x11 x12"
n_as0: shows "0 \<notin> fst ` set n_as"
ring_simprules_6: shows "(1::'a) \<in> UNIV"
hyps: shows "set ats' \<Turnstile>\<^sub>i \<B>\<I> (assert_bound_loop ats' (init t))"
pax: shows "\<exists>b. \<forall>a. p b a \<and> p 0 0 \<and> p 1 a \<or> p 0 1 \<and> p 1 0 \<and> p a b"
add_mset_remove_trivial_iff: fixes N :: "'b multiset"   and a :: "'b"   and b :: "'b" shows "(N = add_mset a (remove1_mset b N)) = (a \<in># N \<and> a = b)"
Close1: fixes P :: "pi"   and a :: "name"   and x :: "name"   and P' :: "pi"   and Q :: "pi"   and Q' :: "pi" assumes "P \<Longrightarrow>a<x> \<prec> P'"   and "Q \<Longrightarrow>a<\<nu>x> \<prec> Q'"   and "x \<sharp> P" shows "P \<parallel> Q \<Longrightarrow>\<tau> \<prec> <\<nu>x>(P' \<parallel> Q')"
vars_iff_subtermeq: fixes x :: "'d"   and t :: "('e, 'd) Term.term" shows "(x \<in> fv t) = (Var x \<sqsubseteq> t)"
pred_sexp_subset_Sigma: shows "pred_sexp \<subseteq> sexp \<times> sexp"
times_set_def: fixes x :: "'a set"   and y :: "'a set" shows "x \<oplus> y = x \<union> y"
edge_preserving: fixes R :: "('a \<times> 'b) set"   and E :: "('c \<times> 'a \<times> 'a) set"   and G :: "('c \<times> 'b \<times> 'b) set" shows "edge_preserving R E G = (on_triple R `` E \<subseteq> G)"
dickson_gradingD1: fixes d :: "'c \<Rightarrow> nat"   and s :: "'c"   and t :: "'c" assumes "dickson_grading d" shows "d (s + t) = max (d s) (d t)"
assms_6: shows "f \<star> e \<cong> f' \<star> e"
att_or_empty: fixes s' :: "'a set"   and s'' :: "'a set" assumes "\<turnstile>[] \<oplus>\<^sub>\<or>\<^bsup>(s', s'')\<^esup>" shows "s' \<subseteq> s''"
cod_eqI': fixes a :: "'a"   and f :: "'a" assumes "a \<in> codomains f" shows "a = cod f"
current_memory: fixes s :: "s"   and i :: "nat"   and j :: "nat"   and m :: "mem"   and n :: "nat"   and vs :: "v list" assumes "smem_ind s i = Some j"   and "s.mem s ! j = m"   and "mem_size m = n" shows "\<lparr>s;vs;[$Current_memory]\<rparr> \<leadsto>_ i \<lparr>s;vs;[$C ConstInt32 (int_of_nat n)]\<rparr>"
currentD_IN: fixes \<Gamma> :: "('a, 'b) web_scheme"   and f :: "'a \<times> 'a \<Rightarrow> ennreal"   and a :: "'a" assumes "current \<Gamma> f"   and "a \<in> A \<Gamma>" shows "d_IN f a = 0"
matrix_mult_index: fixes id :: "'b"   and f :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zer :: "'b"   and g :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inver :: "'b \<Rightarrow> 'b"   and m1 :: "'b list list"   and nr :: "nat"   and n :: "nat"   and nc :: "nat"   and m2 :: "'b list list"   and i :: "nat"   and j :: "nat" assumes "plus_mult id f zer g inver"   and "m1 \<noteq> []"   and "mat nr n m1"   and "mat n nc m2"   and "i < nr"   and "j < nc" shows "mat_multI zer g f (mult.row_length m1) m1 m2 ! j ! i = plus_mult.scalar_product f zer g (row m1 i) (col m2 j)"
simps_1: fixes P :: "'c \<Rightarrow> bool"   and x :: "'c" shows "enforce_option P (Some x) = (if P x then Some x else None)"
rules: shows "R1 \<subseteq> upRules \<and> R2 \<subseteq> modRules2 \<and> R3 \<subseteq> modRules2 \<and> R = Ax \<union> R1 \<union> p_e R2 M1 M2 \<union> R3 \<and> R' = Ax \<union> R1 \<union> R2 \<union> R3"
det_col: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "det A = (\<Sum>p | p permutes {0..<n}. signof p * (\<Prod>j<n. A $$ (p j, j)))"
ex_qfp: fixes CC :: "'a set set" assumes "CC-complete A (\<sqsubseteq>)"   and "C \<in> CC"   and "\<forall>a\<in>A. a \<sqsubseteq> f a \<or> (\<forall>b\<in>A. b \<sqsubseteq> a \<longrightarrow> f b \<sqsubseteq> f a)" shows "\<exists>s\<in>A. f s \<sim> s"
\<R>\<^sub>1_elim: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma> :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" obtains \<Gamma> :: "'Var \<Rightarrow> Sec option" where "c\<^sub>2 = c\<^sub>1" and "\<turnstile> \<Gamma> {c\<^sub>1} \<Gamma>''''__" and "mds_consistent mds \<Gamma>" and "mem\<^sub>1 =\<^bsub>\<Gamma>\<^esub> mem\<^sub>2"
induct: fixes x1 :: "cmd"   and x2 :: "label"   and x3 :: "p_edge_kind"   and x4 :: "label"   and P :: "cmd \<Rightarrow> label \<Rightarrow> p_edge_kind \<Rightarrow> label \<Rightarrow> bool" assumes "x1 \<turnstile> x2 -x3\<rightarrow>\<^sub>p x4"   and "\<And>prog. P prog Entry (IEdge (\<lambda>s. False)\<^sub>\<surd>) Exit"   and "\<And>prog. P prog Entry (IEdge (\<lambda>s. True)\<^sub>\<surd>) (Label 0)"   and "P Skip (Label 0) (IEdge \<Up>id) Exit"   and "\<And>V e. P (V:=e) (Label 0) (IEdge \<Up>\<lambda>cf. update cf V e) (Label 1)"   and "\<And>V e. P (V:=e) (Label 1) (IEdge \<Up>id) Exit"   and "\<And>c\<^sub>1 n et n' c\<^sub>2. \<lbrakk>c\<^sub>1 \<turnstile> n -et\<rightarrow>\<^sub>p n'; P c\<^sub>1 n et n'; n' \<noteq> Exit\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) n et n'"   and "\<And>c\<^sub>1 n et c\<^sub>2. \<lbrakk>c\<^sub>1 \<turnstile> n -et\<rightarrow>\<^sub>p Exit; P c\<^sub>1 n et Exit; n \<noteq> Entry\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) n et (Label #:c\<^sub>1)"   and "\<And>c\<^sub>2 n et n' c\<^sub>1. \<lbrakk>c\<^sub>2 \<turnstile> n -et\<rightarrow>\<^sub>p n'; P c\<^sub>2 n et n'; n \<noteq> Entry\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) (n \<oplus> #:c\<^sub>1) et (n' \<oplus> #:c\<^sub>1)"   and "\<And>b c\<^sub>1 c\<^sub>2. P (if (b) c\<^sub>1 else c\<^sub>2) (Label 0) (IEdge (\<lambda>cf. state_check cf b (Some true))\<^sub>\<surd>) (Label 1)"   and "\<And>b c\<^sub>1 c\<^sub>2. P (if (b) c\<^sub>1 else c\<^sub>2) (Label 0) (IEdge (\<lambda>cf. state_check cf b (Some false))\<^sub>\<surd>) (Label (#:c\<^sub>1 + 1))"   and "\<And>c\<^sub>1 n et n' b c\<^sub>2. \<lbrakk>c\<^sub>1 \<turnstile> n -et\<rightarrow>\<^sub>p n'; P c\<^sub>1 n et n'; n \<noteq> Entry\<rbrakk> \<Longrightarrow> P (if (b) c\<^sub>1 else c\<^sub>2) (n \<oplus> 1) et (n' \<oplus> 1)"   and "\<And>c\<^sub>2 n et n' b c\<^sub>1. \<lbrakk>c\<^sub>2 \<turnstile> n -et\<rightarrow>\<^sub>p n'; P c\<^sub>2 n et n'; n \<noteq> Entry\<rbrakk> \<Longrightarrow> P (if (b) c\<^sub>1 else c\<^sub>2) (n \<oplus> #:c\<^sub>1 + 1) et (n' \<oplus> #:c\<^sub>1 + 1)"   and "\<And>b c'. P (while (b) c') (Label 0) (IEdge (\<lambda>cf. state_check cf b (Some true))\<^sub>\<surd>) (Label 2)"   and "\<And>b c'. P (while (b) c') (Label 0) (IEdge (\<lambda>cf. state_check cf b (Some false))\<^sub>\<surd>) (Label 1)"   and "\<And>b c'. P (while (b) c') (Label 1) (IEdge \<Up>id) Exit"   and "\<And>c' n et n' b. \<lbrakk>c' \<turnstile> n -et\<rightarrow>\<^sub>p n'; P c' n et n'; n \<noteq> Entry; n' \<noteq> Exit\<rbrakk> \<Longrightarrow> P (while (b) c') (n \<oplus> 2) et (n' \<oplus> 2)"   and "\<And>c' n et b. \<lbrakk>c' \<turnstile> n -et\<rightarrow>\<^sub>p Exit; P c' n et Exit; n \<noteq> Entry\<rbrakk> \<Longrightarrow> P (while (b) c') (n \<oplus> 2) et (Label 0)"   and "\<And>p es rets. P (Call p es rets) (Label 0) (CEdge (p, es, rets)) (Label 1)"   and "\<And>p es rets. P (Call p es rets) (Label 1) (IEdge \<Up>id) Exit" shows "P x1 x2 x3 x4"
mono_onI: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>r s. \<lbrakk>r \<in> A; s \<in> A; r \<le> s\<rbrakk> \<Longrightarrow> f r \<le> f s" shows "Fun.mono_on f A"
M: shows "l1 @ x # l2 = l1' @ y # l2'"
quotient_diff1: fixes r :: "('a \<times> 'a) set"   and A :: "'a set"   and a :: "'a" assumes "inj_on (\<lambda>a. {a} // r) A"   and "a \<in> A" shows "(A - {a}) // r = A // r - {a} // r"
homeomorphic_compact: fixes s :: "'a set"   and f :: "'a \<Rightarrow> 'b"   and t :: "'b set" assumes "compact s"   and "continuous_on s f"   and "f ` s = t"   and "inj_on f s" shows "s homeomorphic t"
to_woot_character_keeps_i: fixes M :: "('a, 'b) message set"   and m :: "('a, 'b) insert_message" shows "I (to_woot_character M m) = I m"
mem_sphere_0: fixes x :: "'c"   and e :: "real" shows "(x \<in> sphere (0::'c) e) = (norm x = e)"
dom_llp: fixes x :: "'a"   and y :: "'a" shows "(x \<le> d y \<cdot> x) = (d x \<le> d y)"
con: shows "weak_reduction_contrasimulation (Rel\<inverse>) Cal"
sch_to_pr_into_pr: fixes sch :: "PrimScheme" shows "sch_to_pr sch \<in> PrimRec1"
simps_1: fixes ty_ent :: "'b \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and p :: "predicate"   and vs :: "'b list" shows "wf_atom ty_ent (predAtm p vs) = wf_pred_atom ty_ent (p, vs)"
apply_cltn2_compose: fixes p :: "proj2"   and A :: "cltn2"   and B :: "cltn2" shows "apply_cltn2 (apply_cltn2 p A) B = apply_cltn2 p (cltn2_compose A B)"
f2: fixes r :: "real"   and ra :: "real" shows "r - r * ra = r * (1 - ra)"
pmf_mix_induct: fixes A :: "'b set"   and p :: "'b pmf"   and P :: "'b pmf \<Rightarrow> bool" assumes "finite A"   and "set_pmf p \<subseteq> A"   and "\<And>x. x \<in> A \<Longrightarrow> P (return_pmf x)"   and "\<And>p a y. \<lbrakk>set_pmf p \<subseteq> A; a \<in> {0<..<1}; y \<in> A; P p\<rbrakk> \<Longrightarrow> P (mix_pmf a p (return_pmf y))" shows "P p"
inj_ord_of: fixes A :: "nat set" shows "inj_on ord_of A"
monomial_is_monomial: fixes c :: "'a"   and t :: "'b" assumes "c \<noteq> (0::'a)" shows "is_monomial (monomial c t)"
infinite: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'a" assumes "infinite A" shows "sum g A = (0::'a)"
swap_atom_simps_3: fixes c :: "atom"   and a :: "atom"   and b :: "atom" assumes "c \<noteq> a"   and "c \<noteq> b" shows "(a \<rightleftharpoons> b) \<bullet> c = c"
intermediate_1: shows "R \<subseteq> R\<^sub>d"
vsubst_compose1: fixes X :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and y2 :: "'var"   and y :: "'var"   and y1 :: "'var" assumes "good X" shows "X #[y1 // y]_ys #[y2 // y]_ys = X #[y1 @ys[y2 / y]_ys // y]_ys"
\<Delta>_ffact: fixes n :: "nat"   and k :: "int" shows "\<Delta> (ffact (Suc n)) k = of_nat (Suc n) * ffact n (of_int k)"
rel_mono_strong: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'b list"   and y :: "'c list"   and Ra :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "list_all2 R x y"   and "\<And>z yb. \<lbrakk>z \<in> set x; yb \<in> set y; R z yb\<rbrakk> \<Longrightarrow> Ra z yb" shows "list_all2 Ra x y"
inverse_cancel: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter"   and g :: "'b \<Rightarrow> 'c" assumes "\<forall>\<^sub>F x in F. f x \<noteq> (0::'c)"   and "\<forall>\<^sub>F x in F. g x \<noteq> (0::'c)" shows "((\<lambda>x. inverse (f x)) \<in> o[F](\<lambda>x. inverse (g x))) = (g \<in> o[F](f))"